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 holds (A \or (B \or C)) \imp ((A \or B) \or 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 holds (A \or (B \or C)) \imp ((A \or B) \or 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 holds (A \or (B \or C)) \imp ((A \or B) \or C) in F
let F be PC-theory of L; :: thesis: for A, B, C being Formula of L holds (A \or (B \or C)) \imp ((A \or B) \or C) in F
let A, B, C be Formula of L; :: thesis: (A \or (B \or C)) \imp ((A \or B) \or C) in F
A1: (A \imp ((A \or B) \or C)) \imp (((B \or C) \imp ((A \or B) \or C)) \imp ((A \or (B \or C)) \imp ((A \or B) \or C))) in F by Def38;
( A \imp (A \or B) in F & (A \or B) \imp ((A \or B) \or C) in F ) by Def38;
then A \imp ((A \or B) \or C) in F by Th45;
then A2: ((B \or C) \imp ((A \or B) \or C)) \imp ((A \or (B \or C)) \imp ((A \or B) \or C)) in F by A1, Def38;
( B \imp (A \or B) in F & C \imp C in F ) by Def38, Th34;
then (B \or C) \imp ((A \or B) \or C) in F by Th59;
hence (A \or (B \or C)) \imp ((A \or B) \or C) in F by A2, Def38; :: thesis: verum