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 Th15:
  TrivOrthoRelStr is QuasiOrthocomplemented
proof
  set O = TrivOrthoRelStr;
  set C = the Compl of O;
  set S = the carrier of O;
  C QuasiOrthoComplement_on O
  proof
    reconsider f = C as Function of O,O;
A1: for x being Element of S holds {x,op1.x} = {x}
    by Lm2,PARTIT_2:19,ENUMSET1:29;
    for x being Element of O holds sup {x,f.x} = x & inf {x,f.x} = x &
    ex_sup_of {x,f.x},O & ex_inf_of {x,f.x},O
    proof
      let a be Element of O;
      {a,f.a} = {a} by A1;
      hence thesis by YELLOW_0:38,39;
    end;
    hence thesis by Th14;
  end;
  hence thesis;
end;
