reserve i for object, I for set,
  f for Function,
  x, x1, x2, y, A, B, X, Y, Z for ManySortedSet of I;

theorem Th1:
  for X be object for M be ManySortedSet of I st i in I
  holds dom (M +* (i .--> X)) = I
proof
  let X be object, M be ManySortedSet of I such that
A1: i in I;
  thus dom (M +* (i .--> X)) = dom M \/ dom (i .--> X) by FUNCT_4:def 1
    .= I \/ dom (i .--> X) by PARTFUN1:def 2
    .= I \/ {i}
    .= I by A1,ZFMISC_1:40;
end;
