let n be non empty Nat; 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 ; 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; 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; 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; ( 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
; 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; verum