Ltac not_exist_hyp_perm_par A B C D := not_exist_hyp (Par A B C D); not_exist_hyp (Par A B D C);
                                       not_exist_hyp (Par B A C D); not_exist_hyp (Par B A D C);
                                       not_exist_hyp (Par C D A B); not_exist_hyp (Par C D B A);
                                       not_exist_hyp (Par D C A B); not_exist_hyp (Par D C B A).Ltac not_exist_hyp_perm_pars A B C D := not_exist_hyp (Par_strict A B C D); not_exist_hyp (Par_strict A B D C);
                                        not_exist_hyp (Par_strict B A C D); not_exist_hyp (Par_strict B A D C);
                                        not_exist_hyp (Par_strict C D A B); not_exist_hyp (Par_strict C D B A);
                                        not_exist_hyp (Par_strict D C A B); not_exist_hyp (Par_strict D C B A).Ltac assert_pars_1 :=
 repeat
 match goal with
      | H:Par_strict ?X1 ?X2 ?X3 ?X4 |- _ =>
      let h := fresh in
      not_exist_hyp_perm_par X1 X2 X3 X4;
      assert (h := par_strict_par X1 X2 X3 X4 H)

      | H:Par ?X1 ?X2 ?X3 ?X4, H2:Col ?X1 ?X2 ?X5, H3:(~Col ?X3 ?X4 ?X5) |- _ =>
      let h := fresh in
      not_exist_hyp_perm_pars X1 X2 X3 X4;
      assert (h := par_not_col_strict X1 X2 X3 X4 X5 H H2 H3)

      | H: ?X1 <> ?X2, H2:?X2 <> ?X3, H3:Parallelogram ?X1 ?X2 ?X3 ?X4 |- _ =>
      let h := fresh in
      not_exist_hyp_perm_par X1 X2 X3 X4;
      assert (h := plg_par_1 X1 X2 X3 X4 H H2 H3)

      | H:Parallelogram_strict ?X1 ?X2 ?X3 ?X4 |- _ =>
      let h := fresh in
      not_exist_hyp_perm_pars X1 X2 X3 X4;
      assert (h := plgs_pars_1 X1 X2 X3 X4 H)
  end.Ltac assert_pars_2 :=
 repeat
 match goal with
      | H: ?X1 <> ?X2, H2:?X2 <> ?X3, H3:Parallelogram ?X1 ?X2 ?X3 ?X4 |- _ =>
      let h := fresh in
      not_exist_hyp_perm_par X1 X4 X2 X3;
      assert (h := plg_par_2 X1 X2 X3 X4 H H2 H3)

      | H:Parallelogram_strict ?X1 ?X2 ?X3 ?X4 |- _ =>
      let h := fresh in
      not_exist_hyp_perm_pars X1 X4 X2 X3;
      assert (h := plgs_pars_2 X1 X2 X3 X4 H)
  end.Ltac assert_pars_perm := permutation_intro_in_hyps; assert_pars_1; assert_pars_2; clean_reap_hyps.

Lemma perp3__rect : forall A B C D, Perp A B B C -> Perp B C C D -> Perp C D D A -> Rectangle A B C D.