
theorem
  for X being non empty set,
      R being total reflexive Relation of X holds
    R` /\ R~` is irreflexive symmetric
  proof
    let X be non empty set,
        R be total reflexive Relation of X;
A0: id field R c= R by RELAT_2:1;
A1: dom R = X by PARTFUN1:def 2; then
    dom (R~) = X by RELAT_2:12; then
    rng R = X by RELAT_1:20; then
A3: id X /\ R` = {} by XBOOLE_0:def 7,SUBSET_1:24,A0,A1;
    id X /\ (R` /\ R~`) = (id X /\ R`) /\ R~` by XBOOLE_1:16; then
    id X misses (R` /\ R~`) by XBOOLE_0:def 7,A3; then
z1: id field (R` /\ R~`) misses R` /\ R~` by XBOOLE_1:63,SYSREL:15;
    for x, y being object st [x,y] in R` /\ R~` holds [y,x] in R` /\ R~`
    proof
      let x, y be object;
      assume
B0:   [x,y] in R` /\ R~`; then
B1:   x is Element of X & y is Element of X by ZFMISC_1:87;
      [x,y] in R` & [x,y] in R~` by XBOOLE_0:def 4,B0; then
      not [x,y] in R & not [x,y] in R~ by XBOOLE_0:def 5; then
      not [y,x] in R~ & not [y,x] in R by RELAT_1:def 7; then
      [y,x] in R~` & [y,x] in R` by Lemma12b,B1;
      hence [y,x] in R` /\ R~` by XBOOLE_0:def 4;
    end;
    hence thesis by z1,LemSym,RELAT_2:2;
  end;
