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

( A \and B in F iff ( A in F & 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

( A \and B in F iff ( A in F & 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

( A \and B in F iff ( A in F & B in F ) )

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds

( A \and B in F iff ( A in F & B in F ) )

let A, B be Formula of L; :: thesis: ( A \and B in F iff ( A in F & B in F ) )

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

hence ( A \and B in F implies ( A in F & B in F ) ) by Def38; :: thesis: ( A in F & B in F implies A \and B in F )

A \imp (B \imp (A \and B)) in F by Def38;

then ( A in F implies B \imp (A \and B) in F ) by Def38;

hence ( A in F & B in F implies A \and B in F ) by 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

( A \and B in F iff ( A in F & 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

( A \and B in F iff ( A in F & 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

( A \and B in F iff ( A in F & B in F ) )

let F be PC-theory of L; :: thesis: for A, B being Formula of L holds

( A \and B in F iff ( A in F & B in F ) )

let A, B be Formula of L; :: thesis: ( A \and B in F iff ( A in F & B in F ) )

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

hence ( A \and B in F implies ( A in F & B in F ) ) by Def38; :: thesis: ( A in F & B in F implies A \and B in F )

A \imp (B \imp (A \and B)) in F by Def38;

then ( A in F implies B \imp (A \and B) in F ) by Def38;

hence ( A in F & B in F implies A \and B in F ) by Def38; :: thesis: verum