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Theorem List for Metamath Proof Explorer - 9801-9900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem9lt10 9801 9 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  9  <  10
 
Theorem8lt10 9802 8 is less than 10. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  8  <  10
 
Theorem7lt10 9803 7 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  7  <  10
 
Theorem6lt10 9804 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  6  <  10
 
Theorem5lt10 9805 5 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  5  <  10
 
Theorem4lt10 9806 4 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  4  <  10
 
Theorem3lt10 9807 3 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  3  <  10
 
Theorem2lt10 9808 2 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  2  <  10
 
Theorem1lt10 9809 1 is less than 10. (Contributed by NM, 7-Nov-2012.) (Revised by Mario Carneiro, 9-Mar-2015.)
 |-  1  <  10
 
Theorem1ne2 9810 1 is not equal to 2. (Contributed by NM, 19-Oct-2012.)
 |-  1  =/=  2
 
Theoremhalfgt0 9811 One-half is greater than zero. (Contributed by NM, 24-Feb-2005.)
 |-  0  <  ( 1 
 /  2 )
 
Theoremhalflt1 9812 One-half is less than one. (Contributed by NM, 24-Feb-2005.)
 |-  ( 1  /  2
 )  <  1
 
Theorem1mhlfehlf 9813 Prove that 1 - 1/2 = 1/2. (Contributed by David A. Wheeler, 4-Jan-2017.)
 |-  ( 1  -  (
 1  /  2 )
 )  =  ( 1 
 /  2 )
 
Theorem8th4div3 9814 An eighth of four thirds is a sixth. (Contributed by Paul Chapman, 24-Nov-2007.)
 |-  ( ( 1  / 
 8 )  x.  (
 4  /  3 )
 )  =  ( 1 
 /  6 )
 
Theoremhalfpm6th 9815 One half plus or minus one sixth. (Contributed by Paul Chapman, 17-Jan-2008.)
 |-  ( ( ( 1 
 /  2 )  -  ( 1  /  6
 ) )  =  ( 1  /  3 ) 
 /\  ( ( 1 
 /  2 )  +  ( 1  /  6
 ) )  =  ( 2  /  3 ) )
 
Theoremhalfcl 9816 Closure of half of a number (frequently used special case). (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  CC  ->  ( A  /  2
 )  e.  CC )
 
Theoremrehalfcl 9817 Real closure of half. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  RR  ->  ( A  /  2
 )  e.  RR )
 
Theoremhalf0 9818 Half of a number is zero iff the number is zero. (Contributed by NM, 20-Apr-2006.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  =  0  <->  A  =  0 )
 )
 
Theorem2halves 9819 Two halves make a whole. (Contributed by NM, 11-Apr-2005.)
 |-  ( A  e.  CC  ->  ( ( A  / 
 2 )  +  ( A  /  2 ) )  =  A )
 
Theoremhalfpos2 9820 A number is positive iff its half is positive. (Contributed by NM, 10-Apr-2005.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  0  <  ( A  / 
 2 ) ) )
 
Theoremhalfpos 9821 A positive number is greater than its half. (Contributed by NM, 28-Oct-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  RR  ->  ( 0  <  A  <->  ( A  /  2 )  <  A ) )
 
Theoremhalfnneg2 9822 A number is nonnegative iff its half is nonnegative. (Contributed by NM, 9-Dec-2005.)
 |-  ( A  e.  RR  ->  ( 0  <_  A  <->  0 
 <_  ( A  /  2
 ) ) )
 
Theoremhalfaddsubcl 9823 Closure of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  +  B ) 
 /  2 )  e. 
 CC  /\  ( ( A  -  B )  / 
 2 )  e.  CC ) )
 
Theoremhalfaddsub 9824 Sum and difference of half-sum and half-difference. (Contributed by Paul Chapman, 12-Oct-2007.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( A  +  B )  /  2 )  +  ( ( A  -  B )  /  2
 ) )  =  A  /\  ( ( ( A  +  B )  / 
 2 )  -  (
 ( A  -  B )  /  2 ) )  =  B ) )
 
Theoremlt2halves 9825 A sum is less than the whole if each term is less than half. (Contributed by NM, 13-Dec-2006.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <  ( C  /  2 ) 
 /\  B  <  ( C  /  2 ) ) 
 ->  ( A  +  B )  <  C ) )
 
Theoremaddltmul 9826 Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
 |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
 2  <  A  /\  2  <  B ) ) 
 ->  ( A  +  B )  <  ( A  x.  B ) )
 
Theoremnominpos 9827* There is no smallest positive real number. (Contributed by NM, 28-Oct-2004.)
 |- 
 -.  E. x  e.  RR  ( 0  <  x  /\  -.  E. y  e. 
 RR  ( 0  < 
 y  /\  y  <  x ) )
 
Theoremavglt1 9828 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A  <  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavglt2 9829 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <-> 
 ( ( A  +  B )  /  2
 )  <  B )
 )
 
Theoremavgle1 9830 Ordering property for average. (Contributed by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <->  A  <_  ( ( A  +  B )  / 
 2 ) ) )
 
Theoremavgle2 9831 Ordering property for average. (Contributed by Jeff Hankins, 15-Sep-2013.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <_  B  <-> 
 ( ( A  +  B )  /  2
 )  <_  B )
 )
 
Theoremavgle 9832 The average of two numbers is less than or equal to at least one of them. (Contributed by NM, 9-Dec-2005.) (Revised by Mario Carneiro, 28-May-2014.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B ) 
 /  2 )  <_  A  \/  ( ( A  +  B )  / 
 2 )  <_  B ) )
 
Theorem2timesd 9833 Two times a number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 2  x.  A )  =  ( A  +  A ) )
 
Theoremtimes2d 9834 A number times 2. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  2 )  =  ( A  +  A ) )
 
Theoremhalfcld 9835 Closure of half of a number (frequently used special case). (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 CC )
 
Theorem2halvesd 9836 Two halves make a whole. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 ( A  /  2
 )  +  ( A 
 /  2 ) )  =  A )
 
Theoremrehalfcld 9837 Real closure of half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( A  /  2 )  e. 
 RR )
 
Theoremlt2halvesd 9838 A sum is less than the whole if each term is less than half. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( C  /  2
 ) )   &    |-  ( ph  ->  B  <  ( C  / 
 2 ) )   =>    |-  ( ph  ->  ( A  +  B )  <  C )
 
Theoremrehalfcli 9839 Half a real number is real. Inference form. (Contributed by David Moews, 28-Feb-2017.)
 |-  A  e.  RR   =>    |-  ( A  / 
 2 )  e.  RR
 
5.4.5  The Archimedean property
 
Theoremnnunb 9840* The set of natural numbers is unbounded above. Theorem I.28 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
 |- 
 -.  E. x  e.  RR  A. y  e.  NN  (
 y  <  x  \/  y  =  x )
 
Theoremarch 9841* Archimedean property of real numbers. For any real number, there is an integer greater than it. Theorem I.29 of [Apostol] p. 26. (Contributed by NM, 21-Jan-1997.)
 |-  ( A  e.  RR  ->  E. n  e.  NN  A  <  n )
 
Theoremnnrecl 9842* There exists a natural number whose reciprocal is less than a given positive real. Exercise 3 of [Apostol] p. 28. (Contributed by NM, 8-Nov-2004.)
 |-  ( ( A  e.  RR  /\  0  <  A )  ->  E. n  e.  NN  ( 1  /  n )  <  A )
 
Theorembndndx 9843* A bounded real sequence  A ( k ) is less than or equal to at least one of its indices. (Contributed by NM, 18-Jan-2008.)
 |-  ( E. x  e. 
 RR  A. k  e.  NN  ( A  e.  RR  /\  A  <_  x )  ->  E. k  e.  NN  A  <_  k )
 
5.4.6  Nonnegative integers (as a subset of complex numbers)
 
Syntaxcn0 9844 Extend class notation to include the class of nonnegative integers.
 class  NN0
 
Definitiondf-n0 9845 Define the set of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  =  ( NN  u.  { 0 } )
 
Theoremelnn0 9846 Nonnegative integers expressed in terms of naturals and zero. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  ( A  e.  NN0  <->  ( A  e.  NN  \/  A  =  0 )
 )
 
Theoremnnssnn0 9847 Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN  C_  NN0
 
Theoremnn0ssre 9848 Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.)
 |- 
 NN0  C_  RR
 
Theoremnn0sscn 9849 Nonnegative integers are a subset of the complex numbers.) (Contributed by NM, 9-May-2004.)
 |- 
 NN0  C_  CC
 
Theoremnn0ex 9850 The set of nonnegative integers exists. (Contributed by NM, 18-Jul-2004.)
 |- 
 NN0  e.  _V
 
Theoremnnnn0 9851 A natural number is a nonnegative integer. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN  ->  A  e.  NN0 )
 
Theoremnnnn0i 9852 A natural number is a nonnegative integer. (Contributed by NM, 20-Jun-2005.)
 |-  N  e.  NN   =>    |-  N  e.  NN0
 
Theoremnn0re 9853 A nonnegative integer is a real number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  RR )
 
Theoremnn0cn 9854 A nonnegative integer is a complex number. (Contributed by NM, 9-May-2004.)
 |-  ( A  e.  NN0  ->  A  e.  CC )
 
Theoremnn0rei 9855 A nonnegative integer is a real number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  RR
 
Theoremnn0cni 9856 A nonnegative integer is a complex number. (Contributed by NM, 14-May-2003.)
 |-  A  e.  NN0   =>    |-  A  e.  CC
 
Theoremdfn2 9857 The set of natural numbers (positive integers) defined in terms of nonnegative integers. (Contributed by NM, 23-Sep-2007.) (Proof shortened by Mario Carneiro, 13-Feb-2013.)
 |- 
 NN  =  ( NN0  \  { 0 } )
 
Theoremelnnne0 9858 The natural number property expressed in terms of difference from zero. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  N  =/=  0 ) )
 
Theorem0nn0 9859 0 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  0  e.  NN0
 
Theorem1nn0 9860 1 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  1  e.  NN0
 
Theorem2nn0 9861 2 is a nonnegative integer. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  2  e.  NN0
 
Theorem3nn0 9862 3 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  3  e.  NN0
 
Theorem4nn0 9863 4 is a nonnegative integer. (Contributed by Mario Carneiro, 18-Feb-2014.)
 |-  4  e.  NN0
 
Theorem5nn0 9864 5 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  5  e.  NN0
 
Theorem6nn0 9865 6 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  6  e.  NN0
 
Theorem7nn0 9866 7 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  7  e.  NN0
 
Theorem8nn0 9867 8 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  8  e.  NN0
 
Theorem9nn0 9868 9 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |-  9  e.  NN0
 
Theorem10nn0 9869 10 is a nonnegative integer. (Contributed by Mario Carneiro, 19-Apr-2015.)
 |- 
 10  e.  NN0
 
Theoremnn0ge0 9870 A nonnegative integer is greater than or equal to zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  -> 
 0  <_  N )
 
Theoremnn0nlt0 9871 A nonnegative integer is not less than zero. (Contributed by NM, 9-May-2004.) (Revised by Mario Carneiro, 27-May-2016.)
 |-  ( A  e.  NN0  ->  -.  A  <  0 )
 
Theoremnn0ge0i 9872 Nonnegative integers are nonnegative. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  0  <_  N
 
Theoremnn0le0eq0 9873 A nonnegative integer is less than or equal to zero iff it is equal to zero. (Contributed by NM, 9-Dec-2005.)
 |-  ( N  e.  NN0  ->  ( N  <_  0  <->  N  =  0
 ) )
 
Theoremnnnn0addcl 9874 A natural number plus a nonnegative integer is a natural number. (Contributed by NM, 20-Apr-2005.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN )
 
Theoremnn0nnaddcl 9875 A nonnegative integer plus a natural number is a natural number. (Contributed by NM, 22-Dec-2005.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN )  ->  ( M  +  N )  e.  NN )
 
Theoremun0addcl 9876 If  S is closed under addition, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  +  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  +  N )  e.  T )
 
Theoremun0mulcl 9877 If  S is closed under multiplication, then so is  S  u.  { 0 }. (Contributed by Mario Carneiro, 17-Jul-2014.)
 |-  ( ph  ->  S  C_ 
 CC )   &    |-  T  =  ( S  u.  { 0 } )   &    |-  ( ( ph  /\  ( M  e.  S  /\  N  e.  S ) )  ->  ( M  x.  N )  e.  S )   =>    |-  ( ( ph  /\  ( M  e.  T  /\  N  e.  T )
 )  ->  ( M  x.  N )  e.  T )
 
Theoremnn0addcl 9878 Closure of addition of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  +  N )  e.  NN0 )
 
Theoremnn0mulcl 9879 Closure of multiplication of nonnegative integers. (Contributed by NM, 22-Jul-2004.) (Proof shortened by Mario Carneiro, 17-Jul-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  x.  N )  e.  NN0 )
 
Theoremnn0addcli 9880 Closure of addition of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  +  N )  e.  NN0
 
Theoremnn0mulcli 9881 Closure of multiplication of nonnegative integers, inference form. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( M  x.  N )  e.  NN0
 
Theoremnn0p1nn 9882 A nonnegative integer plus 1 is a natural number. (Contributed by Raph Levien, 30-Jun-2006.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN )
 
Theorempeano2nn0 9883 Second Peano postulate for nonnegative integers. (Contributed by NM, 9-May-2004.)
 |-  ( N  e.  NN0  ->  ( N  +  1
 )  e.  NN0 )
 
Theoremnnm1nn0 9884 A natural number minus 1 is a nonnegative integer. (Contributed by Jason Orendorff, 24-Jan-2007.) (Revised by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN  ->  ( N  -  1
 )  e.  NN0 )
 
Theoremelnn0nn 9885 The nonnegative integer property expressed in terms of natural numbers. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( N  e.  NN0  <->  ( N  e.  CC  /\  ( N  +  1 )  e.  NN ) )
 
Theoremelnnnn0 9886 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-May-2004.)
 |-  ( N  e.  NN  <->  ( N  e.  CC  /\  ( N  -  1 )  e. 
 NN0 ) )
 
Theoremelnnnn0b 9887 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 1-Sep-2005.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  0  <  N ) )
 
Theoremelnnnn0c 9888 The natural number property expressed in terms of nonnegative integers. (Contributed by NM, 10-Jan-2006.)
 |-  ( N  e.  NN  <->  ( N  e.  NN0  /\  1  <_  N ) )
 
Theoremnn0addge1 9889 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( A  +  N )
 )
 
Theoremnn0addge2 9890 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  ( ( A  e.  RR  /\  N  e.  NN0 )  ->  A  <_  ( N  +  A )
 )
 
Theoremnn0addge1i 9891 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( A  +  N )
 
Theoremnn0addge2i 9892 A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005.)
 |-  A  e.  RR   &    |-  N  e.  NN0   =>    |-  A  <_  ( N  +  A )
 
Theoremnn0sub 9893 Subtraction of nonnegative integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.)
 |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  ->  ( M  <_  N  <->  ( N  -  M )  e.  NN0 ) )
 
Theoremnn0le2xi 9894 A nonnegative integer is less than or equal to twice itself. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  N  e.  NN0   =>    |-  N  <_  ( 2  x.  N )
 
Theoremnn0lele2xi 9895 'Less than or equal to' implies 'less than or equal to twice' for nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.)
 |-  M  e.  NN0   &    |-  N  e.  NN0   =>    |-  ( N  <_  M  ->  N  <_  ( 2  x.  M ) )
 
Theoremnn0supp 9896 Two ways to write the support of a function on  NN0. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  ( F : I --> NN0  ->  ( `' F " ( _V  \  {
 0 } ) )  =  ( `' F " NN ) )
 
Theoremnnnn0d 9897 A natural number is a nonnegative integer. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN )   =>    |-  ( ph  ->  A  e.  NN0 )
 
Theoremnn0red 9898 A nonnegative integer is a real number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  RR )
 
Theoremnn0cnd 9899 A nonnegative integer is a complex number. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremnn0ge0d 9900 A nonnegative integer is greater than or equal to zero. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  NN0 )   =>    |-  ( ph  ->  0  <_  A )
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