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 holds ((A \or B) \or C) \iff (A \or (B \or 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 holds ((A \or B) \or C) \iff (A \or (B \or 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 holds ((A \or B) \or C) \iff (A \or (B \or C)) in F
let F be PC-theory of L; for A, B, C being Formula of L holds ((A \or B) \or C) \iff (A \or (B \or C)) in F
let A, B, C be Formula of L; ((A \or B) \or C) \iff (A \or (B \or C)) in F
A1:
((A \or B) \imp (A \or (B \or C))) \imp ((C \imp (A \or (B \or C))) \imp (((A \or B) \or C) \imp (A \or (B \or C)))) in F
by Def38;
( B \imp (B \or C) in F & A \imp A in F )
by Def38, Th34;
then
(A \or B) \imp (A \or (B \or C)) in F
by Th59;
then A2:
(C \imp (A \or (B \or C))) \imp (((A \or B) \or C) \imp (A \or (B \or C))) in F
by A1, Def38;
( C \imp (B \or C) in F & (B \or C) \imp (A \or (B \or C)) in F )
by Def38;
then
C \imp (A \or (B \or C)) in F
by Th45;
then A3:
((A \or B) \or C) \imp (A \or (B \or C)) in F
by A2, Def38;
A4:
(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;
( 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;
then A5:
(A \imp ((A \or B) \or C)) \imp ((A \or (B \or C)) \imp ((A \or B) \or C)) in F
by A4, Th46;
( 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
(A \or (B \or C)) \imp ((A \or B) \or C) in F
by A5, Def38;
hence
((A \or B) \or C) \iff (A \or (B \or C)) in F
by A3, Th43; verum