
theorem LemAntisym:
  for X being object, R being Relation holds
    R is antisymmetric iff
      for x, y being object st [x,y] in R & [y,x] in R holds x = y
  proof
    let X be object, R be Relation;
    thus R is antisymmetric implies
      for x, y being object st [x,y] in R & [y,x] in R holds x = y
    proof
      assume
A0:   R is antisymmetric;
      let x, y be object;
      assume A1: [x,y] in R & [y,x] in R; then
      x in field R & y in field R by RELAT_1:15;
      hence x = y by A0, A1, RELAT_2:def 4,def 12;
    end;
    assume
A1: for x, y being object st [x,y] in R & [y,x] in R holds x = y;
    set X = field R;
    for x,y being object st
      x in X & y in X & [x,y] in R & [y,x] in R holds x = y by A1;
    hence R is antisymmetric by RELAT_2:def 12,def 4;
  end;
