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 Th16:
  for p being Point of TOP-REAL 2 st p in C & p <> W-min C
  holds Segment(p,W-min C,C) is_an_arc_of p,W-min C
proof
  set q = W-min C;
  let p be Point of TOP-REAL 2 such that
A1: p in C and
A2: p <> W-min C;
A3: E-max C in C by SPRECT_1:14;
A4: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:50;
A5: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:50;
A6: q <> E-max C by TOPREAL5:19;
  per cases by A1,A3,JORDAN16:7;
  suppose that
A7: p <> E-max C and
A8: LE p, E-max C, C;
A9: now
      assume W-min C in R_Segment(Upper_Arc C,W-min C,E-max C,p)
      /\ L_Segment(Lower_Arc C,E-max C,W-min C,q);
      then W-min C in R_Segment(Upper_Arc C,W-min C,E-max C,p) by
XBOOLE_0:def 4;
      hence contradiction by A2,A4,JORDAN6:60;
    end;
    p in Upper_Arc C by A8,JORDAN17:3;
    then
A10: LE p,E-max C,Upper_Arc C,W-min C,E-max C by A4,JORDAN5C:10;
A11: R_Segment(Upper_Arc C,W-min C,E-max C,p) =
    Segment(Upper_Arc C,W-min C,E-max C,p,E-max C) by Th14;
    then
A12: E-max C in R_Segment(Upper_Arc C,W-min C,E-max C,p) by A10,JORDAN16:5;
    q in Lower_Arc C by JORDAN7:1;
    then
A13: LE E-max C,q,Lower_Arc C,E-max C,W-min C by A5,JORDAN5C:10;
A14: L_Segment(Lower_Arc C,E-max C,W-min C,q) =
    Segment(Lower_Arc C,E-max C,W-min C,E-max C,q) by Th15;
    then E-max C in L_Segment(Lower_Arc C,E-max C,W-min C,q) by A13,JORDAN16:5;
    then
A15: E-max C in R_Segment(Upper_Arc C,W-min C,E-max C,p)
    /\ L_Segment(Lower_Arc C,E-max C,W-min C,q) by A12,XBOOLE_0:def 4;
A16: R_Segment(Upper_Arc C,W-min C,E-max C,p) c= Upper_Arc C by JORDAN6:20;
A17: L_Segment(Lower_Arc C,E-max C,W-min C,q) c= Lower_Arc C by JORDAN6:19;
    Upper_Arc C /\ Lower_Arc C = {W-min C, E-max C} by JORDAN6:def 9;
    then
A18: R_Segment(Upper_Arc C,W-min C,E-max C,p)
    /\ L_Segment(Lower_Arc C,E-max C,W-min C,q) = {E-max C}
    by A9,A15,A16,A17,TOPREAL8:1,XBOOLE_1:27;
A19: R_Segment(Upper_Arc C,W-min C,E-max C,p) is_an_arc_of p, E-max C
    by A4,A7,A10,A11,JORDAN16:21;
A20: L_Segment(Lower_Arc C,E-max C,W-min C,q) is_an_arc_of E-max C,q
    by A5,A6,A13,A14,JORDAN16:21;
    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) by A8,Th13;
    hence thesis by A18,A19,A20,TOPREAL1:2;
  end;
  suppose
A21: p = E-max C;
    then Segment(p,q,C) = Lower_Arc C by JORDAN7:4;
    hence thesis by A21,JORDAN6:50;
  end;
  suppose that p <> E-max C and
A22: LE E-max C,p, C;
A23: Segment(p,q,C) = Segment(Lower_Arc C,E-max C,W-min C,p,q) by A22,Th10;
    p in Lower_Arc C by A22,JORDAN17:4;
    then LE p, q, Lower_Arc C, E-max C, W-min C by A5,JORDAN5C:10;
    hence thesis by A2,A5,A23,JORDAN16:21;
  end;
end;
