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 \iff B in F & B \iff C in F holds

A \iff 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 \iff B in F & B \iff C in F holds

A \iff 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 \iff B in F & B \iff C in F holds

A \iff C in F

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

A \iff C in F

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

assume A1: ( A \iff B in F & B \iff C in F ) ; :: thesis: A \iff C in F

( (A \iff B) \imp (A \imp B) in F & (A \iff B) \imp (B \imp A) in F & (B \iff C) \imp (B \imp C) in F & (B \iff C) \imp (C \imp B) in F ) by Def38;

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

then ( A \imp C in F & C \imp A in F ) by Th45;

then A2: (A \imp C) \and (C \imp A) in F by Th35;

((A \imp C) \and (C \imp A)) \imp (A \iff C) in F by Def38;

hence A \iff 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 \iff B in F & B \iff C in F holds

A \iff 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 \iff B in F & B \iff C in F holds

A \iff 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 \iff B in F & B \iff C in F holds

A \iff C in F

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

A \iff C in F

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

assume A1: ( A \iff B in F & B \iff C in F ) ; :: thesis: A \iff C in F

( (A \iff B) \imp (A \imp B) in F & (A \iff B) \imp (B \imp A) in F & (B \iff C) \imp (B \imp C) in F & (B \iff C) \imp (C \imp B) in F ) by Def38;

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

then ( A \imp C in F & C \imp A in F ) by Th45;

then A2: (A \imp C) \and (C \imp A) in F by Th35;

((A \imp C) \and (C \imp A)) \imp (A \iff C) in F by Def38;

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