reserve i, j, k, n for Nat,
  P for Subset of TOP-REAL 2,
  C for connected compact non vertical non horizontal Subset of TOP-REAL 2;

theorem Th14:
  union UBD-Family C = UBD C
proof
A1: UBD L~Cage(C,0) c= UBD C & UBD L~Cage(C,0) = LeftComp Cage(C,0) by Th13,
GOBRD14:36;
  for X being set st X in UBD-Family C holds X c= UBD C
  proof
    let X be set;
    assume X in UBD-Family C;
    then ex n st X = UBD L~Cage(C,n);
    hence thesis by Th13;
  end;
  hence union UBD-Family C c= UBD C by ZFMISC_1:76;
  let x be object such that
A2: x in UBD C;
  UBD C = union {B where B is Subset of TOP-REAL 2: B
  is_outside_component_of C} by JORDAN2C:def 5;
  then consider A being set such that
A3: x in A and
A4: A in {B where B is Subset of TOP-REAL 2: B is_outside_component_of C
  } by A2,TARSKI:def 4;
  ex B being Subset of TOP-REAL 2 st A = B & B is_outside_component_of C by A4;
  then reconsider p = x as Point of TOP-REAL 2 by A3;
  consider q being Point of TOP-REAL 2 such that
A5: q`2 > N-bound L~Cage(C,0) and
A6: p <> q by TOPREAL6:33;
  Cage(C,0)/.1 = N-min L~Cage(C,0) by JORDAN9:32;
  then q in LeftComp Cage(C,0) by A5,JORDAN2C:113;
  then consider P such that
A7: P is_S-P_arc_joining p,q and
A8: P c= UBD C by A2,A6,A1,TOPREAL4:29;
  consider f being FinSequence of TOP-REAL 2 such that
A9: f is being_S-Seq and
A10: P = L~f and
A11: p = f/.1 and
  q = f/.len f by A7,TOPREAL4:def 1;
  reconsider f as being_S-Seq FinSequence of TOP-REAL 2 by A9;
  2 <= len f by NAT_D:60;
  then
A12: x in P by A10,A11,JORDAN3:1;
  L~f is non empty & the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2
  by EUCLID:def 8;
  then reconsider
  P1 = P, C1 = C as non empty compact Subset of TopSpaceMetr Euclid
  2 by A10,COMPTS_1:23;
  set d = min_dist_min(P1,C1);
  UBD C is_outside_component_of C by JORDAN2C:68;
  then UBD C is_a_component_of C` by JORDAN2C:def 3;
  then C misses UBD C by JORDAN2C:117;
  then P misses C by A8,XBOOLE_1:63;
  then d > 0 by JGRAPH_1:38;
  then d/2 > 0 by XREAL_1:139;
  then consider n such that
  1 < n and
A13: dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < d/2 & dist(Gauge(C,n)*(1,
  1),Gauge( C,n)*(2,1)) < d/2 by GOBRD14:11;
  set G = Gauge(C,n), g = Cage(C,n);
A14: UBD L~g in UBD-Family C;
A15: now
    assume L~g /\ P <> {};
    then consider a being object such that
A16: a in L~g /\ P by XBOOLE_0:def 1;
    a in L~g by A16,XBOOLE_0:def 4;
    then consider i being Nat such that
A17: 1 <= i & i+1 <= len g and
A18: a in LSeg(g,i) by SPPOL_2:13;
    right_cell(g,i,G) meets C by A17,JORDAN9:31;
    then consider c being object such that
A19: c in right_cell(g,i,G) /\ C by XBOOLE_0:4;
    reconsider c as Point of Euclid 2 by A19,TOPREAL3:8;
    reconsider a9 = a as Point of Euclid 2 by A16,TOPREAL3:8;
A20: c in C by A19,XBOOLE_0:def 4;
    reconsider c9 = c as Point of TOP-REAL 2 by A19;
A21: g is_sequence_on G by JORDAN9:def 1;
    then consider i1, j1, i2, j2 being Nat such that
A22: [i1,j1] in Indices G and
A23: g/.i = G*(i1,j1) and
A24: [i2,j2] in Indices G and
A25: g/.(i+1) = G*(i2,j2) and
A26: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2
    or i1 = i2 & j1 = j2+1 by A17,JORDAN8:3;
    left_cell(g,i,G) /\ right_cell(g,i,G) = LSeg(g,i) by A17,A21,GOBRD13:29;
    then
A27: a in right_cell(g,i,G) by A18,XBOOLE_0:def 4;
    a in P by A16,XBOOLE_0:def 4;
    then
A28: dist(a9,c) >= d by A20,WEIERSTR:34;
    reconsider A = a as Point of TOP-REAL 2 by A16;
    set e = E-bound C, w = W-bound C, p = N-bound C, s = S-bound C;
A29: 4 <= len G by JORDAN8:10;
    then
A30: 1 <= len G by XXREAL_0:2;
A31: len G = width G by JORDAN8:def 1;
    then
A32: 1 <= width G by A29,XXREAL_0:2;
A33: [1,1] in Indices G by A31,A30,MATRIX_0:30;
A34: c in right_cell(g,i,G) by A19,XBOOLE_0:def 4;
    now
      per cases by A26;
      case
A35:    i1 = i2 & j1+1 = j2;
        then
A36:    i1 < len G by A17,A22,A23,A24,A25,Th1;
        then
A37:    i1+1 <= len G by NAT_1:13;
A38:    1 <= i1 by A22,MATRIX_0:32;
        then 1 <= i1+1 by NAT_1:13;
        then
A39:    [i1+1,1] in Indices G by A32,A37,MATRIX_0:30;
        [i1,1] in Indices G by A32,A38,A36,MATRIX_0:30;
        then
A40:    dist(G*(i1,1),G*(i1+1,1)) = G*(i1+1,1)`1 - G*(i1,1)`1 & dist(G*(
        i1,1),G*(i1+ 1,1)) = (e-w)/2|^n by A39,GOBRD14:5,10;
A41:    j1+1 <= width G by A24,A35,MATRIX_0:32;
        then
A42:    j1 < width G by NAT_1:13;
A43:    1 <= j1 by A22,MATRIX_0:32;
        then 1 <= j1+1 by NAT_1:13;
        then
A44:    [1,j1+1] in Indices G by A30,A41,MATRIX_0:30;
A45:    right_cell(g,i,G) = cell(G,i1,j1) by A17,A21,A22,A23,A24,A25,A35,
GOBRD13:22
          .= { |[r,t]| where r, t is Real:
    G*(i1,1)`1 <= r & r <= G*(i1+1,1
)`1 & G*(1,j1)`2 <= t & t <= G*(1,j1+1)`2 } by A38,A36,A43,A42,GOBRD11:32;
        then consider aa, ab being Real such that
A46:    a = |[aa,ab]| and
A47:    G*(i1,1)`1 <= aa & aa <= G*(i1+1,1)`1 & G*(1,j1)`2 <= ab & ab
        <= G*(1 ,j1+1)`2 by A27;
A48:    A`1 = aa & A`2 = ab by A46,EUCLID:52;
        [1,j1] in Indices G by A30,A43,A42,MATRIX_0:30;
        then
A49:    dist(G*(1,j1),G*(1,j1+1)) = G*(1,j1+1)`2 - G*(1,j1)`2 & dist(G*(1
        ,j1),G*(1, j1+1)) = (p-s)/2|^n by A44,GOBRD14:6,9;
        consider r, t being Real such that
A50:    c = |[r,t]| and
A51:    G*(i1,1)`1 <= r & r <= G*(i1+1,1)`1 & G*(1,j1)`2 <= t & t <=
        G*(1,j1+ 1)`2 by A34,A45;
        c9`1 = r & c9`2 = t by A50,EUCLID:52;
        hence dist(A,c9) <= (p-s)/2|^n + (e-w)/2|^n by A51,A47,A48,A49,A40,
TOPREAL6:95;
      end;
      case
A52:    i1+1 = i2 & j1 = j2;
        then
A53:    i1+1 <= len G by A24,MATRIX_0:32;
        then
A54:    i1 < len G by NAT_1:13;
A55:    1 <= i1 by A22,MATRIX_0:32;
        then 1 <= i1+1 by NAT_1:13;
        then
A56:    [i1+1,1] in Indices G by A32,A53,MATRIX_0:30;
        [i1,1] in Indices G by A32,A55,A54,MATRIX_0:30;
        then
A57:    dist(G*(i1,1),G*(i1+1,1)) = G*(i1+1,1)`1 - G*(i1,1)`1 & dist(G*(
        i1,1),G*(i1+ 1,1)) = (e-w)/2|^n by A56,GOBRD14:5,10;
A58:    j2 <= width G by A24,MATRIX_0:32;
        j2 > 1 by A17,A22,A23,A24,A25,A52,Th3;
        then
A59:    j2-1 > 0 by XREAL_1:50;
        then
A60:    j2-1 = j2-'1 by XREAL_0:def 2;
        then
A61:    1 <= j2-'1 by A59,NAT_1:14;
        width G - 1 < width G - 0 by XREAL_1:15;
        then
A62:    j2-'1 < width G by A60,A58,XREAL_1:15;
        then
A63:    [1,j2-'1] in Indices G by A30,A61,MATRIX_0:30;
A64:    right_cell(g,i,G) = cell(G,i1,j2-'1) by A17,A21,A22,A23,A24,A25,A52,
GOBRD13:24
          .= { |[r,t]| where r, t is Real:
     G*(i1,1)`1 <= r & r <= G*(i1+1,1
)`1 & G*(1,j2-'1)`2 <= t & t <= G*(1,j2-'1+1)`2 } by A55,A54,A61,A62,GOBRD11:32
;
        then consider aa, ab being Real such that
A65:    a = |[aa,ab]| and
A66:    G*(i1,1)`1 <= aa & aa <= G*(i1+1,1)`1 & G*(1,j2-'1)`2 <= ab &
        ab <= G *(1,j2-'1+1)`2 by A27;
A67:    A`1 = aa & A`2 = ab by A65,EUCLID:52;
        1 <= j2-'1+1 by A61,NAT_1:13;
        then [1,j2-'1+1] in Indices G by A30,A60,A58,MATRIX_0:30;
        then
A68:    dist(G*(1,j2-'1),G*(1,j2-'1+1)) = G*(1,j2-'1+1)`2 - G*(1,j2-'1)
`2 & dist(G*( 1,j2-'1),G*(1,j2-'1+1)) = (p-s)/2|^n by A63,GOBRD14:6,9;
        consider r, t being Real such that
A69:    c = |[r,t]| and
A70:    G*(i1,1)`1 <= r & r <= G*(i1+1,1)`1 & G*(1,j2-'1)`2 <= t & t
        <= G*(1, j2-'1+1)`2 by A34,A64;
        c9`1 = r & c9`2 = t by A69,EUCLID:52;
        hence dist(A,c9) <= (p-s)/2|^n + (e-w)/2|^n by A70,A66,A67,A68,A57,
TOPREAL6:95;
      end;
      case
A71:    i1 = i2+1 & j1 = j2;
A72:    1 <= j1+1 by NAT_1:12;
A73:    j1 < width G by A17,A22,A23,A24,A25,A71,Th4;
        then j1+1 <= width G by NAT_1:13;
        then
A74:    [1,j1+1] in Indices G by A30,A72,MATRIX_0:30;
A75:    1 <= j1 by A22,MATRIX_0:32;
        then [1,j1] in Indices G by A30,A73,MATRIX_0:30;
        then
A76:    dist(G*(1,j1),G*(1,j1+1)) = G*(1,j1+1)`2 - G*(1,j1)`2 & dist(G*(
        1,j1),G*(1, j1+1)) = (p-s)/2|^n by A74,GOBRD14:6,9;
A77:    i2+1 <= len G by A22,A71,MATRIX_0:32;
        then
A78:    i2 < len G by NAT_1:13;
A79:    1 <= i2 by A24,MATRIX_0:32;
        then 1 <= i2+1 by NAT_1:13;
        then
A80:    [i2+1,1] in Indices G by A32,A77,MATRIX_0:30;
A81:    right_cell(g,i,G) = cell(G,i2,j1) by A17,A21,A22,A23,A24,A25,A71,
GOBRD13:26
          .= { |[r,t]| where r, t is Real:
   G*(i2,1)`1 <= r & r <= G*(i2+1,1
)`1 & G*(1,j1)`2 <= t & t <= G*(1,j1+1)`2 } by A79,A78,A75,A73,GOBRD11:32;
        then consider aa, ab being Real such that
A82:    a = |[aa,ab]| and
A83:    G*(i2,1)`1 <= aa & aa <= G*(i2+1,1)`1 & G*(1,j1)`2 <= ab &
        ab <= G*(1 ,j1+1)`2 by A27;
A84:    A`1 = aa & A`2 = ab by A82,EUCLID:52;
        [i2,1] in Indices G by A32,A79,A78,MATRIX_0:30;
        then
A85:    dist(G*(i2,1),G*(i2+1,1)) = G*(i2+1,1)`1 - G*(i2,1)`1 & dist(G*(
        i2,1),G*(i2+ 1,1)) = (e-w)/2|^n by A80,GOBRD14:5,10;
        consider r, t being Real such that
A86:    c = |[r,t]| and
A87:    G*(i2,1)`1 <= r & r <= G*(i2+1,1)`1 & G*(1,j1)`2 <= t & t <=
        G*(1,j1+ 1)`2 by A34,A81;
        c9`1 = r & c9`2 = t by A86,EUCLID:52;
        hence dist(A,c9) <= (p-s)/2|^n + (e-w)/2|^n by A87,A83,A84,A76,A85,
TOPREAL6:95;
      end;
      case
A88:    i1 = i2 & j1 = j2+1;
        then
A89:    j2+1 <= width G by A22,MATRIX_0:32;
        then
A90:    j2 < width G by NAT_1:13;
A91:    1 <= j2 by A24,MATRIX_0:32;
        then 1 <= j2+1 by NAT_1:13;
        then
A92:    [1,j2+1] in Indices G by A30,A89,MATRIX_0:30;
        [1,j2] in Indices G by A30,A91,A90,MATRIX_0:30;
        then
A93:    dist(G*(1,j2),G*(1,j2+1)) = G*(1,j2+1)`2 - G*(1,j2)`2 & dist(G*(
        1,j2),G*(1, j2+1)) = (p-s)/2|^n by A92,GOBRD14:6,9;
A94:    i1 <= len G by A22,MATRIX_0:32;
        i1 > 1 by A17,A22,A23,A24,A25,A88,Th2;
        then
A95:    i1-1 > 0 by XREAL_1:50;
        then
A96:    i1-1 = i1-'1 by XREAL_0:def 2;
        then
A97:    1 <= i1-'1 by A95,NAT_1:14;
        len G - 1 < len G - 0 by XREAL_1:15;
        then
A98:    i1-'1 < len G by A96,A94,XREAL_1:15;
        then
A99:    [i1-'1,1] in Indices G by A32,A97,MATRIX_0:30;
A100:   right_cell(g,i,G) = cell(G,i1-'1,j2) by A17,A21,A22,A23,A24,A25,A88,
GOBRD13:28
          .= { |[r,t]| where r, t is Real:
G*(i1-'1,1)`1 <= r & r <= G*(i1
        -'1+1,1)`1 & G*(1,j2)`2 <= t & t <= G*(1,j2+1)`2 } by A97,A98,A91,A90,
GOBRD11:32;
        then consider aa, ab being Real such that
A101:   a = |[aa,ab]| and
A102:   G*(i1-'1,1)`1 <= aa & aa <= G*(i1-'1+1,1)`1 & G*(1,j2)`2 <=
        ab & ab <= G*(1,j2+1)`2 by A27;
A103:   A`1 = aa & A`2 = ab by A101,EUCLID:52;
        1 <= i1-'1+1 by A97,NAT_1:13;
        then [i1-'1+1,1] in Indices G by A32,A96,A94,MATRIX_0:30;
        then
A104:   dist(G*(i1-'1,1),G*(i1-'1+1,1)) = G*(i1-'1+1,1)`1 - G*(i1 -'1,1)
`1 & dist(G* (i1-'1,1),G* (i1-'1+1,1)) = (e-w)/2|^n by A99,GOBRD14:5,10;
        consider r, t being Real such that
A105:   c = |[r,t]| and
A106:   G*(i1-'1,1)`1 <= r & r <= G*(i1-'1+1,1)`1 & G*(1,j2)`2 <= t
        & t <= G* (1,j2+1)`2 by A34,A100;
        c9`1 = r & c9`2 = t by A105,EUCLID:52;
        hence dist(A,c9) <= (p-s)/2|^n + (e-w)/2|^n by A106,A102,A103,A93,A104,
TOPREAL6:95;
      end;
    end;
    then
A107: dist(a9,c) <= (p-s)/2|^n + (e-w)/2|^n by TOPREAL6:def 1;
    1+1 <= len G by A29,XXREAL_0:2;
    then [1+1,1] in Indices G by A32,MATRIX_0:30;
    then
A108: dist(G*(1,1),G*(1+1,1)) = (e-w)/2|^n by A33,GOBRD14:10;
    1+1 <= width G by A31,A29,XXREAL_0:2;
    then [1,1+1] in Indices G by A30,MATRIX_0:30;
    then dist(G*(1,1),G*(1,1+1)) = (p-s)/2|^n by A33,GOBRD14:9;
    then (p-s)/2|^n + (e-w)/2|^n < d/2 + d/2 by A13,A108,XREAL_1:8;
    hence contradiction by A28,A107,XXREAL_0:2;
  end;
A109: P c= (L~g)`
  proof
    let a be object;
    assume
A110: a in P;
    then not a in L~g by A15,XBOOLE_0:def 4;
    hence thesis by A110,SUBSET_1:29;
  end;
  0 < n or 0 = n;
  then N-bound L~(Cage(C,0)) >= N-bound L~g by Th7;
  then g/.1 = N-min L~g & q`2 > N-bound L~g by A5,JORDAN9:32,XXREAL_0:2;
  then q in LeftComp g by JORDAN2C:113;
  then q in UBD L~g by GOBRD14:36;
  then
A111: {q} c= UBD L~g by ZFMISC_1:31;
A112: P is_an_arc_of p,q by A7,TOPREAL4:2;
  now
    take a = q;
    thus a in {q} & a in P by A112,TARSKI:def 1,TOPREAL1:1;
  end;
  then
A113: {q} meets P by XBOOLE_0:3;
  UBD L~g is_outside_component_of L~g by JORDAN2C:68;
  then ex E being Subset of (TOP-REAL 2)|(L~g)` st E = UBD L~g & E
is a_component & not E is bounded Subset of Euclid 2 by JORDAN2C:14;
  then UBD L~g is_a_component_of (L~g)` by CONNSP_1:def 6;
  then P c= UBD L~g by A109,A112,A111,A113,GOBOARD9:4,JORDAN6:10;
  hence thesis by A12,A14,TARSKI:def 4;
end;
