reserve A, B, X, Y for set;

theorem Th13:
  for L1 being antisymmetric up-complete non empty reflexive
  RelStr, L2 being non empty reflexive RelStr, x being Element of L1, y being
Element of L2 st the RelStr of L1 = the RelStr of L2 & x = y holds waybelow x =
  waybelow y & wayabove x = wayabove y
proof
  let L1 be antisymmetric up-complete non empty reflexive RelStr, L2 be non
  empty reflexive RelStr, x be Element of L1, y be Element of L2 such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: x = y;
  hereby
    hereby
      let a be object;
      assume a in waybelow x;
      then consider z being Element of L1 such that
A3:   a = z and
A4:   z << x;
      reconsider w = z as Element of L2 by A1;
      L2 is antisymmetric by A1,WAYBEL_8:14;
      then w << y by A1,A2,A4,WAYBEL_8:8;
      hence a in waybelow y by A3;
    end;
    let a be object;
    assume a in waybelow y;
    then consider z being Element of L2 such that
A5: a = z and
A6: z << y;
    reconsider w = z as Element of L1 by A1;
    L2 is up-complete antisymmetric by A1,WAYBEL_8:14,15;
    then w << x by A1,A2,A6,WAYBEL_8:8;
    hence a in waybelow x by A5;
  end;
  hereby
    let a be object;
    assume a in wayabove x;
    then consider z being Element of L1 such that
A7: a = z and
A8: z >> x;
    reconsider w = z as Element of L2 by A1;
    L2 is antisymmetric by A1,WAYBEL_8:14;
    then w >> y by A1,A2,A8,WAYBEL_8:8;
    hence a in wayabove y by A7;
  end;
  let a be object;
  assume a in wayabove y;
  then consider z being Element of L2 such that
A9: a = z and
A10: z >> y;
  reconsider w = z as Element of L1 by A1;
  L2 is up-complete antisymmetric by A1,WAYBEL_8:14,15;
  then w >> x by A1,A2,A10,WAYBEL_8:8;
  hence thesis by A9;
end;
