let n be non empty Nat; for S being non empty non void n PC-correct PCLangSignature
for L being language MSAlgebra over S
for F being PC-theory of L
for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds
(A \and C) \imp (B \and D) in F
let S be non empty non void n PC-correct PCLangSignature ; for L being language MSAlgebra over S
for F being PC-theory of L
for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds
(A \and C) \imp (B \and D) in F
let L be language MSAlgebra over S; for F being PC-theory of L
for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds
(A \and C) \imp (B \and D) in F
let F be PC-theory of L; for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds
(A \and C) \imp (B \and D) in F
let A, B, C, D be Formula of L; ( A \imp B in F & C \imp D in F implies (A \and C) \imp (B \and D) in F )
assume A1:
A \imp B in F
; ( not C \imp D in F or (A \and C) \imp (B \and D) in F )
assume A2:
C \imp D in F
; (A \and C) \imp (B \and D) in F
A3:
((A \and C) \imp B) \imp (((A \and C) \imp D) \imp ((A \and C) \imp (B \and D))) in F
by Th49;
( (A \and C) \imp A in F & (A \and C) \imp C in F )
by Def38;
then A4:
( (A \and C) \imp B in F & (A \and C) \imp D in F )
by A1, A2, Th45;
then
((A \and C) \imp D) \imp ((A \and C) \imp (B \and D)) in F
by A3, Def38;
hence
(A \and C) \imp (B \and D) in F
by A4, Def38; verum