
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 h.i is Homomorphism of F.i,G.i
  holds
    for i be Element of I,
        f be Element of F.i,
        hi be Homomorphism of F.i,G.i
    st hi = h.i
    holds SumMap(F,G,h).(1ProdHom(F,i).f) = 1ProdHom(G,i).(hi.f)
  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 h.i is Homomorphism of F.i,G.i;
    let i be Element of I,
        f be Element of F.i,
        hi be Homomorphism of F.i,G.i;
    assume
    A3: hi = h.i;
    set x = 1ProdHom(F,i).f;
    set y = SumMap(F,G,h).x;
    x in ProjGroup(F,i); then
    A6: x in sum F by GROUP_2:40;
    reconsider x as Element of product F by GROUP_2:42;
    y in sum G by A6,FUNCT_2:5; then
    reconsider y as Element of product G by GROUP_2:42;
    A8: x in dom SumMap(F,G,h) by A6,FUNCT_2:def 1;
    SumMap(F,G,h) = ProductMap(F,G,h) | sum F by A1,A2,Def7; then
    A10: y = ProductMap(F,G,h).x by A8,FUNCT_1:47;
    A11: dom y = I by Th3;
    A13: x = (1_product F) +* (i,f) by GROUP_12:def 3;
    consider hi0 be Homomorphism of F.i,G.i such that
    A15: hi0 = h.i & y.i = hi0.(x.i) by A1,A2,A10,Th39;
    A16: y.i = hi.f by A3,A13,A15,GROUP_12:1;
    A17: for j be Element of I st j <> i holds y.j = 1_G.j
    proof
      let j be Element of I;
      assume j <> i; then
      A18: x.j = 1_F.j by A13,GROUP_12:1;
      consider hj be Homomorphism of F.j,G.j such that
      A19: hj = h.j & y.j = hj.(x.j) by A1,A2,A10,Th39;
      thus thesis by A18,A19,GROUP_6:31;
    end;
    y = (1_product G) +* (i,hi.f) by A11,A16,A17,GROUP_12:1;
    hence thesis by GROUP_12:def 3;
  end;
