assert(A' <> B').
intro.
apply H.
assert(Cong A' B' A B).
eapply l7_13.
apply H0.
assumption.
apply cong_symmetry in H3.
subst B'.
apply cong_identity in H3.
assumption.
induction(col_dec A B B').
assert(A' <> B' /\ Col A A' B' /\ Col B A' B').
eapply (midpoint_col _ _ _ _ M); auto.
unfold Par.
right.
split; auto.
assert(HH0:= H0).
assert(HH1:= H1).
unfold Midpoint in HH0.
unfold Midpoint in HH1.
spliter.
assert(Col M A A').
apply bet_col in H6.
Col.
assert(Col M B B').
apply bet_col in H4.
Col.
unfold Par.
left.
unfold Par_strict.
repeat split; auto; try apply all_coplanar.
intro.
ex_and H10 X.
prolong X M X' M X.
assert(Col A' B' X').
eapply mid_preserves_col.
2: apply H0.
2: apply H1.
apply col_permutation_1.
apply H10.
unfold Midpoint.
split.
assumption.
apply cong_left_commutativity.
Cong.
assert(Col B' X X').
eapply (col_transitivity_1 _ A').
auto.
Col.
Col.
induction(eq_dec_points X X').
subst X'.
apply between_identity in H12.
subst X.
apply H3.
induction(eq_dec_points B M).
subst M.
apply cong_symmetry in H5.
apply cong_identity in H5.
subst B'.
Col.
apply col_permutation_2.
apply (col_transitivity_1 _ M); Col.
assert(Col X M B').
apply bet_col in H12.
apply (col_transitivity_1 _ X'); Col.
assert(Col X' M B').
apply bet_col in H12.
apply (col_transitivity_1 _ X); Col.
assert(Col M B X).
eapply (col_transitivity_1 ).
2: apply col_permutation_5.
2: apply H9.
intro.
subst B'.
apply cong_identity in H5.
subst B.
apply H3.
Col.
Col.
assert(Col X M A).
eapply (col_transitivity_1 ).
2: apply col_permutation_3.
2:apply H19.
intro.
subst X.
assert(Cong M X' M B').
eapply cong_transitivity.
apply H13.
Cong.
assert (HH:=l7_20 M X' B' H18 H20).
induction HH.
subst X'.
apply H3.
apply col_permutation_2.
apply (col_transitivity_1 _ M).
intro.
subst M.
apply cong_identity in H13.
contradiction.
Col.
assert(Col M B A').
ColR.
induction(eq_dec_points M A').
subst A'.
apply cong_identity in H7.
subst A.
Col.
ColR.
assert(X'= B).
eapply l7_9.
apply H21.
assumption.
subst X'.
tauto.
Col.
apply H3.
eapply col3.
2: apply H20.
intro.
subst X.
apply cong_identity in H13.
subst X'.
tauto.
Col.
Col.
Qed.

Lemma midpoint_par_strict : forall A B A' B' M, A <> B -> ~Col A B B' -> Midpoint M A A' -> Midpoint M B B' -> Par_strict A B A' B'.
intros.