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 Th8:
  LE p,q,C & LE q, E-max C, C & p <> q implies
  Segment(p,q,C) = Segment(Upper_Arc C,W-min C,E-max C,p,q)
proof
  assume that
A1: LE p,q,C and
A2: LE q, E-max C, C and
A3: p <> q;
A4: Upper_Arc C is_an_arc_of W-min C,E-max C by JORDAN6:50;
A5: LE p, E-max C, C by A1,A2,JORDAN6:58;
A6: p in Upper_Arc C by A1,A2,JORDAN17:3,JORDAN6:58;
A7: q in Upper_Arc C by A2,JORDAN17:3;
A8: Upper_Arc C c= C by JORDAN6:61;
A9: now
    assume q = W-min C;
    then LE q,p,C by A6,A8,JORDAN7:3;
    hence contradiction by A1,A3,JORDAN6:57;
  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,Upper_Arc C,W-min C,E-max C &
  LE $1,q,Upper_Arc C,W-min C,E-max C;
A10: P[p1] iff Q[p1]
  proof
    hereby
      assume that
A11:  LE p,p1,C and
A12:  LE p1,q,C;
      hereby per cases;
        suppose p1 = E-max C;
          hence LE p,p1,Upper_Arc C,W-min C,E-max C by A4,A6,JORDAN5C:10;
        end;
        suppose p1 = W-min C;
          then LE p1,p,C by A6,A8,JORDAN7:3;
          then p = p1 by A11,JORDAN6:57;
hence LE p,p1,Upper_Arc C,W-min C,E-max C by A5,JORDAN17:3,JORDAN5C:9;
        end;
        suppose that
A13:      p1 <> E-max C and
A14:      p1 <> W-min C;
          now
            assume
A15:        p1 in Lower_Arc C;
            p1 in Upper_Arc C by A2,A12,JORDAN17:3,JORDAN6:58;
            then p1 in Upper_Arc C /\ Lower_Arc C by A15,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 p,p1,Upper_Arc C,W-min C,E-max C by A11,JORDAN6:def 10;
        end;
      end;
      per cases;
      suppose
A16:    q = E-max C;
        then p1 in Upper_Arc C by A12,JORDAN17:3;
        hence LE p1,q,Upper_Arc C,W-min C,E-max C by A4,A16,JORDAN5C:10;
      end;
      suppose
A17:    q <> E-max C;
        now
          assume q in Lower_Arc C;
          then q in Upper_Arc C /\ Lower_Arc C by A7,XBOOLE_0:def 4;
          then q in {W-min C,E-max C} by JORDAN6:50;
          hence contradiction by A9,A17,TARSKI:def 2;
        end;
        hence LE p1,q,Upper_Arc C,W-min C,E-max C by A12,JORDAN6:def 10;
      end;
    end;
    assume that
A18: LE p,p1,Upper_Arc C,W-min C,E-max C and
A19: LE p1,q,Upper_Arc C,W-min C,E-max C;
A20: p1 in Upper_Arc C by A18,JORDAN5C:def 3;
    hence LE p,p1,C by A6,A18,JORDAN6:def 10;
    thus thesis by A7,A19,A20,JORDAN6:def 10;
  end;
  deffunc F(set) = $1;
  set X = {F(p1): P[p1]}, Y = {F(p1): Q[p1]};
A21: X = Y from FRAENKEL:sch 3(A10);
  Segment(p,q,C) = X by A9,JORDAN7:def 1;
  hence thesis by A21,JORDAN6:26;
end;
