reserve C, P for Simple_closed_curve,
  a, b, c, d, e for Point of TOP-REAL 2;

theorem Th8:
  a <> b & LE a,b,P implies ex c st c <> a & c <> b & LE a,c,P & LE c,b,P
proof
  assume that
A1: a <> b and
A2: LE a,b,P;
A3: Upper_Arc(P) is_an_arc_of W-min(P),E-max(P) by JORDAN6:def 8;
A4: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
  per cases by A2,JORDAN6:def 10;
  suppose that
A5: a in Upper_Arc(P) and
A6: b in Lower_Arc(P) and
A7: not b = W-min(P);
A8: E-max(P) in Lower_Arc(P) by JORDAN7:1;
A9: E-max(P) in Upper_Arc(P) & E-max(P) <> W-min(P) by JORDAN7:1,TOPREAL5:19;
    thus thesis
    proof
      per cases;
      suppose that
A10:    a <> E-max(P) & b <> E-max(P);
        take e = E-max(P);
        thus a <> e & b <> e by A10;
        thus thesis by A5,A6,A7,A8,A9,JORDAN6:def 10;
      end;
      suppose
A11:    a = E-max(P);
        then LE a,b,Lower_Arc(P),E-max(P),W-min(P) by A4,A6,JORDAN5C:10;
        then consider e such that
A12:    a <> e & b <> e and
A13:    LE a,e,Lower_Arc(P),E-max(P),W-min(P) & LE e,b,Lower_Arc(P),
        E-max(P),W-min(P ) by A1,A4,Th6;
        take e;
        thus e <> a & e <> b by A12;
        e in Lower_Arc(P) & e <> W-min(P) by A4,A7,A13,JORDAN5C:def 3
,JORDAN6:55;
        hence thesis by A6,A7,A8,A11,A13,JORDAN6:def 10;
      end;
      suppose
A14:    b = E-max(P);
        then LE a,b,Upper_Arc(P),W-min(P),E-max(P) by A3,A5,JORDAN5C:10;
        then consider e such that
A15:    a <> e & b <> e and
A16:    LE a,e,Upper_Arc(P),W-min(P),E-max(P) and
        LE e,b,Upper_Arc(P),W-min(P),E-max(P) by A1,A3,Th6;
        take e;
        thus e <> a & e <> b by A15;
        e in Upper_Arc(P) by A16,JORDAN5C:def 3;
        hence thesis by A5,A7,A8,A14,A16,JORDAN6:def 10;
      end;
    end;
  end;
  suppose that
A17: a in Upper_Arc(P) & b in Upper_Arc(P) and
A18: LE a,b,Upper_Arc(P),W-min(P),E-max(P);
    consider e such that
A19: a <> e & b <> e and
A20: LE a,e,Upper_Arc(P),W-min(P),E-max(P) and
A21: LE e,b,Upper_Arc(P),W-min(P),E-max(P) by A1,A3,A18,Th6;
    take e;
    thus e <> a & e <> b by A19;
    e in Upper_Arc(P) by A20,JORDAN5C:def 3;
    hence thesis by A17,A20,A21,JORDAN6:def 10;
  end;
  suppose that
A22: a in Lower_Arc(P) & b in Lower_Arc(P) and
A23: not b = W-min(P) and
A24: LE a,b,Lower_Arc(P),E-max(P),W-min(P);
    consider e such that
A25: a <> e & b <> e and
A26: LE a,e,Lower_Arc(P),E-max(P),W-min(P) and
A27: LE e,b,Lower_Arc(P),E-max(P),W-min(P) by A1,A4,A24,Th6;
    take e;
    thus e <> a & e <> b by A25;
A28: e in Lower_Arc(P) by A26,JORDAN5C:def 3;
    e <> W-min(P) by A4,A23,A27,JORDAN6:55;
    hence thesis by A22,A23,A26,A27,A28,JORDAN6:def 10;
  end;
end;
