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_preserves_bet :
forall A B C A' B' C' X Y,
Bet A B C ->
Projp A A' X Y ->
Projp B B' X Y ->
Projp C C' X Y ->
Bet A' B' C'.