let p, q be XFinSequence; :: thesis: rng p c= rng (p ^ q)
A1: dom p c= dom (p ^ q) by Th19;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng p or x in rng (p ^ q) )
assume x in rng p ; :: thesis: x in rng (p ^ q)
then consider y being object such that
A2: y in dom p and
A3: x = p . y by FUNCT_1:def 3;
reconsider k = y as Element of NAT by A2;
(p ^ q) . k = p . k by A2, Def3;
hence x in rng (p ^ q) by A2, A3, A1, FUNCT_1:3; :: thesis: verum