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

theorem Th10:
  for M being non-empty ManySortedSet of I for X, Y being Element
of M st X (\/) Y is Element of M
  holds (id M)..(X (\/) Y) = ((id M)..X) (\/) ((id M) ..Y)
proof
  let M be non-empty ManySortedSet of I;
  let X, Y be Element of M;
  set F = id M;
  assume X (\/) Y is Element of M;
  hence F..(X (\/) Y) = X (\/) Y by Th8
    .= (F..X) (\/) Y by Th8
    .= (F..X) (\/) (F..Y) by Th8;
end;
