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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let L be non empty IfWhileAlgebra of X,S; :: thesis: for M being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let M be Algorithm of L; :: thesis: for A, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let A, V be Formula of L; :: thesis: for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let H be AL-theory of V,L; :: thesis: (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

A1: (\Cap (M,A)) \iff (A \and (\Cap (M,(M * A)))) in H by Def43;

(\Cap (M,(M * A))) \iff ((M * A) \and (\Cap (M,(M * (M * A))))) in H by Def43;

then A2: (\Cap (M,A)) \iff (A \and ((M * A) \and (\Cap (M,(M * (M * A)))))) in H by A1, Th97;

((A \and (M * A)) \and (\Cap (M,(M * (M * A))))) \iff (A \and ((M * A) \and (\Cap (M,(M * (M * A)))))) in H by Th78;

then (A \and ((M * A) \and (\Cap (M,(M * (M * A)))))) \iff ((A \and (M * A)) \and (\Cap (M,(M * (M * A))))) in H by Th90;

then A3: (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,(M * (M * A))))) in H by A2, Th91;

((M \; M) * A) \iff (M * (M * A)) in H by Def43;

then (M * (M * A)) \iff ((M \; M) * A) in H by Th90;

then (\Cap (M,(M * (M * A)))) \iff (\Cap (M,((M \; M) * A))) in H by Th148;

hence (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H by A3, Th97; :: 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) 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, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let L be non empty IfWhileAlgebra of X,S; :: thesis: for M being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let M be Algorithm of L; :: thesis: for A, V being Formula of L

for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let A, V be Formula of L; :: thesis: for H being AL-theory of V,L holds (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

let H be AL-theory of V,L; :: thesis: (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H

A1: (\Cap (M,A)) \iff (A \and (\Cap (M,(M * A)))) in H by Def43;

(\Cap (M,(M * A))) \iff ((M * A) \and (\Cap (M,(M * (M * A))))) in H by Def43;

then A2: (\Cap (M,A)) \iff (A \and ((M * A) \and (\Cap (M,(M * (M * A)))))) in H by A1, Th97;

((A \and (M * A)) \and (\Cap (M,(M * (M * A))))) \iff (A \and ((M * A) \and (\Cap (M,(M * (M * A)))))) in H by Th78;

then (A \and ((M * A) \and (\Cap (M,(M * (M * A)))))) \iff ((A \and (M * A)) \and (\Cap (M,(M * (M * A))))) in H by Th90;

then A3: (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,(M * (M * A))))) in H by A2, Th91;

((M \; M) * A) \iff (M * (M * A)) in H by Def43;

then (M * (M * A)) \iff ((M \; M) * A) in H by Th90;

then (\Cap (M,(M * (M * A)))) \iff (\Cap (M,((M \; M) * A))) in H by Th148;

hence (\Cap (M,A)) \iff ((A \and (M * A)) \and (\Cap (M,((M \; M) * A)))) in H by A3, Th97; :: thesis: verum