reserve C for Simple_closed_curve,
  P for Subset of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  n for Element of NAT;
reserve D for compact with_the_max_arc Subset of TOP-REAL 2;

theorem
  not (LMP C in Lower_Arc C & UMP C in Lower_Arc C)
proof
  assume that
A1: LMP C in Lower_Arc C and
A2: UMP C in Lower_Arc C;
A3: Upper_Arc(C) /\ Lower_Arc(C) = {W-min(C),E-max(C)} by JORDAN6:def 9;
  set n = |[ (W-bound C + E-bound C) / 2, N-bound C ]|;
  set S1 = LSeg(n, UMP C);
A4: Lower_Arc C c= C by JORDAN6:61;
A5: n`2 = N-bound C by EUCLID:52;
  set s = |[ (W-bound C + E-bound C) / 2, S-bound C ]|;
  set S2 = LSeg(LMP C, s);
A6: Lower_Arc C is_an_arc_of E-max C,W-min C by JORDAN6:def 9;
A7: (W-min C)`1 = W-bound C & (E-max C)`1 = E-bound C by EUCLID:52;
A8: Upper_Arc C c= C by JORDAN6:61;
  then
A9: for p being Point of TOP-REAL 2 st p in Upper_Arc C holds (W-min C)`1<=
  p`1 & p`1<=(E-max C)`1 by A7,PSCOMP_1:24;
A10: UMP C <> E-max C by Th25;
A11: UMP C <> W-min C by Th24;
A12: now
    assume UMP C in Upper_Arc C;
    then UMP C in Upper_Arc C /\ Lower_Arc C by A2,XBOOLE_0:def 4;
    hence contradiction by A11,A10,A3,TARSKI:def 2;
  end;
A13: W-bound C < E-bound C by SPRECT_1:31;
A14: Upper_Arc C misses S1
  proof
A15: S1 /\ C = {UMP C} by Th34;
    assume Upper_Arc C meets S1;
    then consider x being object such that
A16: x in Upper_Arc C and
A17: x in S1 by XBOOLE_0:3;
    x in S1 /\ C by A8,A16,A17,XBOOLE_0:def 4;
    then
A18: x = UMP C by A15,TARSKI:def 1;
    then x in Lower_Arc C /\ Upper_Arc C by A2,A16,XBOOLE_0:def 4;
    hence contradiction by A11,A10,A3,A18,TARSKI:def 2;
  end;
A19: UMP C in C & LMP C in C by Th30,Th31;
A20: (UMP C)`1 = (W-bound C + E-bound C)/2 by EUCLID:52;
  n`1 = (W-bound C + E-bound C)/2 by EUCLID:52;
  then
A21: S1 is vertical by A20,SPPOL_1:16;
A22: UMP C in S1 by RLTOPSP1:68;
A23: LMP C <> E-max C by Th27;
A24: LMP C <> W-min C by Th26;
A25: now
    assume LMP C in Upper_Arc C;
    then LMP C in Upper_Arc C /\ Lower_Arc C by A1,XBOOLE_0:def 4;
    hence contradiction by A24,A23,A3,TARSKI:def 2;
  end;
A26: LMP C <> UMP C by Th37;
A27: (LMP C)`1 = (W-bound C + E-bound C)/2 by EUCLID:52;
A28: Upper_Arc C misses S2
  proof
A29: S2 /\ C = {LMP C} by Th35;
    assume Upper_Arc C meets S2;
    then consider x being object such that
A30: x in Upper_Arc C and
A31: x in S2 by XBOOLE_0:3;
    x in S2 /\ C by A8,A30,A31,XBOOLE_0:def 4;
    then
A32: x = LMP C by A29,TARSKI:def 1;
    then x in Lower_Arc C /\ Upper_Arc C by A1,A30,XBOOLE_0:def 4;
    hence contradiction by A24,A23,A3,A32,TARSKI:def 2;
  end;
  s`1 = (W-bound C + E-bound C)/2 by EUCLID:52;
  then
A33: S2 is vertical by A27,SPPOL_1:16;
A34: LMP C in S2 by RLTOPSP1:68;
A35: S1 misses S2 by Th42;
A36: s`2 = S-bound C by EUCLID:52;
  then
A37: Upper_Arc C is_an_arc_of W-min C,E-max C & for p being Point of
  TOP-REAL 2 st p in Upper_Arc C holds s`2<=p`2 & p`2<=n`2 by A8,A5,
JORDAN6:def 8,PSCOMP_1:24;
  per cases by A19,JORDAN16:7;
  suppose
A38: LE LMP C, UMP C, C;
    set S = Segment(Lower_Arc C, E-max C, W-min C, LMP C, UMP C);
A39: S c= Lower_Arc C by JORDAN16:2;
    then
A40: S c= C by A4;
A41: now
      let p be Point of TOP-REAL 2;
      assume
A42:  p in S1 \/ S \/ S2;
      per cases by A42,Lm3;
      suppose
        p in S1;
        then p`1 = (UMP C)`1 by A21,A22,SPPOL_1:def 3;
        hence (W-min C)`1<=p`1 & p`1<=(E-max C)`1 by A20,A7,A13,XREAL_1:226;
      end;
      suppose
        p in S;
        hence (W-min C)`1<=p`1 & p`1<=(E-max C)`1 by A7,A40,PSCOMP_1:24;
      end;
      suppose
        p in S2;
        then p`1 = (LMP C)`1 by A33,A34,SPPOL_1:def 3;
        hence (W-min C)`1<=p`1 & p`1<=(E-max C)`1 by A27,A7,A13,XREAL_1:226;
      end;
    end;
A43: now
      let p be Point of TOP-REAL 2;
      assume
A44:  p in S1 \/ S \/ S2;
      per cases by A44,Lm3;
      suppose
A45:    p in S1;
A46:    s`2<=(UMP C)`2 by A36,Th38;
A47:    (UMP C)`2<=n`2 by A5,Th39;
        then (UMP C)`2<=p`2 by A45,TOPREAL1:4;
        hence s`2<=p`2 & p`2<=n`2 by A45,A47,A46,TOPREAL1:4,XXREAL_0:2;
      end;
      suppose
        p in S;
        hence s`2<=p`2 & p`2<=n`2 by A36,A5,A40,PSCOMP_1:24;
      end;
      suppose
A48:    p in S2;
A49:    s`2<=(LMP C)`2 by A36,Th40;
        hence s`2<=p`2 by A48,TOPREAL1:4;
        p`2<=(LMP C)`2 by A48,A49,TOPREAL1:4;
        hence p`2<=n`2 by A5,Th41,XXREAL_0:2;
      end;
    end;
A50: S c= Lower_Arc C \ {W-min C,E-max C}
    proof
      let s be object;
      assume
A51:  s in S;
      now
        assume s in {W-min C,E-max C};
        then s = W-min C or s = E-max C by TARSKI:def 2;
        hence contradiction by A11,A23,A6,A51,Th11;
      end;
      hence thesis by A39,A51,XBOOLE_0:def 5;
    end;
    Upper_Arc C misses S
    proof
      assume Upper_Arc C meets S;
      then consider x being object such that
A52:  x in Upper_Arc C and
A53:  x in S by XBOOLE_0:3;
      x in Lower_Arc C by A50,A53,XBOOLE_0:def 5;
      then x in Lower_Arc C /\ Upper_Arc C by A52,XBOOLE_0:def 4;
      hence contradiction by A3,A50,A53,XBOOLE_0:def 5;
    end;
    then
A54: Upper_Arc C misses S1 \/ S \/ S2 by A14,A28,Lm4;
A55: LE LMP C, UMP C, Lower_Arc C, E-max C, W-min C by A25,A38,JORDAN6:def 10;
    then
A56: UMP C in S by JORDAN16:5;
A57: S1 /\ S = {UMP C}
    proof
      S1 /\ C = {UMP C} by Th34;
      hence S1 /\ S c= {UMP C} by A40,XBOOLE_1:26;
      let x be object;
      assume x in {UMP C};
      then
A58:  x = UMP C by TARSKI:def 1;
      UMP C in S1 by RLTOPSP1:68;
      hence thesis by A56,A58,XBOOLE_0:def 4;
    end;
A59: LMP C in S by A55,JORDAN16:5;
A60: S2 /\ S = {LMP C}
    proof
      S2 /\ C = {LMP C} by Th35;
      hence S2 /\ S c= {LMP C} by A40,XBOOLE_1:26;
      let x be object;
      assume x in {LMP C};
      then x = LMP C by TARSKI:def 1;
      hence thesis by A34,A59,XBOOLE_0:def 4;
    end;
    S is_an_arc_of LMP C, UMP C by A26,A6,A55,JORDAN16:21;
    then S is_an_arc_of UMP C, LMP C by JORDAN5B:14;
    then
A61: S1 \/ S is_an_arc_of n,LMP C by A57,TOPREAL1:11;
    (S1 \/ S) /\ S2 = S1 /\ S2 \/ S /\ S2 by XBOOLE_1:23
      .= {} \/ S /\ S2 by A35;
    then S1 \/ S \/ S2 is_an_arc_of n,s by A60,A61,TOPREAL1:10;
    then S1 \/ S \/ S2 is_an_arc_of s,n by JORDAN5B:14;
    hence thesis by A37,A9,A54,A43,A41,JGRAPH_1:44;
  end;
  suppose
A62: LE UMP C, LMP C, C;
    set S = Segment(Lower_Arc C, E-max C, W-min C, UMP C, LMP C);
A63: S c= Lower_Arc C by JORDAN16:2;
    then
A64: S c= C by A4;
A65: now
      let p be Point of TOP-REAL 2;
      assume
A66:  p in S1 \/ S \/ S2;
      per cases by A66,Lm3;
      suppose
        p in S1;
        then p`1 = (UMP C)`1 by A21,A22,SPPOL_1:def 3;
        hence (W-min C)`1<=p`1 & p`1<=(E-max C)`1 by A20,A7,A13,XREAL_1:226;
      end;
      suppose
        p in S;
        hence (W-min C)`1<=p`1 & p`1<=(E-max C)`1 by A7,A64,PSCOMP_1:24;
      end;
      suppose
        p in S2;
        then p`1 = (LMP C)`1 by A33,A34,SPPOL_1:def 3;
        hence (W-min C)`1<=p`1 & p`1<=(E-max C)`1 by A27,A7,A13,XREAL_1:226;
      end;
    end;
A67: now
      let p be Point of TOP-REAL 2;
      assume
A68:  p in S1 \/ S \/ S2;
      per cases by A68,Lm3;
      suppose
A69:    p in S1;
A70:    s`2<=(UMP C)`2 by A36,Th38;
A71:    (UMP C)`2<=n`2 by A5,Th39;
        then (UMP C)`2<=p`2 by A69,TOPREAL1:4;
        hence s`2<=p`2 & p`2<=n`2 by A69,A71,A70,TOPREAL1:4,XXREAL_0:2;
      end;
      suppose
        p in S;
        hence s`2<=p`2 & p`2<=n`2 by A36,A5,A64,PSCOMP_1:24;
      end;
      suppose
A72:    p in S2;
A73:    s`2<=(LMP C)`2 by A36,Th40;
        hence s`2<=p`2 by A72,TOPREAL1:4;
        p`2<=(LMP C)`2 by A72,A73,TOPREAL1:4;
        hence p`2<=n`2 by A5,Th41,XXREAL_0:2;
      end;
    end;
A74: S c= Lower_Arc C \ {W-min C,E-max C}
    proof
      let s be object;
      assume
A75:  s in S;
      now
        assume s in {W-min C,E-max C};
        then s = W-min C or s = E-max C by TARSKI:def 2;
        hence contradiction by A10,A24,A6,A75,Th11;
      end;
      hence thesis by A63,A75,XBOOLE_0:def 5;
    end;
    Upper_Arc C misses S
    proof
      assume Upper_Arc C meets S;
      then consider x being object such that
A76:  x in Upper_Arc C and
A77:  x in S by XBOOLE_0:3;
      x in Lower_Arc C by A74,A77,XBOOLE_0:def 5;
      then x in Lower_Arc C /\ Upper_Arc C by A76,XBOOLE_0:def 4;
      hence contradiction by A3,A74,A77,XBOOLE_0:def 5;
    end;
    then
A78: Upper_Arc C misses S1 \/ S \/ S2 by A14,A28,Lm4;
A79: LE UMP C, LMP C, Lower_Arc C, E-max C, W-min C by A12,A62,JORDAN6:def 10;
    then
A80: UMP C in S by JORDAN16:5;
A81: S1 /\ S = {UMP C}
    proof
      S1 /\ C = {UMP C} by Th34;
      hence S1 /\ S c= {UMP C} by A64,XBOOLE_1:26;
      let x be object;
      assume x in {UMP C};
      then
A82:  x = UMP C by TARSKI:def 1;
      UMP C in S1 by RLTOPSP1:68;
      hence thesis by A80,A82,XBOOLE_0:def 4;
    end;
A83: LMP C in S by A79,JORDAN16:5;
A84: S2 /\ S = {LMP C}
    proof
      S2 /\ C = {LMP C} by Th35;
      hence S2 /\ S c= {LMP C} by A64,XBOOLE_1:26;
      let x be object;
      assume x in {LMP C};
      then x = LMP C by TARSKI:def 1;
      hence thesis by A34,A83,XBOOLE_0:def 4;
    end;
    S is_an_arc_of UMP C, LMP C by A6,A79,Th37,JORDAN16:21;
    then
A85: S1 \/ S is_an_arc_of n,LMP C by A81,TOPREAL1:11;
    (S1 \/ S) /\ S2 = S1 /\ S2 \/ S /\ S2 by XBOOLE_1:23
      .= {} \/ S /\ S2 by A35;
    then S1 \/ S \/ S2 is_an_arc_of n,s by A84,A85,TOPREAL1:10;
    then S1 \/ S \/ S2 is_an_arc_of s,n by JORDAN5B:14;
    hence thesis by A37,A9,A78,A67,A65,JGRAPH_1:44;
  end;
end;
