Unset Regular Subst Tactic.

(* Specifications: *)
Lemma characterization_of_betweenness :
forall O E E' S U1 U2
A AX AY B BX BY C CX CY
BXMAX BYMAY CXMAX CYMAY,
Cd O E S U1 U2 A AX AY -> Cd O E S U1 U2 B BX BY -> Cd O E S U1 U2 C CX CY ->
Diff O E E' BX AX BXMAX -> Diff O E E' BY AY BYMAY ->
Diff O E E' CX AX CXMAX -> Diff O E E' CY AY CYMAY ->
(Bet A B C <-> exists T, O <> E /\ Col O E T /\
LeP O E E' O T /\ LeP O E E' T E /\
Prod O E E' T CXMAX BXMAX /\
Prod O E E' T CYMAY BYMAY).
Lemma same_abscissa_col : forall O E S U1 U2 A AX AY B BY C CY,
Cd O E S U1 U2 A AX AY -> Cd O E S U1 U2 B AX BY -> Cd O E S U1 U2 C AX CY ->
Col A B C.
Lemma characterization_of_collinearity :
forall O E E' S U1 U2
A AX AY B BX BY C CX CY
AXMBX AYMBY BXMCX BYMCY XProd YProd,
Cd O E S U1 U2 A AX AY -> Cd O E S U1 U2 B BX BY -> Cd O E S U1 U2 C CX CY ->
Diff O E E' AX BX AXMBX -> Diff O E E' AY BY AYMBY ->
Diff O E E' BX CX BXMCX -> Diff O E E' BY CY BYMCY ->
Prod O E E' AXMBX BYMCY XProd -> Prod O E E' AYMBY BXMCX YProd ->
(Col A B C <-> XProd = YProd).
Lemma le_pos_prod_le : forall O E E' A B C AC BC,
LeP O E E' A B -> LeP O E E' O C ->
Prod O E E' A C AC -> Prod O E E' B C BC ->
LeP O E E' AC BC.
Lemma bet_lt12_le23 : forall O E E' A B C,
Bet A B C -> LtP O E E' A B -> LeP O E E' B C.
Lemma bet_lt12_le13 : forall O E E' A B C,
Bet A B C -> LtP O E E' A B -> LeP O E E' A C.
Lemma bet_lt21_le32 : forall O E E' A B C,
Bet A B C -> LtP O E E' B A -> LeP O E E' C B.
Lemma bet_lt21_le31 : forall O E E' A B C,
Bet A B C -> LtP O E E' B A -> LeP O E E' C A.
Lemma opp_2_le_le : forall O E E' A MA B MB,
Opp O E E' A MA -> Opp O E E' B MB -> LeP O E E' A B -> LeP O E E' MB MA.
Lemma diff_2_le_le : forall O E E' A B C AMC BMC,
Diff O E E' A C AMC -> Diff O E E' B C BMC -> LeP O E E' A B ->
LeP O E E' AMC BMC.