reserve n for Nat;

theorem Th46:
  for C be compact connected non vertical non horizontal Subset of
TOP-REAL 2 for i,j be Nat st 1 <= i & i <= len Gauge(C,n) & 1 <= j &
j <= width Gauge(C,n) & Gauge(C,n)*(i,j) in L~Cage(C,n) holds LSeg(Gauge(C,n)*(
  i,1),Gauge(C,n)*(i,j)) meets L~Lower_Seq(C,n)
proof
  let C be compact connected non vertical non horizontal Subset of TOP-REAL 2;
  let i,j be Nat;
  set Gij = Gauge(C,n)*(i,j);
  assume that
A1: 1 <= i and
A2: i <= len Gauge(C,n) and
A3: 1 <= j & j <= width Gauge(C,n) and
A4: Gij in L~Cage(C,n);
A5: Lower_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n)):- E-max L~Cage(C,n)
  by JORDAN1E:def 2;
  set Wmi = W-min L~Cage(C,n);
  set h = mid(Lower_Seq(C,n),2,len Lower_Seq(C,n));
  set v1 = L_Cut(Upper_Seq(C,n),Gij);
  set NE = NE-corner L~Cage(C,n);
  set Gv1 = <*Gauge(C,n)*(i,1)*>^v1;
  set v = Gv1^<*NE*>;
A6: L~Cage(C,n) = L~Upper_Seq(C,n) \/ L~Lower_Seq(C,n) by JORDAN1E:13;
A7: Upper_Seq(C,n) = Rotate(Cage(C,n),W-min L~Cage(C,n))-: E-max L~Cage(C,n)
  by JORDAN1E:def 1;
A8: len Upper_Seq(C,n) >= 3 by JORDAN1E:15;
  then
A9: len Upper_Seq(C,n) >= 1 by XXREAL_0:2;
A10: len Lower_Seq(C,n) >= 3 by JORDAN1E:15;
  then
A11: len Lower_Seq(C,n) >= 2 & len Lower_Seq(C,n) >= 1 by XXREAL_0:2;
A12: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1;
A13: Gauge(C,n)*(i,1)`2 = S-bound L~Cage(C,n) by A1,A2,JORDAN1A:72;
  now
    per cases by A1,A4,A6,XBOOLE_0:def 3,XXREAL_0:1;
    suppose
A14:  Gij in L~Upper_Seq(C,n) & i = 1;
      set G11 = Gauge(C,n)*(1,1);
A15:  Wmi in L~Cage(C,n) by SPRECT_1:13;
      S-bound L~Cage(C,n) = G11`2 by A2,A14,JORDAN1A:72;
      then
A16:  Wmi`1 = W-bound L~Cage(C,n) & G11`2 <= Wmi`2 by A15,EUCLID:52,PSCOMP_1:24
;
A17:  rng Lower_Seq(C,n) c= L~Lower_Seq(C,n) by A10,SPPOL_2:18,XXREAL_0:2;
A18:  Gij`1 = W-bound L~Cage(C,n) by A3,A12,A14,JORDAN1A:73;
      then Gij in W-most L~Cage(C,n) by A4,SPRECT_2:12;
      then
A19:  Wmi`2 <= Gij`2 by PSCOMP_1:31;
      Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = Wmi by JORDAN1F:8;
      then
A20:  Wmi in rng Lower_Seq(C,n) by FINSEQ_6:168;
      G11`1 = W-bound L~Cage(C,n) by A2,A14,JORDAN1A:73;
      then Wmi in LSeg(Gauge(C,n)*(1,1),Gauge(C,n)*(1,j)) by A14,A16,A18,A19,
GOBOARD7:7;
      hence thesis by A14,A17,A20,XBOOLE_0:3;
    end;
    suppose
A21:  Gij in L~Upper_Seq(C,n) & Gij <> Upper_Seq(C,n).len Upper_Seq(C
      ,n) & E-max L~Cage(C,n) <> NE & i > 1;
      len Cage(C,n) > 4 by GOBOARD7:34;
      then
A22:  rng Cage(C,n) c= L~Cage(C,n) by SPPOL_2:18,XXREAL_0:2;
A23:  not NE in rng Cage(C,n)
      proof
A24:    NE`2 = N-bound L~Cage(C,n) by EUCLID:52;
        then NE`1 = E-bound L~Cage(C,n) & NE`2 >= S-bound L~Cage(C,n) by
EUCLID:52,SPRECT_1:22;
        then
        NE in { p where p is Point of TOP-REAL 2 : p`1 = E-bound L~Cage(C
        ,n) & p`2 <= N-bound L~Cage(C,n) & p`2 >= S-bound L~Cage(C,n) } by A24;
        then
A25:    NE in LSeg(SE-corner L~Cage(C,n), NE-corner L~Cage(C,n)) by SPRECT_1:23
;
        assume NE in rng Cage(C,n);
        then NE in LSeg(SE-corner L~Cage(C,n), NE-corner L~Cage(C,n)) /\ L~
        Cage(C,n) by A22,A25,XBOOLE_0:def 4;
        then
A26:    NE`2 <= (E-max L~Cage(C,n))`2 by PSCOMP_1:47;
A27:    (E-max L~Cage(C,n))`1 = NE`1 by PSCOMP_1:45;
        (E-max L~Cage(C,n))`2 <= NE`2 by PSCOMP_1:46;
        then (E-max L~Cage(C,n))`2 = NE`2 by A26,XXREAL_0:1;
        hence contradiction by A21,A27,TOPREAL3:6;
      end;
A28:  now
        per cases;
        suppose
          Gij <> Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          then v1 = <*Gij*>^mid(Upper_Seq(C,n), Index(Gij,Upper_Seq(C,n))+1,
          len Upper_Seq(C,n)) by JORDAN3:def 3;
          then rng v1 = rng <*Gij*> \/ rng mid(Upper_Seq(C,n), Index(Gij,
          Upper_Seq(C,n))+1,len Upper_Seq(C,n)) by FINSEQ_1:31;
          then
A29:      rng v1 = {Gij} \/ rng mid(Upper_Seq(C,n), Index(Gij,Upper_Seq(C
          ,n))+1,len Upper_Seq(C,n)) by FINSEQ_1:38;
          not NE in L~Cage(C,n)
          proof
            assume NE in L~Cage(C,n);
            then consider i be Nat such that
A30:        1 <= i and
A31:        i+1 <= len Cage(C,n) and
A32:        NE in LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by SPPOL_2:14;
            per cases by A30,A31,TOPREAL1:def 5;
            suppose
A33:          (Cage(C,n)/.i)`1 = (Cage(C,n)/.(i+1))`1;
              (Cage(C,n)/.i)`2 <= (Cage(C,n)/.(i+1))`2 or (Cage(C,n)/.i)
              `2 >= (Cage(C,n)/.(i+1))`2;
              then
A34:          NE`2 <= (Cage(C,n)/.(i+1))`2 or NE`2 <= (Cage(C,n)/.i)`2 by A32,
TOPREAL1:4;
A35:          NE`1 = (Cage(C,n)/.i)`1 by A32,A33,GOBOARD7:5;
A36:          1 <= i+1 by NAT_1:11;
              then
A37:          i+1 in dom Cage(C,n) by A31,FINSEQ_3:25;
A38:          NE`2 = N-bound L~Cage(C,n) by EUCLID:52;
              then
A39:          (Cage(C,n)/.(i+1))`2 <= NE`2 by A31,A36,JORDAN5D:11;
A40:          i < len Cage(C,n) by A31,NAT_1:13;
              then (Cage(C,n)/.i)`2 <= NE`2 by A30,A38,JORDAN5D:11;
              then NE`2 = (Cage(C,n)/.(i+1))`2 or NE`2 = (Cage(C,n)/.i)`2 by
A39,A34,XXREAL_0:1;
              then
A41:          NE = (Cage(C,n)/.(i+1)) or NE = (Cage(C,n)/.i) by A33,A35,
TOPREAL3:6;
              i in dom Cage(C,n) by A30,A40,FINSEQ_3:25;
              hence contradiction by A23,A37,A41,PARTFUN2:2;
            end;
            suppose
A42:          (Cage(C,n)/.i)`2 = (Cage(C,n)/.(i+1))`2;
              (Cage(C,n)/.i)`1 <= (Cage(C,n)/.(i+1))`1 or (Cage(C,n)/.i)
              `1 >= (Cage(C,n)/.(i+1))`1;
              then
A43:          NE`1 <= (Cage(C,n)/.(i+1))`1 or NE`1 <= (Cage(C,n)/.i)`1 by A32,
TOPREAL1:3;
A44:          NE`2 = (Cage(C,n)/.i)`2 by A32,A42,GOBOARD7:6;
A45:          1 <= i+1 by NAT_1:11;
              then
A46:          i+1 in dom Cage(C,n) by A31,FINSEQ_3:25;
A47:          NE`1 = E-bound L~Cage(C,n) by EUCLID:52;
              then
A48:          (Cage(C,n)/.(i+1))`1 <= NE`1 by A31,A45,JORDAN5D:12;
A49:          i < len Cage(C,n) by A31,NAT_1:13;
              then (Cage(C,n)/.i)`1 <= NE`1 by A30,A47,JORDAN5D:12;
              then NE`1 = (Cage(C,n)/.(i+1))`1 or NE`1 = (Cage(C,n)/.i)`1 by
A48,A43,XXREAL_0:1;
              then
A50:          NE = (Cage(C,n)/.(i+1)) or NE = (Cage(C,n)/.i) by A42,A44,
TOPREAL3:6;
              i in dom Cage(C,n) by A30,A49,FINSEQ_3:25;
              hence contradiction by A23,A46,A50,PARTFUN2:2;
            end;
          end;
          then
A51:      not NE in {Gij} by A4,TARSKI:def 1;
          rng mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1, len
Upper_Seq(C,n)) c= rng Upper_Seq(C,n) & rng Upper_Seq(C,n) c= rng Cage(C,n) by
Th39,FINSEQ_6:119;
          then rng mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1, len
          Upper_Seq(C,n)) c= rng Cage(C,n);
          then not NE in rng mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1,
          len Upper_Seq(C,n)) by A23;
          hence not NE in rng v1 by A29,A51,XBOOLE_0:def 3;
        end;
        suppose
          Gij = Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          then v1 = mid(Upper_Seq(C,n), Index(Gij,Upper_Seq(C,n))+1,len
          Upper_Seq(C,n)) by JORDAN3:def 3;
          then
A52:      rng v1 c= rng Upper_Seq(C,n) by FINSEQ_6:119;
          rng Upper_Seq(C,n) c= rng Cage(C,n) by Th39;
          then rng v1 c= rng Cage(C,n) by A52;
          hence not NE in rng v1 by A23;
        end;
      end;
      S-bound L~Cage(C,n) < N-bound L~Cage(C,n) by SPRECT_1:32;
      then NE <> Gauge(C,n)*(i,1) by A13,EUCLID:52;
      then not NE in {Gauge(C,n)*(i,1)} by TARSKI:def 1;
      then not NE in rng <*Gauge(C,n)*(i,1)*> by FINSEQ_1:39;
      then not NE in rng <*Gauge(C,n)*(i,1)*> \/ rng v1 by A28,XBOOLE_0:def 3;
      then not NE in rng Gv1 by FINSEQ_1:31;
      then rng Gv1 misses {NE} by ZFMISC_1:50;
      then
A53:  rng Gv1 misses rng <*NE*> by FINSEQ_1:38;
A54:  len v = len Gv1 + 1 by FINSEQ_2:16
        .= 1 + len v1 + 1 by FINSEQ_5:8;
A55:  v1 is non empty by A21,JORDAN1E:3;
      then
A56:  0+1 <= len v1 by NAT_1:13;
      then 1 in dom v1 by FINSEQ_3:25;
      then
A57:  v1/.1 = v1.1 by PARTFUN1:def 6
        .= Gij by A21,JORDAN3:23;
      then
A58:  (v1^<*NE*>)/.1 = Gij by A56,BOOLMARK:7;
      1+len v1 >= 1+1 by A56,XREAL_1:7;
      then
A59:  2 < len v by A54,NAT_1:13;
A60:  v1 is being_S-Seq by A21,JORDAN3:34;
      v = <*Gauge(C,n)*(i,1)*>^(v1^<*NE*>) by FINSEQ_1:32;
      then v/.1 = Gauge(C,n)*(i,1) by FINSEQ_5:15;
      then
A61:  (v/.1)`2 = S-bound L~Cage(C,n) by A1,A2,JORDAN1A:72;
      len v = len Gv1 + 1 by FINSEQ_2:16;
      then v/.(len v) = NE by FINSEQ_4:67;
      then
A62:  (v/.len v)`2 = N-bound L~Cage(C,n) by EUCLID:52;
A63:  Cage(C,n)/.1 = N-min L~Cage(C,n) by JORDAN9:32;
      then (N-max L~Cage(C,n))..Cage(C,n) <= (E-max L~Cage(C,n))..Cage(C,n)
      by SPRECT_2:70;
      then
A64:  (E-max L~Cage(C,n))..Cage(C,n) > 1 by A63,SPRECT_2:69,XXREAL_0:2;
      (E-min L~Cage(C,n))..Cage(C,n) <= (S-max L~Cage(C,n))..Cage(C,n)
      by A63,SPRECT_2:72;
      then (E-max L~Cage(C,n))..Cage(C,n) < (S-max L~Cage(C,n))..Cage(C,n) by
A63,SPRECT_2:71,XXREAL_0:2;
      then (E-max L~Cage(C,n))..Cage(C,n) < (S-min L~Cage(C,n))..Cage(C,n) by
A63,SPRECT_2:73,XXREAL_0:2;
      then
A65:  (E-max L~Cage(C,n))..Cage(C,n) < (W-min L~Cage(C,n))..Cage(C,n )
      by A63,SPRECT_2:74,XXREAL_0:2;
      then (E-max L~Cage(C,n))..Cage(C,n) < len Cage(C,n) by A63,SPRECT_2:76
,XXREAL_0:2;
      then
A66:  (E-max L~Cage(C,n))..Cage(C,n)+1 <= len Cage(C,n) by NAT_1:13;
A67:  E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
      then Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)) = E-max L~Cage(C,n) by
FINSEQ_5:38;
      then
A68:  (Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1))`1 = E-bound L~Cage
      (C,n) by A64,A66,JORDAN1E:20;
A69:  W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
      then
A70:  (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) = len
Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)- (W-min L~Cage(C,n))..Cage(C,n) by A67
,A65,SPRECT_5:9;
      now
        let m be Nat;
        assume
A71:    m in dom <*Gauge(C,n)*(i,1)*>;
        then m in Seg 1 by FINSEQ_1:38;
        then m = 1 by FINSEQ_1:2,TARSKI:def 1;
        then <*Gauge(C,n)*(i,1)*>.m = Gauge(C,n)*(i,1);
        then
A72:    <*Gauge(C,n)*(i,1)*>/.m = Gauge(C,n)*(i,1) by A71,PARTFUN1:def 6;
        width Gauge(C,n) >= 4 by A12,JORDAN8:10;
        then
A73:    1 <= width Gauge(C,n) by XXREAL_0:2;
        then Gauge(C,n)*(1,1)`1 <= Gauge(C,n)*(i,1)`1 by A1,A2,SPRECT_3:13;
        hence W-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`1 by A12,A72,A73
,JORDAN1A:73;
        (Gauge(C,n)*(i,1))`1 <= Gauge(C,n)*(len Gauge(C,n),1)`1 by A1,A2,A73,
SPRECT_3:13;
        hence (<*Gauge(C,n)*(i,1)*>/.m)`1 <= E-bound L~Cage(C,n) by A12,A72,A73
,JORDAN1A:71;
        thus S-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`2 by A1,A2,A72,
JORDAN1A:72;
        S-bound L~Cage(C,n) = Gauge(C,n)*(i,1)`2 by A1,A2,JORDAN1A:72;
        hence (<*Gauge(C,n)*(i,1)*>/.m)`2 <= N-bound L~Cage(C,n) by A72,
SPRECT_1:22;
      end;
      then
A74:  <*Gauge(C,n)*(i,1)*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
A75:  <*NE*> is_in_the_area_of Cage(C,n) by SPRECT_2:25;
      3 <= len Lower_Seq(C,n) by JORDAN1E:15;
      then 2 <= len Lower_Seq(C,n) by XXREAL_0:2;
      then
A76:  2 in dom Lower_Seq(C,n) by FINSEQ_3:25;
      <*Gij*> is_in_the_area_of Cage(C,n) by A21,JORDAN1E:17,SPRECT_3:46;
      then v1 is_in_the_area_of Cage(C,n) by A21,JORDAN1E:17,SPRECT_3:56;
      then Gv1 is_in_the_area_of Cage(C,n) by A74,SPRECT_2:24;
      then v is_in_the_area_of Cage(C,n) by A75,SPRECT_2:24;
      then
A77:  v is_a_v.c._for Cage(C,n) by A61,A62,SPRECT_2:def 3;
A78:  1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)- (W-min L~Cage(C,n
))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)- len Cage(C,n) = 1+(E-max L~Cage(
      C,n))..Cage(C,n);
A79:  len Lower_Seq(C,n) in dom Lower_Seq(C,n) by FINSEQ_5:6;
      then h is_in_the_area_of Cage(C,n) by A76,JORDAN1E:18,SPRECT_2:22;
      then
A80:  Rev h is_in_the_area_of Cage(C,n) by SPRECT_3:51;
      1+(E-max L~Cage(C,n))..Cage(C,n)<=0+(W-min L~Cage(C,n))..Cage(C,n)
      by A65,NAT_1:13;
      then 1+(E-max L~Cage(C,n))..Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)<=0
      by XREAL_1:20;
      then
A81:  len Cage(C,n)+(1+(E-max L~Cage(C,n))..Cage(C,n)- (W-min L~Cage(C,n
      ))..Cage(C,n)) <= len Cage(C,n)+0 by XREAL_1:6;
A82:  len Lower_Seq(C,n) >= 2+1 by JORDAN1E:15;
      then
A83:  len Lower_Seq(C,n) > 2 by NAT_1:13;
      len Cage(C,n)+0 <= len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n) by
XREAL_1:6;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n) <= (E-max L~Cage(C,n)
      )..Rotate(Cage(C,n),W-min L~Cage(C,n)) by A70,XREAL_1:9;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)+1 <= 1+(E-max L~Cage(
      C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) by XREAL_1:6;
      then
A84:  len(Cage(C,n):-W-min L~Cage(C,n)) <= 1+(E-max L~Cage(C,n))..Rotate
      (Cage(C,n),W-min L~Cage(C,n)) by A69,FINSEQ_5:50;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
      then
A85:  E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by
FINSEQ_6:90,SPRECT_2:43;
A86:  L~v1 c= L~Upper_Seq(C,n) by A21,JORDAN3:42;
A87:  len Lower_Seq(C,n) > 1 by A82,XXREAL_0:2;
      then
A88:  h is non empty by A83,JORDAN1B:2;
A89:  E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by A67,
FINSEQ_6:90,SPRECT_2:43;
      then Lower_Seq(C,n)/.(1+1) = Rotate(Cage(C,n),W-min L~Cage(C,n))/. (1+(
      E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))) by A5,A76,
FINSEQ_5:52
        .= Cage(C,n)/.(1+(E-max L~Cage(C,n))..Rotate(Cage(C,n), W-min L~Cage
      (C,n))+(W-min L~Cage(C,n))..Cage(C,n)-'len Cage(C,n)) by A69,A70,A84,A81,
FINSEQ_6:182
        .= Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1) by A70,A78,
XREAL_0:def 2;
      then (h/.1)`1 = E-bound L~Cage(C,n) by A76,A79,A68,SPRECT_2:8;
      then (Rev h/.len h)`1 = E-bound L~Cage(C,n) by A88,FINSEQ_5:65;
      then
A90:  (Rev h/.len Rev h)`1 = E-bound L~Cage(C,n) by FINSEQ_5:def 3;
      Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = Rotate(Cage(C,n),W-min L~
Cage(C,n))/. (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A5,A89,FINSEQ_5:54
        .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
        .= W-min L~Cage(C,n) by A69,FINSEQ_6:92;
      then (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 = W-bound L~Cage(C,n) by
EUCLID:52;
      then (h/.len h)`1 = W-bound L~Cage(C,n) by A76,A79,SPRECT_2:9;
      then (Rev h/.1)`1 = W-bound L~Cage(C,n) by A88,FINSEQ_5:65;
      then
A91:  Rev h is_a_h.c._for Cage(C,n) by A80,A90,SPRECT_2:def 2;
A92:  len Upper_Seq(C,n) in dom Upper_Seq(C,n) by A9,FINSEQ_3:25;
      set ci = mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1, len Upper_Seq(C
      ,n));
      rng Upper_Seq(C,n) c= L~Upper_Seq(C,n) by A8,SPPOL_2:18,XXREAL_0:2;
      then
A93:  not Gauge(C,n)*(i,1) in rng Upper_Seq(C,n) by A2,A21,Th44;
      not Gauge(C,n)*(i,1) in L~Upper_Seq(C,n) by A2,A21,Th44;
      then not Gauge(C,n)*(i,1) in {Gij} by A21,TARSKI:def 1;
      then
A94:  not Gauge(C,n)*(i,1) in rng <*Gij*> by FINSEQ_1:38;
      now
        per cases;
        suppose
A95:      Gij <> Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          rng ci c= rng Upper_Seq(C,n) by FINSEQ_6:119;
          then not Gauge(C,n)*(i,1) in rng ci by A93;
          then not Gauge(C,n)*(i,1) in rng <*Gij*> \/ rng ci by A94,
XBOOLE_0:def 3;
          then not Gauge(C,n)*(i,1) in rng(<*Gij*>^ci) by FINSEQ_1:31;
          hence not Gauge(C,n)*(i,1) in rng v1 by A95,JORDAN3:def 3;
        end;
        suppose
          Gij = Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          then v1 = ci by JORDAN3:def 3;
          then rng v1 c= rng Upper_Seq(C,n) by FINSEQ_6:119;
          hence not Gauge(C,n)*(i,1) in rng v1 by A93;
        end;
      end;
      then {Gauge(C,n)*(i,1)} misses rng v1 by ZFMISC_1:50;
      then
A96:  rng <*Gauge(C,n)*(i,1)*> misses rng v1 by FINSEQ_1:38;
A97:  <*NE*> is one-to-one by FINSEQ_3:93;
      Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):- E-max
      L~Cage(C,n))/.1 by JORDAN1E:def 2
        .= E-max L~Cage(C,n) by FINSEQ_5:53;
      then
A98: not E-max L~Cage(C,n) in L~h by A83,JORDAN5B:16;
      <*Gauge(C,n)*(i,1)*> is one-to-one by FINSEQ_3:93;
      then Gv1 is one-to-one by A96,A60,FINSEQ_3:91;
      then
A99: v is one-to-one by A53,A97,FINSEQ_3:91;
A100: L~h c= L~Lower_Seq(C,n) by A11,JORDAN4:35;
      (<*Gauge(C,n)*(i,1)*>/.len <*Gauge(C,n)*(i,1)*>)`1 = (<*Gauge(C,n)
      *(i,1)*>/.1)`1 by FINSEQ_1:39
        .= Gauge(C,n)*(i,1)`1 by FINSEQ_4:16
        .= (v1/.1)`1 by A1,A2,A3,A57,GOBOARD5:2;
      then
A101: Gv1 is special by A60,GOBOARD2:8;
      len v1 in dom v1 by A56,FINSEQ_3:25;
      then
A102: v1/.(len v1) = v1.(len v1) by PARTFUN1:def 6
        .= Upper_Seq(C,n).len Upper_Seq(C,n) by A21,JORDAN1B:4
        .= Upper_Seq(C,n)/.len Upper_Seq(C,n) by A92,PARTFUN1:def 6
        .= (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/. ((
E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))) by A7,A85,FINSEQ_5:42
        .= E-max L~Cage(C,n) by A85,FINSEQ_5:45;
      then Gv1/.len Gv1 = E-max L~Cage(C,n) by A55,SPRECT_3:1;
      then (Gv1/.len Gv1)`1 = NE`1 by PSCOMP_1:45
        .= (<*NE*>/.1)`1 by FINSEQ_4:16;
      then
A103: v is special by A101,GOBOARD2:8;
      h is S-Sequence_in_R2 by A83,A87,JORDAN3:6;
      then
A104: Rev h is S-Sequence_in_R2;
      then 2 <= len Rev h by TOPREAL1:def 8;
      then L~Rev h meets L~v by A59,A99,A103,A104,A91,A77,SPRECT_2:29;
      then L~h meets L~v by SPPOL_2:22;
      then consider x be object such that
A105: x in L~h and
A106: x in L~v by XBOOLE_0:3;
A107: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
      L~v = L~(<*Gauge(C,n)*(i,1)*>^(v1^<*NE*>)) by FINSEQ_1:32
        .= LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1) \/ L~(v1^<*NE*>) by SPPOL_2:20
        .= LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1) \/ (L~v1 \/ LSeg(v1/.(len
      v1),NE)) by A55,SPPOL_2:19;
      then
A108: x in LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1) or x in L~v1 \/ LSeg(v1
      /.(len v1),NE) by A106,XBOOLE_0:def 3;
      now
        per cases by A108,XBOOLE_0:def 3;
        suppose
          x in LSeg(Gauge(C,n)*(i,1),(v1^<*NE*>)/.1);
          then x in L~<*Gauge(C,n)*(i,1),Gij*> by A58,SPPOL_2:21;
          hence
          L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*> by A105,A100,
XBOOLE_0:3;
        end;
        suppose
A109:     x in L~v1;
          then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A105,A100,A86,
XBOOLE_0:def 4;
          then x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:16;
          then
A110:     x = W-min L~Cage(C,n) by A105,A98,TARSKI:def 2;
          1 in dom Upper_Seq(C,n) by A9,FINSEQ_3:25;
          then Upper_Seq(C,n).1 = Upper_Seq(C,n)/.1 by PARTFUN1:def 6
            .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by A7,A85,FINSEQ_5:44
            .= W-min L~Cage(C,n) by A107,FINSEQ_6:92;
          then x = Gij by A21,A109,A110,JORDAN1E:7;
          then x in LSeg(Gauge(C,n)*(i,1),Gij) by RLTOPSP1:68;
          then x in L~<*Gauge(C,n)*(i,1),Gij*> by SPPOL_2:21;
          hence
          L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*> by A105,A100,
XBOOLE_0:3;
        end;
        suppose
A111:     x in LSeg(v1/.(len v1),NE);
          x in L~Cage(C,n) by A6,A105,A100,XBOOLE_0:def 3;
          then x in LSeg(E-max L~Cage(C,n), NE) /\ L~Cage(C,n) by A102,A111,
XBOOLE_0:def 4;
          then x in {E-max L~Cage(C,n)} by PSCOMP_1:51;
          hence
          L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*> by A105,A98,
TARSKI:def 1;
        end;
      end;
      then L~<*Gauge(C,n)*(i,1),Gij*> meets L~Lower_Seq(C,n);
      hence thesis by SPPOL_2:21;
    end;
    suppose
A112: Gij in L~Upper_Seq(C,n) & Gij <> Upper_Seq(C,n).len Upper_Seq(
      C,n) & E-max L~Cage(C,n) = NE & i > 1;
      now
        let m be Nat;
        assume
A113:   m in dom <*Gauge(C,n)*(i,1)*>;
        then m in Seg 1 by FINSEQ_1:38;
        then m = 1 by FINSEQ_1:2,TARSKI:def 1;
        then <*Gauge(C,n)*(i,1)*>.m = Gauge(C,n)*(i,1);
        then
A114:   <*Gauge(C,n)*(i,1)*>/.m = Gauge(C,n)*(i,1) by A113,PARTFUN1:def 6;
        width Gauge(C,n) >= 4 by A12,JORDAN8:10;
        then
A115:   1 <= width Gauge(C,n) by XXREAL_0:2;
        then Gauge(C,n)*(1,1)`1 <= Gauge(C,n)*(i,1)`1 by A1,A2,SPRECT_3:13;
        hence W-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`1 by A12,A114
,A115,JORDAN1A:73;
        (Gauge(C,n)*(i,1))`1 <= Gauge(C,n)*(len Gauge(C,n),1)`1 by A1,A2,A115,
SPRECT_3:13;
        hence (<*Gauge(C,n)*(i,1)*>/.m)`1 <= E-bound L~Cage(C,n) by A12,A114
,A115,JORDAN1A:71;
        thus S-bound L~Cage(C,n) <= (<*Gauge(C,n)*(i,1)*>/.m)`2 by A1,A2,A114,
JORDAN1A:72;
        S-bound L~Cage(C,n) = Gauge(C,n)*(i,1)`2 by A1,A2,JORDAN1A:72;
        hence (<*Gauge(C,n)*(i,1)*>/.m)`2 <= N-bound L~Cage(C,n) by A114,
SPRECT_1:22;
      end;
      then
A116: <*Gauge(C,n)*(i,1)*> is_in_the_area_of Cage(C,n) by SPRECT_2:def 1;
      <*Gij*> is_in_the_area_of Cage(C,n) by A112,JORDAN1E:17,SPRECT_3:46;
      then v1 is_in_the_area_of Cage(C,n) by A112,JORDAN1E:17,SPRECT_3:56;
      then
A117: Gv1 is_in_the_area_of Cage(C,n) by A116,SPRECT_2:24;
A118: len Upper_Seq(C,n) in dom Upper_Seq(C,n) by A9,FINSEQ_3:25;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
      then
A119: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by
FINSEQ_6:90,SPRECT_2:43;
A120: v1 is non empty by A112,JORDAN1E:3;
      then
A121: 0+1 <= len v1 by NAT_1:13;
      then 1 in dom v1 by FINSEQ_3:25;
      then
A122: v1/.1 = v1.1 by PARTFUN1:def 6
        .= Gij by A112,JORDAN3:23;
      len v1 in dom v1 by A121,FINSEQ_3:25;
      then v1/.(len v1) = v1.(len v1) by PARTFUN1:def 6
        .= Upper_Seq(C,n).len Upper_Seq(C,n) by A112,JORDAN1B:4
        .= Upper_Seq(C,n)/.len Upper_Seq(C,n) by A118,PARTFUN1:def 6
        .= (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/. ((
E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))) by A7,A119,FINSEQ_5:42
        .= E-max L~Cage(C,n) by A119,FINSEQ_5:45;
      then Gv1/.len Gv1 = E-max L~Cage(C,n) by A120,SPRECT_3:1;
      then
A123: (Gv1/.len Gv1)`2 = N-bound L~Cage(C,n) by A112,EUCLID:52;
      Gv1/.1 = Gauge(C,n)*(i,1) by FINSEQ_5:15;
      then (Gv1/.1)`2 = S-bound L~Cage(C,n) by A1,A2,JORDAN1A:72;
      then
A124: Gv1 is_a_v.c._for Cage(C,n) by A117,A123,SPRECT_2:def 3;
A125: Cage(C,n)/.1 = N-min L~Cage(C,n) by JORDAN9:32;
      then (N-max L~Cage(C,n))..Cage(C,n) <= (E-max L~Cage(C,n))..Cage(C,n)
      by SPRECT_2:70;
      then
A126: (E-max L~Cage(C,n))..Cage(C,n) > 1 by A125,SPRECT_2:69,XXREAL_0:2;
      (E-min L~Cage(C,n))..Cage(C,n) <= (S-max L~Cage(C,n))..Cage(C,n)
      by A125,SPRECT_2:72;
      then (E-max L~Cage(C,n))..Cage(C,n) < (S-max L~Cage(C,n))..Cage(C,n) by
A125,SPRECT_2:71,XXREAL_0:2;
      then (E-max L~Cage(C,n))..Cage(C,n) < (S-min L~Cage(C,n))..Cage(C,n) by
A125,SPRECT_2:73,XXREAL_0:2;
      then
A127: (E-max L~Cage(C,n))..Cage(C,n) < (W-min L~Cage(C,n))..Cage(C,n )
      by A125,SPRECT_2:74,XXREAL_0:2;
      then (E-max L~Cage(C,n))..Cage(C,n) < len Cage(C,n) by A125,SPRECT_2:76
,XXREAL_0:2;
      then
A128: (E-max L~Cage(C,n))..Cage(C,n)+1 <= len Cage(C,n) by NAT_1:13;
A129: E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
      then Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)) = E-max L~Cage(C,n) by
FINSEQ_5:38;
      then
A130: (Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1))`1 = E-bound L~Cage
      (C,n) by A126,A128,JORDAN1E:20;
      1+(E-max L~Cage(C,n))..Cage(C,n)<=0+(W-min L~Cage(C,n))..Cage(C,n)
      by A127,NAT_1:13;
      then 1+(E-max L~Cage(C,n))..Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)<=0
      by XREAL_1:20;
      then
A131: len Cage(C,n)+(1+(E-max L~Cage(C,n))..Cage(C,n)- (W-min L~Cage(C,n
      ))..Cage(C,n)) <= len Cage(C,n)+0 by XREAL_1:6;
A132: len Lower_Seq(C,n) >= 2+1 by JORDAN1E:15;
      then
A133: len Lower_Seq(C,n) > 2 by NAT_1:13;
      set ci = mid(Upper_Seq(C,n),Index(Gij,Upper_Seq(C,n))+1, len Upper_Seq(C
      ,n));
      rng Upper_Seq(C,n) c= L~Upper_Seq(C,n) by A8,SPPOL_2:18,XXREAL_0:2;
      then
A134: not Gauge(C,n)*(i,1) in rng Upper_Seq(C,n) by A2,A112,Th44;
      not Gauge(C,n)*(i,1) in L~Upper_Seq(C,n) by A2,A112,Th44;
      then not Gauge(C,n)*(i,1) in {Gij} by A112,TARSKI:def 1;
      then
A135: not Gauge(C,n)*(i,1) in rng <*Gij*> by FINSEQ_1:38;
      now
        per cases;
        suppose
A136:     Gij <> Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          rng ci c= rng Upper_Seq(C,n) by FINSEQ_6:119;
          then not Gauge(C,n)*(i,1) in rng ci by A134;
          then not Gauge(C,n)*(i,1) in rng <*Gij*> \/ rng ci by A135,
XBOOLE_0:def 3;
          then not Gauge(C,n)*(i,1) in rng(<*Gij*>^ci) by FINSEQ_1:31;
          hence not Gauge(C,n)*(i,1) in rng v1 by A136,JORDAN3:def 3;
        end;
        suppose
          Gij = Upper_Seq(C,n).(Index(Gij,Upper_Seq(C,n))+1);
          then v1 = ci by JORDAN3:def 3;
          then rng v1 c= rng Upper_Seq(C,n) by FINSEQ_6:119;
          hence not Gauge(C,n)*(i,1) in rng v1 by A134;
        end;
      end;
      then {Gauge(C,n)*(i,1)} misses rng v1 by ZFMISC_1:50;
      then
A137: rng <*Gauge(C,n)*(i,1)*> misses rng v1 by FINSEQ_1:38;
A138: 1+(len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)- (W-min L~Cage(C,n
))..Cage(C,n))+(W-min L~Cage(C,n))..Cage(C,n)- len Cage(C,n) = 1+(E-max L~Cage(
      C,n))..Cage(C,n);
      1+len v1 >= 1+1 by A121,XREAL_1:7;
      then
A139: len Gv1 >= 2 by FINSEQ_5:8;
      3 <= len Lower_Seq(C,n) by JORDAN1E:15;
      then 2 <= len Lower_Seq(C,n) by XXREAL_0:2;
      then
A140: 2 in dom Lower_Seq(C,n) by FINSEQ_3:25;
      Lower_Seq(C,n)/.1 = (Rotate(Cage(C,n),W-min L~Cage(C,n)):- E-max
      L~Cage(C,n))/.1 by JORDAN1E:def 2
        .= E-max L~Cage(C,n) by FINSEQ_5:53;
      then
A141: not E-max L~Cage(C,n) in L~h by A133,JORDAN5B:16;
A142: v1 is being_S-Seq by A112,JORDAN3:34;
      (<*Gauge(C,n)*(i,1)*>/.len <*Gauge(C,n)*(i,1)*>)`1 = (<*Gauge(C,n)
      *(i,1)*>/.1)`1 by FINSEQ_1:39
        .= Gauge(C,n)*(i,1)`1 by FINSEQ_4:16
        .= (v1/.1)`1 by A1,A2,A3,A122,GOBOARD5:2;
      then
A143: Gv1 is special by A142,GOBOARD2:8;
A144: L~Gv1 = LSeg(Gauge(C,n)*(i,1),v1/.1) \/ L~v1 by A120,SPPOL_2:20;
A145: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
      then
A146: (E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) = len
Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n)- (W-min L~Cage(C,n))..Cage(C,n) by
A129,A127,SPRECT_5:9;
      len Cage(C,n)+0 <= len Cage(C,n)+(E-max L~Cage(C,n))..Cage(C,n) by
XREAL_1:6;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n) <= (E-max L~Cage(C,n)
      )..Rotate(Cage(C,n),W-min L~Cage(C,n)) by A146,XREAL_1:9;
      then len Cage(C,n)-(W-min L~Cage(C,n))..Cage(C,n)+1 <= 1+(E-max L~Cage(
      C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n)) by XREAL_1:6;
      then
A147: len(Cage(C,n):-W-min L~Cage(C,n)) <= 1+(E-max L~Cage(C,n))..Rotate
      (Cage(C,n),W-min L~Cage(C,n)) by A145,FINSEQ_5:50;
A148: len Lower_Seq(C,n) > 1 by A132,XXREAL_0:2;
      then
A149: h is non empty by A133,JORDAN1B:2;
A150: len Lower_Seq(C,n) in dom Lower_Seq(C,n) by FINSEQ_5:6;
      then h is_in_the_area_of Cage(C,n) by A140,JORDAN1E:18,SPRECT_2:22;
      then
A151: Rev h is_in_the_area_of Cage(C,n) by SPRECT_3:51;
A152: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by A129,
FINSEQ_6:90,SPRECT_2:43;
      then Lower_Seq(C,n)/.(1+1) = Rotate(Cage(C,n),W-min L~Cage(C,n))/. (1+(
      E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))) by A5,A140,
FINSEQ_5:52
        .= Cage(C,n)/.(1+(E-max L~Cage(C,n))..Rotate(Cage(C,n), W-min L~Cage
(C,n))+(W-min L~Cage(C,n))..Cage(C,n)-'len Cage(C,n)) by A145,A146,A147,A131,
FINSEQ_6:182
        .= Cage(C,n)/.((E-max L~Cage(C,n))..Cage(C,n)+1) by A146,A138,
XREAL_0:def 2;
      then (h/.1)`1 = E-bound L~Cage(C,n) by A140,A150,A130,SPRECT_2:8;
      then (Rev h/.len h)`1 = E-bound L~Cage(C,n) by A149,FINSEQ_5:65;
      then
A153: (Rev h/.len Rev h)`1 = E-bound L~Cage(C,n) by FINSEQ_5:def 3;
      Lower_Seq(C,n)/.(len Lower_Seq(C,n)) = Rotate(Cage(C,n),W-min L~
Cage(C,n))/. (len Rotate(Cage(C,n),W-min L~Cage(C,n))) by A5,A152,FINSEQ_5:54
        .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by FINSEQ_6:def 1
        .= W-min L~Cage(C,n) by A145,FINSEQ_6:92;
      then (Lower_Seq(C,n)/.len Lower_Seq(C,n))`1 = W-bound L~Cage(C,n) by
EUCLID:52;
      then (h/.len h)`1 = W-bound L~Cage(C,n) by A140,A150,SPRECT_2:9;
      then (Rev h/.1)`1 = W-bound L~Cage(C,n) by A149,FINSEQ_5:65;
      then
A154: Rev h is_a_h.c._for Cage(C,n) by A151,A153,SPRECT_2:def 2;
      <*Gauge(C,n)*(i,1)*> is one-to-one by FINSEQ_3:93;
      then
A155: Gv1 is one-to-one by A137,A142,FINSEQ_3:91;
A156: L~h c= L~Lower_Seq(C,n) by A11,JORDAN4:35;
      h is S-Sequence_in_R2 by A133,A148,JORDAN3:6;
      then
A157: Rev h is S-Sequence_in_R2;
      then 2 <= len Rev h by TOPREAL1:def 8;
      then L~Rev h meets L~Gv1 by A139,A155,A143,A157,A154,A124,SPRECT_2:29;
      then L~h meets L~Gv1 by SPPOL_2:22;
      then consider x be object such that
A158: x in L~h and
A159: x in L~Gv1 by XBOOLE_0:3;
A160: W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43;
A161: L~v1 c= L~Upper_Seq(C,n) by A112,JORDAN3:42;
      now
        per cases by A159,A144,XBOOLE_0:def 3;
        suppose
          x in LSeg(Gauge(C,n)*(i,1),v1/.1);
          then x in L~<*Gauge(C,n)*(i,1),Gij*> by A122,SPPOL_2:21;
          hence
          L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*> by A158,A156,
XBOOLE_0:3;
        end;
        suppose
A162:     x in L~v1;
          then x in L~Upper_Seq(C,n) /\ L~Lower_Seq(C,n) by A158,A156,A161,
XBOOLE_0:def 4;
          then x in {W-min L~Cage(C,n),E-max L~Cage(C,n)} by JORDAN1E:16;
          then
A163:     x = W-min L~Cage(C,n) by A158,A141,TARSKI:def 2;
          1 in dom Upper_Seq(C,n) by A9,FINSEQ_3:25;
          then Upper_Seq(C,n).1 = Upper_Seq(C,n)/.1 by PARTFUN1:def 6
            .= Rotate(Cage(C,n),W-min L~Cage(C,n))/.1 by A7,A119,FINSEQ_5:44
            .= W-min L~Cage(C,n) by A160,FINSEQ_6:92;
          then x = Gij by A112,A162,A163,JORDAN1E:7;
          then x in LSeg(Gauge(C,n)*(i,1),Gij) by RLTOPSP1:68;
          then x in L~<*Gauge(C,n)*(i,1),Gij*> by SPPOL_2:21;
          hence
          L~Lower_Seq(C,n) meets L~<*Gauge(C,n)*(i,1),Gij*> by A158,A156,
XBOOLE_0:3;
        end;
      end;
      then L~<*Gauge(C,n)*(i,1),Gij*> meets L~Lower_Seq(C,n);
      hence thesis by SPPOL_2:21;
    end;
    suppose
A164: Gij in L~Lower_Seq(C,n);
      Gij in LSeg(Gauge(C,n)*(i,1),Gij) by RLTOPSP1:68;
      hence thesis by A164,XBOOLE_0:3;
    end;
    suppose
A165: Gij in L~Upper_Seq(C,n) & Gij = Upper_Seq(C,n).len Upper_Seq(C ,n);
A166: Gij in LSeg(Gauge(C,n)*(i,1),Gij) by RLTOPSP1:68;
A167: rng Lower_Seq(C,n) c= L~Lower_Seq(C,n) & E-max L~Cage(C,n) in rng
      Lower_Seq( C,n) by A5,A10,FINSEQ_6:61,SPPOL_2:18,XXREAL_0:2;
      E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46;
      then
A168: E-max L~Cage(C,n) in rng Rotate(Cage(C,n),W-min L~Cage(C,n)) by
FINSEQ_6:90,SPRECT_2:43;
      len Upper_Seq(C,n) in dom Upper_Seq(C,n) by A9,FINSEQ_3:25;
      then Upper_Seq(C,n).len Upper_Seq(C,n) = Upper_Seq(C,n)/.len Upper_Seq(
      C,n) by PARTFUN1:def 6
        .= (Rotate(Cage(C,n),W-min L~Cage(C,n))-:E-max L~Cage(C,n))/. ((
E-max L~Cage(C,n))..Rotate(Cage(C,n),W-min L~Cage(C,n))) by A7,A168,FINSEQ_5:42
        .= E-max L~Cage(C,n) by A168,FINSEQ_5:45;
      hence thesis by A165,A167,A166,XBOOLE_0:3;
    end;
  end;
  hence thesis;
end;
