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 M being Algorithm of L
for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) 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 M being Algorithm of L
for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) 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 M being Algorithm of L
for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) 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 M being Algorithm of L
for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) 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 M being Algorithm of L
for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) in H
let L be non empty IfWhileAlgebra of X,S; for M being Algorithm of L
for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) in H
let M be Algorithm of L; for A, B, V being Formula of L
for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) in H
let A, B, V be Formula of L; for H being AL-theory of V,L st A \iff B in H holds
(\Cap (M,A)) \iff (\Cap (M,B)) in H
let H be AL-theory of V,L; ( A \iff B in H implies (\Cap (M,A)) \iff (\Cap (M,B)) in H )
assume
A \iff B in H
; (\Cap (M,A)) \iff (\Cap (M,B)) in H
then
( A \imp B in H & B \imp A in H )
by Th43;
then
( (\Cap (M,A)) \imp (\Cap (M,B)) in H & (\Cap (M,B)) \imp (\Cap (M,A)) in H )
by Def43;
hence
(\Cap (M,A)) \iff (\Cap (M,B)) in H
by Th43; verum