Section bachmann_s_lotschnittaxiom_variant.
Context `{TnEQD:Tarski_neutral_dimensionless_with_decidable_point_equality}.

Lemma bachmann_s_lotschnittaxiom_aux : bachmann_s_lotschnittaxiom <->
forall A1 A2 B1 B2 C1 C2 D1 D2 IAB IAC IBD, IAB <> IAC -> IAB <> IBD ->
Perp A1 A2 B1 B2 -> Perp A1 A2 C1 C2 -> Perp B1 B2 D1 D2 ->
Col A1 A2 IAB -> Col B1 B2 IAB -> Col A1 A2 IAC ->
Col C1 C2 IAC -> Col B1 B2 IBD -> Col D1 D2 IBD ->
Coplanar IAB IAC IBD C1 -> Coplanar IAB IAC IBD C2 ->
Coplanar IAB IAC IBD D1 -> Coplanar IAB IAC IBD D2 ->
exists I, Col C1 C2 I /\ Col D1 D2 I.