let s1, s2 be boolean-valued Function; :: thesis: ( dom s1 = (dom p) /\ (dom q) & ( for x being set st x in dom s1 holds
s1 . x = (p . x) '&' (q . x) ) & dom s2 = (dom p) /\ (dom q) & ( for x being set st x in dom s2 holds
s2 . x = (p . x) '&' (q . x) ) implies s1 = s2 )
assume that
A16: dom s1 = (dom p) /\ (dom q) and
A17: for x being set st x in dom s1 holds
s1 . x = (p . x) '&' (q . x) and
A18: dom s2 = (dom p) /\ (dom q) and
A19: for x being set st x in dom s2 holds
s2 . x = (p . x) '&' (q . x) ; :: thesis: s1 = s2
for x being set st x in dom s1 holds
s1 . x = s2 . x

proof

let x be set ; :: thesis: ( x in dom s1 implies s1 . x = s2 . x )
assume x in dom s1 ; :: thesis: s1 . x = s2 . x
then ( s1 . x = (p . x) '&' (q . x) & s2 . x = (p . x) '&' (q . x) ) by A16, A17, A18, A19;
hence s1 . x = s2 . x ; :: thesis: verum

end;

hence s1 = s2 by A16, A18, FUNCT_1:9; :: thesis: verum