
theorem
  for I be set, X be ManySortedSet of I, i be set st i in I holds
  X.i <> {} iff (coprod X).i <> {}
proof
  let I be set, X be ManySortedSet of I, i be set;
  assume
A1: i in I;
  then
A2: (coprod X).i = coprod(i,X) by Def3;
  thus X.i <> {} implies (coprod X).i <> {}
  proof
    assume X.i <> {};
    then consider x be object such that
A3: x in X.i by XBOOLE_0:def 1;
    [x,i] in (coprod X).i by A1,A2,A3,Def2;
    hence thesis;
  end;
  assume (coprod X).i <> {};
  then consider a be object such that
A4: a in coprod(i,X) by A2,XBOOLE_0:def 1;
  ex x be set st x in X.i & a = [x,i] by A1,A4,Def2;
  hence thesis;
end;
