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