
theorem :: Exercise 4.29.iii
  for R being Relation, a, b being set st R is antisymmetric
  holds [a,b] in R\~ iff [a,b] in R & a <> b
proof
  let R be Relation, a, b be set;
  assume R is antisymmetric;
  then
A1: R is_antisymmetric_in field R;
A2: R\~ is_asymmetric_in field (R\~) by RELAT_2:def 13;
  hereby
    assume
A3: [a,b] in R\~;
    hence [a,b] in R;
    now
      assume
A4:   a = b;
      a in field (R\~) by A3,RELAT_1:15;
      hence contradiction by A2,A3,A4;
    end;
    hence a <> b;
  end;
  assume that
A5: [a,b] in R and
A6: a <> b;
A7: a in field R by A5,RELAT_1:15;
  b in field R by A5,RELAT_1:15;
  then not [b,a] in R by A1,A5,A6,A7;
  then not [a,b] in R~ by RELAT_1:def 7;
  hence thesis by A5,XBOOLE_0:def 5;
end;
