unfold Projp in *.
spliter.
induction H1; induction H2.
spliter.
apply (l8_18_uniqueness A B P); Col.
apply perp_not_col2 in H4.
induction H4.
assumption.
contradiction.
spliter.
subst P'.
apply perp_not_col2 in H3.
induction H3; contradiction.
spliter.
subst Q'.
apply perp_not_col2 in H4.
induction H4; contradiction.
spliter.
subst P'.
subst Q'.
auto.
Qed.

Lemma projp_projp_perp : forall P P1 P2 Q1 Q2,
P1 <> P2 -> Projp P1 P Q1 Q2 -> Projp P2 P Q1 Q2 -> Perp P1 P2 Q1 Q2.