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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) in F

let A, B, C, D be Formula of L; :: thesis: ( A \imp B in F & C \imp D in F & (\not B) \or (\not D) in F implies (\not A) \or (\not C) in F )

assume A1: A \imp B in F ; :: thesis: ( not C \imp D in F or not (\not B) \or (\not D) in F or (\not A) \or (\not C) in F )

assume A2: C \imp D in F ; :: thesis: ( not (\not B) \or (\not D) in F or (\not A) \or (\not C) in F )

assume A3: (\not B) \or (\not D) in F ; :: thesis: (\not A) \or (\not C) in F

( (A \imp B) \imp ((\not B) \imp (\not A)) in F & (C \imp D) \imp ((\not D) \imp (\not C)) in F ) by Th57;

then ( (\not B) \imp (\not A) in F & (\not D) \imp (\not C) in F ) by A1, A2, Def38;

then ((\not B) \or (\not D)) \imp ((\not A) \or (\not C)) in F by Th59;

hence (\not A) \or (\not C) in F by A3, 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) 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 & (\not B) \or (\not D) in F holds

(\not A) \or (\not C) in F

let A, B, C, D be Formula of L; :: thesis: ( A \imp B in F & C \imp D in F & (\not B) \or (\not D) in F implies (\not A) \or (\not C) in F )

assume A1: A \imp B in F ; :: thesis: ( not C \imp D in F or not (\not B) \or (\not D) in F or (\not A) \or (\not C) in F )

assume A2: C \imp D in F ; :: thesis: ( not (\not B) \or (\not D) in F or (\not A) \or (\not C) in F )

assume A3: (\not B) \or (\not D) in F ; :: thesis: (\not A) \or (\not C) in F

( (A \imp B) \imp ((\not B) \imp (\not A)) in F & (C \imp D) \imp ((\not D) \imp (\not C)) in F ) by Th57;

then ( (\not B) \imp (\not A) in F & (\not D) \imp (\not C) in F ) by A1, A2, Def38;

then ((\not B) \or (\not D)) \imp ((\not A) \or (\not C)) in F by Th59;

hence (\not A) \or (\not C) in F by A3, Def38; :: thesis: verum