reserve C for Simple_closed_curve,
  A,A1,A2 for Subset of TOP-REAL 2,
  p,p1,p2,q ,q1,q2 for Point of TOP-REAL 2,
  n for Element of NAT;

theorem
  Segment(p,q,C) c= C
proof
  set S =Segment(p,q,C);
  let e be object such that
A1: e in S;
  Upper_Arc C \/ Lower_Arc C = C by JORDAN6:50;
  then
A2: Upper_Arc C c= C & Lower_Arc C c= C by XBOOLE_1:7;
  per cases;
  suppose
    q = W-min C;
    then S = {p1: LE p,p1,C or p in C & p1=W-min C} by JORDAN7:def 1;
    then consider p1 such that
A3: e = p1 &( LE p,p1,C or p in C & p1=W-min C) by A1;
    now
      assume LE p,p1,C;
      then p1 in Upper_Arc C or p1 in Lower_Arc C by JORDAN6:def 10;
      hence p1 in C by A2;
    end;
    hence thesis by A3,SPRECT_1:13;
  end;
  suppose
    q <> W-min C;
    then S = {p1: LE p,p1,C & LE p1,q,C} by JORDAN7:def 1;
    then consider p1 such that
A4: e = p1 and
A5: LE p,p1,C and
    LE p1,q,C by A1;
    p1 in Upper_Arc C or p1 in Lower_Arc C by A5,JORDAN6:def 10;
    hence thesis by A2,A4;
  end;
end;
