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, M2 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * 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, M2 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * 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, M2 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * 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, M2 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * 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, M2 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * A in H
let L be non empty IfWhileAlgebra of X,S; :: thesis: for M, M1, M2 being Algorithm of L
for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * A in H
let M, M1, M2 be Algorithm of L; :: thesis: for A, V being Formula of L
for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * A in H
let A, V be Formula of L; :: thesis: for H being AL-theory of V,L st ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) holds
(if-then-else (M,M1,M2)) * A in H
let H be AL-theory of V,L; :: thesis: ( ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) implies (if-then-else (M,M1,M2)) * A in H )
assume ( ( M * V in H & M * (M1 * A) in H ) or ( M * (\not V) in H & M * (M2 * A) in H ) ) ; :: thesis: (if-then-else (M,M1,M2)) * A in H
then ( ( (M * V) \and (M * (M1 * A)) in H or (M * (\not V)) \and (M * (M2 * A)) in H ) & ((M * V) \and (M * (M1 * A))) \imp (((M * V) \and (M * (M1 * A))) \or ((M * (\not V)) \and (M * (M2 * A)))) in H & ((M * (\not V)) \and (M * (M2 * A))) \imp (((M * V) \and (M * (M1 * A))) \or ((M * (\not V)) \and (M * (M2 * A)))) in H ) by Def38, Th35;
then A1: ((M * V) \and (M * (M1 * A))) \or ((M * (\not V)) \and (M * (M2 * A))) in H by Def38;
((if-then-else (M,M1,M2)) * A) \iff (((M * V) \and (M * (M1 * A))) \or ((M * (\not V)) \and (M * (M2 * A)))) in H by Def43;
then (((M * V) \and (M * (M1 * A))) \or ((M * (\not V)) \and (M * (M2 * A)))) \imp ((if-then-else (M,M1,M2)) * A) in H by Th43;
hence (if-then-else (M,M1,M2)) * A in H by Def38, A1; :: thesis: verum