reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th13:
  LE p,E-max C,C implies
  Segment(p,W-min C,C) = R_Segment(Upper_Arc C,W-min C,E-max C,p)
  \/ L_Segment(Lower_Arc C,E-max C,W-min C,W-min C)
proof
  set q = W-min C;
  assume LE p,E-max C,C;
  then
A1: p in Upper_Arc C by JORDAN17:3;
A2: q in Lower_Arc C by JORDAN7:1;
A3: Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:50;
  defpred P[Point of TOP-REAL 2] means LE p,$1,C or p in C & $1 = W-min C;
  defpred Q1[Point of TOP-REAL 2] means LE p,$1,Upper_Arc C,W-min C,E-max C;
  defpred Q2[Point of TOP-REAL 2] means LE $1,q,Lower_Arc C,E-max C,W-min C;
  defpred Q[Point of TOP-REAL 2] means Q1[$1] or Q2[$1];
A4: P[p1] iff Q[p1]
  proof
    thus LE p,p1,C or p in C & p1=W-min C implies
    LE p,p1,Upper_Arc C,W-min C,E-max C or LE p1,q,Lower_Arc C,E-max C,W-min C
    proof
      assume
A5:   LE p,p1,C or p in C & p1=W-min C;
A6:   now
        assume that
A7:     p1 in Lower_Arc C and
A8:     p1 in Upper_Arc C;
        p1 in Lower_Arc C /\ Upper_Arc C by A7,A8,XBOOLE_0:def 4;
        then p1 in {W-min C,E-max C} by JORDAN6:def 9;
        hence p1 = W-min C or p1 = E-max C by TARSKI:def 2;
      end;
      per cases by A6;
      suppose p1 = W-min C;
        hence thesis by A2,JORDAN5C:9;
      end;
      suppose p1 = E-max C;
        hence thesis by A2,A3,JORDAN5C:10;
      end;
      suppose not p1 in Lower_Arc C & p1 <> W-min C;
        hence thesis by A5,JORDAN6:def 10;
      end;
      suppose that
A9:     not p1 in Upper_Arc C and
A10:    p1 <> W-min C;
A11:    p1 in C by A5,A10,JORDAN7:5;
        C = Lower_Arc C \/ Upper_Arc C by JORDAN6:def 9;
        then p1 in Lower_Arc C by A9,A11,XBOOLE_0:def 3;
        hence thesis by A3,JORDAN5C:10;
      end;
    end;
    assume that
A12: LE p,p1,Upper_Arc C,W-min C,E-max C or
    LE p1,q,Lower_Arc C,E-max C,W-min C;
A13: Upper_Arc C c= C by JORDAN6:61;
    per cases by A12;
    suppose
A14:  LE p,p1,Upper_Arc C,W-min C,E-max C;
      then p1 in Upper_Arc C by JORDAN5C:def 3;
      hence thesis by A1,A14,JORDAN6:def 10;
    end;
    suppose
      LE p1,q,Lower_Arc C,E-max C,W-min C;
      then
A15:  p1 in Lower_Arc C by JORDAN5C:def 3;
      now per cases;
        suppose p1 = W-min C;
          hence thesis by A1,A13;
        end;
        suppose p1 <> W-min C;
          hence thesis by A1,A15,JORDAN6:def 10;
        end;
      end;
      hence thesis;
    end;
  end;
  set Y1 = {p1: Q1[p1]}, Y2 = {p1: Q2[p1]};
  deffunc F(set) = $1;
  set X = {F(p1): P[p1]}, Y = {F(p1): Q[p1]}, Y9 = {p1: Q1[p1] or Q2[p1]};
A16: X = Y from FRAENKEL:sch 3(A4);
A17: Segment(p,q,C) = X by JORDAN7:def 1;
A18: L_Segment(Lower_Arc C,E-max C,W-min C,q) = Y2 by JORDAN6:def 3;
  Y9 = Y1 \/ Y2 from TOPREAL1:sch 1;
  hence thesis by A16,A17,A18,JORDAN6:def 4;
end;
