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 Th9:
  LE E-max C, q, C implies
  Segment(E-max C,q,C) = Segment(Lower_Arc C,E-max C,W-min C,E-max C,q)
proof
  set p = E-max C;
  assume
A1: LE E-max C, q, C;
A2: Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:50;
A3: p in Lower_Arc C by JORDAN7:1;
A4: q in Lower_Arc C by A1,JORDAN17:4;
A5: Lower_Arc C c= C by JORDAN6:61;
A6: now
    assume
A7: q = W-min C;
    then p = q by A1,JORDAN7:2;
    hence contradiction by A7,TOPREAL5:19;
  end;
  defpred P[Point of TOP-REAL 2] means LE p,$1,C & LE $1,q,C;
  defpred Q[Point of TOP-REAL 2] means LE p,$1,Lower_Arc C,E-max C,W-min C &
  LE $1,q,Lower_Arc C,E-max C, W-min C;
A8: P[p1] iff Q[p1]
  proof
    hereby
      assume that
A9:   LE p,p1,C and
A10:  LE p1,q,C;
A11:  p1 in Lower_Arc C by A9,JORDAN17:4;
      hence LE p,p1,Lower_Arc C,E-max C,W-min C by A2,JORDAN5C:10;
      per cases;
      suppose
A12:    p1 = E-max C;
        then q in Lower_Arc C by A10,JORDAN17:4;
        hence LE p1,q,Lower_Arc C,E-max C,W-min C by A2,A12,JORDAN5C:10;
      end;
      suppose
A13:    p1 <> E-max C;
A14:    now
          assume
A15:      p1 = W-min C;
          then LE p1,p, C by A3,A5,JORDAN7:3;
          hence contradiction by A9,A15,JORDAN6:57,TOPREAL5:19;
        end;
        now
          assume p1 in Upper_Arc C;
          then p1 in Upper_Arc C /\ Lower_Arc C by A11,XBOOLE_0:def 4;
          then p1 in {W-min C,E-max C} by JORDAN6:50;
          hence contradiction by A13,A14,TARSKI:def 2;
        end;
        hence LE p1,q,Lower_Arc C,E-max C,W-min C by A10,JORDAN6:def 10;
      end;
    end;
    assume that
A16: LE p,p1,Lower_Arc C,E-max C,W-min C and
A17: LE p1,q,Lower_Arc C,E-max C,W-min C;
A18: p1 in Lower_Arc C by A16,JORDAN5C:def 3;
    p1 <> W-min C by A2,A6,A17,JORDAN6:55;
    hence LE p,p1,C by A3,A16,A18,JORDAN6:def 10;
    thus thesis by A4,A6,A17,A18,JORDAN6:def 10;
  end;
  deffunc F(set) = $1;
  set X = {F(p1): P[p1]}, Y = {F(p1): Q[p1]};
A19: X = Y from FRAENKEL:sch 3(A8);
  Segment(p,q,C) = X by A6,JORDAN7:def 1;
  hence thesis by A19,JORDAN6:26;
end;
