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 \or C)) \imp ((A \and B) \or (A \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 \or C)) \imp ((A \and B) \or (A \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 \or C)) \imp ((A \and B) \or (A \and C)) in F

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

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

set AB = A \and B;

set AC = A \and C;

set BC = B \or C;

set ABC = A \and (B \or C);

A1: (((\not A) \or (\not B)) \and ((\not A) \or (\not C))) \imp ((\not A) \or ((\not B) \and (\not C))) in F by Th80;

( (\not (A \and B)) \imp ((\not A) \or (\not B)) in F & (\not (A \and C)) \imp ((\not A) \or (\not C)) in F ) by Th70;

then A2: ((\not (A \and B)) \and (\not (A \and C))) \imp (((\not A) \or (\not B)) \and ((\not A) \or (\not C))) in F by Th72;

(\not ((A \and B) \or (A \and C))) \imp ((\not (A \and B)) \and (\not (A \and C))) in F by Th71;

then (\not ((A \and B) \or (A \and C))) \imp (((\not A) \or (\not B)) \and ((\not A) \or (\not C))) in F by A2, Th45;

then A3: (\not ((A \and B) \or (A \and C))) \imp ((\not A) \or ((\not B) \and (\not C))) in F by A1, Th45;

( (\not A) \imp (\not A) in F & ((\not B) \and (\not C)) \imp (\not (B \or C)) in F ) by Th34, Th74;

then ( ((\not A) \or ((\not B) \and (\not C))) \imp ((\not A) \or (\not (B \or C))) in F & ((\not A) \or (\not (B \or C))) \imp (\not (A \and (B \or C))) in F ) by Th73, Th59;

then ((\not A) \or ((\not B) \and (\not C))) \imp (\not (A \and (B \or C))) in F by Th45;

then (\not ((A \and B) \or (A \and C))) \imp (\not (A \and (B \or C))) in F by A3, Th45;

hence (A \and (B \or C)) \imp ((A \and B) \or (A \and C)) in F by Th58; :: 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 \or C)) \imp ((A \and B) \or (A \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 \or C)) \imp ((A \and B) \or (A \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 \or C)) \imp ((A \and B) \or (A \and C)) in F

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

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

set AB = A \and B;

set AC = A \and C;

set BC = B \or C;

set ABC = A \and (B \or C);

A1: (((\not A) \or (\not B)) \and ((\not A) \or (\not C))) \imp ((\not A) \or ((\not B) \and (\not C))) in F by Th80;

( (\not (A \and B)) \imp ((\not A) \or (\not B)) in F & (\not (A \and C)) \imp ((\not A) \or (\not C)) in F ) by Th70;

then A2: ((\not (A \and B)) \and (\not (A \and C))) \imp (((\not A) \or (\not B)) \and ((\not A) \or (\not C))) in F by Th72;

(\not ((A \and B) \or (A \and C))) \imp ((\not (A \and B)) \and (\not (A \and C))) in F by Th71;

then (\not ((A \and B) \or (A \and C))) \imp (((\not A) \or (\not B)) \and ((\not A) \or (\not C))) in F by A2, Th45;

then A3: (\not ((A \and B) \or (A \and C))) \imp ((\not A) \or ((\not B) \and (\not C))) in F by A1, Th45;

( (\not A) \imp (\not A) in F & ((\not B) \and (\not C)) \imp (\not (B \or C)) in F ) by Th34, Th74;

then ( ((\not A) \or ((\not B) \and (\not C))) \imp ((\not A) \or (\not (B \or C))) in F & ((\not A) \or (\not (B \or C))) \imp (\not (A \and (B \or C))) in F ) by Th73, Th59;

then ((\not A) \or ((\not B) \and (\not C))) \imp (\not (A \and (B \or C))) in F by Th45;

then (\not ((A \and B) \or (A \and C))) \imp (\not (A \and (B \or C))) in F by A3, Th45;

hence (A \and (B \or C)) \imp ((A \and B) \or (A \and C)) in F by Th58; :: thesis: verum