
theorem AuxEq:
  for X being non empty set,
      R being total reflexive Relation of X holds
    Aux R = (R~` /\ R`) \/ (R`~ /\ (R` \/ R~))
  proof
    let X be non empty set;
    let R be total reflexive Relation of X;
    set Z1 = ((R /\ R~`) \/ (R /\ R~`)~ \/ (R /\ R~))`;
z1: Z1 = ((R /\ R~`) \/ (R /\ R~`)~)` /\ (R /\ R~)` by XBOOLE_1:53
    .= ((R /\ R~`) \/ (R /\ R~`)~)` /\ (R` \/ R~`) by XBOOLE_1:54
    .= ((R /\ R~`)` /\ (R /\ R~`)~`) /\ (R` \/ R~`) by XBOOLE_1:53
    .= ((R` \/ R~``) /\ (R /\ R~`)~`) /\ (R` \/ R~`) by XBOOLE_1:54
    .= ((R` \/ R~``) /\ (R~ /\ R~`~)`) /\ (R` \/ R~`) by RELAT_1:22
    .= (R~` \/ R~`~`) /\ (R` \/ R~) /\ (R` \/ R~`) by XBOOLE_1:54
    .= (R~` \/ R~`~`) /\ ((R` \/ R~) /\ (R` \/ R~`)) by XBOOLE_1:16
    .= (R~` \/ R~`~`) /\ (R` \/ (R~ /\ R~`)) by XBOOLE_1:24
    .= (R~` \/ R~`~`) /\ (R` \/ {}) by XBOOLE_0:def 7,SUBSET_1:23
    .= (R~` \/ R) /\ R` by Tilde1;
    set Z2 = ((R /\ R~`) \/ (R /\ R~`)~ \/ (R /\ R~))`~;
z2: Z2 = (((R /\ R~`) \/ (R /\ R~`)~)` /\ (R /\ R~)`)~ by XBOOLE_1:53
    .= (((R /\ R~`)` /\ (R /\ R~`)~`) /\ (R /\ R~)`)~ by XBOOLE_1:53
    .= (((R` \/ R~``) /\ (R /\ R~`)~`) /\ (R /\ R~)`)~ by XBOOLE_1:54
    .= (((R` \/ R~) /\ (R /\ R~`)~`))~ /\ (R /\ R~)`~ by RELAT_1:22
    .= ((R` \/ R~)~ /\ (R /\ R~`)~`~) /\ (R /\ R~)`~ by RELAT_1:22
    .= (R`~ \/ R~~) /\ (R /\ R~`)~`~ /\ (R /\ R~)`~ by RELAT_1:23
    .= (R`~ \/ R) /\ (R /\ R~`)~`~ /\ (R` \/ R~`)~ by XBOOLE_1:54
    .= (R`~ \/ R) /\ (R /\ R~`)~`~ /\ (R`~ \/ R~`~) by RELAT_1:23
    .= (R`~ \/ R) /\ (R~ /\ R~`~)`~ /\ (R`~ \/ R~`~) by RELAT_1:22
    .= (R`~ \/ R) /\ (R~` \/ R~`~`)~ /\ (R`~ \/ R~`~) by XBOOLE_1:54
    .= (R`~ \/ R) /\ (R~`~ \/ R~`~`~) /\ (R`~ \/ R~`~) by RELAT_1:23
    .= (R`~ \/ R) /\ (R~`~ \/ R~) /\ (R`~ \/ R~`~) by Tilde1
    .= (R`~ \/ R) /\ (R` \/ R~) /\ (R`~ \/ R~`~) by Tilde3
    .= (R`~ \/ R) /\ (R` \/ R~) /\ (R`~ \/ R`) by Tilde3
    .= (R`~ \/ R`) /\ (R`~ \/ R) /\ (R` \/ R~) by XBOOLE_1:16
    .= (R`~ \/ (R` /\ R)) /\ (R` \/ R~) by XBOOLE_1:24
    .= (R`~ \/ {}) /\ (R` \/ R~) by XBOOLE_0:def 7,SUBSET_1:23
    .= R`~ /\ (R` \/ R~);
    Aux R = ((R~` /\ R`) \/ (R /\ R`)) \/ (R`~ /\ (R` \/ R~))
        by z1,z2,XBOOLE_1:23
      .= ((R~` /\ R`) \/ {}) \/ (R`~ /\ (R` \/ R~))
        by XBOOLE_0:def 7,SUBSET_1:23
      .= (R~` /\ R`) \/ (R`~ /\ (R` \/ R~));
    hence thesis;
  end;
