Hint Resolve perp_sym perp_left_comm perp_right_comm perp_comm per_perp_in per_perp
perp_in_per perp_in_left_comm perp_in_right_comm perp_in_comm perp_in_sym : perp.
Ltac double A B A' :=
assert (mp:= symmetric_point_construction A B);
elim mp; intros A' ; intro; clear mp.
Section T8_4.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

Lemma midpoint_existence_aux : forall A B P Q T,
A<>B -> Perp A B Q B -> Perp A B P A ->
Col A B T -> Bet Q T P -> Le A P B Q ->
exists X : Tpoint, Midpoint X A B.