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

theorem Th7:
  a in P implies ex e st a <> e & LE a,e,P
proof
  assume
A1: a in P;
A2: E-max(P) <> W-min(P) by TOPREAL5:19;
A3: Lower_Arc(P) is_an_arc_of E-max(P),W-min(P) by JORDAN6:def 9;
  per cases;
  suppose
A4: a = W-min(P);
    take E-max(P);
    thus thesis by A4,JORDAN7:3,SPRECT_1:14,TOPREAL5:19;
  end;
  suppose
A5: a <> W-min(P);
    thus thesis
    proof
      per cases;
      suppose
A6:     a in Upper_Arc(P);
        thus thesis
        proof
          per cases;
          suppose
A7:         a = E-max(P);
            consider e such that
A8:         e in Lower_Arc(P) & e <> E-max(P) & e <> W-min(P) by A3,JORDAN6:42;
            take e;
            thus thesis by A6,A7,A8,JORDAN6:def 10;
          end;
          suppose
A9:         a <> E-max(P);
            take E-max(P);
            E-max(P) in Lower_Arc(P) by JORDAN7:1;
            hence thesis by A2,A6,A9,JORDAN6:def 10;
          end;
        end;
      end;
      suppose
A10:    not a in Upper_Arc(P);
        Upper_Arc(P) \/ Lower_Arc(P) = P by JORDAN6:def 9;
        then
A11:    a in Lower_Arc(P) by A1,A10,XBOOLE_0:def 3;
        then consider
        f being Function of I[01], (TOP-REAL 2)|Lower_Arc(P), r being
        Real such that
A12:    f is being_homeomorphism and
A13:    f.0 = E-max(P) and
A14:    f.1 = W-min(P) and
A15:    0 <= r and
A16:    r <= 1 and
A17:    f.r = a by A3,Th1;
A18:    f is one-to-one by A12,TOPS_2:def 5;
        r < 1 by A5,A14,A16,A17,XXREAL_0:1;
        then consider s being Real such that
A19:    r < s and
A20:    s < 1 by XREAL_1:5;
A21:    dom f = [#]I[01] by A12,TOPS_2:def 5
          .= [.0,1.] by BORSUK_1:40;
A22:    rng f = [#]((TOP-REAL 2)|Lower_Arc(P)) by A12,TOPS_2:def 5
          .= the carrier of (TOP-REAL 2)|Lower_Arc(P)
          .= Lower_Arc(P) by PRE_TOPC:8;
A23:    0 <= s by A15,A19,XXREAL_0:2;
        then
A24:    s in dom f by A21,A20,XXREAL_1:1;
        then
A25:    f.s in rng f by FUNCT_1:def 3;
        then reconsider e = f.s as Point of TOP-REAL 2 by A22;
        1 in dom f by A21,XXREAL_1:1;
        then
A26:    e <> W-min(P) by A14,A18,A20,A24,FUNCT_1:def 4;
        take e;
        r in dom f by A15,A16,A21,XXREAL_1:1;
        hence a <> e by A17,A18,A19,A24,FUNCT_1:def 4;
        LE a,e,Lower_Arc(P),E-max(P),W-min(P) by A3,A12,A13,A14,A15,A16
,A17,A19,A20,A23,JORDAN5C:8;
        hence thesis by A11,A22,A25,A26,JORDAN6:def 10;
      end;
    end;
  end;
end;
