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) \iff (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) \iff (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) \iff (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) \iff (A \or (B \or C)) in F

let A, B, C be Formula of L; :: thesis: ((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; :: thesis: verum

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 ; :: 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) \iff (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) \iff (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) \iff (A \or (B \or C)) in F

let A, B, C be Formula of L; :: thesis: ((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; :: thesis: verum