let n be non empty Nat; :: thesis: 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 ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( not C \imp D in F or (A \and C) \imp (B \and D) in F )

assume A2: C \imp D in F ; :: thesis: (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; :: thesis: verum

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 ; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: ( not C \imp D in F or (A \and C) \imp (B \and D) in F )

assume A2: C \imp D in F ; :: thesis: (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; :: thesis: verum