Ltac image_6 A B P' H P:=
 let T:= fresh in assert (T:= l10_6_existence A B P' H);
 ex_and T P.Ltac image A B P P':=
 let T:= fresh in assert (T:= l10_2_existence A B P);
 ex_and T P'.Ltac perp A B C X :=
 match goal with
   | H:(~Col A B C) |- _ =>
    let T:= fresh in assert (T:= l8_18_existence A B C H);
    ex_and T X
 end.Ltac parallel A B C D P :=
 match goal with
   | H:(A <> B) |- _ =>
    let T := fresh in assert(T:= parallel_existence A B P H);
    ex_and T C
 end.Ltac par_strict :=
repeat
 match goal with
      | H: Par_strict ?A ?B ?C ?D |- _ =>
       let T := fresh in not_exist_hyp (Par_strict B A D C); assert (T := par_strict_comm A B C D H)  
      | H: Par_strict ?A ?B ?C ?D |- _ =>
       let T := fresh in not_exist_hyp (Par_strict C D A B); assert (T := par_strict_symmetry A B C D H)  
      | H: Par_strict ?A ?B ?C ?D |- _ =>
       let T := fresh in not_exist_hyp (Par_strict B A C D); assert (T := par_strict_left_comm A B C D H)   
      | H: Par_strict ?A ?B ?C ?D |- _ =>
       let T := fresh in not_exist_hyp (Par_strict A B D C); assert (T := par_strict_right_comm A B C D H) 
 end.Ltac clean_trivial_hyps :=
  repeat
  match goal with
   | H:(Cong ?X1 ?X1 ?X2 ?X2) |- _ => clear H
   | H:(Cong ?X1 ?X2 ?X2 ?X1) |- _ => clear H
   | H:(Cong ?X1 ?X2 ?X1 ?X2) |- _ => clear H
   | H:(Bet ?X1 ?X1 ?X2) |- _ => clear H
   | H:(Bet ?X2 ?X1 ?X1) |- _ => clear H
   | H:(Col ?X1 ?X1 ?X2) |- _ => clear H
   | H:(Col ?X2 ?X1 ?X1) |- _ => clear H
   | H:(Col ?X1 ?X2 ?X1) |- _ => clear H
   | H:(Par ?X1 ?X2 ?X1 ?X2) |- _ => clear H
   | H:(Par ?X1 ?X2 ?X2 ?X1) |- _ => clear H
   | H:(Per ?X1 ?X2 ?X2)     |- _ => clear H
   | H:(Per ?X1 ?X1 ?X2)     |- _ => clear H
   | H:(Midpoint ?X1 ?X1 ?X1) |- _ => clear H
end.Ltac show_distinct2 := unfold not;intro;treat_equalities; try (solve [intuition]).Ltac symmetric A B A' :=
 let T := fresh in assert(T:= symmetric_point_construction A B);
 ex_and T A'.Ltac sfinish := repeat match goal with
 | |- Bet ?A ?B ?C => Between; eBetween
 | |- Col ?A ?B ?C => Col;ColR
 | |- ~ Col ?A ?B ?C => Col
 | |- ~ Col ?A ?B ?C => intro;search_contradiction
 | |- Par ?A ?B ?C ?D => Par
 | |- Par_strict ?A ?B ?C ?D => Par
 | |- Perp ?A ?B ?C ?D => Perp
 | |- Perp_at ?A ?B ?C ?D ?E => Perp
 | |- Per ?A ?B ?C => Perp
 | |- Cong ?A ?B ?C ?D => Cong;eCong
 | |- Midpoint ?A ?B ?C => Midpoint
 | |- ?A<>?B => apply swap_diff;assumption
 | |- ?A<>?B => intro;treat_equalities; solve [search_contradiction]
 | |- ?G1 /\ ?G2 => split
 | |- _ => try assumption
end.Ltac clean_reap_hyps :=
  clean_duplicated_hyps;
  repeat
  match goal with
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?B ?C ?D ?A |- _ => clear H2
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?C ?D ?A ?B |- _ => clear H2
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?D ?A ?B ?C |- _ => clear H2
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?D ?C ?B ?A |- _ => clear H2
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?C ?B ?A ?D |- _ => clear H2
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?B ?A ?D ?C |- _ => clear H2
   | H:(Parallelogram ?A ?B ?C ?D), H2 : Parallelogram ?A ?D ?C ?B |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?A ?B ?D ?C |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?C ?D ?A ?B |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?C ?D ?B ?A |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?D ?C ?B ?A |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?D ?C ?A ?B |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?B ?A ?C ?D |- _ => clear H2
   | H:(Par ?A ?B ?C ?D), H2 : Par ?B ?A ?D ?C |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?A ?B ?D ?C |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?C ?D ?A ?B |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?C ?D ?B ?A |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?D ?C ?B ?A |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?D ?C ?A ?B |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?B ?A ?C ?D |- _ => clear H2
   | H:(Par_strict ?A ?B ?C ?D), H2 : Par_strict ?B ?A ?D ?C |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?A ?B ?D ?C |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?C ?D ?A ?B |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?C ?D ?B ?A |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?D ?C ?B ?A |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?D ?C ?A ?B |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?B ?A ?C ?D |- _ => clear H2
   | H:(Perp ?A ?B ?C ?D), H2 : Perp ?B ?A ?D ?C |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?A ?B ?D ?C |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?C ?D ?A ?B |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?C ?D ?B ?A |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?D ?C ?B ?A |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?D ?C ?A ?B |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?B ?A ?C ?D |- _ => clear H2
   | H:(Perp_at ?X ?A ?B ?C ?D), H2 : Perp_at ?X ?B ?A ?D ?C |- _ => clear H2
   | H:(Per ?A ?B ?C), H2 : Per ?C ?B ?A |- _ => clear H2
   | H:(Midpoint ?A ?B ?C), H2 : Midpoint ?A ?C ?B |- _ => clear H2
   | H:(~Col ?A ?B ?C), H2 : (~Col ?B ?A ?C) |- _ => clear H2
   | H:(~Col ?A ?B ?C), H2 : (~Col ?B ?C ?A) |- _ => clear H2
   | H:(~Col ?A ?B ?C), H2 : (~Col ?C ?B ?A) |- _ => clear H2
   | H:(~Col ?A ?B ?C), H2 : (~Col ?C ?A ?B) |- _ => clear H2
   | H:(~Col ?A ?B ?C), H2 : (~Col ?A ?C ?B) |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?A ?C ?B |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?B ?A ?C |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?B ?C ?A |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?C ?B ?A |- _ => clear H2
   | H:(Col ?A ?B ?C), H2 : Col ?C ?A ?B |- _ => clear H2
   | H:(Bet ?A ?B ?C), H2 : Bet ?C ?B ?A |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?A ?B ?D ?C |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?C ?D ?A ?B |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?C ?D ?B ?A |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?D ?C ?B ?A |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?D ?C ?A ?B |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?B ?A ?C ?D |- _ => clear H2
   | H:(Cong ?A ?B ?C ?D), H2 : Cong ?B ?A ?D ?C |- _ => clear H2
   | H:(?A<>?B), H2 : (?B<>?A) |- _ => clear H2
end.Ltac tag_hyps :=
  repeat
  match goal with
    | H : ?A <> ?B |- _ => apply Diff_Diff_tagged in H
    | H : Cong ?A ?B ?C ?D |- _ => apply Cong_Cong_tagged in H
    | H : Bet ?A ?B ?C |- _ => apply Bet_Bet_tagged in H
    | H : Col ?A ?B ?C |- _ => apply Col_Col_tagged in H
    | H : ~ Col ?A ?B ?C |- _ => apply NCol_NCol_tagged in H
    | H : Midpoint ?A ?B ?C |- _ => apply Mid_Mid_tagged in H
    | H : Per ?A ?B ?C |- _ => apply Per_Per_tagged in H
    | H : Perp_at ?X ?A ?B ?C ?D |- _ => apply Perp_in_Perp_in_tagged in H
    | H : Perp ?A ?B ?C ?D |- _ => apply Perp_Perp_tagged in H
    | H : Par_strict ?A ?B ?C ?D |- _ => apply Par_strict_Par_strict_tagged in H
    | H : Par ?A ?B ?C ?D |- _ => apply Par_Par_tagged in H
    | H : Parallelogram ?A ?B ?C ?D |- _ => apply Plg_Plg_tagged in H
  end.Ltac permutation_intro_in_goal :=
 match goal with
 | |- Par ?A ?B ?C ?D => apply Par_cases
 | |- Par_strict ?A ?B ?C ?D => apply Par_strict_cases
 | |- Perp ?A ?B ?C ?D => apply Perp_cases
 | |- Perp_at ?X ?A ?B ?C ?D => apply Perp_in_cases
 | |- Per ?A ?B ?C => apply Per_cases
 | |- Midpoint ?A ?B ?C => apply Mid_cases
 | |- ~ Col ?A ?B ?C => apply NCol_cases
 | |- Col ?A ?B ?C => apply Col_cases
 | |- Bet ?A ?B ?C => apply Bet_cases
 | |- Cong ?A ?B ?C ?D => apply Cong_cases
 end.Ltac perm_apply t :=
 permutation_intro_in_goal;
 try_or ltac:(apply t;solve [finish]).Ltac assert_ncols :=
repeat
  match goal with
      | H:OS ?A ?B ?X ?Y |- _ =>
     not_exist_hyp_perm_ncol A B X;assert (~ Col A B X) by (apply(one_side_not_col123 A B X Y);finish)

      | H:OS ?A ?B ?X ?Y |- _ =>
     not_exist_hyp_perm_ncol A B Y;assert (~ Col A B Y) by (apply(one_side_not_col124 A B X Y);finish)

      | H:TS ?A ?B ?X ?Y |- _ =>
     not_exist_hyp_perm_ncol A B X;assert (~ Col A B X) by (apply(two_sides_not_col A B X Y);finish)

      | H:TS ?A ?B ?X ?Y |- _ =>
     not_exist_hyp_perm_ncol A B Y;assert (~ Col A B Y) by (apply(two_sides_not_col A B Y X);finish)

      | H:~ Coplanar ?A ?B ?C ?D |- _ =>
      let h := fresh in
      not_exist_hyp_perm4 A B C D;
      assert (h := ncop__ncols A B C D H);decompose [and] h;clear h;clean_reap_hyps

      | H:Par_strict ?A ?B ?C ?D |- _ =>
      let h := fresh in
      not_exist_hyp_perm4 A B C D;
      assert (h := par_strict_not_cols A B C D H);decompose [and] h;clear h;clean_reap_hyps

      | H:Parallelogram_strict ?A ?B ?C ?D |- _ =>
      let h := fresh in
      not_exist_hyp_perm4 A B C D;
      assert (h := plgs_not_col A B C D H);decompose [and] h;clear h;clean_reap_hyps
  end.

Lemma par_preserves_conga_os : forall A B C D P , Par A B C D -> Bet A D P -> D <> P -> OS A D B C -> CongA B A P C D P.