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 B)) \imp ((\not A) \or (\not 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 B)) \imp ((\not A) \or (\not 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 B)) \imp ((\not A) \or (\not B)) in F

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

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

( (\not A) \imp ((\not A) \or (\not B)) in F & (\not B) \imp ((\not A) \or (\not B)) in F ) by Def38;

then ( (\not ((\not A) \or (\not B))) \imp (\not (\not A)) in F & (\not (\not A)) \imp A in F & (\not ((\not A) \or (\not B))) \imp (\not (\not B)) in F & (\not (\not B)) \imp B in F ) by Th58, Th65;

then A1: ( (\not ((\not A) \or (\not B))) \imp A in F & (\not ((\not A) \or (\not B))) \imp B in F ) by Th45;

((\not ((\not A) \or (\not B))) \imp A) \imp (((\not ((\not A) \or (\not B))) \imp B) \imp ((\not ((\not A) \or (\not B))) \imp (A \and B))) in F by Th49;

then ((\not ((\not A) \or (\not B))) \imp B) \imp ((\not ((\not A) \or (\not B))) \imp (A \and B)) in F by A1, Def38;

then (\not ((\not A) \or (\not B))) \imp (A \and B) in F by A1, Def38;

then ( (\not (A \and B)) \imp (\not (\not ((\not A) \or (\not B)))) in F & (\not (\not ((\not A) \or (\not B)))) \imp ((\not A) \or (\not B)) in F ) by Th58, Th65;

hence (\not (A \and B)) \imp ((\not A) \or (\not 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 B)) \imp ((\not A) \or (\not 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 B)) \imp ((\not A) \or (\not 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 B)) \imp ((\not A) \or (\not B)) in F

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

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

( (\not A) \imp ((\not A) \or (\not B)) in F & (\not B) \imp ((\not A) \or (\not B)) in F ) by Def38;

then ( (\not ((\not A) \or (\not B))) \imp (\not (\not A)) in F & (\not (\not A)) \imp A in F & (\not ((\not A) \or (\not B))) \imp (\not (\not B)) in F & (\not (\not B)) \imp B in F ) by Th58, Th65;

then A1: ( (\not ((\not A) \or (\not B))) \imp A in F & (\not ((\not A) \or (\not B))) \imp B in F ) by Th45;

((\not ((\not A) \or (\not B))) \imp A) \imp (((\not ((\not A) \or (\not B))) \imp B) \imp ((\not ((\not A) \or (\not B))) \imp (A \and B))) in F by Th49;

then ((\not ((\not A) \or (\not B))) \imp B) \imp ((\not ((\not A) \or (\not B))) \imp (A \and B)) in F by A1, Def38;

then (\not ((\not A) \or (\not B))) \imp (A \and B) in F by A1, Def38;

then ( (\not (A \and B)) \imp (\not (\not ((\not A) \or (\not B)))) in F & (\not (\not ((\not A) \or (\not B)))) \imp ((\not A) \or (\not B)) in F ) by Th58, Th65;

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