unfold proj in *.
spliter.
induction H2;
spliter; split;auto.
induction(col_dec P Q B).
right.
split.
assumption.
induction(eq_dec_points A' P).
subst P.
assert(Perp_at A' A' Q A A').
eapply perp_perp_in.
assumption.
eapply l8_14_2_1b.
apply H6.
Col.
Col.
assert(Perp P A' A A').
eapply perp_col.
auto.
apply H3.
assumption.
assert(Perp_at A' A' P A A').
eapply perp_perp_in.
Perp.
induction(eq_dec_points B P).
subst P.
assert(Perp B Q A B).
apply perp_sym.
eapply perp_col.
intro.
subst B.
apply H2.
Col.
apply perp_sym.
apply H3.
Col.
eapply l8_14_2_1b.
apply H8.
Col.
Col.
assert(Perp P B A B).
eapply perp_col.
auto.
apply perp_sym.
eapply perp_col.
intro.
subst B.
contradiction.
apply perp_sym.
apply H3.
Col.
Col.
eapply l8_14_2_1b.
apply H8.
ColR.
Col.
left.
repeat split; auto.
apply perp_sym.
apply perp_left_comm.
eapply perp_col.
intro.
subst A'.
contradiction.
apply perp_sym.
apply perp_right_comm.
apply H3.
Col.
subst A'.
tauto.
Qed.

Lemma proj_id : forall A B A' B' P Q, A <> A' -> Col A B A' -> proj A P Q A' -> proj B P Q B' -> A'= B'.
intros.