
theorem :: Exercise 4.29.i
  for R being Relation holds R = R\~ iff R is asymmetric
proof
  let R be Relation;
  thus R = R\~ implies R is asymmetric;
  assume R is asymmetric;
  then
A1: R is_asymmetric_in field R;
  now
    let a,b be object;
    hereby
      assume
A2:   [a,b] in R;
      then
A3:   a in field R by RELAT_1:15;
      b in field R by A2,RELAT_1:15;
      then not [b,a] in R by A1,A2,A3;
      then not [a,b] in R~ by RELAT_1:def 7;
      hence [a,b] in R\~ by A2,XBOOLE_0:def 5;
    end;
    assume [a,b] in R\~;
    hence [a,b] in R;
  end;
  hence thesis by RELAT_1:def 2;
end;
