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

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

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

A1: (((A \and B) \and C) \imp A) \imp ((((A \and B) \and C) \imp (B \and C)) \imp (((A \and B) \and C) \imp (A \and (B \and C)))) in F by Th49;

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

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

then A2: (((A \and B) \and C) \imp (B \and C)) \imp (((A \and B) \and C) \imp (A \and (B \and C))) in F by A1, Def38;

( (A \and B) \imp B in F & C \imp C in F ) by Def38, Th34;

then ((A \and B) \and C) \imp (B \and C) in F by Th72;

then A3: ((A \and B) \and C) \imp (A \and (B \and C)) in F by A2, Def38;

A4: ((A \and (B \and C)) \imp (A \and B)) \imp (((A \and (B \and C)) \imp C) \imp ((A \and (B \and C)) \imp ((A \and B) \and C))) in F by Th49;

( (B \and C) \imp B in F & A \imp A in F ) by Def38, Th34;

then (A \and (B \and C)) \imp (A \and B) in F by Th72;

then A5: ((A \and (B \and C)) \imp C) \imp ((A \and (B \and C)) \imp ((A \and B) \and C)) in F by A4, Def38;

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

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

then (A \and (B \and C)) \imp ((A \and B) \and C) in F by A5, Def38;

hence ((A \and B) \and C) \iff (A \and (B \and 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 \and B) \and C) \iff (A \and (B \and 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 \and B) \and C) \iff (A \and (B \and 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 \and B) \and C) \iff (A \and (B \and C)) in F

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

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

A1: (((A \and B) \and C) \imp A) \imp ((((A \and B) \and C) \imp (B \and C)) \imp (((A \and B) \and C) \imp (A \and (B \and C)))) in F by Th49;

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

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

then A2: (((A \and B) \and C) \imp (B \and C)) \imp (((A \and B) \and C) \imp (A \and (B \and C))) in F by A1, Def38;

( (A \and B) \imp B in F & C \imp C in F ) by Def38, Th34;

then ((A \and B) \and C) \imp (B \and C) in F by Th72;

then A3: ((A \and B) \and C) \imp (A \and (B \and C)) in F by A2, Def38;

A4: ((A \and (B \and C)) \imp (A \and B)) \imp (((A \and (B \and C)) \imp C) \imp ((A \and (B \and C)) \imp ((A \and B) \and C))) in F by Th49;

( (B \and C) \imp B in F & A \imp A in F ) by Def38, Th34;

then (A \and (B \and C)) \imp (A \and B) in F by Th72;

then A5: ((A \and (B \and C)) \imp C) \imp ((A \and (B \and C)) \imp ((A \and B) \and C)) in F by A4, Def38;

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

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

then (A \and (B \and C)) \imp ((A \and B) \and C) in F by A5, Def38;

hence ((A \and B) \and C) \iff (A \and (B \and C)) in F by A3, Th43; :: thesis: verum