reserve n for Nat;

theorem Th54:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for n be Nat st n > 0 holds First_Point(L~Upper_Seq(C,n),
W-min L~Cage(C,n),E-max L~Cage(C,n), Vertical_Line((W-bound L~Cage(C,n)+E-bound
L~Cage(C,n))/2))`2 > Last_Point(L~Lower_Seq(C,n),E-max L~Cage(C,n),W-min L~Cage
  (C,n), Vertical_Line((W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2))`2
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let n be Nat;
  set sr = (W-bound L~Cage(C,n)+E-bound L~Cage(C,n))/2;
  set Wmin = W-min L~Cage(C,n);
  set Emax = E-max L~Cage(C,n);
  set Nbo = N-bound L~Cage(C,n);
  set Ebo = E-bound L~Cage(C,n);
  set Sbo = S-bound L~Cage(C,n);
  set Wbo = W-bound L~Cage(C,n);
  set SW = SW-corner L~Cage(C,n);
  set FiP = First_Point(L~Upper_Seq(C,n),Wmin,Emax,Vertical_Line sr);
  set LaP = Last_Point(L~Lower_Seq(C,n),Emax,Wmin,Vertical_Line sr);
  set g = mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))^<*|[Ebo,FiP`2]|*>;
  set h = <*SW*>^((Rev Lower_Seq(C,n))-:LaP)^<*|[sr,Nbo]|*>;
A1: Upper_Seq(C,n)/.1 = W-min L~Cage(C,n) by JORDAN1F:5;
A2: Wbo <= Ebo by SPRECT_1:21;
  then Wbo <= sr by JORDAN6:1;
  then
A3: Wmin`1 <= sr by EUCLID:52;
  sr <= Ebo by A2,JORDAN6:1;
  then
A4: sr <= Emax`1 by EUCLID:52;
  set GCw = Gauge(C,n)*(Center Gauge(C,n),width Gauge(C,n));
A5: 1 <= Center Gauge(C,n) by JORDAN1B:11;
  len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
  then
A6: GCw`2 = Nbo by A5,JORDAN1A:70,JORDAN1B:13;
A7: SW`2 <= Wmin`2 by PSCOMP_1:30;
A8: |[sr,Nbo]|`2 = Nbo by EUCLID:52;
  set RevL = (Rev Lower_Seq(C,n))-:LaP;
A9: <*|[Ebo,FiP`2]|*> is one-to-one & <*|[Ebo,FiP`2]|*> is special by
FINSEQ_3:93;
A10: rng ((Rev Lower_Seq(C,n))-:LaP) c= rng (Rev Lower_Seq(C,n)) by FINSEQ_5:48
;
A11: Lower_Seq(C,n)/.1 = E-max L~Cage(C,n) & Lower_Seq(C,n)/.len Lower_Seq(C
  ,n) = W-min L~Cage(C,n) by JORDAN1F:6,8;
  then
A12: L~Lower_Seq(C,n) is_an_arc_of Emax,Wmin by TOPREAL1:25;
A13: 4 <= len Gauge(C,n) by JORDAN8:10;
  then
A14: len Gauge(C,n) >= 3 by XXREAL_0:2;
A15: Wbo < Ebo by SPRECT_1:31;
  then
A16: Wbo < sr by XREAL_1:226;
  L~Lower_Seq(C,n) is_an_arc_of Wmin,Emax by A11,JORDAN5B:14,TOPREAL1:25;
  then
A17: L~Lower_Seq(C,n) meets Vertical_Line(sr) & L~Lower_Seq(C,n) /\
  Vertical_Line (sr) is closed by A3,A4,JORDAN6:49;
  then
A18: LaP in L~Lower_Seq(C,n) /\ Vertical_Line sr by A12,JORDAN5C:def 2;
  then
A19: LaP in L~Lower_Seq(C,n) by XBOOLE_0:def 4;
  then LaP in L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by XBOOLE_0:def 3;
  then
A20: LaP in L~Cage(C,n) by JORDAN1E:13;
  assume
A21: n > 0;
  then
A22: FiP in rng Upper_Seq(C,n) by Th47;
  then
A23: FiP..Upper_Seq(C,n) in dom Upper_Seq(C,n) by FINSEQ_4:20;
  then
A24: 1<=FiP..Upper_Seq(C,n) by FINSEQ_3:25;
  1 <= len Gauge(C,n) by A13,XXREAL_0:2;
  then 1 <= width Gauge(C,n) by JORDAN8:def 1;
  then GCw`1 = (W-bound C + E-bound C)/2 by A21,Th35
    .= sr by Th33;
  then GCw = |[sr,Nbo]| by A6,EUCLID:53;
  then not |[sr,Nbo]| in rng Lower_Seq(C,n) by A5,A14,Th43,JORDAN1B:15;
  then not |[sr,Nbo]| in rng (Rev Lower_Seq(C,n)) by FINSEQ_5:57;
  then
A25: not |[sr,Nbo]| in rng ((Rev Lower_Seq(C,n))-:LaP) by A10;
  SW`2 = Sbo by EUCLID:52;
  then |[sr,Nbo]| <> SW by A8,SPRECT_1:32;
  then not |[sr,Nbo]| in {SW} by TARSKI:def 1;
  then not |[sr,Nbo]| in rng <*SW*> by FINSEQ_1:38;
  then
  not |[sr,Nbo]| in rng <*SW*> \/ rng ((Rev Lower_Seq(C,n))-:LaP) by A25,
XBOOLE_0:def 3;
  then not |[sr,Nbo]| in rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) by
FINSEQ_1:31;
  then rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) misses {|[sr,Nbo]|} by
ZFMISC_1:50;
  then rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) /\ {|[sr,Nbo]|} = {};
  then rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) /\ rng <*|[sr,Nbo]|*> = {} by
FINSEQ_1:38;
  then
A26: rng (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) misses rng <*|[sr,Nbo]| *>;
  LaP in rng Lower_Seq(C,n) by A21,Th48;
  then
A27: LaP in rng Rev Lower_Seq(C,n) by FINSEQ_5:57;
  then
A28: RevL is non empty by FINSEQ_5:47;
A29: len RevL = LaP..Rev Lower_Seq(C,n) by A27,FINSEQ_5:42;
A30: Upper_Seq(C,n)/.len Upper_Seq(C,n) = E-max L~Cage(C,n) by JORDAN1F:7;
  then
A31: L~Upper_Seq(C,n) is_an_arc_of Wmin,Emax by A1,TOPREAL1:25;
A32: sr < Ebo by A15,XREAL_1:226;
  then
A33: sr <= Emax`1 by EUCLID:52;
  Wmin`1 <= sr by A16,EUCLID:52;
  then L~Upper_Seq(C,n) meets Vertical_Line(sr) & L~Upper_Seq(C,n) /\
  Vertical_Line (sr) is closed by A31,A33,JORDAN6:49;
  then
A34: FiP in L~Upper_Seq(C,n) /\ Vertical_Line sr by A31,JORDAN5C:def 1;
  then
A35: FiP in L~Upper_Seq(C,n) by XBOOLE_0:def 4;
  then FiP in L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by XBOOLE_0:def 3;
  then
A36: FiP in L~Cage(C,n) by JORDAN1E:13;
  now
    let m be Nat;
    assume m in dom <*|[Ebo,FiP`2]|*>;
    then m in Seg 1 by FINSEQ_1:38;
    then m = 1 by FINSEQ_1:2,TARSKI:def 1;
    then
A37: <*|[Ebo,FiP`2]|*>/.m = |[Ebo,FiP`2]| by FINSEQ_4:16;
    then (<*|[Ebo,FiP`2]|*>/.m)`1 = Ebo by EUCLID:52;
    hence W-bound L~Cage(C,n) <= (<*|[Ebo,FiP`2]|*>/.m)`1 & (<*|[Ebo,FiP`2]|*>
    /.m)`1 <= E-bound L~Cage(C,n) by SPRECT_1:21;
    (<*|[Ebo,FiP`2]|*>/.m)`2 = FiP`2 by A37,EUCLID:52;
    hence S-bound L~Cage(C,n) <= (<*|[Ebo,FiP`2]|*>/.m)`2 & (<*|[Ebo,FiP`2]|*>
    /.m)`2 <= N-bound L~Cage(C,n) by A36,PSCOMP_1:24;
  end;
  then
A38: <*|[Ebo,FiP`2]|*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
A39: FiP in Vertical_Line sr by A34,XBOOLE_0:def 4;
  then
A40: FiP`1 = sr by JORDAN6:31;
  now
    assume
    rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) /\ {|[Ebo,FiP`2]|} <> {};
    then consider x be object such that
A41: x in rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) /\ {|[Ebo,FiP
    `2]|} by XBOOLE_0:def 1;
    x in rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) & x in {|[Ebo,FiP
    `2 ]| } by A41,XBOOLE_0:def 4;
    then |[Ebo,FiP`2]| in rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) by
TARSKI:def 1;
    then |[Ebo,FiP`2]|`1 <= sr by A21,Th53;
    hence contradiction by A32,EUCLID:52;
  end;
  then
  rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) misses {|[Ebo,FiP`2]|};
  then
A42: rng mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) misses rng <*|[Ebo,FiP`2
  ]|*> by FINSEQ_1:38;
A43: FiP..Upper_Seq(C,n)<=len Upper_Seq(C,n) by A23,FINSEQ_3:25;
  LaP in Vertical_Line sr by A18,XBOOLE_0:def 4;
  then
A44: LaP`1 = sr by JORDAN6:31;
A45: now
    assume FiP`2 = LaP`2;
    then FiP = LaP by A40,A44,TOPREAL3:6;
    then FiP in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A35,A19,XBOOLE_0:def 4;
    then FiP in {Wmin,Emax} by JORDAN1E:16;
    then FiP = Wmin or FiP = Emax by TARSKI:def 2;
    hence contradiction by A16,A32,A40,EUCLID:52;
  end;
  len Upper_Seq(C,n) >= 3 by JORDAN1E:15;
  then
A46: len Upper_Seq(C,n) > 2 by XXREAL_0:2;
  then
A47: 2 in dom Upper_Seq(C,n) by FINSEQ_3:25;
  then
A48: (mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))/.len mid(Upper_Seq(C,n),2,
  FiP..Upper_Seq(C,n)))`2 = (Upper_Seq(C,n)/.(FiP..Upper_Seq(C,n)))`2 by A23,
SPRECT_2:9
    .= FiP`2 by A22,FINSEQ_5:38
    .= |[Ebo,FiP`2]|`2 by EUCLID:52
    .= (<*|[Ebo,FiP`2]|*>/.1)`2 by FINSEQ_4:16;
  2 <> FiP..Upper_Seq(C,n)
  proof
    assume 2 = FiP..Upper_Seq(C,n);
    then Upper_Seq(C,n)/.2 = FiP by A22,FINSEQ_5:38;
    then FiP`1 = Wbo by Th31;
    then Wbo = sr by A39,JORDAN6:31;
    hence contradiction by SPRECT_1:31;
  end;
  then
  mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) is being_S-Seq by A46,A24,A43,
JORDAN3:6;
  then reconsider g as one-to-one special FinSequence of TOP-REAL 2 by A42,A48
,A9,FINSEQ_3:91,GOBOARD2:8;
  mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) is_in_the_area_of Cage(C,n)
  by A47,A23,JORDAN1E:17,SPRECT_2:22;
  then
A49: g is_in_the_area_of Cage(C,n) by A38,SPRECT_2:24;
A50: (g/.len g)`1 = (<*|[Ebo,FiP`2]|*>/.len <*|[Ebo,FiP`2]|*>)`1 by SPRECT_3:1
    .= (<*|[Ebo,FiP`2]|*>/.1)`1 by FINSEQ_1:39
    .= (|[Ebo,FiP`2]|)`1 by FINSEQ_4:16
    .= E-bound L~Cage(C,n) by EUCLID:52;
A51: 1 <= len mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) by A47,A23,SPRECT_2:5;
  then 1 in dom mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n)) by FINSEQ_3:25;
  then (g/.1)`1 = (mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))/.1)`1 by
FINSEQ_4:68
    .= (Upper_Seq(C,n)/.2)`1 by A47,A23,SPRECT_2:8
    .= W-bound L~Cage(C,n) by Th31;
  then
A52: g is_a_h.c._for Cage(C,n) by A49,A50,SPRECT_2:def 2;
  assume FiP`2 <= LaP`2;
  then
A53: FiP`2 < LaP`2 by A45,XXREAL_0:1;
A54: rng Lower_Seq(C,n) c= rng Cage(C,n) by Th39;
  now
    per cases;
    suppose
A55:  SW <> Wmin;
      not SW in rng Lower_Seq(C,n)
      proof
        SW`1 = Wmin`1 by PSCOMP_1:29;
        then
A56:    SW`2 <> Wmin`2 by A55,TOPREAL3:6;
        assume SW in rng Lower_Seq(C,n);
        then
A57:    SW in rng Cage(C,n) by A54;
        len Cage(C,n) > 4 by GOBOARD7:34;
        then
A58:    rng Cage(C,n) c= L~Cage(C,n) by SPPOL_2:18,XXREAL_0:2;
        SW`1 = W-bound L~Cage(C,n) by EUCLID:52;
        then SW in W-most L~Cage(C,n) by A57,A58,SPRECT_2:12;
        then Wmin`2 <= SW`2 by PSCOMP_1:31;
        hence contradiction by A7,A56,XXREAL_0:1;
      end;
      then not SW in rng (Rev Lower_Seq(C,n)) by FINSEQ_5:57;
      then not SW in rng ((Rev Lower_Seq(C,n))-:LaP) by A10;
      then {SW} misses rng ((Rev Lower_Seq(C,n))-:LaP) by ZFMISC_1:50;
      then {SW} /\ rng ((Rev Lower_Seq(C,n))-:LaP) = {};
      then rng <*SW*> /\ rng ((Rev Lower_Seq(C,n))-:LaP) = {} by FINSEQ_1:38;
      then
A59:  rng <*SW*> misses rng ((Rev Lower_Seq(C,n))-:LaP);
      <*SW*> is one-to-one by FINSEQ_3:93;
      then
A60:  <*SW*>^((Rev Lower_Seq(C,n))-:LaP) is one-to-one by A59,FINSEQ_3:91;
      set FiP2 = First_Point(L~Lower_Seq(C,n),Wmin,Emax,Vertical_Line sr);
      set midU = mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n));
      reconsider RevLS = Rev Lower_Seq(C,n) as special FinSequence of TOP-REAL
      2;
      (<*SW*>/.len <*SW*>)`1 = (<*SW*>/.1)`1 by FINSEQ_1:39
        .= SW`1 by FINSEQ_4:16
        .= W-bound L~Cage(C,n) by EUCLID:52
        .= (W-min L~Cage(C,n))`1 by EUCLID:52
        .= (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 by JORDAN1F:8
        .= ((Rev Lower_Seq(C,n))/.1)`1 by FINSEQ_5:65
        .= (((Rev Lower_Seq(C,n))-:LaP)/.1)`1 by A27,FINSEQ_5:44;
      then
A61:  <*SW*>^(RevLS-:LaP) is special by GOBOARD2:8;
      (Rev Lower_Seq(C,n))-:LaP is non empty by A27,FINSEQ_5:47;
      then
A62:  ((<*SW*>^((Rev Lower_Seq(C,n))-:LaP))/. len (<*SW*>^((Rev
Lower_Seq(C,n))-:LaP)))`1 = (((Rev Lower_Seq(C,n))-:LaP)/.len ((Rev Lower_Seq(C
      ,n))-:LaP))`1 by SPRECT_3:1
        .= (((Rev Lower_Seq(C,n))-:LaP)/.(LaP..Rev Lower_Seq(C,n)))`1 by A27,
FINSEQ_5:42
        .= LaP`1 by A27,FINSEQ_5:45
        .= |[sr,Nbo]|`1 by A44,EUCLID:52
        .= (<*|[sr,Nbo]|*>/.1)`1 by FINSEQ_4:16;
      <*|[sr,Nbo]|*> is one-to-one & <*|[sr,Nbo]|*> is special by FINSEQ_3:93;
      then reconsider
      h as one-to-one special FinSequence of TOP-REAL 2 by A26,A60,A62,A61,
FINSEQ_3:91,GOBOARD2:8;
A63:  |[Ebo,FiP`2]|`1 = Ebo by EUCLID:52;
      now
        let m be Nat;
        assume m in dom <*SW*>;
        then m in Seg 1 by FINSEQ_1:38;
        then m = 1 by FINSEQ_1:2,TARSKI:def 1;
        then
A64:    <*SW*>/.m = SW by FINSEQ_4:16;
        then (<*SW*>/.m)`1 = Wbo by EUCLID:52;
        hence W-bound L~Cage(C,n) <= (<*SW*>/.m)`1 & (<*SW*>/.m)`1 <= E-bound
        L~Cage(C,n) by SPRECT_1:21;
        (<*SW*>/.m)`2 = Sbo by A64,EUCLID:52;
        hence S-bound L~Cage(C,n) <= (<*SW*>/.m)`2 & (<*SW*>/.m)`2 <= N-bound
        L~Cage(C,n) by SPRECT_1:22;
      end;
      then
A65:  <*SW*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
A66:  RevL/.len RevL = RevL/.(LaP..Rev Lower_Seq(C,n)) by A27,FINSEQ_5:42
        .= LaP by A27,FINSEQ_5:45;
      now
        let m be Nat;
A67:    W-bound L~Cage(C,n) <= E-bound L~Cage(C,n) by SPRECT_1:21;
        assume m in dom <*|[sr,Nbo]|*>;
        then m in Seg 1 by FINSEQ_1:38;
        then m = 1 by FINSEQ_1:2,TARSKI:def 1;
        then
A68:    <*|[sr,Nbo]|*>/.m = |[sr,Nbo]| by FINSEQ_4:16;
        then (<*|[sr,Nbo]|*>/.m)`1 = sr by EUCLID:52;
        hence
        W-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`1 & (<*|[sr,Nbo]|*>/.m
        )`1 <= E-bound L~Cage(C,n) by A67,JORDAN6:1;
        (<*|[sr,Nbo]|*>/.m)`2 = Nbo by A68,EUCLID:52;
        hence
        S-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`2 & (<*|[sr,Nbo]|*>/.m
        )`2 <= N-bound L~Cage(C,n) by SPRECT_1:22;
      end;
      then
A69:  <*|[sr,Nbo]|*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
A70:  L~Rev Lower_Seq(C,n) = L~Lower_Seq(C,n) & FiP2 = LaP by A12,A17,
JORDAN5C:18,SPPOL_2:22;
      Rev Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:18,SPRECT_3:51
;
      then ((Rev Lower_Seq(C,n))-:LaP) is_in_the_area_of Cage(C,n) by A27,
JORDAN1E:1;
      then <*SW*>^((Rev Lower_Seq(C,n))-:LaP) is_in_the_area_of Cage(C,n) by
A65,SPRECT_2:24;
      then
A71:  h is_in_the_area_of Cage(C,n) by A69,SPRECT_2:24;
      len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) = 1 + len ((Rev Lower_Seq
      (C,n))-:LaP) by FINSEQ_5:8;
      then
A72:  len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) >= 1 by NAT_1:11;
      1 in dom h by FINSEQ_5:6;
      then h/.1 = h.1 by PARTFUN1:def 6;
      then
A73:  (h/.1)`2 = ((<*SW*>^((Rev Lower_Seq(C,n))-:LaP))/.1)`2 by A72,
FINSEQ_6:109
        .= SW`2 by FINSEQ_5:15
        .= S-bound L~Cage(C,n) by EUCLID:52;
A74:  len h = len (<*SW*>^((Rev Lower_Seq(C,n))-:LaP)) + 1 by FINSEQ_2:16;
      then
A75:  1+1 <= len h by A72,XREAL_1:7;
      L~Cage(C,n) = L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by JORDAN1E:13;
      then
A76:  L~Upper_Seq(C,n) c= L~Cage(C,n) by XBOOLE_1:7;
A77:  midU/.len midU = Upper_Seq(C,n)/.(FiP..Upper_Seq(C,n)) by A47,A23,
SPRECT_2:9
        .= FiP by A22,FINSEQ_5:38;
A78:  Wmin in rng Upper_Seq(C,n) by A1,FINSEQ_6:42;
      now
        assume FiP..Upper_Seq(C,n) = 1;
        then FiP..Upper_Seq(C,n) = (Upper_Seq(C,n)/.1)..Upper_Seq(C,n) by
FINSEQ_6:43
          .= Wmin..Upper_Seq(C,n) by JORDAN1F:5;
        then FiP = Wmin by A22,A78,FINSEQ_5:9;
        hence contradiction by A16,A40,EUCLID:52;
      end;
      then FiP..Upper_Seq(C,n) > 1 by A24,XXREAL_0:1;
      then
A79:  1+1+0 <= FiP..Upper_Seq(C,n) by NAT_1:13;
      then FiP..Upper_Seq(C,n)-2 >= 0 by XREAL_1:19;
      then FiP..Upper_Seq(C,n)-'2 = FiP..Upper_Seq(C,n)-2 by XREAL_0:def 2;
      then
A80:  len midU = FiP..Upper_Seq(C,n)-2+1 by A43,A79,FINSEQ_6:186
        .= FiP..Upper_Seq(C,n)-(2-1);
      1 in dom RevL by A28,FINSEQ_5:6;
      then
A81:  (RevL^<*|[sr,Nbo]|*>)/.1 = RevL/.1 by FINSEQ_4:68
        .= (Rev Lower_Seq(C,n))/.1 by A27,FINSEQ_5:44
        .= Lower_Seq(C,n)/.len Lower_Seq(C,n) by FINSEQ_5:65
        .= Wmin by JORDAN1F:8;
A82:  SW`2 <= Wmin`2 by PSCOMP_1:30;
      len g = len mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))+1 by FINSEQ_2:16;
      then
A83:  1+1 <= len g by A51,XREAL_1:7;
A84:  L~g = L~midU \/ LSeg(midU/.len midU,|[Ebo,FiP`2]|) by A47,A23,SPPOL_2:19
,SPRECT_2:7;
      L~Rev Lower_Seq(C,n) = L~RevL \/ L~((Rev Lower_Seq(C,n)):-LaP) by A27,
SPPOL_2:24;
      then L~RevL c= L~Rev Lower_Seq(C,n) by XBOOLE_1:7;
      then
A85:  L~RevL c= L~Lower_Seq(C,n) by SPPOL_2:22;
A86:  LaP`2 <= Nbo by A20,PSCOMP_1:24;
A87:  |[Ebo,FiP`2]|`2 = FiP`2 by EUCLID:52;
      then
A88:  LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by SPPOL_1:15;
      LaP`1 = |[sr,Nbo]|`1 by A44,EUCLID:52;
      then
A89:  LSeg(LaP,|[sr,Nbo]|) is vertical by SPPOL_1:16;
A90:  L~midU c= L~Upper_Seq(C,n) by A47,A23,SPRECT_3:18;
      (h/.len h)`2 = |[sr,Nbo]|`2 by A74,FINSEQ_4:67
        .= N-bound L~Cage(C,n) by EUCLID:52;
      then h is_a_v.c._for Cage(C,n) by A71,A73,SPRECT_2:def 3;
      then L~g meets L~h by A52,A75,A83,SPRECT_2:29;
      then consider x be object such that
A91:  x in L~g and
A92:  x in L~h by XBOOLE_0:3;
      reconsider x as Point of TOP-REAL 2 by A91;
      L~h = L~(<*SW*>^(((Rev Lower_Seq(C,n))-:LaP)^<*|[sr,Nbo]|*>)) by
FINSEQ_1:32
        .= LSeg(SW,(RevL^<*|[sr,Nbo]|*>)/.1) \/ L~(RevL^<*|[sr,Nbo]|*>) by
SPPOL_2:20
        .= LSeg(SW,(RevL^<*|[sr,Nbo]|*>)/.1) \/ (L~RevL \/ LSeg(RevL/.len
      RevL,|[sr,Nbo]|)) by A27,FINSEQ_5:47,SPPOL_2:19;
      then
A93:  x in LSeg(SW,(RevL^<*|[sr,Nbo]|*>)/.1) or x in L~RevL \/ LSeg(RevL
      /.len RevL,|[sr,Nbo]|) by A92,XBOOLE_0:def 3;
A94:  SW`1 = Wmin`1 by PSCOMP_1:29;
      then
A95:  LSeg(SW,Wmin) is vertical by SPPOL_1:16;
      now
        per cases by A93,A81,A66,XBOOLE_0:def 3;
        suppose
A96:      x in LSeg(SW,Wmin);
          then
A97:      x`2 <= Wmin`2 by A82,TOPREAL1:4;
A98:      x`1 = SW`1 by A95,A96,SPPOL_1:41;
          then
A99:     x`1 = Wbo by EUCLID:52;
          now
            per cases by A91,A84,A77,XBOOLE_0:def 3;
            suppose
A100:         x in L~midU;
              then x in L~Upper_Seq(C,n) by A90;
              then x in W-most L~Cage(C,n) by A76,A98,EUCLID:52,SPRECT_2:12;
              then x`2 >= Wmin`2 by PSCOMP_1:31;
              then x`2 = Wmin`2 by A97,XXREAL_0:1;
              then x = Wmin by A94,A98,TOPREAL3:6;
              then FiP..Upper_Seq(C,n) = 1 by A46,A1,A24,A43,A100,Th37;
              then Wmin = FiP by A1,A22,FINSEQ_5:38;
              hence contradiction by A16,A40,EUCLID:52;
            end;
            suppose
              x in LSeg(FiP,|[Ebo,FiP`2]|);
              hence contradiction by A16,A32,A40,A63,A99,TOPREAL1:3;
            end;
          end;
          hence contradiction;
        end;
        suppose
A101:     x in L~RevL;
          now
            per cases by A91,A84,A77,XBOOLE_0:def 3;
            suppose
A102:         x in L~midU;
              then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A90,A85,A101,
XBOOLE_0:def 4;
              then
A103:         x in {Wmin,Emax} by JORDAN1E:16;
              now
                per cases by A103,TARSKI:def 2;
                suppose
                  x = Wmin;
                  then FiP..Upper_Seq(C,n) = 1 by A46,A1,A24,A43,A102,Th37;
                  then Wmin = FiP by A1,A22,FINSEQ_5:38;
                  hence contradiction by A16,A40,EUCLID:52;
                end;
                suppose
                  x = Emax;
                  then FiP..Upper_Seq(C,n) = len Upper_Seq(C,n) by A46,A30,A24
,A43,A102,Th38;
                  then Emax = FiP by A30,A22,FINSEQ_5:38;
                  hence contradiction by A32,A40,EUCLID:52;
                end;
              end;
              hence contradiction;
            end;
            suppose
A104:         x in LSeg(FiP,|[Ebo,FiP`2]|);
              LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by A87,SPPOL_1:15;
              then
A105:         x`2 = FiP`2 by A104,SPPOL_1:40;
              consider i be Nat such that
A106:         1 <= i and
A107:         i+1 <= len RevL and
A108:         x in LSeg(RevL/.i,RevL/.(i+1)) by A101,SPPOL_2:14;
A109:         i < len RevL by A107,NAT_1:13;
              then
A110:         (Rev Lower_Seq(C,n)/.i)`1 < sr by A21,A29,A70,A106,Th52;
              i in Seg (LaP..Rev Lower_Seq(C,n)) by A29,A106,A109,FINSEQ_1:1;
              then
A111:         RevL/.i = (Rev Lower_Seq(C,n))/.i by A27,FINSEQ_5:43;
              i+1 >= 1 by NAT_1:11;
              then i+1 in Seg (LaP..Rev Lower_Seq(C,n)) by A29,A107,FINSEQ_1:1;
              then
A112:         RevL/.(i+1) = (Rev Lower_Seq(C,n))/.(i+1) by A27,FINSEQ_5:43;
A113:         FiP`1 <= x`1 by A32,A40,A63,A104,TOPREAL1:3;
              now
                per cases by A107,XXREAL_0:1;
                suppose
A114:             i+1 < len RevL;
                  (RevL/.i)`1 <= (RevL/.(i+1))`1 or (RevL/.(i+1))`1 <= (
                  RevL/.i) `1;
                  then
A115:             x`1 <= (RevL/.(i+1))`1 or x`1 <= (RevL/.i)`1 by A108,
TOPREAL1:3;
                  (Rev Lower_Seq(C,n)/.(i+1))`1 < sr by A21,A29,A70,A114,Th52,
NAT_1:11;
                  hence contradiction by A40,A113,A111,A112,A110,A115,
XXREAL_0:2;
                end;
                suppose
A116:             i+1 = len RevL;
                  then i+1 <= len Rev Lower_Seq(C,n) by A27,A29,FINSEQ_4:21;
                  then LSeg(Rev Lower_Seq(C,n)/.i,Rev Lower_Seq(C,n)/.(i+1))
                  = LSeg(Rev Lower_Seq(C,n),i) by A106,TOPREAL1:def 3;
                  then LSeg(RevL/.i,RevL/.(i+1)) is vertical or LSeg(RevL/.i,
RevL/.(i+1)) is horizontal by A111,A112,SPPOL_1:19;
                  hence contradiction by A44,A45,A66,A105,A108,A111,A110,A116,
SPPOL_1:16,40;
                end;
              end;
              hence contradiction;
            end;
          end;
          hence contradiction;
        end;
        suppose
A117:     x in LSeg(LaP,|[sr,Nbo]|);
          then
A118:     LaP`2 <= x`2 by A8,A86,TOPREAL1:4;
A119:     x`1 = LaP`1 by A89,A117,SPPOL_1:41;
          now
            per cases by A91,A84,A77,XBOOLE_0:def 3;
            suppose
              x in L~midU;
              then consider i be Nat such that
A120:         1 <= i and
A121:         i+1 <= len midU and
A122:         x in LSeg(midU/.i,midU/.(i+1)) by SPPOL_2:14;
              i+2 >= 1+1 by NAT_1:11;
              then
A123:         i+2-1 >= 1+1-1 by XREAL_1:9;
              i < len midU by A121,NAT_1:13;
              then i in dom midU by A120,FINSEQ_3:25;
              then
A124:         midU/.i = Upper_Seq(C,n)/.(i+2-'1) by A47,A23,A79,SPRECT_2:3
                .= Upper_Seq(C,n)/.(i+(2-1)) by A123,XREAL_0:def 2;
              i+1+2 >= 1+1 by NAT_1:11;
              then
A125:         i+1+2-1 >= 1+1-1 by XREAL_1:9;
A126:         1 <= i+1 by NAT_1:11;
              then i+1 in dom midU by A121,FINSEQ_3:25;
              then
A127:         midU/.(i+1) = Upper_Seq(C,n)/.(i+1+2-'1) by A47,A23,A79,
SPRECT_2:3
                .= Upper_Seq(C,n)/.(i+1+(2-1)) by A125,XREAL_0:def 2;
A128:         i+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A80,A121,XREAL_1:7;
              then i+1 < FiP..Upper_Seq(C,n) by NAT_1:13;
              then
A129:         (midU/.i)`1 < sr by A21,A124,Th51,NAT_1:11;
              i+1+1 <= len Upper_Seq(C,n) by A43,A128,XXREAL_0:2;
              then LSeg(midU/.i,midU/.(i+1)) = LSeg(Upper_Seq(C,n),i+1) by A124
,A126,A127,TOPREAL1:def 3;
              then
A130:         LSeg(midU/.i,midU/.(i+1)) is vertical or LSeg(midU/.i,midU
              /.(i+1)) is horizontal by SPPOL_1:19;
              now
                per cases by A121,XXREAL_0:1;
                suppose
                  i+1 < len midU;
                  then i+1+1 <= len midU by NAT_1:13;
                  then i+1+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A80,XREAL_1:7;
                  then i+1+1 < FiP..Upper_Seq(C,n) by NAT_1:13;
                  then
A131:             (midU/.(i+1))`1 < sr by A21,A127,Th51,NAT_1:11;
                  (midU/.i)`1 <= (midU/.(i+1))`1 or (midU/.(i+1))`1 <= (
                  midU/.i) `1;
                  hence contradiction by A44,A119,A122,A129,A131,TOPREAL1:3;
                end;
                suppose
A132:             i+1 = len midU;
                  then (midU/.i)`2 = (midU/.(i+1))`2 by A40,A77,A129,A130,
SPPOL_1:15,16;
                  hence contradiction by A53,A77,A118,A122,A132,GOBOARD7:6;
                end;
              end;
              hence contradiction;
            end;
            suppose
              x in LSeg(FiP,|[Ebo,FiP`2]|);
              hence contradiction by A53,A88,A118,SPPOL_1:40;
            end;
          end;
          hence contradiction;
        end;
      end;
      hence contradiction;
    end;
    suppose
A133: SW = Wmin;
      reconsider RevLS = Rev Lower_Seq(C,n) as special FinSequence of TOP-REAL
      2;
      set h = ((Rev Lower_Seq(C,n))-:LaP)^<*|[sr,Nbo]|*>;
A134: <*|[sr,Nbo]|*> is one-to-one & RevLS-:LaP is special by FINSEQ_3:93;
      rng ((Rev Lower_Seq(C,n))-:LaP) misses {|[sr,Nbo]|} by A25,ZFMISC_1:50;
      then rng ((Rev Lower_Seq(C,n))-:LaP) /\ {|[sr,Nbo]|} = {};
      then
rng ((Rev Lower_Seq(C,n))-:LaP) /\ rng <*|[sr,Nbo]|*> = {} by FINSEQ_1:38;
      then
A135: rng ((Rev Lower_Seq(C,n))-:LaP) misses rng <*|[sr,Nbo]|*>;
      (((Rev Lower_Seq(C,n))-:LaP)/.len ((Rev Lower_Seq(C,n))-:LaP))`1 =
(((Rev Lower_Seq(C,n))-:LaP)/.(LaP..Rev Lower_Seq(C,n)))`1 by A27,FINSEQ_5:42
        .= LaP`1 by A27,FINSEQ_5:45
        .= |[sr,Nbo]|`1 by A44,EUCLID:52
        .= (<*|[sr,Nbo]|*>/.1)`1 by FINSEQ_4:16;
      then reconsider
      h as one-to-one special FinSequence of TOP-REAL 2 by A135,A134,
FINSEQ_3:91,GOBOARD2:8;
      now
        let m be Nat;
A136:   W-bound L~Cage(C,n) <= E-bound L~Cage(C,n) by SPRECT_1:21;
        assume m in dom <*|[sr,Nbo]|*>;
        then m in Seg 1 by FINSEQ_1:38;
        then m = 1 by FINSEQ_1:2,TARSKI:def 1;
        then
A137:   <*|[sr,Nbo]|*>/.m = |[sr,Nbo]| by FINSEQ_4:16;
        then (<*|[sr,Nbo]|*>/.m)`1 = sr by EUCLID:52;
        hence
        W-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`1 & (<*|[sr,Nbo]|*>/.m
        )`1 <= E-bound L~Cage(C,n) by A136,JORDAN6:1;
        (<*|[sr,Nbo]|*>/.m)`2 = Nbo by A137,EUCLID:52;
        hence
        S-bound L~Cage(C,n) <= (<*|[sr,Nbo]|*>/.m)`2 & (<*|[sr,Nbo]|*>/.m
        )`2 <= N-bound L~Cage(C,n) by SPRECT_1:22;
      end;
      then
A138: <*|[sr,Nbo]|*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
      Rev Lower_Seq(C,n) is_in_the_area_of Cage(C,n) by JORDAN1E:18,SPRECT_3:51
;
      then ((Rev Lower_Seq(C,n))-:LaP) is_in_the_area_of Cage(C,n) by A27,
JORDAN1E:1;
      then
A139: h is_in_the_area_of Cage(C,n) by A138,SPRECT_2:24;
A140: len h = len ((Rev Lower_Seq(C,n))-:LaP) + 1 by FINSEQ_2:16;
      then
A141: (h/.len h)`2 = |[sr,Nbo]|`2 by FINSEQ_4:67
        .= N-bound L~Cage(C,n) by EUCLID:52;
      L~Rev Lower_Seq(C,n) = L~RevL \/ L~((Rev Lower_Seq(C,n)):-LaP) by A27,
SPPOL_2:24;
      then L~RevL c= L~Rev Lower_Seq(C,n) by XBOOLE_1:7;
      then
A142: L~RevL c= L~Lower_Seq(C,n) by SPPOL_2:22;
A143: LaP`2 <= Nbo by A20,PSCOMP_1:24;
      LaP..(Rev Lower_Seq(C,n)) >= 0+1 by A28,A29,NAT_1:13;
      then
A144: len ((Rev Lower_Seq(C,n))-:LaP) >= 1 by A27,FINSEQ_5:42;
      1 in dom h by FINSEQ_5:6;
      then h/.1 = h.1 by PARTFUN1:def 6;
      then (h/.1)`2 = (((Rev Lower_Seq(C,n))-:LaP)/.1)`2 by A144,FINSEQ_6:109
        .= ((Rev Lower_Seq(C,n))/.1)`2 by A27,FINSEQ_5:44
        .= (Lower_Seq(C,n)/.len Lower_Seq(C,n))`2 by FINSEQ_5:65
        .= Wmin`2 by JORDAN1F:8
        .= S-bound L~Cage(C,n) by A133,EUCLID:52;
      then
A145: h is_a_v.c._for Cage(C,n) by A139,A141,SPRECT_2:def 3;
      set FiP2 = First_Point(L~Lower_Seq(C,n),Wmin,Emax,Vertical_Line sr);
      set midU = mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n));
A146: |[Ebo,FiP`2]|`1 = Ebo by EUCLID:52;
A147: L~g = L~midU \/ LSeg(midU/.len midU,|[Ebo,FiP`2]|) by A47,A23,SPPOL_2:19
,SPRECT_2:7;
A148: Wmin in rng Upper_Seq(C,n) by A1,FINSEQ_6:42;
      now
        assume FiP..Upper_Seq(C,n) = 1;
        then FiP..Upper_Seq(C,n) = (Upper_Seq(C,n)/.1)..Upper_Seq(C,n) by
FINSEQ_6:43
          .= Wmin..Upper_Seq(C,n) by JORDAN1F:5;
        then FiP = Wmin by A22,A148,FINSEQ_5:9;
        hence contradiction by A16,A40,EUCLID:52;
      end;
      then FiP..Upper_Seq(C,n) > 1 by A24,XXREAL_0:1;
      then
A149: 1+1+0 <= FiP..Upper_Seq(C,n) by NAT_1:13;
      then FiP..Upper_Seq(C,n)-2 >= 0 by XREAL_1:19;
      then FiP..Upper_Seq(C,n)-'2 = FiP..Upper_Seq(C,n)-2 by XREAL_0:def 2;
      then
A150: len midU = FiP..Upper_Seq(C,n)-2+1 by A43,A149,FINSEQ_6:186
        .= FiP..Upper_Seq(C,n)-(2-1);
      LaP`1 = |[sr,Nbo]|`1 by A44,EUCLID:52;
      then
A151: LSeg(LaP,|[sr,Nbo]|) is vertical by SPPOL_1:16;
      len g = len mid(Upper_Seq(C,n),2,FiP..Upper_Seq(C,n))+1 by FINSEQ_2:16;
      then
A152: 1+1 <= len g by A51,XREAL_1:7;
A153: RevL/.len RevL = RevL/.(LaP..Rev Lower_Seq(C,n)) by A27,FINSEQ_5:42
        .= LaP by A27,FINSEQ_5:45;
      1+1 <= len h by A144,A140,XREAL_1:7;
      then L~g meets L~h by A52,A145,A152,SPRECT_2:29;
      then consider x be object such that
A154: x in L~g and
A155: x in L~h by XBOOLE_0:3;
      reconsider x as Point of TOP-REAL 2 by A154;
A156: L~h = L~RevL \/ LSeg(RevL/.len RevL,|[sr,Nbo]|) by A27,FINSEQ_5:47
,SPPOL_2:19;
A157: midU/.len midU = Upper_Seq(C,n)/.(FiP..Upper_Seq(C,n)) by A47,A23,
SPRECT_2:9
        .= FiP by A22,FINSEQ_5:38;
A158: L~midU c= L~Upper_Seq(C,n) by A47,A23,SPRECT_3:18;
A159: L~Rev Lower_Seq(C,n) = L~Lower_Seq(C,n) & FiP2 = LaP by A12,A17,
JORDAN5C:18,SPPOL_2:22;
A160: |[Ebo,FiP`2]|`2 = FiP`2 by EUCLID:52;
      then
A161: LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by SPPOL_1:15;
      now
        per cases by A155,A156,A153,XBOOLE_0:def 3;
        suppose
A162:     x in L~RevL;
          now
            per cases by A154,A147,A157,XBOOLE_0:def 3;
            suppose
A163:         x in L~midU;
              then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A158,A142,A162,
XBOOLE_0:def 4;
              then
A164:         x in {Wmin,Emax} by JORDAN1E:16;
              now
                per cases by A164,TARSKI:def 2;
                suppose
                  x = Wmin;
                  then FiP..Upper_Seq(C,n) = 1 by A46,A1,A24,A43,A163,Th37;
                  then Wmin = FiP by A1,A22,FINSEQ_5:38;
                  hence contradiction by A16,A40,EUCLID:52;
                end;
                suppose
                  x = Emax;
                  then FiP..Upper_Seq(C,n) = len Upper_Seq(C,n) by A46,A30,A24
,A43,A163,Th38;
                  then Emax = FiP by A30,A22,FINSEQ_5:38;
                  hence contradiction by A32,A40,EUCLID:52;
                end;
              end;
              hence contradiction;
            end;
            suppose
A165:         x in LSeg(FiP,|[Ebo,FiP`2]|);
              LSeg(FiP,|[Ebo,FiP`2]|) is horizontal by A160,SPPOL_1:15;
              then
A166:         x`2 = FiP`2 by A165,SPPOL_1:40;
              consider i be Nat such that
A167:         1 <= i and
A168:         i+1 <= len RevL and
A169:         x in LSeg(RevL/.i,RevL/.(i+1)) by A162,SPPOL_2:14;
A170:         i < len RevL by A168,NAT_1:13;
              then
A171:         (Rev Lower_Seq(C,n)/.i)`1 < sr by A21,A29,A159,A167,Th52;
              i in Seg (LaP..Rev Lower_Seq(C,n)) by A29,A167,A170,FINSEQ_1:1;
              then
A172:         RevL/.i = (Rev Lower_Seq(C,n))/.i by A27,FINSEQ_5:43;
              i+1 >= 1 by NAT_1:11;
              then i+1 in Seg (LaP..Rev Lower_Seq(C,n)) by A29,A168,FINSEQ_1:1;
              then
A173:         RevL/.(i+1) = (Rev Lower_Seq(C,n))/.(i+1) by A27,FINSEQ_5:43;
A174:         FiP`1 <= x`1 by A32,A40,A146,A165,TOPREAL1:3;
              now
                per cases by A168,XXREAL_0:1;
                suppose
A175:             i+1 < len RevL;
                  (RevL/.i)`1 <= (RevL/.(i+1))`1 or (RevL/.(i+1))`1 <= (
                  RevL/.i) `1;
                  then
A176:             x`1 <= (RevL/.(i+1))`1 or x`1 <= (RevL/.i)`1 by A169,
TOPREAL1:3;
                  (Rev Lower_Seq(C,n)/.(i+1))`1 < sr by A21,A29,A159,A175,Th52,
NAT_1:11;
                  hence contradiction by A40,A174,A172,A173,A171,A176,
XXREAL_0:2;
                end;
                suppose
A177:             i+1 = len RevL;
                  then i+1 <= len Rev Lower_Seq(C,n) by A27,A29,FINSEQ_4:21;
                  then LSeg(Rev Lower_Seq(C,n)/.i,Rev Lower_Seq(C,n)/.(i+1))
                  = LSeg(Rev Lower_Seq(C,n),i) by A167,TOPREAL1:def 3;
                  then LSeg(RevL/.i,RevL/.(i+1)) is vertical or LSeg(RevL/.i,
RevL/.(i+1)) is horizontal by A172,A173,SPPOL_1:19;
                  hence contradiction by A44,A45,A153,A166,A169,A172,A171,A177,
SPPOL_1:16,40;
                end;
              end;
              hence contradiction;
            end;
          end;
          hence contradiction;
        end;
        suppose
A178:     x in LSeg(LaP,|[sr,Nbo]|);
          then
A179:     LaP`2 <= x`2 by A8,A143,TOPREAL1:4;
A180:     x`1 = LaP`1 by A151,A178,SPPOL_1:41;
          now
            per cases by A154,A147,A157,XBOOLE_0:def 3;
            suppose
              x in L~midU;
              then consider i be Nat such that
A181:         1 <= i and
A182:         i+1 <= len midU and
A183:         x in LSeg(midU/.i,midU/.(i+1)) by SPPOL_2:14;
              i+2 >= 1+1 by NAT_1:11;
              then
A184:         i+2-1 >= 1+1-1 by XREAL_1:9;
              i < len midU by A182,NAT_1:13;
              then i in dom midU by A181,FINSEQ_3:25;
              then
A185:         midU/.i = Upper_Seq(C,n)/.(i+2-'1) by A47,A23,A149,SPRECT_2:3
                .= Upper_Seq(C,n)/.(i+(2-1)) by A184,XREAL_0:def 2;
              i+1+2 >= 1+1 by NAT_1:11;
              then
A186:         i+1+2-1 >= 1+1-1 by XREAL_1:9;
A187:         1 <= i+1 by NAT_1:11;
              then i+1 in dom midU by A182,FINSEQ_3:25;
              then
A188:         midU/.(i+1) = Upper_Seq(C,n)/.(i+1+2-'1) by A47,A23,A149,
SPRECT_2:3
                .= Upper_Seq(C,n)/.(i+1+(2-1)) by A186,XREAL_0:def 2;
A189:         i+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A150,A182,XREAL_1:7;
              then i+1 < FiP..Upper_Seq(C,n) by NAT_1:13;
              then
A190:         (midU/.i)`1 < sr by A21,A185,Th51,NAT_1:11;
              i+1+1 <= len Upper_Seq(C,n) by A43,A189,XXREAL_0:2;
              then LSeg(midU/.i,midU/.(i+1)) = LSeg(Upper_Seq(C,n),i+1) by A185
,A187,A188,TOPREAL1:def 3;
              then
A191:         LSeg(midU/.i,midU/.(i+1)) is vertical or LSeg(midU/.i,midU
              /.(i+1)) is horizontal by SPPOL_1:19;
              now
                per cases by A182,XXREAL_0:1;
                suppose
                  i+1 < len midU;
                  then i+1+1 <= len midU by NAT_1:13;
                  then i+1+1+1 <= FiP..Upper_Seq(C,n)-1+1 by A150,XREAL_1:7;
                  then i+1+1 < FiP..Upper_Seq(C,n) by NAT_1:13;
                  then
A192:             (midU/.(i+1))`1 < sr by A21,A188,Th51,NAT_1:11;
                  (midU/.i)`1 <= (midU/.(i+1))`1 or (midU/.(i+1))`1 <= (
                  midU/.i) `1;
                  hence contradiction by A44,A180,A183,A190,A192,TOPREAL1:3;
                end;
                suppose
A193:             i+1 = len midU;
                  then (midU/.i)`2 = (midU/.(i+1))`2 by A40,A157,A190,A191,
SPPOL_1:15,16;
                  hence contradiction by A53,A157,A179,A183,A193,GOBOARD7:6;
                end;
              end;
              hence contradiction;
            end;
            suppose
              x in LSeg(FiP,|[Ebo,FiP`2]|);
              hence contradiction by A53,A161,A179,SPPOL_1:40;
            end;
          end;
          hence contradiction;
        end;
      end;
      hence contradiction;
    end;
  end;
  hence contradiction;
end;
