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

now

let x be set ; :: thesis: ( x in rng q implies x in rng (p ^ q) )
assume x in rng q ; :: thesis: x in rng (p ^ q)
then consider y being set such that
A1: y in dom q and
A2: x = q . y by FUNCT_1:def 3;
reconsider k = y as Element of NAT by A1;
( (len p) + k in dom (p ^ q) & (p ^ q) . ((len p) + k) = q . k ) by A1, Def4, Th26;
hence x in rng (p ^ q) by A2, FUNCT_1:3; :: thesis: verum

end;

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