
theorem LemAsym:
  for R being Relation holds
    R is asymmetric iff
      for x, y being object st [x,y] in R holds not [y,x] in R
  proof
    let R be Relation;
A1: R is asymmetric implies
      for x, y being object st [x,y] in R holds not [y,x] in R
    proof
      assume
Z0:   R is asymmetric;
      let x, y be object;
      assume
B1:   [x,y] in R; then
      x in field R & y in field R by RELAT_1:15;
      hence not [y,x] in R by Z0, RELAT_2:def 5,def 13,B1;
    end;
    (for x, y being object st [x,y] in R holds not [y,x] in R) implies
      R is asymmetric
    proof
      assume
Z1:   for x, y being object st [x,y] in R holds not [y,x] in R;
      set X = field R;
      for x,y being object st
        x in X & y in X & [x,y] in R holds not [y,x] in R by Z1;
      hence thesis by RELAT_2:def 13,RELAT_2:def 5;
    end;
    hence thesis by A1;
  end;
