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

theorem Th1:
  for F being multMagma-Family of I, G being non empty multMagma,
      p, q being Element of product F, x, y being Element of G st
    i in I & G = F.i & f = p & g = q & h = p * q & f.i = x & g.i = y
      holds x * y = h.i
proof
  let F be multMagma-Family of I, G be non empty multMagma,
      p, q be Element of product F, x, y be Element of G such that
A1: i in I and
A2: G = F.i and
A3: f = p and
A4: g = q and
A5: h = p * q and
A6: f.i = x and
A7: g.i = y;
  set GP = product F;
  q in the carrier of GP;
  then
A8: g in product Carrier F by A4,Def2;
  p in the carrier of GP;
  then f in product Carrier F by A3,Def2;
  then
  ex Fi being non empty multMagma, m being Function st Fi = F.i & m = (
  the multF of GP).(f,g) & m.i = (the multF of Fi).(f.i,g.i) by A1,A8,Def2;
  hence thesis by A2,A3,A4,A5,A6,A7;
end;
