
theorem AuxEq3:
  for X being non empty set,
      R being total reflexive Relation of X holds
    (R /\ R~) misses Aux R
  proof
    let X be non empty set;
    let R be total reflexive Relation of X;
    set A = R /\ R~;
    A /\ Aux R = R /\ R~ /\ ((R~` /\ R`) \/ (R`~ /\ (R` \/ R~))) by AuxEq
    .= (R /\ R~ /\ (R~` /\ R`)) \/
     (R /\ R~ /\ (R`~ /\ (R` \/ R~))) by XBOOLE_1:23
    .= (R /\ R~ /\ R~` /\ R`) \/
     (R /\ R~ /\ (R`~ /\ (R` \/ R~))) by XBOOLE_1:16
    .= (R /\ (R~ /\ R~`) /\ R`) \/
     (R /\ R~ /\ (R`~ /\ (R` \/ R~))) by XBOOLE_1:16
    .= (R /\ {} /\ R`) \/
     (R /\ R~ /\ (R`~ /\ (R` \/ R~))) by XBOOLE_0:def 7,SUBSET_1:23
    .= (R /\ (R~ /\ (R`~ /\ (R` \/ R~)))) by XBOOLE_1:16
    .= (R /\ (R~ /\ R`~ /\ (R` \/ R~))) by XBOOLE_1:16
    .= (R /\ (R~ /\ R`~) /\ (R` \/ R~)) by XBOOLE_1:16
    .= (R /\ (R~ /\ R`~) /\ R`) \/
         (R /\ (R~ /\ R`~) /\ R~) by XBOOLE_1:23
    .= (R` /\ R /\ (R~ /\ R`~)) \/
         (R /\ (R~ /\ R`~) /\ R~) by XBOOLE_1:16
    .= ({} /\ (R~ /\ R`~)) \/
         (R /\ (R~ /\ R`~) /\ R~) by XBOOLE_0:def 7,SUBSET_1:23
    .= (R /\ ((R`~ /\ R~) /\ R~)) by XBOOLE_1:16
    .= (R /\ (R`~ /\ (R~ /\ R~))) by XBOOLE_1:16
    .= R /\ (R` /\ R)~ by RELAT_1:22
    .= R /\ ({} qua Relation)~ by XBOOLE_0:def 7,SUBSET_1:23
    .= {};
    hence thesis by XBOOLE_0:def 7;
  end;
