
theorem Lemma22:
  for X being set, R being asymmetric Relation of X holds
    R * R misses id X
  proof
    let X be set, R be asymmetric Relation of X;
    for x,y being object holds not [x,y] in R*R /\ id X
    proof
      let x,y be object;
      assume [x,y] in R*R /\ id X; then
A1:   [x,y] in R*R & [x,y] in id X by XBOOLE_0:def 4; then
      consider z being object such that
A2:   [x,z] in R & [z,y] in R by RELAT_1:def 8;
      x = y by RELAT_1:def 10,A1;
      hence contradiction by LemAsym,A2;
    end;
    hence thesis by XBOOLE_0:def 7,RELAT_1:37;
  end;
