reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem Th42:
  for f, g being Function of X,TOP-REAL n holds
  f<//>g is Function of X,TOP-REAL n
  proof
    let f, g be Function of X,TOP-REAL n;
    set h = f<//>g;
A1: dom f = X by FUNCT_2:def 1;
 dom g = X by FUNCT_2:def 1;
then A2: dom h = X by A1,VALUED_2:def 48;
    for x st x in X holds h.x in the carrier of TOP-REAL n
    proof
      let x;
      assume
A3:   x in X;
      then reconsider X as non empty set;
      reconsider x as Element of X by A3;
      reconsider f, g as Function of X,TOP-REAL n;
      h.x = f.x /" g.x by A2,VALUED_2:def 48;
      hence thesis;
    end;
    hence thesis by A2,FUNCT_2:3;
  end;
