Hint Resolve thm4_1c thm4_1d.

(* Specifications: *)
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))).