reserve  X for non empty set,
  R for Relation of X;
reserve O for non empty RelStr;
reserve O for non empty OrthoRelStr;
reserve QO for QuasiOrdered non empty OrthoRelStr;

theorem Th14:
  the Compl of TrivOrthoRelStr is Orderinvolutive
proof
  set O = TrivOrthoRelStr;
  set C = the Compl of O;
  reconsider Emp = {} as Element of O by TARSKI:def 1;
  C is antitone Function of O,O
  proof
    reconsider f = C as Function of O,O;
    for x1,x2 being Element of O st x1 <= x2 for y1,y2 being Element of O
    st y1 = f.x1 & y2 = f.x2 holds y1 >= y2
    proof
      let a1,a2 be Element of O;
      set b1 = f.a1;
      b1 = Emp by FUNCT_2:50;
      then f.a2 <= b1 by FUNCT_2:50;
      hence thesis;
    end;
    hence thesis by WAYBEL_0:def 5;
  end;
  hence thesis;
end;
