
theorem Th38:
  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(Carrier F,Carrier 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(Carrier F,Carrier G,h).x and
    A3: for i be Element of I holds h.i is Homomorphism of F.i,G.i;
    set p = ProductMap(Carrier F,Carrier G,h);
    dom Carrier G = I by PARTFUN1:def 2; then
    A6: dom Carrier F = dom Carrier G = dom h by A1,PARTFUN1:def 2;
    A7: for i be object st i in dom h holds
        h.i is Function of (Carrier F).i,(Carrier G).i
    proof
      let i be object;
      assume i in dom h; then
      reconsider i as Element of I by A1;
      A8: [#](F.i) = (Carrier F).i by PENCIL_3:7;
      [#](G.i) = (Carrier G).i by PENCIL_3:7;
      hence thesis by A3,A8;
    end;
    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 Element of I;
      reconsider x as Element of product Carrier F by GROUP_7:def 2;
      consider hi be Function of (Carrier F).i,(Carrier G).i such that
      A13: hi = h.i and
      A14: (p.x).i = hi.(x.i) by A1,A6,A7,Def5;
      hi is Homomorphism of F.i, G.i by A3,A13;
      hence thesis by A2,A13,A14;
    end;
    hence thesis;
  end;
