
theorem SumNabla2:
  for X being non empty set,
      R being Relation of X holds
    (R /\ (R~)`) \/ (R /\ (R~)`)~ \/ (R /\ R~) \/ Aux R = nabla X
  proof
    let X be non empty set,
        R be Relation of X;
    set P = R /\ (R~)`;
    set J = R /\ R~;
    (P \/ P~ \/ J)` c= Aux R by XBOOLE_1:7; then
    P \/ P~ \/ J \/ (P \/ P~ \/ J)` c= P \/ P~ \/ J \/ Aux R
      by XBOOLE_1:13; then
    [#][:X,X:] c= P \/ P~ \/ J \/ Aux R by SUBSET_1:10;
    hence thesis by XBOOLE_0:def 10;
  end;
