let A be non empty set ; :: thesis: for v being Element of Valuations_in A
for p, q being Element of CQC-WFF
for J being interpretation of A holds
( J,v |= p '&' q iff ( J,v |= p & J,v |= q ) )
let v be Element of Valuations_in A; :: thesis: for p, q being Element of CQC-WFF
for J being interpretation of A holds
( J,v |= p '&' q iff ( J,v |= p & J,v |= q ) )
let p, q be Element of CQC-WFF ; :: thesis: for J being interpretation of A holds
( J,v |= p '&' q iff ( J,v |= p & J,v |= q ) )
let J be interpretation of A; :: thesis: ( J,v |= p '&' q iff ( J,v |= p & J,v |= q ) )

A1: now

assume ( J,v |= p & J,v |= q ) ; :: thesis: J,v |= p '&' q
then ( (Valid (p,J)) . v = TRUE & (Valid (q,J)) . v = TRUE ) by Def12;
then ((Valid (p,J)) . v) '&' ((Valid (q,J)) . v) = TRUE ;
then ((Valid (p,J)) '&' (Valid (q,J))) . v = TRUE by MARGREL1:def 20;
then (Valid ((p '&' q),J)) . v = TRUE by Lm1;
hence J,v |= p '&' q by Def12; :: thesis: verum

end;

now

assume J,v |= p '&' q ; :: thesis: ( J,v |= p & J,v |= q )
then (Valid ((p '&' q),J)) . v = TRUE by Def12;
then ((Valid (p,J)) '&' (Valid (q,J))) . v = TRUE by Lm1;
then ((Valid (p,J)) . v) '&' ((Valid (q,J)) . v) = TRUE by MARGREL1:def 20;
then ( (Valid (p,J)) . v = TRUE & (Valid (q,J)) . v = TRUE ) by MARGREL1:12;
hence ( J,v |= p & J,v |= q ) by Def12; :: thesis: verum

end;

hence ( J,v |= p '&' q iff ( J,v |= p & J,v |= q ) ) by A1; :: thesis: verum