reserve Y for non empty set,
  a for Function of Y,BOOLEAN,
  G for Subset of PARTITIONS(Y),
  P,Q for a_partition of Y;
reserve x,y,z for set,
  S, X for non empty set,
  R for Relation of X;

theorem
  R is_asymmetric_in X implies R is asymmetric
proof
  assume
A1: R is_asymmetric_in X;
  let x,y be object;
  field R c= X \/ X by RELSET_1:8;
  hence thesis by A1;
end;
