theorem
  for f being Function of X,Y st
    x in X holds (Flatten f).x in Y
  proof
    let f be Function of X,Y;
    assume
A1: x in X; then
    reconsider xx = x as set;
A2: not xx in xx;
    set FX = FlatPoset(X);
    reconsider x as Element of FX by A1,XBOOLE_0:def 3;
    f.x in Y by A1,FUNCT_2:5;
    hence thesis by DefFlatten04,A1,A2;
  end;
