reserve A for preIfWhileAlgebra;

theorem
  for X being non empty set for x being Element of X holds .x = .(^x)
proof
  let X be non empty set;
  let x be Element of X;
  for s being Element of Funcs(X, INT) holds .(^x).s = s.x
  proof
    let s be Element of Funcs(X, INT);
    thus .(^x).s = s.((^x).s) by Def19
      .= s.x;
  end;
  hence thesis by Def18;
end;
