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 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L holds (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (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 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 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L holds (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (M,((M \; M) * 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 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L holds (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (M,((M \; M) * 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 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L holds (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (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 (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (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 (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (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 (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (M,((M \; M) * A)))) in H
let A, V be Formula of L; :: thesis: for H being AL-theory of V,L holds (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (M,((M \; M) * A)))) in H
let H be AL-theory of V,L; :: thesis: (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (M,((M \; M) * A)))) in H
A1: (\Cup (M,A)) \iff (A \or (\Cup (M,(M * A)))) in H by Def43;
(\Cup (M,(M * A))) \iff ((M * A) \or (\Cup (M,(M * (M * A))))) in H by Def43;
then A2: (\Cup (M,A)) \iff (A \or ((M * A) \or (\Cup (M,(M * (M * A)))))) in H by A1, Th99;
((A \or (M * A)) \or (\Cup (M,(M * (M * A))))) \iff (A \or ((M * A) \or (\Cup (M,(M * (M * A)))))) in H by Th76;
then (A \or ((M * A) \or (\Cup (M,(M * (M * A)))))) \iff ((A \or (M * A)) \or (\Cup (M,(M * (M * A))))) in H by Th90;
then A3: (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (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 (\Cup (M,(M * (M * A)))) \iff (\Cup (M,((M \; M) * A))) in H by Th147;
hence (\Cup (M,A)) \iff ((A \or (M * A)) \or (\Cup (M,((M \; M) * A)))) in H by A3, Th99; :: thesis: verum