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 being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and 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 being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and C) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B, C being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and C) in F

let F be PC-theory of L; :: thesis: for A, B, C being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and C) in F

let A, B, C be Formula of L; :: thesis: ( A \imp B in F & A \imp C in F implies A \imp (B \and C) in F )

assume Z0: ( A \imp B in F & A \imp C in F ) ; :: thesis: A \imp (B \and C) in F

(A \imp B) \imp ((A \imp C) \imp (A \imp (B \and C))) in F by Th49;

then (A \imp C) \imp (A \imp (B \and C)) in F by Z0, Def38;

hence A \imp (B \and C) in F by Z0, Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and 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 being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and C) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B, C being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and C) in F

let F be PC-theory of L; :: thesis: for A, B, C being Formula of L st A \imp B in F & A \imp C in F holds

A \imp (B \and C) in F

let A, B, C be Formula of L; :: thesis: ( A \imp B in F & A \imp C in F implies A \imp (B \and C) in F )

assume Z0: ( A \imp B in F & A \imp C in F ) ; :: thesis: A \imp (B \and C) in F

(A \imp B) \imp ((A \imp C) \imp (A \imp (B \and C))) in F by Th49;

then (A \imp C) \imp (A \imp (B \and C)) in F by Z0, Def38;

hence A \imp (B \and C) in F by Z0, Def38; :: thesis: verum