
theorem AuxIrr:
  for X being non empty set,
      R being total reflexive Relation of X holds
    Aux R is irreflexive symmetric
  proof
    let X be non empty set;
    let R be total reflexive Relation of X;
    set P = R /\ R~`,
        T = R /\ R~,
        C = ((R /\ R~`) \/ (R /\ R~`)~ \/ (R /\ R~))`;
    set Y = field Aux R;
    for x,y being object st [x,y] in id Y holds [x,y] in T
    proof
      let x,y be object;
      assume [x,y] in id Y; then
A0:   x in Y & x = y by RELAT_1:def 10;
      field R = X by ORDERS_1:12; then
A1:   [x,x] in R by A0,RELAT_2:def 1,def 9; then
      [x,x] in R~ by RELAT_1:def 7;
      hence thesis by A0,A1,XBOOLE_0:def 4;
    end; then
x4: id Y c= P \/ P~ \/ T by XBOOLE_1:10,RELAT_1:def 3; then
Y2: id Y /\ (P \/ P~ \/ T)` = {} by XBOOLE_0:def 7,XBOOLE_1:85;
    (id Y qua Relation)~ misses (P \/ P~ \/ T)`~
      by x4,LemmaAuxIrr,XBOOLE_1:85; then
Y3: (id Y qua Relation) /\ (P \/ P~ \/ T)`~ = {} by XBOOLE_0:def 7;
    id Y /\ Aux R = {} \/ {} by Y2,Y3,XBOOLE_1:23;
    hence thesis by RELAT_2:2,XBOOLE_0:def 7;
  end;
