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;

theorem
  x in X & y in X & z in X & [x,y] in O & [y,z] in O implies [x,z] in O
proof
  field O = X by Lm4;
  then O is_transitive_in X by RELAT_2:def 16;
  hence thesis;
end;
