reserve I for set;

theorem
  for A, B, C being ManySortedSet of I, F being ManySortedFunction of A,
  B holds F""C is ManySortedSubset of A
proof
  let A, B, C be ManySortedSet of I, F be ManySortedFunction of A, B;
  let i be object;
  assume
A1: i in I;
  then reconsider J = I as non empty set;
  reconsider j = i as Element of J by A1;
  reconsider A1 = A, B1 = B as ManySortedSet of J;
  reconsider F1 = F as ManySortedFunction of A1, B1;
  (F1.j)"(C.j) c= A.j;
  hence thesis by Def1;
end;
