
theorem MutuDis2:
  for X being non empty set,
      R being total reflexive Relation of X holds
    R /\ (R~)`, R /\ R~, Aux R are_mutually_disjoint
  proof
    let X be non empty set;
    let R be total reflexive Relation of X;
    set A = R /\ (R~)`,
        B = R /\ R~,
        C = Aux R;
    (R /\ (R~)`) /\ (R /\ R~) = R /\ (R~` /\ R~) by XBOOLE_1:116
      .= R /\ {} by XBOOLE_0:def 7,SUBSET_1:23
      .= {};
    hence thesis by AuxEq3,XBOOLE_0:def 7,AuxEq2;
  end;
