reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;

theorem
  rng p c= rng(p^q)
proof
A1: dom p c= dom(p^q) by Th19;
    let x be object;
    assume x in rng p;
    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;
end;
