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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let L be non empty IfWhileAlgebra of X,S; :: thesis: for M, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let M, M1 be Algorithm of L; :: thesis: for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let A, V be Formula of L; :: thesis: for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let H be AL-theory of V,L; :: thesis: ( ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) implies (while (M,M1)) * A in H )

assume ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) ; :: thesis: (while (M,M1)) * A in H

then A1: ( (M * (\not V)) \and A in H or (M * V) \and (M * (M1 * ((while (M,M1)) * A))) in H ) by Th35;

( ((M * (\not V)) \and A) \imp (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) in H & ((M * V) \and (M * (M1 * ((while (M,M1)) * A)))) \imp (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) in H ) by Def38;

then A2: ((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A)))) in H by A1, Def38;

((while (M,M1)) * A) \iff (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) in H by Def43;

then (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) \imp ((while (M,M1)) * A) in H by Th43;

hence (while (M,M1)) * A in H by Def38, A2; :: 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * 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, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let L be non empty IfWhileAlgebra of X,S; :: thesis: for M, M1 being Algorithm of L

for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let M, M1 be Algorithm of L; :: thesis: for A, V being Formula of L

for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let A, V be Formula of L; :: thesis: for H being AL-theory of V,L st ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) holds

(while (M,M1)) * A in H

let H be AL-theory of V,L; :: thesis: ( ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) implies (while (M,M1)) * A in H )

assume ( ( M * (\not V) in H & A in H ) or ( M * V in H & M * (M1 * ((while (M,M1)) * A)) in H ) ) ; :: thesis: (while (M,M1)) * A in H

then A1: ( (M * (\not V)) \and A in H or (M * V) \and (M * (M1 * ((while (M,M1)) * A))) in H ) by Th35;

( ((M * (\not V)) \and A) \imp (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) in H & ((M * V) \and (M * (M1 * ((while (M,M1)) * A)))) \imp (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) in H ) by Def38;

then A2: ((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A)))) in H by A1, Def38;

((while (M,M1)) * A) \iff (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) in H by Def43;

then (((M * (\not V)) \and A) \or ((M * V) \and (M * (M1 * ((while (M,M1)) * A))))) \imp ((while (M,M1)) * A) in H by Th43;

hence (while (M,M1)) * A in H by Def38, A2; :: thesis: verum