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

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

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

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

then A1: ( (\not (A \or B)) \imp (\not A) in F & (\not (A \or B)) \imp (\not B) in F ) by Th58;

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

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

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

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

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

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

then A1: ( (\not (A \or B)) \imp (\not A) in F & (\not (A \or B)) \imp (\not B) in F ) by Th58;

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

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

hence (\not (A \or B)) \imp ((\not A) \and (\not B)) in F by A1, Def38; :: thesis: verum