
theorem Th40:
  for I be non empty set,
      F,G be Group-Family of I,
      h be non empty Function
  st I = dom h
   & for i be Element of I holds
     ex hi be Homomorphism of F.i,G.i
     st hi = h.i & hi is bijective
  holds ProductMap(F,G,h) is bijective
  proof
    let I be non empty set,
        F,G be Group-Family of I,
        h be non empty Function;
    assume
    A1: I = dom h
      & for i be Element of I holds
        ex hi be Homomorphism of F.i,G.i st hi = h.i & hi is bijective;
    set p = ProductMap(Carrier F,Carrier G,h);
    A3: dom(Carrier F) = I & dom(Carrier G) = I by PARTFUN1:def 2;
    for i be object st i in I holds
    ex hi be Function of (Carrier F).i,(Carrier G).i
    st hi = h.i & hi is bijective
    proof
      let i be object;
      assume i in I;
      then reconsider j = i as Element of I;
      consider hi be Homomorphism of F.j,G.j such that
      A5: hi = h.j & hi is bijective by A1;
      A6: (Carrier F).j = [#](F.j) by PENCIL_3:7;
      A7: (Carrier G).j = [#](G.j) by PENCIL_3:7; then
      reconsider hi as Function of (Carrier F).i,(Carrier G).i by A6;
      take hi;
      thus thesis by A5,A7;
    end; then
    A8: p is bijective by A1,A3,Th37;
    A9: [#] product F = product Carrier F by GROUP_7:def 2;
    A10: [#] product G = product Carrier G by GROUP_7:def 2;
    reconsider p as Function of product F,product G by A9,GROUP_7:def 2;
    for i be Element of I holds h.i is Homomorphism of F.i,G.i
    proof
      let i be Element of I;
      ex hi be Homomorphism of F.i,G.i st
      hi = h.i & hi is bijective by A1;
      hence thesis;
    end; then
    p = ProductMap(F,G,h) by A1,Def6;
    hence thesis by A8,A10;
  end;
