
theorem
  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 SumMap(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 that
    A1: I = dom h and
    A2: for i be Element of I holds
        ex hi be Homomorphism of F.i,G.i st hi = h.i & hi is bijective;
    A3: for i be Element of I holds h.i is Homomorphism of F.i,G.i
    proof
      let i be Element of I;
      consider hi be Homomorphism of F.i,G.i such that
      A4: hi = h.i & hi is bijective by A2;
      thus thesis by A4;
    end;
    set p = ProductMap(F,G,h);
    set s = SumMap(F,G,h);
    A5: p is bijective by A1,A2,Th40;
    A6: s = p | sum F by A1,A3,Def7; then
    A7: s is one-to-one by A5,FUNCT_1:52;
    A9: rng s c= [#] sum G;
    for y be object holds y in [#] sum G implies y in rng s
    proof
      let y be object;
      assume
      A10: y in [#] sum G; then
      y in sum G; then
      A11: y in product G by GROUP_2:40; then
      y in rng p by A5,GROUP_6:61; then
      consider x be object such that
      A12: x in dom p and
      A13: y = p.x by FUNCT_1:def 3;
      reconsider x as Element of product F by A12;
      reconsider y as Element of product G by A11;
      A15: x in sum F
      proof
        assume
        A16: not x in sum F;
        for i be Element of I holds
        ex ki be Homomorphism of G.i,F.i st x.i = ki.(y.i)
        proof
          let i be Element of I;
          consider hi be Homomorphism of F.i,G.i such that
          A18: hi = h.i and
          A19: y.i = hi.(x.i) by A1,A3,A13,Th39;
          consider li be Homomorphism of F.i,G.i such that
          A20: li = h.i & li is bijective by A2;
          reconsider ki = hi" as Homomorphism of G.i,F.i by A18,A20,GROUP_6:62;
          take ki;
          x in product F; then
          x.i in F.i by Th5;
          hence thesis by A18,A19,A20,FUNCT_2:26;
        end; then
        support(x,F) c= support(y,G) by Th34;
        hence contradiction by A10,A16,Th8;
      end;
      A22: x in dom s by A15,FUNCT_2:def 1; then
      s.x = p.x by A6,FUNCT_1:47;
      hence thesis by A13,A22,FUNCT_1:3;
    end;
    hence thesis by A7,A9,TARSKI:2,GROUP_6:60;
  end;
