
theorem Th39:
  for I be non empty set,
      F,G be Group-Family of I,
      h be non empty Function,
      x be Element of product F,
      y be Element of product G
  st I = dom h
   & y = ProductMap(F,G,h).x
   & for i be Element of I holds
     h.i is Homomorphism of F.i,G.i
  holds
    for i be Element of I holds
    ex hi be Homomorphism of F.i,G.i
    st hi = h.i & y.i = hi.(x.i)
  proof
    let I be non empty set,
        F,G be Group-Family of I,
        h be non empty Function,
        x be Element of product F,
        y be Element of product G;
    assume that
    A1: I = dom h and
    A2: y = ProductMap(F,G,h).x and
    A3: for i be Element of I holds
        h.i is Homomorphism of F.i,G.i;
    y = ProductMap(Carrier F,Carrier G,h).x by A1,A2,A3,Def6;
    hence thesis by A1,A3,Th38;
  end;
