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 Th17:
  for p,q being Point of TOP-REAL 2 st p<> q & LE p,q,C
  holds Segment(p,q,C) is_an_arc_of p,q
proof
  let p,q be Point of TOP-REAL 2 such that
A1: p <> q and
A2: LE p,q,C;
A3: E-max C in C by SPRECT_1:14;
A4: p in C by A2,JORDAN7:5;
A5: q in C by A2,JORDAN7:5;
A6: Upper_Arc C is_an_arc_of W-min C, E-max C by JORDAN6:50;
A7: Lower_Arc C is_an_arc_of E-max C, W-min C by JORDAN6:50;
  per cases by A3,A4,A5,JORDAN16:7;
  suppose q = W-min C;
    hence thesis by A1,A4,Th16;
  end;
  suppose that
A8: LE q, E-max C, C and
A9: not LE E-max C, q, C and
A10: q <> W-min C;
A11: Segment(p,q,C) = Segment(Upper_Arc C,W-min C,E-max C,p,q) by A1,A2,A8,Th8;
    not q in Lower_Arc C by A9,A10,Th5;
    then LE p, q, Upper_Arc C, W-min C, E-max C by A2,JORDAN6:def 10;
    hence thesis by A1,A6,A11,JORDAN16:21;
  end;
  suppose that
A12: p <> E-max C and
A13: LE p, E-max C, C and
A14: LE E-max C,q, C and
A15: q <> E-max C;
A16: now
      assume
A17:  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;
      then
A18:  p = W-min C by A6,JORDAN6:60;
W-min C in L_Segment(Lower_Arc C,E-max C,W-min C,q) by A17,XBOOLE_0:def 4;
      hence contradiction by A1,A7,A18,JORDAN6:59;
    end;
    p in Upper_Arc C by A13,JORDAN17:3;
    then
A19: LE p,E-max C,Upper_Arc C,W-min C,E-max C by A6,JORDAN5C:10;
A20: 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
A21: E-max C in R_Segment(Upper_Arc C,W-min C,E-max C,p) by A19,JORDAN16:5;
    q in Lower_Arc C by A14,JORDAN17:4;
    then
A22: LE E-max C,q,Lower_Arc C,E-max C,W-min C by A7,JORDAN5C:10;
A23: 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 A22,JORDAN16:5;
    then
A24: 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 A21,XBOOLE_0:def 4;
A25: R_Segment(Upper_Arc C,W-min C,E-max C,p) c= Upper_Arc C by JORDAN6:20;
A26: 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
A27: 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 A16,A24,A25,A26,TOPREAL8:1,XBOOLE_1:27;
A28: R_Segment(Upper_Arc C,W-min C,E-max C,p) is_an_arc_of p, E-max C
    by A6,A12,A19,A20,JORDAN16:21;
A29: L_Segment(Lower_Arc C,E-max C,W-min C,q) is_an_arc_of E-max C,q
    by A7,A15,A22,A23,JORDAN16:21;
    Segment(p,q,C) = 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 A13,A14,Th12;
    hence thesis by A27,A28,A29,TOPREAL1:2;
  end;
  suppose that
A30: q = E-max C and
A31: LE p, E-max C, C and
A32: LE E-max C,q, C;
    p in Upper_Arc C by A31,JORDAN17:3;
    then
A33: LE p,E-max C,Upper_Arc C,W-min C,E-max C by A6,JORDAN5C:10;
A34: 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
A35: E-max C in R_Segment(Upper_Arc C,W-min C,E-max C,p) by A33,JORDAN16:5;
A36: L_Segment(Lower_Arc C,E-max C,W-min C,q) = {E-max C} by A7,A30,JORDAN6:21;
    Segment(p,q,C) = 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 A31,A32,Th12
      .= R_Segment(Upper_Arc C,W-min C,E-max C,p) by A35,A36,ZFMISC_1:40;
    hence thesis by A1,A6,A30,A33,A34,JORDAN16:21;
  end;
  suppose that
A37: p = E-max C and
A38: LE p, E-max C, C and
A39: LE E-max C,q, C;
    q in Lower_Arc C by A39,JORDAN17:4;
    then
A40: LE E-max C,q,Lower_Arc C,E-max C,W-min C by A7,JORDAN5C:10;
A41: 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
A42: E-max C in L_Segment(Lower_Arc C,E-max C,W-min C,q) by A40,JORDAN16:5;
A43: R_Segment(Upper_Arc C,W-min C,E-max C,p) = {E-max C} by A6,A37,JORDAN6:23;
    Segment(p,q,C) = 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 A38,A39,Th12
      .= L_Segment(Lower_Arc C,E-max C,W-min C,q) by A42,A43,ZFMISC_1:40;
    hence thesis by A1,A7,A37,A40,A41,JORDAN16:21;
  end;
  suppose that
A44: LE E-max C,p, C and
A45: not LE p, E-max C, C;
A46: Segment(p,q,C) = Segment(Lower_Arc C,E-max C,W-min C,p,q) by A2,A44,Th11;
    not p in Upper_Arc C by A45,Th6;
    then LE p, q, Lower_Arc C, E-max C, W-min C by A2,JORDAN6:def 10;
    hence thesis by A1,A7,A46,JORDAN16:21;
  end;
end;
