unfold proj in H1.
spliter.
induction H3; spliter.
assert(Col P Q B').
apply col_permutation_1.
eapply proj_col.
apply H2.
induction(eq_dec_points P A').
subst P.
assert(Perp A' B' A A').
eapply perp_col.
assumption.
apply perp_left_comm.
eapply perp_col.
2: apply perp_left_comm.
2: apply H4.
auto.
Col.
Col.
assert(~Col A' A B').
eapply perp_not_col.
apply perp_comm.
Perp.
intro.
apply H8.
Col.
assert(Perp A' B' A A').
eapply perp_col.
assumption.
apply perp_left_comm.
eapply perp_col.
2: apply H4.
assumption.
assumption.
ColR.
assert(~Col A' A B').
eapply perp_not_col.
apply perp_comm.
Perp.
intro.
apply H9.
Col.
subst A'.
tauto.
Qed.

Lemma proj_par_strict : forall A B A' B' P Q, A <> A' -> B <> B' -> A' <> B' -> proj A P Q A' -> proj B P Q B' -> Par_strict A A' B B'.
intros.