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, C, V being Formula of L

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

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

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

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

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

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

for A, B, C, V being Formula of L

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

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

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

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

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

A1: (M * ((A \and B) \and C)) \iff ((M * (A \and B)) \and (M * C)) in H by Def43;

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

hence (M * ((A \and B) \and C)) \iff (((M * A) \and (M * B)) \and (M * C)) in H by A1, Th98; :: 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, C, V being Formula of L

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

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

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

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

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

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

for A, B, C, V being Formula of L

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

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

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

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

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

A1: (M * ((A \and B) \and C)) \iff ((M * (A \and B)) \and (M * C)) in H by Def43;

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

hence (M * ((A \and B) \and C)) \iff (((M * A) \and (M * B)) \and (M * C)) in H by A1, Th98; :: thesis: verum