let n be natural non empty number ; :: thesis: 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 b_{1} -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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( A \iff B in H implies (\Cap (M,A)) \iff (\Cap (M,B)) in H )

assume A \iff B in H ; :: thesis: (\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; :: thesis: verum

for T being non-empty VarMSAlgebra over J

for X being V2() GeneratorSet of T

for S being non empty non void b

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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( A \iff B in H implies (\Cap (M,A)) \iff (\Cap (M,B)) in H )

assume A \iff B in H ; :: thesis: (\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; :: thesis: verum