
theorem
  for X being set, R being Relation, Y being Subset of X st
    R is_asymmetric_in X holds R is_asymmetric_in Y
proof
  let X be set, R be Relation, Y be Subset of X;
  assume A1: R is_asymmetric_in X;
  for x,y being object holds x in Y & y in Y & [x,y] in R implies
    not [y,x] in R by A1, RELAT_2:def 5;
  hence thesis by RELAT_2:def 5;
end;
