reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th35:
  for f,g being real-valued Function st rng g c= rng f & f <= n holds g <= n
  proof
    let f,g be real-valued Function such that
A1: rng g c= rng f and
A2: f <= n;
    let x be object;
    assume x in dom g;
    then g.x in rng g by FUNCT_1:def 3;
    then ex a being object st a in dom f & f.a = g.x by A1,FUNCT_1:def 3;
    hence g.x <= n by A2;
  end;
