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