reserve X,Y for set,
  x,x1,x2,y,y1,y2,z for set,
  f,g,h for Function;
reserve M for non empty set;
reserve D for non empty set;
reserve P for Relation;
reserve O for Order of X;
reserve R,P for Relation,
  X,X1,X2,Y,Z,x,y,z,u for set,
  g,h for Function,
  O for Order of X,
  D for non empty set,
  d,d1,d2 for Element of D,
  A1,A2,B for Ordinal,
  L,L1,L2 for Sequence;

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;
