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

theorem Th7:
  for F being associative Group-like multMagma-Family of I,
      x being Element of product F st
    x = g & for i being set st i in I
    ex G being Group, y being Element of G st
      G = F.i & s.i = y" & y = g.i holds s = x"
proof
  let F be associative Group-like multMagma-Family of I, x be Element of
  product F such that
A1: x = g and
A2: for i being set st i in I ex G being Group, y being Element of G st
  G = F.i & s.i = y" & y = g.i;
  set GP = product F;
A3: dom Carrier F = I by PARTFUN1:def 2;
A4: dom s = I by PARTFUN1:def 2;
  now
    let i be object;
    assume
A5: i in dom s;
    then
A6: 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, y being Element of G st G = F.i & s.i = y" & y =
    g.i by A2,A5;
    hence s.i in (Carrier F).i by A6;
  end;
  then
A7: s in product Carrier F by A3,A4,CARD_3:9;
  then reconsider f1 = s as Element of GP by Def2;
  reconsider II = 1_GP, xf = x * f1, fx = f1 * x, x1 = x as Element of product
  Carrier F by Def2;
A8: dom II = I by A3,CARD_3:9;
A9: now
    let i be object;
    assume
A10: i in I;
    then consider G being Group, y being Element of G such that
A11: G = F.i and
A12: s.i = y" and
A13: y = g.i by A2;
A14: ex Fi being non empty multMagma, m being Function st Fi = F.i & m =
(the multF of GP).(s,x) & m.i = (the multF of Fi).(s.i,x1.i) by A7,A10,Def2;
    y" * y = 1_G by GROUP_1:def 5;
    hence fx.i = II.i by A1,A10,A14,A11,A12,A13,Th6;
  end;
  dom fx = I by A3,CARD_3:9;
  then
A15: f1 * x = 1_GP by A8,A9;
A16: now
    let i be object;
    assume
A17: i in I;
    then consider G being Group, y being Element of G such that
A18: G = F.i and
A19: s.i = y" and
A20: y = g.i by A2;
A21: ex Fi being non empty multMagma, m being Function st Fi = F.i & m =
(the multF of GP).(x,s) & m.i = (the multF of Fi).(x1.i,s.i) by A7,A17,Def2;
    y * y" = 1_G by GROUP_1:def 5;
    hence xf.i = II.i by A1,A17,A21,A18,A19,A20,Th6;
  end;
  dom xf = I by A3,CARD_3:9;
  then x * f1 = 1_GP by A8,A16;
  hence thesis by A15,GROUP_1:def 5;
end;
