reserve i, I for set,
  f, g, h for Function,
  s for ManySortedSet of I;

theorem Th5:
  for F being Group-like multMagma-Family of I st for i being set
  st i in I ex G being Group-like non empty multMagma st G = F.i & s.i = 1_G
  holds s = 1_product F
proof
  let F be Group-like multMagma-Family of I such that
A1: for i being set st i in I ex G being Group-like non empty multMagma
   st G = F.i & s.i = 1_G;
  set GP = product F;
A2: dom Carrier F = I by PARTFUN1:def 2;
A3: dom s = I by PARTFUN1:def 2;
  now
    let i be object;
    assume
A4: i in dom s;
    then
A5: ex R being 1-sorted st R = F.i & (Carrier F).i = the carrier of R
    by PRALG_1:def 15;
    ex G being Group-like non empty multMagma st ( G = F.i)&( s.i = 1_G)
    by A1,A4;
    hence s.i in (Carrier F).i by A5;
  end;
  then
A6: s in product Carrier F by A3,A2,CARD_3:9;
  then reconsider e = s as Element of GP by Def2;
  now
    let h be Element of GP;
    reconsider h1 = h, he = h*e, eh = e*h as Element of product Carrier F by
Def2;
A7: dom h1 = I by A2,CARD_3:9;
A8: now
      let i be object;
      assume
A9:   i in I;
      then consider G being Group-like non empty multMagma such that
A10:  G = F.i and
A11:  s.i = 1_G by A1;
      reconsider b = h1.i, c = s.i as Element of G by A9,A10,A11,Lm1;
A12:  ex Fi being non empty multMagma, m being Function st Fi = F.i & m
= (the multF of GP).(h,s) & m.i = (the multF of Fi).(h1.i,s.i) by A6,A9,Def2;
      b * c = b by A11,GROUP_1:def 4;
      hence he.i = h1.i by A12,A10;
    end;
    dom he = I by A2,CARD_3:9;
    hence h * e = h by A7,A8;
A13: now
      let i be object;
      assume
A14:  i in I;
      then consider G being Group-like non empty multMagma such that
A15:  G = F.i and
A16:  s.i = 1_G by A1;
      reconsider b = h1.i, c = s.i as Element of G by A14,A15,A16,Lm1;
A17:  ex Fi being non empty multMagma, m being Function st Fi = F.i & m
= (the multF of GP).(s,h) & m.i = (the multF of Fi).(s.i,h1.i) by A6,A14,Def2;
      c * b = b by A16,GROUP_1:def 4;
      hence eh.i = h1.i by A17,A15;
    end;
    dom eh = I by A2,CARD_3:9;
    hence e * h = h by A7,A13;
  end;
  hence thesis by GROUP_1:4;
end;
