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 non-empty 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, 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 non-empty 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 non-empty 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 non-empty 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