
theorem AuxEq2:
  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` \/ R~))) by XBOOLE_1:16
    .= ({} /\ R~`) \/ (R /\ R~` /\ (R`~ /\ (R` \/ R~)))
      by XBOOLE_0:def 7,SUBSET_1:23
    .= R /\ R~` /\ ((R`~ /\ R`) \/ (R`~ /\ R~)) by XBOOLE_1:23
    .= ((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~` /\ (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`~) by XBOOLE_0:def 7,SUBSET_1:23
    .= {};
    hence thesis by XBOOLE_0:def 7;
  end;
