reserve A for preIfWhileAlgebra;

theorem
  for X being non empty set, t being INT-Expression of X holds t+ .(0,X)
  = t & t(#).(1,X) = t
proof
  let X be non empty set;
  let t be INT-Expression of X;
  now
    let s be Element of Funcs(X,INT);
    dom (t+ .(0,X)) = Funcs(X,INT) by FUNCT_2:def 1;
    hence (t+ .(0,X)).s = (t.s)+(.(0,X).s) by VALUED_1:def 1
      .= t.s;
  end;
  hence t+ .(0,X) = t;
  now
    let s be Element of Funcs(X,INT);
    dom (t(#).(1,X)) = Funcs(X,INT) by FUNCT_2:def 1;
    hence (t(#).(1,X)).s = (t.s)*(.(1,X).s) by VALUED_1:def 4
      .= (t.s)*1
      .= t.s;
  end;
  hence thesis;
end;
