Definition playfair_ter := forall A1 A2 B1 B2 C1 C2 P,
A1 <> A2 -> B1 <> B2 -> C1 <> C2 ->
Col P B1 B2 -> Col P C1 C2 ->
~ (Col A1 B1 B2 /\ Col A2 B1 B2) ->
~ (Col A1 C1 C2 /\ Col A2 C1 C2) ->
~ (Col C1 B1 B2 /\ Col C2 B1 B2) ->
(exists I, ~ (~ (Col A1 A2 I /\ Col B1 B2 I) /\
~ (Col A1 B1 I /\ Col A2 B2 I) /\
~ (Col A1 B2 I /\ Col A2 B1 I))) ->
(exists I, ~ (~ (Col A1 A2 I /\ Col C1 C2 I) /\
~ (Col A1 C1 I /\ Col A2 C2 I) /\
~ (Col A1 C2 I /\ Col A2 C1 I))) ->
~ (~ (exists I, Col I A1 A2 /\ Col I B1 B2) /\
~ (exists I, Col I A1 A2 /\ Col I C1 C2)).
Definition playfair_quater_qf A1 A2 B1 B2 C1 C2 P :=
A1 <> A2 /\ B1 <> B2 /\ C1 <> C2 /\
Col P B1 B2 /\ Col P C1 C2 /\
~ (Col A1 B1 B2 /\ Col A2 B1 B2) /\
~ (Col A1 C1 C2 /\ Col A2 C1 C2) /\
~ (Col C1 B1 B2 /\ Col C2 B1 B2) /\
(exists I, ~ (~ (Col A1 A2 I /\ Col B1 B2 I) /\
~ (Col A1 B1 I /\ Col A2 B2 I) /\
~ (Col A1 B2 I /\ Col A2 B1 I))) /\
(exists I, ~ (~ (Col A1 A2 I /\ Col C1 C2 I) /\
~ (Col A1 C1 I /\ Col A2 C2 I) /\
~ (Col A1 C2 I /\ Col A2 C1 I))) /\
~ (exists I, Col I A1 A2 /\ Col I B1 B2) /\
~ (exists I, Col I A1 A2 /\ Col I C1 C2).
Definition playfair_quater := ~ exists A1 A2 B1 B2 C1 C2 P,
playfair_quater_qf A1 A2 B1 B2 C1 C2 P.

(* Specifications: *)
Lemma playfair__playfair_ter :
playfair_s_postulate -> playfair_ter.
Lemma playfair__playfair_quater :
playfair_s_postulate -> playfair_quater.
Lemma playfair_ter__playfair :
playfair_ter -> playfair_s_postulate.
Lemma not_ex_forall_not_7 :
forall (T : Type) (P : T -> T -> T -> T -> T -> T -> T -> Prop),
~(exists x1 : T, exists x2 : T, exists x3 : T,
exists x4 : T, exists x5 : T, exists x6 : T,
exists x7 :T, P x1 x2 x3 x4 x5 x6 x7) <->
forall x1 : T, forall x2 : T, forall x3 : T,
forall x4 : T, forall x5 : T, forall x6 : T,
forall x7 : T, ~ P x1 x2 x3 x4 x5 x6 x7.
Lemma playfair_quater__playfair :
playfair_quater -> playfair_s_postulate.