thus rng (a * p) c= INT :: according to RELAT_1:def 19 :: thesis: verum

proof

let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (a * p) or y in INT )
assume A1: y in rng (a * p) ; :: thesis: y in INT
consider x being object such that
A2: ( x in dom (a * p) & (a * p) . x = y ) by A1, FUNCT_1:def 3;
reconsider x = x as bag of X by A2;
(a * p) . x = a * (p . x) by POLYNOM7:def 9;
hence y in INT by A2, INT_1:def 2; :: thesis: verum

end;