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

theorem Th8:
  for F being associative Group-like multMagma-Family of I,
      x being Element of product F, G being Group,
      y being Element of G st
    i in I & G = F.i & f = x & g = x" & f.i = y holds g.i = y"
proof
  let F be associative Group-like multMagma-Family of I, x be Element of
  product F, G be Group, y be Element of G such that
A1: i in I and
A2: G = F.i and
A3: f = x and
A4: g = x" and
A5: f.i = y;
  set GP = product F;
A6: (the multF of GP).(g,f) = x" * x by A3,A4
    .= 1_GP by GROUP_1:def 5;
  x" in the carrier of GP;
  then
A7: g in product Carrier F by A4,Def2;
  then reconsider gi = g.i as Element of G by A1,A2,Lm1;
  x in the carrier of GP;
  then
A8: f in product Carrier F by A3,Def2;
  then
  ex Fi being non empty multMagma, h being Function st Fi = F.i & h = (
  the multF of GP).(g,f) & h.i = (the multF of Fi).(g.i,f.i) by A1,A7,Def2;
  then
A9: gi * y = 1_G by A1,A2,A5,A6,Th6;
A10: (the multF of GP).(f,g) = x * x" by A3,A4
    .= 1_GP by GROUP_1:def 5;
  ex Fi being non empty multMagma, h being Function st Fi = F.i & h = (
  the multF of GP).(f,g) & h.i = (the multF of Fi).(f.i,g.i) by A1,A8,A7,Def2;
  then y * gi = 1_G by A1,A2,A5,A10,Th6;
  hence thesis by A9,GROUP_1:def 5;
end;
