reserve i, x, I for set,
  A, M for ManySortedSet of I,
  f for Function,
  F for ManySortedFunction of I;

theorem Th8:
  for M being non-empty ManySortedSet of I for X being Element of M
  holds X = (id M)..X
proof
  let M be non-empty ManySortedSet of I;
  let X be Element of M;
  set F = id M;
  now
    let i be object;
    reconsider g = F.i as Function;
    assume
A2: i in I;
    then X.i is Element of M.i & F.i = id (M.i)
          by MSUALG_3:def 1,PBOOLE:def 14;
    then g.(X.i) = X.i;
    hence X.i = (F..X).i by A2,PRALG_1:def 20;
  end;
  hence thesis;
end;
