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Parameter Apart : Point -> Line -> Prop.
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Definition Incident (a : Point) (l : Line) := ~ Apart a l.
Record Segment : Set := Seg
{origin : Point; extremity : Point; Seg_cond : DiPt origin extremity}.
Record Twolines : Set := Twol
{line1 : Line; line2 : Line; Twol_cond : ConLn line1 line2}.
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Axiom
line :
forall x : Segment,
{l : Line | Incident (origin x) l /\ Incident (extremity x) l}.
Axiom
point :
forall x : Twolines,
{a : Point | Incident a (line1 x) /\ Incident a (line2 x)}.
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Definition ln : Segment -> Line.

(* Specifications: *)
Theorem inc_ln2 : forall x : Segment, Incident (extremity x) (ln x).
Theorem inc_pt1 : forall x : Twolines, Incident (pt x) (line1 x).
Theorem inc_pt2 : forall x : Twolines, Incident (pt x) (line2 x).
Hint Resolve inc_ln1 inc_ln2 inc_pt1 inc_pt2.