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 Th54:
  for f, g being Function of T,TOP-REAL n holds f <##> g = TIMES(n).:(f,g)
  proof
    set R = TOP-REAL n;
    set F = TIMES(n);
    let f, g be Function of T,R;
A1: dom(f<##>g) = dom f /\ dom g by VALUED_2:def 47;
    dom F = the carrier of [:R,R:] by FUNCT_2:def 1
    .= [:the carrier of R,the carrier of R:] by BORSUK_1:def 2;
    then [:rng f, rng g:] c= dom F by ZFMISC_1:96;
    then
A2: dom(F.:(f,g)) = dom f /\ dom g by FUNCOP_1:69;
    now
      let x be object;
      assume
A3:   x in dom (f<##>g);
      then
A4:   f.x is Point of R & g.x is Point of R by FUNCT_2:5;
      thus (f<##>g).x = f.x(#)g.x by A3,VALUED_2:def 47
      .= F.(f.x,g.x) by A4,Def5
      .= F.:(f,g).x by A1,A2,A3,FUNCOP_1:22;
    end;
    hence thesis by A1,A2;
  end;
