reserve X, Y for non empty set;

theorem
  for r being Relation of X st r is antisymmetric holds chi(r,[:X,X:])
  is antisymmetric
proof
  let r be Relation of X;
  assume r is antisymmetric;
  then
A1: r is_antisymmetric_in field r by RELAT_2:def 12;
  for x,y being Element of X holds chi(r,[:X,X:]).(x,y) <> 0 & chi(r,[:X,X
  :]).(y,x) <> 0 implies x = y
  proof
    let x,y be Element of X;
    assume that
A2: chi(r,[:X,X:]).(x,y) <> 0 and
A3: chi(r,[:X,X:]).(y,x) <> 0;
A4: x in field r & y in field r & [x,y] in r & [y,x] in r implies x = y by A1,
RELAT_2:def 4;
    [x,y] in r by A2,FUNCT_3:def 3;
    hence thesis by A3,A4,FUNCT_3:def 3,RELAT_1:15;
  end;
  hence thesis;
end;
