theorem
  O partially_orders X
proof
A1: field O = X by Lm4;
  then
A2: O is_antisymmetric_in X by RELAT_2:def 12;
A3: O is_transitive_in X by A1,RELAT_2:def 16;
  O is_reflexive_in X by A1,RELAT_2:def 9;
  hence thesis by A2,A3;
end;
