
theorem Tilde1:
  for X being non empty set,
      R being Relation of X holds
    R = R~`~`
  proof
    let X be non empty set,
        R be Relation of X;
    for x,y being object st [x,y] in R holds [x,y] in R~`~`
    proof
      let x,y be object;
      assume
X0:   [x,y] in R; then
x1:   x in field R & y in field R by RELAT_1:15;
      [y,x] in R~ by X0,RELAT_1:def 7; then
      not [y,x] in R~` by XBOOLE_0:def 5; then
      not [x,y] in R~`~ by RELAT_1:def 7;
      hence thesis by x1,Lemma12b;
    end; then
n1: R c= R~`~` by RELAT_1:def 3;
    for x,y being object st [x,y] in R~`~` holds [x,y] in R
    proof
      let x,y be object;
      assume
X0:   [x,y] in R~`~`; then
x1:   x in field (R~`~`) & y in field (R~`~`) by RELAT_1:15;
      not [x,y] in R~`~ by XBOOLE_0:def 5,X0; then
      not [y,x] in R~` by RELAT_1:def 7; then
      [y,x] in R~ by Lemma12b,x1;
      hence thesis by RELAT_1:def 7;
    end; then
    R~`~` c= R by RELAT_1:def 3;
    hence thesis by n1,XBOOLE_0:def 10;
  end;
