let p, q be FinSequence; :: thesis: rng p c= rng (p ^ q)

now

let x be set ; :: thesis: ( x in rng p implies x in rng (p ^ q) )
assume x in rng p ; :: thesis: x in rng (p ^ q)
then consider y being set such that
A1: y in dom p and
A2: x = p . y by FUNCT_1:def 5;
reconsider k = y as Element of NAT by A1;
A3: dom p c= dom (p ^ q) by Th39;
(p ^ q) . k = p . k by A1, Def7;
hence x in rng (p ^ q) by A1, A2, A3, FUNCT_1:def 5; :: thesis: verum

end;

hence rng p c= rng (p ^ q) by TARSKI:def 3; :: thesis: verum