let n be natural non empty number ; for J being non empty non void Signature
for T being non-empty VarMSAlgebra over J
for X being V2() GeneratorSet of T
for S being non empty non void b1 -extension n PC-correct QC-correct n AL-correct essential AlgLangSignature over Union X
for L being non empty IfWhileAlgebra of X,S
for A, V being Formula of L
for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let J be non empty non void Signature; for T being non-empty VarMSAlgebra over J
for X being V2() GeneratorSet of T
for S being non empty non void J -extension n PC-correct QC-correct n AL-correct essential AlgLangSignature over Union X
for L being non empty IfWhileAlgebra of X,S
for A, V being Formula of L
for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let T be non-empty VarMSAlgebra over J; for X being V2() GeneratorSet of T
for S being non empty non void J -extension n PC-correct QC-correct n AL-correct essential AlgLangSignature over Union X
for L being non empty IfWhileAlgebra of X,S
for A, V being Formula of L
for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let X be V2() GeneratorSet of T; for S being non empty non void J -extension n PC-correct QC-correct n AL-correct essential AlgLangSignature over Union X
for L being non empty IfWhileAlgebra of X,S
for A, V being Formula of L
for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let S be non empty non void J -extension n PC-correct QC-correct n AL-correct essential AlgLangSignature over Union X; for L being non empty IfWhileAlgebra of X,S
for A, V being Formula of L
for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let L be non empty IfWhileAlgebra of X,S; for A, V being Formula of L
for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let A, V be Formula of L; for H being AL-theory of V,L
for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let H be AL-theory of V,L; for a being SortSymbol of J
for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let a be SortSymbol of J; for x, y being Element of X . a
for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let x, y be Element of X . a; for x0, y0 being Element of Union (X extended_by ({}, the carrier of S)) st x = x0 & y = y0 holds
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
let x0, y0 be Element of Union (X extended_by ({}, the carrier of S)); ( x = x0 & y = y0 implies ((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H )
reconsider b = a as SortSymbol of S by Th8;
reconsider t = @ y as Element of the Sorts of L . b by Th16;
assume A1:
( x = x0 & y = y0 )
; ((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
then A2:
((x := ((@ y),L)) * A) \iff (A / (x0,t)) in H
by Def43;
A3:
X extended_by ({}, the carrier of S) is ManySortedSubset of the Sorts of L
by Th23;
( a in the carrier of J & the carrier of J = dom X )
by PARTFUN1:def 2;
then
b in dom (X | the carrier of S)
by RELAT_1:57;
then (X extended_by ({}, the carrier of S)) . b =
(X | the carrier of S) . b
by FUNCT_4:13
.=
X . b
by FUNCT_1:49
;
hence
((x := ((@ y),L)) * A) \iff (A / (x0,y0)) in H
by A1, A2, A3, Th14; verum