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 Th17:
  TrivOrthoRelStr is Orthocomplemented
proof
  set O = TrivOrthoRelStr;
  set C = the Compl of O;
  set S = the carrier of O;
  reconsider f = C as Function of O,O;
  f OrthoComplement_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;
A2: for x being Element of O holds ex_sup_of {x,f.x},O & ex_inf_of {x,f.x}
    ,O & sup {x,f.x} = x & inf {x,f.x} = x
    proof
      let a be Element of O;
      {a,f.a} = {a} by A1;
      hence thesis by YELLOW_0:38,39;
    end;
A3: for x being Element of O holds sup {x,f.x} in {x,f.x} & inf {x,f.x} in
    {x,f.x}
    proof
      let a be Element of O;
      sup {a,f.a} = a & inf {a,f.a} = a by A2;
      hence thesis by TARSKI:def 2;
    end;
A4: for x being Element of O holds x is_maximum_of {x,f.x} & x
    is_minimum_of {x,f.x}
    proof
      let a be Element of O;
A5:   sup {a,f.a} = a & ex_sup_of {a,f.a},O by A2;
      sup {a,f.a} in {a,f.a} & inf {a,f.a} = a by A2,A3;
      hence thesis by A5,A2,WAYBEL_1:def 6,def 7;
    end;
    for y being Element of O holds sup {y,f.y} is_maximum_of S & inf {y,f
    .y} is_minimum_of S
    proof
      let a be Element of O;
      reconsider a0 = a as Element of S;
      {a0,f.a0} = {a0} by A1;
      then
A6:   {a0,f.a0} = S by TARSKI:def 1;
      a is_maximum_of {a,f.a} & a is_minimum_of {a,f.a} by A4;
      hence thesis by A2,A6;
    end;
    hence thesis by A2,Th14;
  end;
  hence thesis;
end;
