A7: dom g = X by FUNCT_2:def 1;
A8: dom (f mod g) = (dom f) /\ (dom g) by Def4;
A9: dom f = X by FUNCT_2:def 1;
rng (f mod g) c= INT

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (f mod g) or y in INT )
assume y in rng (f mod g) ; :: thesis: y in INT
then consider a being set such that
A10: a in dom (f mod g) and
A11: y = (f mod g) . a by FUNCT_1:def 5;
A12: g . a in rng g by A8, A7, A10, FUNCT_1:12;
f . a in rng f by A8, A9, A10, FUNCT_1:12;
then reconsider i = f . a, j = g . a as Element of INT by A12;
thus y in INT by A11, INT_1:def 2; :: thesis: verum

end;

hence f mod g is Function of X,INT by A8, A9, A7, FUNCT_2:4; :: thesis: verum