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 & B \imp C in F holds

A \imp 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 & B \imp C in F holds

A \imp 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 & B \imp C in F holds

A \imp 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 & B \imp C in F holds

A \imp C in F

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

assume that

A1: A \imp B in F and

A2: B \imp C in F ; :: thesis: A \imp C in F

(A \imp B) \imp ((B \imp C) \imp (A \imp C)) in F by Th39;

then (B \imp C) \imp (A \imp C) in F by A1, Def38;

hence A \imp C in F by A2, 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 & B \imp C in F holds

A \imp 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 & B \imp C in F holds

A \imp 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 & B \imp C in F holds

A \imp 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 & B \imp C in F holds

A \imp C in F

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

assume that

A1: A \imp B in F and

A2: B \imp C in F ; :: thesis: A \imp C in F

(A \imp B) \imp ((B \imp C) \imp (A \imp C)) in F by Th39;

then (B \imp C) \imp (A \imp C) in F by A1, Def38;

hence A \imp C in F by A2, Def38; :: thesis: verum