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 Th11:
  LE p,q,C & LE E-max C, p, C implies
  Segment(p,q,C) = Segment(Lower_Arc C,E-max C,W-min C,p,q)
proof
  assume that
A1: LE p,q,C and
A2: LE E-max C, p, C;
  per cases;
  suppose p = E-max C;
    hence thesis by A1,Th9;
  end;
  suppose
A3: p <> E-max C;
A4: Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:50;
A5: q in Lower_Arc C by A1,A2,JORDAN17:4,JORDAN6:58;
A6: p in Lower_Arc C by A2,JORDAN17:4;
A7: Lower_Arc C c= C by JORDAN6:61;
A8: now
      assume
A9:   p = W-min C;
      then LE p, E-max C, C by JORDAN17:2;
      hence contradiction by A2,A9,JORDAN6:57,TOPREAL5:19;
    end;
A10: now
      assume
A11:  q = W-min C;
      then LE q,p,C by A6,A7,JORDAN7:3;
      hence contradiction by A1,A8,A11,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,Lower_Arc C,E-max C,W-min C &
    LE $1,q,Lower_Arc C,E-max C,W-min C;
A12: P[p1] iff Q[p1]
    proof
      hereby
        assume that
A13:    LE p,p1,C and
A14:    LE p1,q,C;
A15:    now
          assume
A16:      p1 = W-min C;
          then LE p1,p,C by A6,A7,JORDAN7:3;
          hence contradiction by A8,A13,A16,JORDAN6:57;
        end;
A17:    now
          assume
A18:      p in Upper_Arc C;
          p in Lower_Arc C by A2,JORDAN17:4;
          then p in Lower_Arc C /\ Upper_Arc C by A18,XBOOLE_0:def 4;
          then p in {W-min C,E-max C} by JORDAN6:50;
          hence contradiction by A3,A8,TARSKI:def 2;
        end;
        hence LE p,p1,Lower_Arc C,E-max C,W-min C by A13,JORDAN6:def 10;
A19:    p1 in Lower_Arc C by A13,A17,JORDAN6:def 10;
        per cases;
        suppose q = E-max C;
          hence LE p1,q,Lower_Arc C,E-max C,W-min C by A1,A2,A3,JORDAN6:57;
        end;
        suppose p1 = E-max C;
          hence LE p1,q,Lower_Arc C,E-max C,W-min C by A4,A5,JORDAN5C:10;
        end;
        suppose that
A20:      p1 <> E-max C;
          now
            assume p1 in Upper_Arc C;
            then p1 in Lower_Arc C /\ Upper_Arc C by A19,XBOOLE_0:def 4;
            then p1 in {W-min C,E-max C} by JORDAN6:50;
            hence contradiction by A15,A20,TARSKI:def 2;
          end;
          hence LE p1,q,Lower_Arc C,E-max C,W-min C by A14,JORDAN6:def 10;
        end;
      end;
      assume that
A21:  LE p,p1,Lower_Arc C,E-max C,W-min C and
A22:  LE p1,q,Lower_Arc C,E-max C,W-min C;
A23:  p1 <> W-min C by A4,A10,A22,JORDAN6:55;
A24:  p1 in Lower_Arc C by A21,JORDAN5C:def 3;
      hence LE p,p1,C by A6,A21,A23,JORDAN6:def 10;
      thus thesis by A5,A10,A22,A24,JORDAN6:def 10;
    end;
    deffunc F(set) = $1;
    set X = {F(p1): P[p1]}, Y = {F(p1): Q[p1]};
A25: X = Y from FRAENKEL:sch 3(A12);
    Segment(p,q,C) = X by A10,JORDAN7:def 1;
    hence thesis by A25,JORDAN6:26;
  end;
end;
