(* Specifications: *)
Theorem Uniqueness_of_constructed_lines : forall (x : Segment) (l : Line), Incident (origin x) l -> Incident (extremity x) l -> EqLn l (ln x).
Theorem Convergent_imp_distinct : forall l m : Line, ConLn l m -> DiLn l m.
Theorem Uniqueness_of_constructed_points : forall (x : Twolines) (a : Point), Incident a (line1 x) -> Incident a (line2 x) -> EqPt a (pt x).
Theorem cong_eqpt_apt : forall (a b : Point) (l : Line), Apart a l -> EqPt a b -> Apart b l.
Theorem cong_eqln_apt : forall (a : Point) (l m : Line), Apart a l -> EqLn l m -> Apart a m.
Theorem cong_eqpt_inc : forall (a b : Point) (l : Line), Incident a l -> EqPt a b -> Incident b l.
Theorem cong_eqln_inc : forall (a : Point) (l m : Line), Incident a l -> EqLn l m -> Incident a m.
Theorem cong_eqln_con : forall l m n : Line, ConLn l m -> EqLn m n -> ConLn l n.
Theorem cong_eqln_par : forall l m n : Line, Par l m -> EqLn m n -> Par l n.
Theorem cong_eqpt_dipt : forall a b c : Point, DiPt a b -> EqPt b c -> DiPt a c.
Theorem cong_eqln_diln : forall l m n : Line, DiLn l m -> EqLn m n -> DiLn l n.
Theorem eqln_imp_par : forall l m : Line, EqLn l m -> Par l m.
Theorem cong_par_con : forall l m n : Line, ConLn l m -> Par m n -> ConLn l n.
Theorem sym_SPar : forall x y : Line, SPar x y -> SPar y x.
Theorem cong_eqln_spar : forall l m n : Line, SPar l m -> EqLn m n -> SPar l n.
Theorem pb9_1 : forall (a : Point) (l : Line), ex (fun b : Point => Incident b l).
Theorem thm9_2 : Parallelogram.
Theorem thm4_1a : forall (x : Segment) (l : Line), DiLn l (ln x) -> Apart (origin x) l \/ Apart (extremity x) l.
Theorem thm4_1b : forall (x : Segment) (l : Line), Apart (origin x) l \/ Apart (extremity x) l -> DiLn l (ln x).
Theorem thm4_1c : forall (x : Twolines) (a : Point), DiPt a (pt x) -> Apart a (line1 x) \/ Apart a (line2 x).
Theorem thm4_1d : forall (x : Twolines) (a : Point), Apart a (line1 x) \/ Apart a (line2 x) -> DiPt a (pt x).
Theorem Symmetry_of_Apart : forall x y : Segment, Apart (origin x) (ln y) \/ Apart (extremity x) (ln y) -> Apart (origin y) (ln x) \/ Apart (extremity y) (ln x).
Theorem thm4_3a : forall (x : Segment) (c : Point), Apart c (ln x) -> DiPt c (origin x).
Theorem thm4_3b : forall (x : Segment) (c : Point), Apart c (ln x) -> DiPt c (extremity x).
Theorem Triangle_axioms_i : forall t : Triangle, Apart (origin (base t)) (ln (Side2 t)).
Theorem Triangle_axioms_ii : forall t : Triangle, Apart (extremity (base t)) (ln (reverse (Side1 t))).
Theorem Triangle_axioms_iii : forall t : Triangle, Apart (summit t) (ln (reverse (base t))).
Theorem thm4_5ia : forall (x : Segment) (l : Line), EqLn l (ln x) -> Incident (origin x) l /\ Incident (extremity x) l.
Theorem thm4_5ib : forall (x : Segment) (l : Line), Incident (origin x) l /\ Incident (extremity x) l -> EqLn l (ln x).
Theorem thm4_5iia : forall (x : Twolines) (a : Point), EqPt a (pt x) -> Incident a (line1 x) /\ Incident a (line2 x).
Theorem thm4_5iib : forall (x : Twolines) (a : Point), Incident a (line1 x) /\ Incident a (line2 x) -> EqPt a (pt x).
Theorem thm4_6 : forall x y : Segment, Incident (origin x) (ln y) /\ Incident (extremity x) (ln y) -> Incident (origin y) (ln x) /\ Incident (extremity y) (ln x).
Theorem thm4_7i : forall (a b c : Point) (H1 : DiPt a b) (H2 : DiPt b c), Incident c (ln (Seg a b H1)) -> Incident a (ln (reverse (Seg b c H2))).
Theorem thm4_7ii : forall (a b c : Point) (H1 : DiPt a b) (H2 : DiPt a c), Incident c (ln (Seg a b H1)) -> Incident b (ln (Seg a c H2)).
Theorem thm4_7iii : forall (x : Segment) (c : Point), Incident c (ln x) -> Incident c (ln (reverse x)).
Theorem Symmetry_of_Apart' : forall x y : Twolines, Apart (pt y) (line1 x) \/ Apart (pt y) (line2 x) -> Apart (pt x) (line1 y) \/ Apart (pt x) (line2 y).
Theorem thm4_9a : forall (x : Twolines) (c : Line), Apart (pt x) c -> DiLn c (line1 x).
Theorem thm4_9b : forall (x : Twolines) (c : Line), Apart (pt x) c -> DiLn c (line2 x).
Theorem thm5_3 : forall (x y : Segment) (z : Twolines), origin x = origin y -> line1 z = ln x -> line2 z = ln y -> EqPt (pt z) (origin x).
Theorem thm5_4 : forall (x y : Twolines) (z : Segment), line1 x = line1 y -> origin z = pt x -> extremity z = pt y -> EqLn (ln z) (line1 x).
Theorem thm5_5 : forall (a b c : Point) (H1 : DiPt a b) (H2 : EqPt b c), EqLn (ln (Seg a b H1)) (ln (Seg a c (cong_eqpt_dipt a b c H1 H2))).
Theorem thm5_6 : forall (l m n : Line) (H1 : ConLn l m) (H2 : EqLn m n), EqPt (pt (Twol l m H1)) (pt (Twol l n (cong_eqln_con l m n H1 H2))).