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Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdleme11dN 29701 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q )  /\  ( S  =/=  T 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T ) ) )  ->  ( P  .\/  S )  =/=  ( P  .\/  T ) )
 
Theoremcdleme11e 29702 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 13-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  D  =  ( ( P  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  P  =/=  Q )  /\  ( S  =/=  T 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T ) ) )  ->  C  =/=  D )
 
Theoremcdleme11fN 29703 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  D  =  ( ( P  .\/  T )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  F  =/=  C )
 
Theoremcdleme11g 29704 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 14-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  D  =  ( ( P  .\/  T )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  S  e.  A )  /\  P  =/=  Q )  ->  ( Q  .\/  F )  =  ( Q 
 .\/  C ) )
 
Theoremcdleme11h 29705 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 14-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  D  =  ( ( P  .\/  T )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  F  =/=  Q )
 
Theoremcdleme11j 29706 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 14-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  D  =  ( ( P  .\/  T )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  C  .<_  ( Q  .\/  F ) )
 
Theoremcdleme11k 29707 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 15-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  D  =  ( ( P  .\/  T )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  C  =  ( ( Q  .\/  F )  ./\  W )
 )
 
Theoremcdleme11l 29708 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 29709. (Contributed by NM, 15-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T )
 ) )  ->  F  =/=  G )
 
Theoremcdleme11 29709 Part of proof of Lemma E in [Crawley] p. 113, 1st sentence of 3rd paragraph on p. 114.  F and  G represent f(s) and f(t) respectively. Their proof provides no details of our cdleme11a 29699 through cdleme11 29709, so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T )
 ) )  ->  ( F  .\/  G )  =  ( S  .\/  T ) )
 
Theoremcdleme12 29710 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. 
F and  G represent f(s) and f(t) respectively. (Contributed by NM, 16-Jun-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  Q  e.  A  /\  P  =/=  Q ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( S  =/=  T 
 /\  -.  U  .<_  ( S  .\/  T )
 ) ) )  ->  ( ( S  .\/  F )  ./\  ( T  .\/  G ) )  =  U )
 
Theoremcdleme13 29711 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, "<s,t,p> and <f(s),f(t),q> are centrally perspective."  F and  G represent f(s) and f(t) respectively. (Contributed by NM, 7-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  Q  e.  A  /\  P  =/=  Q ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( S  =/=  T 
 /\  -.  U  .<_  ( S  .\/  T )
 ) ) )  ->  ( ( S  .\/  F )  ./\  ( T  .\/  G ) )  .<_  ( P  .\/  Q )
 )
 
Theoremcdleme14 29712 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, "<s,t,p> and <f(s),f(t),q> ... are axially perspective." We apply dalaw 29325 to cdleme13 29711. 
F and  G represent f(s) and f(t) respectively. (Contributed by NM, 8-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  ./\  ( F  .\/  G ) )  .<_  ( ( ( T  .\/  P )  ./\  ( G  .\/  Q ) )  .\/  ( ( P  .\/  S )  ./\  ( Q  .\/  F ) ) ) )
 
Theoremcdleme15a 29713 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((s  \/ p)  /\ (f(s)  \/ q))  \/ ((t  \/ p)  /\ (f(t)  \/ q))=((p  \/ s1)  /\ (q  \/ s1))  \/ ((p  \/ t1)  /\ (q 
\/ t1)). We represent f(s), f(t), s1, and t1 with  F,  G,  C, and  X respectively. The order of our operations is slightly different. (Contributed by NM, 9-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  X  =  ( ( P  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T )
 ) )  ->  (
 ( ( T  .\/  P )  ./\  ( G  .\/  Q ) )  .\/  ( ( P  .\/  S )  ./\  ( Q  .\/  F ) ) )  =  ( ( ( P  .\/  X )  ./\  ( Q  .\/  X ) )  .\/  ( ( P  .\/  C )  ./\  ( Q  .\/  C ) ) ) )
 
Theoremcdleme15b 29714 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, (p  \/ s1)  /\ (q  \/ s1)=s1. We represent s1 with  C. (Contributed by NM, 10-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  X  =  ( ( P  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T )
 ) )  ->  (
 ( P  .\/  C )  ./\  ( Q  .\/  C ) )  =  C )
 
Theoremcdleme15c 29715 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, ((p  \/ s1)  /\ (q  \/ s1))  \/ ((p  \/ t1)  /\ (q  \/ t1))=s1  \/ t1.  C and  X represent s1 and t1 respectively. The order of our operations is slightly different. (Contributed by NM, 10-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  X  =  ( ( P  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T )
 ) )  ->  (
 ( ( P  .\/  X )  ./\  ( Q  .\/  X ) )  .\/  ( ( P  .\/  C )  ./\  ( Q  .\/  C ) ) )  =  ( X  .\/  C ) )
 
Theoremcdleme15d 29716 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, s1  \/ t1  <_ w.  C and  X represent s1 and t1 respectively. The order of our operations is slightly different. (Contributed by NM, 10-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   &    |-  X  =  ( ( P  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T )
 ) )  ->  ( X  .\/  C )  .<_  W )
 
Theoremcdleme15 29717 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, showing, in their notation, (s  \/ t)  /\ (f(s)  \/ f(t))  <_ w. We use  F,  G for f(s), f(t) respectively. (Contributed by NM, 10-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  ./\  ( F  .\/  G ) )  .<_  W )
 
Theoremcdleme16b 29718 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. 
F and  G represent f(s) and f(t) respectively. It is unclear how this follows from s  \/ u  =/= t  \/ u, as the authors state, and we used a different proof. (Note: the antecedent  -.  T  .<_  ( P 
.\/  Q ) is not used.) (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  F  =/=  G )
 
Theoremcdleme16c 29719 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 2nd part of 3rd sentence.  F and  G represent f(s) and f(t) respectively. We show, in their notation, s  \/ t 
\/ f(s)  \/ f(t)=s  \/ t  \/ u. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  .\/  ( F  .\/  G ) )  =  ( ( S  .\/  T )  .\/  U )
 )
 
Theoremcdleme16d 29720 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence.  F and  G represent f(s) and f(t) respectively. We show, in their notation, (s  \/ t)  /\ (f(s)  \/ f(t)) is an atom. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  ./\  ( F  .\/  G ) )  e.  A )
 
Theoremcdleme16e 29721 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence.  F and  G represent f(s) and f(t) respectively. We show, in their notation, (s  \/ t)  /\ (f(s)  \/ f(t))=(s  \/ t)  /\ w. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  ./\  ( F  .\/  G ) )  =  ( ( S  .\/  T )  ./\  W )
 )
 
Theoremcdleme16f 29722 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, 3rd part of 3rd sentence.  F and  G represent f(s) and f(t) respectively. We show, in their notation, (s  \/ t)  /\ (f(s)  \/ f(t))=(f(s)  \/ f(t))  /\ w. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  ./\  ( F  .\/  G ) )  =  ( ( F  .\/  G )  ./\  W )
 )
 
Theoremcdleme16g 29723 Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph on p. 114, Eq. (1).  F and  G represent f(s) and f(t) respectively. We show, in their notation, (s 
\/ t)  /\ w=(f(s)  \/ f(t))  /\ w. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  ( ( S  .\/  T )  ./\  W )  =  ( ( F  .\/  G )  ./\  W )
 )
 
Theoremcdleme16 29724 Part of proof of Lemma E in [Crawley] p. 113, conclusion of 3rd paragraph on p. 114.  F and  G represent f(s) and f(t) respectively. We show, in their notation, (s  \/ t)  /\ w=(f(s)  \/ f(t))  /\ w, whether or not u  <_ s  \/ t. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  S  =/=  T ) )  /\  ( -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q ) ) )  ->  ( ( S  .\/  T )  ./\  W )  =  ( ( F  .\/  G )  ./\  W )
 )
 
Theoremcdleme17a 29725 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph.  F,  G, and  C represent f(s), fs(p), and s1 respectively. We show, in their notation, fs(p)=(p  \/ q)  /\ (q  \/ s1). (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  G  =  ( ( P  .\/  Q )  ./\  ( Q  .\/  C ) ) )
 
Theoremcdleme17b 29726 Lemma leading to cdleme17c 29727. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
 .\/  Q ) ) ) 
 ->  -.  C  .<_  ( P 
 .\/  Q ) )
 
Theoremcdleme17c 29727 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph.  C represents s1. We show, in their notation, (p  \/ q)  /\ (q  \/ s1)=q. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  C  =  ( ( P  .\/  S )  ./\  W )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  S  e.  A  /\  -.  S  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( ( P  .\/  Q )  ./\  ( Q  .\/  C ) )  =  Q )
 
Theoremcdleme17d1 29728 Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph.  F,  G represent f(s), fs(p) respectively. We show, in their notation, fs(p)=q. (Contributed by NM, 11-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  G  =  Q )
 
Theoremcdleme0nex 29729* Part of proof of Lemma E in [Crawley] p. 114, 4th line of 4th paragraph. Whenever (in their terminology) p  \/ q/0 (i.e. the sublattice from 0 to p  \/ q) contains precisely three atoms, any atom not under w must equal either p or q. (In case of 3 atoms, one of them must be u - see cdleme0a 29650- which is under w, so the only 2 left not under w are p and q themselves.) Note that by cvlsupr2 28783, our  ( P  .\/  r )  =  ( Q  .\/  r ) is a shorter way to express  r  =/=  P  /\  r  =/=  Q  /\  r  .<_  ( P 
.\/  Q ). Thus the negated existential condition states there are no atoms different from p or q that are also not under w. (Contributed by NM, 12-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  R  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) 
 /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  ->  ( R  =  P  \/  R  =  Q )
 )
 
Theoremcdleme18a 29730 Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph.  F,  G represent f(s), fs(q) respectively. We show  -. fs(q)  <_ w. (Contributed by NM, 12-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( Q  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  -.  G  .<_  W )
 
Theoremcdleme18b 29731 Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph.  F,  G represent f(s), fs(q) respectively. We show  -. fs(q)  =/= q. (Contributed by NM, 12-Oct-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( Q  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  G  =/=  Q )
 
Theoremcdleme18c 29732* Part of proof of Lemma E in [Crawley] p. 114, 2nd sentence of 4th paragraph.  F,  G represent f(s), fs(q) respectively. We show  -. fs(q) = p whenever p  \/ q has three atoms under it (implied by the negated existential condition). (Contributed by NM, 10-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( Q  .\/  S )  ./\  W )
 ) )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
 ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P 
 .\/  r )  =  ( Q  .\/  r
 ) ) ) ) 
 ->  G  =  P )
 
Theoremcdleme22gb 29733 Utility lemma for Lemma E in [Crawley] p. 115. (Contributed by NM, 5-Dec-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
 ) )   &    |-  B  =  (
 Base `  K )   =>    |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A ) 
 /\  ( R  e.  A  /\  S  e.  A ) )  ->  G  e.  B )
 
Theoremcdleme18d 29734* Part of proof of Lemma E in [Crawley] p. 114, 4th sentence of 4th paragraph.  F,  G,  D,  E represent f(s), fs(r), f(t), ft(r) respectively. We show fs(r)=ft(r) for all possible r (which must equal p or q in the case of exactly 3 atoms in p  \/ q/0 i.e. when  -.  E. r  e.  A...). (Contributed by NM, 12-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( R  .\/  S )  ./\  W )
 ) )   &    |-  D  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  E  =  ( ( P  .\/  Q )  ./\  ( D  .\/  ( ( R  .\/  T )  ./\  W )
 ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( R  e.  A  /\  -.  R  .<_  W ) 
 /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) ) 
 /\  ( P  =/=  Q 
 /\  ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P 
 .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q ) )  /\  -.  E. r  e.  A  ( -.  r  .<_  W  /\  ( P  .\/  r )  =  ( Q  .\/  r ) ) ) )  ->  G  =  E )
 
Theoremcdlemesner 29735 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 13-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   &    |-  H  =  ( LHyp `  K )   =>    |-  (
 ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A ) 
 /\  ( R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P 
 .\/  Q ) ) ) 
 ->  S  =/=  R )
 
Theoremcdlemedb 29736 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma.  D represents s2. (Contributed by NM, 20-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  B  =  ( Base `  K )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A ) )  ->  D  e.  B )
 
Theoremcdlemeda 29737 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma.  D represents s2. (Contributed by NM, 13-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )
 ) )  ->  D  e.  A )
 
Theoremcdlemednpq 29738 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma.  D represents s2. (Contributed by NM, 18-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  -.  D  .<_  ( P  .\/  Q ) )
 
TheoremcdlemednuN 29739 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma.  D represents s2. (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  R  .<_  ( P  .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  D  =/=  U )
 
Theoremcdleme20zN 29740 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( S  =/=  T  /\  -.  R  .<_  ( S 
 .\/  T ) ) ) 
 ->  ( ( S  .\/  R )  ./\  T )  =  ( 0. `  K ) )
 
Theoremcdleme20y 29741 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma. (Contributed by NM, 17-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   =>    |-  (
 ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( S  =/=  T  /\  -.  R  .<_  ( S 
 .\/  T ) ) ) 
 ->  ( ( S  .\/  R )  ./\  ( T  .\/  R ) )  =  R )
 
Theoremcdleme19a 29742 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line.  D represents s2. In their notation, we prove that if r  <_ s  \/ t, then s2=(s  \/ t)  /\ w. (Contributed by NM, 13-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   =>    |-  (
 ( K  e.  HL  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A )  /\  ( R  .<_  ( P 
 .\/  Q )  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( S  .\/  T )
 ) )  ->  D  =  ( ( S  .\/  T )  ./\  W )
 )
 
Theoremcdleme19b 29743 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line.  D,  F,  G represent s2, f(s), f(t). In their notation, we prove that if r 
<_ s  \/ t, then s2  <_ f(s)  \/ f(t). (Contributed by NM, 13-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  R  e.  A )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  R  .<_  ( S 
 .\/  T ) ) ) )  ->  D  .<_  ( F  .\/  G )
 )
 
Theoremcdleme19c 29744 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, 1st line.  D,  F represent s2, f(s). We prove f(s)  =/= s2. (Contributed by NM, 13-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) 
 /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( R  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) ) )  ->  F  =/=  D )
 
Theoremcdleme19d 29745 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114.  D,  F,  G represent s2, f(s), f(t). We prove f(s)  \/ s2 = f(s)  \/ f(t). (Contributed by NM, 14-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  R  e.  A )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  R  .<_  ( S 
 .\/  T ) ) ) )  ->  ( F  .\/  D )  =  ( F  .\/  G )
 )
 
Theoremcdleme19e 29746 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 2.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We prove f(s)  \/ s2=f(t)  \/ t2. (Contributed by NM, 14-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  R  e.  A )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  R  .<_  ( S 
 .\/  T ) ) ) )  ->  ( F  .\/  D )  =  ( G  .\/  Y )
 )
 
Theoremcdleme19f 29747 Part of proof of Lemma E in [Crawley] p. 113, 5th paragraph on p. 114, line 3.  D,  F,  N,  Y,  G,  O represent s2, f(s), fs(r), t2, f(t), ft(r). We prove that if r  <_ s  \/ t, then ft(r) = ft(r). (Contributed by NM, 14-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  R  e.  A )  /\  ( ( P  =/=  Q  /\  S  =/=  T )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q ) )  /\  ( R 
 .<_  ( P  .\/  Q )  /\  R  .<_  ( S 
 .\/  T ) ) ) )  ->  N  =  O )
 
Theoremcdleme20aN 29748 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 14-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  D )  =  ( ( ( S  .\/  R )  .\/  T )  ./\  W ) )
 
Theoremcdleme20bN 29749 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We show v  \/ s2 = v  \/ t2. (Contributed by NM, 15-Nov-2012.) (New usage is discouraged.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( R  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) ) )  ->  ( V  .\/  D )  =  ( V  .\/  Y ) )
 
Theoremcdleme20c 29750 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 15-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  T  e.  A )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P 
 .\/  Q ) ) ) 
 ->  ( D  .\/  Y )  =  ( (
 ( R  .\/  S )  .\/  T )  ./\  W ) )
 
Theoremcdleme20d 29751 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, second line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 17-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )
 )  /\  R  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( F  .\/  G )  ./\  ( D  .\/  Y ) )  =  V )
 
Theoremcdleme20e 29752 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We show <f(s),s2,s> and <f(t),t2,t> are centrally perspective. (Contributed by NM, 17-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )
 )  /\  R  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( F  .\/  G )  ./\  ( D  .\/  Y ) )  .<_  ( S 
 .\/  T ) )
 
Theoremcdleme20f 29753 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, 4th line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We show <f(s),s2,s> and <f(t),t2,t> are axially perspective. (Contributed by NM, 17-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )
 )  /\  R  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( F  .\/  D )  ./\  ( G  .\/  Y ) )  .<_  ( ( ( D  .\/  S )  ./\  ( Y  .\/  T ) )  .\/  (
 ( S  .\/  F )  ./\  ( T  .\/  G ) ) ) )
 
Theoremcdleme20g 29754 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 18-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )
 )  /\  R  .<_  ( P  .\/  Q )
 ) )  ->  (
 ( ( D  .\/  S )  ./\  ( Y  .\/  T ) )  .\/  ( ( S  .\/  F )  ./\  ( T  .\/  G ) ) )  =  ( ( ( S  .\/  R )  ./\  ( T  .\/  R ) )  .\/  ( ( S  .\/  U )  ./\  ( T  .\/  U ) ) ) )
 
Theoremcdleme20h 29755 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 18-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  ( -.  R  .<_  ( S  .\/  T )  /\  -.  U  .<_  ( S  .\/  T ) ) ) ) 
 ->  ( ( ( S 
 .\/  R )  ./\  ( T  .\/  R ) ) 
 .\/  ( ( S 
 .\/  U )  ./\  ( T  .\/  U ) ) )  =  ( R 
 .\/  U ) )
 
Theoremcdleme20i 29756 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We show (f(s)  \/ s2)  /\ (f(t)  \/ t2)  <_ p  \/ q. (Contributed by NM, 18-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  ( -.  R  .<_  ( S  .\/  T )  /\  -.  U  .<_  ( S  .\/  T ) ) ) ) 
 ->  ( ( F  .\/  D )  ./\  ( G  .\/  Y ) )  .<_  ( P  .\/  Q )
 )
 
Theoremcdleme20j 29757 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114.  D,  F,  Y,  G represent s2, f(s), t2, f(t). We show s2  =/= t2. (Contributed by NM, 18-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  -.  R  .<_  ( S  .\/  T ) ) )  ->  D  =/=  Y )
 
Theoremcdleme20k 29758 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, antepenultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t). (Contributed by NM, 20-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q )
 ) )  ->  ( F  .\/  D )  =/=  ( P  .\/  Q ) )
 
Theoremcdleme20l1 29759 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( R  e.  A  /\  S  e.  A  /\  -.  S  .<_  W )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q )
 ) )  ->  ( F  .\/  D )  e.  ( LLines `  K )
 )
 
Theoremcdleme20l2 29760 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  ( -.  R  .<_  ( S  .\/  T )  /\  -.  U  .<_  ( S  .\/  T ) ) ) ) 
 ->  ( ( F  .\/  D )  ./\  ( G  .\/  Y ) )  e.  A )
 
Theoremcdleme20l 29761 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line.  D,  F,  Y,  G represent s2, f(s), t2, f(t) respectively. (Contributed by NM, 20-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  ( -.  R  .<_  ( S  .\/  T )  /\  -.  U  .<_  ( S  .\/  T ) ) ) ) 
 ->  ( ( F  .\/  D )  ./\  ( G  .\/  Y ) )  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) ) )
 
Theoremcdleme20m 29762 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 114, penultimate line. 
D,  F,  N,  Y,  G,  O represent s2, f(s), fs(r), t2, f(t), ft(r) respectively. We prove that if  -. r  <_ s  \/ t and  -. u  <_ s  \/ t, then fs(r) = ft(r). (Contributed by NM, 20-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  ( -.  R  .<_  ( S  .\/  T )  /\  -.  U  .<_  ( S  .\/  T ) ) ) ) 
 ->  N  =  O )
 
Theoremcdleme20 29763 Combine cdleme19f 29747 and cdleme20m 29762 to eliminate  -.  R  .<_  ( S  .\/  T ) condition. (Contributed by NM, 28-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  V  =  ( ( S  .\/  T )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( S  e.  A  /\  -.  S  .<_  W ) 
 /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  (
 ( P  =/=  Q  /\  S  =/=  T ) 
 /\  ( -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q ) )  /\  -.  U  .<_  ( S  .\/  T ) ) )  ->  N  =  O )
 
Theoremcdleme21a 29764 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  -.  S  .<_  ( P 
 .\/  Q ) )  /\  ( z  e.  A  /\  ( P  .\/  z
 )  =  ( S 
 .\/  z ) ) )  ->  S  =/=  z )
 
Theoremcdleme21b 29765 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  A  =  (
 Atoms `  K )   =>    |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  (
 z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) 
 ->  -.  z  .<_  ( P 
 .\/  Q ) )
 
Theoremcdleme21c 29766 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q ) )  /\  (
 z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) 
 ->  -.  U  .<_  ( S 
 .\/  z ) )
 
Theoremcdleme21at 29767 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( S  e.  A  /\  P  =/=  Q  /\  -.  S  .<_  ( P 
 .\/  Q ) )  /\  U  .<_  ( S  .\/  T ) )  /\  (
 z  e.  A  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) 
 ->  T  =/=  z )
 
Theoremcdleme21ct 29768 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   =>    |-  (
 ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( P  =/=  Q  /\  -.  S  .<_  ( P  .\/  Q )  /\  U  .<_  ( S  .\/  T )
 ) )  /\  (
 ( z  e.  A  /\  -.  z  .<_  W ) 
 /\  ( P  .\/  z )  =  ( S  .\/  z ) ) )  ->  -.  U  .<_  ( T  .\/  z )
 )
 
Theoremcdleme21d 29769 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line.  D,  F,  N,  E,  B,  Z represent s2, f(s), fs(r), z2, f(z), fz(r) respectively. We prove fs(r) = fz(r). (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  B  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  E  =  ( ( R  .\/  z )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( B  .\/  E ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) 
 /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  (
 ( R  e.  A  /\  -.  R  .<_  W ) 
 /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  ( P  =/=  Q  /\  ( -.  S  .<_  ( P  .\/  Q )  /\  R  .<_  ( P  .\/  Q )
 )  /\  ( (
 z  e.  A  /\  -.  z  .<_  W )  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )  ->  N  =  Z )
 
Theoremcdleme21e 29770 Part of proof of Lemma E in [Crawley] p. 113, last paragraph on p. 115, 3rd line.  Y,  G,  O,  E,  B,  Z represent s2, f(s), fs(r), z2, f(z), fz(r) respectively. We prove that if u  <_ s  \/ z, then ft(r) = fz(r). (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  B  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  E  =  ( ( R  .\/  z )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( B  .\/  E ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P 
 .\/  Q ) ) ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P 
 .\/  Q )  /\  U  .<_  ( S  .\/  T ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )  ->  O  =  Z )
 
Theoremcdleme21f 29771 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  B  =  ( ( z  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  z )  ./\  W ) ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  E  =  ( ( R  .\/  z )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  Z  =  ( ( P  .\/  Q )  ./\  ( B  .\/  E ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P 
 .\/  Q ) ) ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P 
 .\/  Q )  /\  U  .<_  ( S  .\/  T ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )  ->  N  =  O )
 
Theoremcdleme21g 29772 Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P 
 .\/  Q ) ) ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P 
 .\/  Q )  /\  U  .<_  ( S  .\/  T ) )  /\  ( ( z  e.  A  /\  -.  z  .<_  W )  /\  ( P  .\/  z )  =  ( S  .\/  z ) ) ) )  ->  N  =  O )
 
Theoremcdleme21h 29773* Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &    |-  U  =  ( ( P  .\/  Q )  ./\  W )   &    |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
 ) )   &    |-  G  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
 ) )   &    |-  D  =  ( ( R  .\/  S )  ./\  W )   &    |-  Y  =  ( ( R  .\/  T )  ./\  W )   &    |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  D ) )   &    |-  O  =  ( ( P  .\/  Q )  ./\  ( G  .\/  Y ) )   =>    |-  ( ( ( ( K  e.  HL  /\  W  e.  H ) 
 /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) 
 /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) 
 /\  ( P  =/=  Q 
 /\  -.  S  .<_  ( P  .\/  Q )  /\  -.  T  .<_  ( P 
 .\/  Q ) ) ) 
 /\  ( ( R  e.  A  /\  -.  R  .<_  W )  /\  ( R  .<_  ( P 
 .\/  Q )  /\  U  .<_  ( S  .\/  T ) ) ) ) 
 ->  ( E. z  e.  A  ( -.  z  .<_  W  /\  ( P 
 .\/  z )  =  ( S  .\/  z
 ) )  ->  N  =  O ) )
 
Theoremcdleme21i 29774* Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 29-Nov-2012.)
 |-  .<_  =  ( le `  K )   &    |- 
 .\/  =  ( join `  K )   &    |-  ./\  =  ( meet `  K )   &    |-  A  =  ( Atoms `  K )   &    |-  H  =  ( LHyp `  K )   &