theorem
  R is_irreflexive_in X implies R is irreflexive
proof
A1: field R c= X \/ X by RELSET_1:8;
  assume R is_irreflexive_in X;
  then for x being object holds x in field R implies not [x,x] in R by A1;
  then R is_irreflexive_in field R;
  hence thesis;
end;
