
theorem
  for X being non empty set,
      R being Relation of X holds
    R /\ (R~)`, R /\ R~, R` /\ (R~)` are_mutually_disjoint
  proof
    let X be non empty set;
    let R be Relation of X;
    set C = R` /\ R~`;
z1: (R /\ (R~)`) /\ (R /\ R~) = R /\ (R~` /\ R~) by XBOOLE_1:116
      .= R /\ {} by XBOOLE_0:def 7,SUBSET_1:23
      .= {};
z2: (R /\ R~`) /\ (R` /\ R~`) = R~` /\ (R /\ R`) by XBOOLE_1:116
      .= R~` /\ {} by XBOOLE_0:def 7,SUBSET_1:23
      .= {};
X0: R /\ R` = {} by XBOOLE_0:def 7,SUBSET_1:23;
    (R /\ R~) /\ (R` /\ R~`) = (R /\ C) /\ R~ by XBOOLE_1:16
      .= ((R /\ R`) /\ R~`) /\ R~ by XBOOLE_1:16
      .= {} by X0;
    hence thesis by z1,z2,XBOOLE_0:def 7;
  end;
