mmtheorems42 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4101-4200 - Page 42 of 123

Type	Label	Description
Statement
Theorem	oprvconst2 4101	The value of a constant operation.
|- C e. V   =>   |- ((R e. A /\ S e. B) -> (R((A X. B) X. {C})S) = C)
Theorem	oprvex 4102	Existence of a class of operation values.
|- A e. V   &   |- B e. V   =>   |- {z | E.x e. A E.y e. B z = (xFy)} e. V
Theorem	oprssdm 4103	Domain of closure of an operation.
|- -. (/) e. S   &   |- ((x e. S /\ y e. S) -> (xFy) e. S)   =>   |- (S X. S) (_ dom F
Theorem	ndmoprg 4104	The value of an operation outside its domain.
|- ((dom F = (R X. S) /\ B e. C /\ -. (A e. R /\ B e. S)) -> (AFB) = (/))
Theorem	ndmoprcl 4105	The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is is dependent on our particular definitions of operation value, function value, and ordered pair.
|- dom F = (S X. S)   &   |- ((A e. S /\ x e. S) -> (AFx) e. S)   &   |- (/) e. S   =>   |- (AFB) e. S
Theorem	ndmopr 4106	The value of an operation outside its domain.
|- B e. V   &   |- dom F = (S X. S)   =>   |- (-. (A e. S /\ B e. S) -> (AFB) = (/))
Theorem	ndmoprrcl 4107	Reverse closure law, when an operation's domain doesn't contain the empty set.
|- B e. V   &   |- dom F = (S X. S)   &   |- -. (/) e. S   =>   |- ((AFB) e. S -> (A e. S /\ B e. S))
Theorem	ndmoprcom 4108	Any operation is commutative outside its domain.
|- B e. V   &   |- dom F = (S X. S)   &   |- A e. V   =>   |- (-. (A e. S /\ B e. S) -> (AFB) = (BFA))
Theorem	ndmoprass 4109	Any operation is associative outside its domain, if the domain doesn't contain the empty set.
|- B e. V   &   |- dom F = (S X. S)   &   |- C e. V   &   |- -. (/) e. S   =>   |- (-. (A e. S /\ B e. S /\ C e. S) -> ((AFB)FC) = (AF(BFC)))
Theorem	ndmoprdistr 4110	Any operation is distributive outside its domain, if the domain doesn't contain the empty set.
|- B e. V   &   |- dom F = (S X. S)   &   |- C e. V   &   |- -. (/) e. S   &   |- dom G = (S X. S)   =>   |- (-. (A e. S /\ B e. S /\ C e. S) -> (AG(BFC)) = ((AGB)F(AGC)))
Theorem	ndmord 4111	Elimination of redundant antecedents in an ordering law.
|- B e. V   &   |- dom F = (S X. S)   &   |- A e. V   &   |- R (_ (S X. S)   &   |- -. (/) e. S   &   |- ((A e. S /\ B e. S /\ C e. S) -> (ARB <-> (CFA)R(CFB)))   =>   |- (C e. S -> (ARB <-> (CFA)R(CFB)))
Theorem	ndmordi 4112	Elimination of redundant antecedent in an ordering law.
|- A e. V   &   |- dom F = (S X. S)   &   |- R (_ (S X. S)   &   |- -. (/) e. S   &   |- (C e. S -> (ARB <-> (CFA)R(CFB)))   =>   |- ((CFA)R(CFB) -> ARB)
Theorem	caoprcl 4113	Convert an operation closure law to class notation.
|- ((x e. S /\ y e. S) -> (xFy) e. S)   =>   |- ((A e. S /\ B e. S) -> (AFB) e. S)
Theorem	caoprcom 4114	Convert an operation commutative law to class notation.
|- A e. V   &   |- B e. V   &   |- (xFy) = (yFx)   =>   |- (AFB) = (BFA)
Theorem	caoprass 4115	Convert an operation associative law to class notation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- ((xFy)Fz) = (xF(yFz))   =>   |- ((AFB)FC) = (AF(BFC))
Theorem	caoprcan 4116	Convert an operation cancellation law to class notation.
|- C e. V   &   |- ((x e. S /\ y e. S) -> ((xFy) = (xFz) -> y = z))   =>   |- ((A e. S /\ B e. S) -> ((AFB) = (AFC) -> B = C))
Theorem	caoprord 4117	Convert an operation ordering law to class notation.
|- A e. V   &   |- B e. V   &   |- (z e. S -> (xRy <-> (zFx)R(zFy)))   =>   |- (C e. S -> (ARB <-> (CFA)R(CFB)))
Theorem	caoprord2 4118	Operation ordering law with commuted arguments.
|- A e. V   &   |- B e. V   &   |- (z e. S -> (xRy <-> (zFx)R(zFy)))   &   |- C e. V   &   |- (xFy) = (yFx)   =>   |- (C e. S -> (ARB <-> (AFC)R(BFC)))
Theorem	caoprord3 4119	Ordering law.
|- A e. V   &   |- B e. V   &   |- (z e. S -> (xRy <-> (zFx)R(zFy)))   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- D e. V   =>   |- (((B e. S /\ C e. S) /\ (AFB) = (CFD)) -> (ARC <-> DRB))
Theorem	caoprdistr 4120	Convert an operation distributive law to class notation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xG(yFz)) = ((xGy)F(xGz))   =>   |- (AG(BFC)) = ((AGB)F(AGC))
Theorem	caopr32 4121	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   =>   |- ((AFB)FC) = ((AFC)FB)
Theorem	caopr12 4122	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   =>   |- (AF(BFC)) = (BF(AFC))
Theorem	caopr31 4123	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   =>   |- ((AFB)FC) = ((CFB)FA)
Theorem	caopr13 4124	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   =>   |- (AF(BFC)) = (CF(BFA))
Theorem	caopr4 4125	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- D e. V   =>   |- ((AFB)F(CFD)) = ((AFC)F(BFD))
Theorem	caopr411 4126	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- D e. V   =>   |- ((AFB)F(CFD)) = ((CFB)F(AFD))
Theorem	caopr42 4127	Rearrange arguments in a commutative, associative operation.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- D e. V   =>   |- ((AFB)F(CFD)) = ((AFC)F(DFB))
Theorem	caoprdistrr 4128	Reverse distributive law.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xGy) = (yGx)   &   |- (xG(yFz)) = ((xGy)F(xGz))   =>   |- ((AFB)GC) = ((AGC)F(BGC))
Theorem	caoprdilem 4129	Lemma used by real number construction.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xGy) = (yGx)   &   |- (xG(yFz)) = ((xGy)F(xGz))   &   |- D e. V   &   |- H e. V   &   |- ((xGy)Gz) = (xG(yGz))   =>   |- (((AGC)F(BGD))GH) = ((AG(CGH))F(BG(DGH)))
Theorem	caoprlem2 4130	Lemma used in real number construction.
|- A e. V   &   |- B e. V   &   |- C e. V   &   |- (xGy) = (yGx)   &   |- (xG(yFz)) = ((xGy)F(xGz))   &   |- D e. V   &   |- H e. V   &   |- ((xGy)Gz) = (xG(yGz))   &   |- R e. V   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   =>   |- ((((AGC)F(BGD))GH)F(((AGD)F(BGC))GR)) = ((AG((CGH)F(DGR)))F(BG((CGR)F(DGH))))
Theorem	caoprmo 4131	Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119.
|- A e. V   &   |- B e. S   &   |- dom F = (S X. S)   &   |- -. (/) e. S   &   |- (xFy) = (yFx)   &   |- ((xFy)Fz) = (xF(yFz))   &   |- (x e. S -> (xFB) = x)   =>   |- E*w(AFw) = B
"Maps to" notation
Syntax	cmpt 4132	Extend the definition of a class to include "maps to" notation for defining a function via a rule.
class (x e. A |-> B)
Syntax	cmpt2 4133	Extend the definition of a class to include "maps to" notation for defining an operation via a rule.
class (x e. A, y e. B |-> C)
Definition	df-mpt 4134	Define "maps to" notation for defining a function via a rule. Read as "the function defined by the map from x (in A) to B(x)." The class expression B is the value of the function at x and normally contains the variable x. An example is the square function for complex numbers, (x e. CC |-> (x^2). Similar to the definition of mapping in [ChoquetDD] p. 2.
|- (x e. A |-> B) = {<.x, y>. | (x e. A /\ y = B)}
Definition	df-mpt2 4135	Define "maps to" notation for defining an operation via a rule. Read as "the operation defined by the map from x, y (in A X. B) to B(x, y)." An extension of df-mpt 4134 for two arguments.
|- (x e. A, y e. B |-> C) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
Theorem	mptg 4136	Value of a function given by a "maps to" rule. Equivalent to fvopab4g 3890.
|- (x = A -> B = C)   &   |- F = (x e. D |-> B)   =>   |- ((A e. D /\ C e. R) -> (F` A) = C)
Theorem	mpt2g 4137	Value of an operation given by a "maps to" rule. Equivalent to oprabval2g 4087.
|- (x = A -> R = G)   &   |- (y = B -> G = S)   &   |- F = (x e. C, y e. D |-> R)   =>   |- ((A e. C /\ B e. D /\ S e. H) -> (AFB) = S)
First and second members of an ordered pair
Syntax	c1st 4138	Extend the definition of a class to include the first member an ordered pair function.
class 1st
Syntax	c2nd 4139	Extend the definition of a class to include the second member an ordered pair function.
class 2nd
Definition	df-1st 4140	Define a function that extracts the first member of an ordered pair. Theorem op1st 4146 proves that it does this. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 3579 and op1stb 3136). The notation is the same as Monk's.
|- 1st = {<.x, y>. | y = U.dom { x}}
Definition	df-2nd 4141	Define a function that extracts the second member of an ordered pair. Theorem op2nd 4147 proves that it does this. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 3584 and op2ndb 3583). The notation is the same as Monk's.
|- 2nd = {<.x, y>. | y = U.ran { x}}
Theorem	1stval 4142	The value of the function that extracts the first member of an ordered pair.
|- (1st` A) = U.dom { A}
Theorem	2ndval 4143	The value of the function that extracts the second member of an ordered pair.
|- (2nd` A) = U.ran { A}
Theorem	1st0 4144	The value of the first-member function at the empty set.
|- (1st` (/)) = (/)
Theorem	2nd0 4145	The value of the second-member function at the empty set.
|- (2nd` (/)) = (/)
Theorem	op1st 4146	Extract the first member of an ordered pair.
|- A e. V   =>   |- (1st` <.A, B>.) = A
Theorem	op2nd 4147	Extract the second member of an ordered pair.
|- A e. V   &   |- B e. V   =>   |- (2nd` <.A, B>.) = B
Theorem	op1stg 4148	Extract the first member of an ordered pair.
|- (A e. C -> (1st` <.A, B>.) = A)
Theorem	op2ndg 4149	Extract the second member of an ordered pair.
|- ((A e. C /\ B e. D) -> (2nd` <.A, B>.) = B)
Theorem	1stval2 4150	Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52.
|- (A e. (V X. V) -> (1st` A) = |^||^|A)
Theorem	2ndval2 4151	Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52.
|- (A e. (V X. V) -> (2nd` A) = |^||^||^|`'{A})
Theorem	fo1st 4152	The 1st function maps the universe onto the universe.
|- 1st:V-onto->V
Theorem	fo2nd 4153	The 2nd function maps the universe onto the universe.
|- 2nd:V-onto->V
Theorem	f1stres 4154	Mapping of a restriction of the 1st (first member of an ordered pair) function.
|- (1st |` (A X. B)):(A X. B)-->A
Theorem	f2ndres 4155	Mapping of a restriction of the 2nd (second member of an ordered pair) function.
|- (2nd |` (A X. B)):(A X. B)-->B
Theorem	fo1stres 4156	Onto mapping of a restriction of the 1st (first member of an ordered pair) function.
|- (B =/= (/) -> (1st |` (A X. B)):(A X. B)-onto->A)
Theorem	fo2ndres 4157	Onto mapping of a restriction of the 2nd (second member of an ordered pair) function.
|- (A =/= (/) -> (2nd |` (A X. B)):(A X. B)-onto->B)
Theorem	1st2val 4158	Value of an alternate definition of the 1st function.
|- ({<.<.x, y>., z>. | z = x}` A) = (1st` A)
Theorem	2nd2val 4159	Value of an alternate definition of the 2nd function.
|- ({<.<.x, y>., z>. | z = y}` A) = (2nd` A)
Theorem	1stcof 4160	Composition of the first member function with another function.
|- (F:A-->(B X. C) -> (1st o. F):A-->B)
Theorem	elxp6 4161	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 3585.
|- (A e. (B X. C) <-> (A = <.(1st` A), (2nd` A)>. /\ ((1st` A) e. B /\ (2nd` A) e. C)))
Theorem	elxp7 4162	Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 3585.
|- (A e. (B X. C) <-> (A e. (V X. V) /\ ((1st` A) e. B /\ (2nd` A) e. C)))
Theorem	eqopi 4163	Equality with an ordered pair.
|- ((A e. (V X. V) /\ ((1st` A) = B /\ (2nd` A) = C)) -> A = <.B, C>.)
Theorem	eqop 4164	Two ways to express equality with an ordered pair.
|- C e. V   =>   |- (A e. (V X. V) -> (A = <.B, C>. <-> ((1st` A) = B /\ (2nd` A) = C)))
Theorem	xp2 4165	Representation of cross product based on ordered pair component functions.
|- (A X. B) = {x e. (V X. V) | ((1st` x) e. A /\ (2nd` x) e. B)}
Theorem	xpopth 4166	An ordered pair theorem for members of cross products.
|- ((A e. (C X. D) /\ B e. (R X. S)) -> (((1st` A) = (1st` B) /\ (2nd` A) = (2nd` B)) <-> A = B))
Theorem	unielxp 4167	The membership relation for a cross product is inherited by union.
|- (A e. (B X. C) -> U.A e. U.(B X. C))
Theorem	1st2nd 4168	Reconstruction of a member of a relation in terms of its ordered pair components.
|- ((Rel B /\ A e. B) -> A = <.(1st` A), (2nd` A)>.)
Theorem	1stdm 4169	The first ordered pair component of a member of a relation belongs to the domain of the relation.
|- ((Rel R /\ A e. R) -> (1st` A) e. dom R)
Theorem	2ndrn 4170	The second ordered pair component of a member of a relation belongs to the range of the relation.
|- ((Rel R /\ A e. R) -> (2nd` A) e. ran R)
Theorem	sbcopeq1a 4171	Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 1989, that avoids the existential quantifiers of copsexg 2868).
|- (<.x, y>. = A -> (ph <-> [(1st` A) / x][(2nd` A) / y]ph))
Theorem	csbopeq1a 4172	Equality theorem for substitution of a class A for an ordered pair <.x, y>. in B (analog of csbeq1a 2057).
|- (<.x, y>. = A -> B = [_(1st` A) / x]_[_(2nd` A) / y]_B)
Theorem	dfopab2 4173	A way to define an ordered-pair class abstraction without using existential quantifiers.
|- {<.x, y>. | ph} = {z | (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph)}
Theorem	dfoprab3 4174	A way to define an operation class abstraction without using existential quantifiers.
|- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
Theorem	dfoprab3s 4175	Operation class abstraction expressed without existential quantifiers.
|- (<.x, y>. = w -> (ph <-> ps))   =>   |- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ ps)}
Theorem	dfoprab4s 4176	Operation class abstraction expressed without existential quantifiers.
|- (<.x, y>. = w -> C = D)   =>   |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}
Theorem	dfoprab5s 4177	Operation class abstraction expressed without existential quantifiers.
|- ((x = (1st` w) /\ y = (2nd` w)) -> C = D)   =>   |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}
Theorem	dfoprab5sf 4178	Operation class abstraction expressed without existential quantifiers.
|- (v e. D -> A.x v e. D)   &   |- (v e. D -> A.y v e. D)   &   |- ((x = (1st` w) /\ y = (2nd` w)) -> C = D)   =>   |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}
Theorem	elopabi 4179	A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors.
|- (x = (1st` A) -> (ph <-> ps))   &   |- (y = (2nd` A) -> (ps <-> ch))   =>   |- (A e. {<.x, y>. | ph} -> ch)
Theorem	eloprabi 4180	A consequence of membership in an operation class abstraction, using ordered pair extractors.
|- (x = (1st` (1st` A)) -> (ph <-> ps))   &   |- (y = (2nd` (1st` A)) -> (ps <-> ch))   &   |- (z = (2nd` A) -> (ch <-> th))   =>   |- (A e. {<.<.x, y>., z>. | ph} -> th)
Theorem	foprab2 4181	Mapping of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (A.x e. A A.y e. B C e. D <-> F:(A X. B)-->D)
Theorem	foprab 4182	Mapping of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   &   |- ((x e. A /\ y e. B) -> C e. D)   =>   |- F:(A X. B)-->D
Theorem	fnoprab2g 4183	Functionality and domain of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (A.x e. A A.y e. B C e. V <-> F Fn (A X. B))
Theorem	fnoprab2 4184	Functionality and domain of an operation class abstraction.
|- C e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- F Fn (A X. B)
Theorem	dmoprab2 4185	Domain of an operation class abstraction.
|- C e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- dom F = (A X. B)
Theorem	elrnoprabg 4186	Membership in the range of an operation class abstraction.
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (A.x e. A A.y e. B C e. R -> (D e. ran F <-> E.x e. A E.y e. B D = C))
Theorem	elrnoprab 4187	Membership in the range of an operation class abstraction.
|- C e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- (D e. ran F <-> E.x e. A E.y e. B D = C)
Theorem	df1st2 4188	An alternate possible definition of the 1st function.
|- {<.<.x, y>., z>. | z = x} = (1st |` (V X. V))
Theorem	df2nd2 4189	An alternate possible definition of the 2nd function.
|- {<.<.x, y>., z>. | z = y} = (2nd |` (V X. V))
Theorem	1stconst 4190	The mapping of a restriction of the 1st function to a constant function.
|- (B e. C -> (1st |` (A X. {B})):(A X. {B})-1-1-onto->A)
Theorem	2ndconst 4191	The mapping of a restriction of the 2nd function to a converse constant function.
|- (A e. C -> (2nd |` ({A} X. B)):({A} X. B)-1-1-onto->B)
Theorem	iunfoprab 4192	Two ways to express an operation as a class of ordered pairs.
|- C e. V   =>   |- U_x e. A U_y e. B {<.<.x, y>., C>.} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
Theorem	curry1 4193	Composition with `'(2nd |` ({C} X. V)) turns any binary operation F with a constant first operand into a function G of the second operand only. This transformation is called "currying."
|- G = (F o. `'(2nd |` ({C} X. V)))   =>   |- ((F Fn (A X. B) /\ C e. A) -> G = {<.x, y>. | (x e. B /\ y = (CFx))})
Theorem	curry1f 4194	Functionality of a curried function with a constant first argument.
|- G = (F o. `'(2nd |` ({C} X. V)))   =>   |- ((F:(A X. B)-->D /\ C e. A) -> G:B-->D)
Theorem	curry1val 4195	The value of a curried function with a constant first argument.
|- G = (F o. `'(2nd |` ({C} X. V)))   =>   |- ((F Fn (A X. B) /\ C e. A /\ D e. U) -> (G` D) = (CFD))
Theorem	curry2 4196	Composition with `'(1st |` (V X. {C})) turns any binary operation F with a constant second operand into a function G of the first operand only. This transformation is called "currying." (If this becomes frequently used, we can introduce a new notation for the hypothesis.)
|- G = (F o. `'(1st |` (V X. {C})))   =>   |- ((F Fn (A X. B) /\ C e. B) -> G = {<.x, y>. | (x e. A /\ y = (xFC))})
Theorem	curry2f 4197	Functionality of a curried function with a constant second argument.
|- G = (F o. `'(1st |` (V X. {C})))   =>   |- ((F:(A X. B)-->D /\ C e. B) -> G:A-->D)
Theorem	curry2val 4198	The value of a curried function with a constant second argument.
|- G = (F o. `'(1st |` (V X. {C})))   =>   |- ((F Fn (A X. B) /\ C e. B /\ D e. U) -> (G` D) = (DFC))
Theorem	fparlem1 4199	Lemma for fpar 4203.
Theorem	fparlem2 4200	Lemma for fpar 4203.