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 being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) 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 being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

let A, B be Formula of L; :: thesis: ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

A1: ((\not (\not A)) \or (\not (\not B))) \imp (\not ((\not A) \and (\not B))) in F by Th73;

( A \imp (\not (\not A)) in F & B \imp (\not (\not B)) in F ) by Th64;

then (A \or B) \imp ((\not (\not A)) \or (\not (\not B))) in F by Th59;

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

then ( ((\not A) \and (\not B)) \imp (\not (\not ((\not A) \and (\not B)))) in F & (\not (\not ((\not A) \and (\not B)))) \imp (\not (A \or B)) in F ) by Th64, Th58;

hence ((\not A) \and (\not B)) \imp (\not (A \or B)) in F by Th45; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) 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 being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

let A, B be Formula of L; :: thesis: ((\not A) \and (\not B)) \imp (\not (A \or B)) in F

A1: ((\not (\not A)) \or (\not (\not B))) \imp (\not ((\not A) \and (\not B))) in F by Th73;

( A \imp (\not (\not A)) in F & B \imp (\not (\not B)) in F ) by Th64;

then (A \or B) \imp ((\not (\not A)) \or (\not (\not B))) in F by Th59;

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

then ( ((\not A) \and (\not B)) \imp (\not (\not ((\not A) \and (\not B)))) in F & (\not (\not ((\not A) \and (\not B)))) \imp (\not (A \or B)) in F ) by Th64, Th58;

hence ((\not A) \and (\not B)) \imp (\not (A \or B)) in F by Th45; :: thesis: verum