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