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 Th8:
  N-min C in RightComp Cage(C,n)
proof
  set f = Cage(C,n), G = Gauge(C,n);
  consider k being Nat such that
A1: 1 <= k and
A2: k+1 <= len G and
A3: f/.1 = G*(k,width G) & f/.2 = G*(k+1,width G) and
A4: N-min C in cell(G,k,width G-'1) and
  N-min C <> G*(k,width G-'1) by JORDAN9:def 1;
A5: len G = width G by JORDAN8:def 1;
A6: 1 <= k+1 by NAT_1:11;
  then
A7: 1 <= len G by A2,XXREAL_0:2;
  then
A8: [k+1,len G] in Indices G by A2,A5,A6,MATRIX_0:30;
  L~f <> {};
  then
A9: f is_sequence_on G & 1+1 <= len f by GOBRD14:2,JORDAN9:def 1;
  then right_cell(f,1,G) is closed by GOBRD13:30;
  then Fr right_cell(f,1,G) = right_cell(f,1,G) \ Int right_cell(f,1,G) by
TOPS_1:43;
  then
A10: Fr right_cell(f,1,G) \/ Int right_cell(f,1,G) = right_cell(f,1,G) by
TOPS_1:16,XBOOLE_1:45;
A11: k < len G by A2,NAT_1:13;
  then [k,len G] in Indices G by A1,A5,A7,MATRIX_0:30;
  then
A12: cell(G,k,len G-'1) = right_cell(f,1,G) by A3,A9,A5,A8,GOBRD13:24;
A13: Int right_cell(f,1) c= RightComp f by GOBOARD9:def 2;
  Int right_cell(f,1,G) c= Int right_cell(f,1) by A9,GOBRD13:33,TOPS_1:19;
  then
A14: Int cell(G,k,len G-'1) c= RightComp f by A12,A13;
  RightComp f misses L~f by SPRECT_3:25;
  then
A15: RightComp f /\ L~f = {};
A16: Fr cell(G,k,len G-'1) c= RightComp f \/ L~f
  proof
    let q be object;
    assume
A17: q in Fr cell(G,k,len G-'1);
    then reconsider s = q as Point of TOP-REAL 2;
    4 <= len G by JORDAN8:10;
    then 4 - 1 <= len G - 1 by XREAL_1:13;
    then 0 <= len G - 1 by XXREAL_0:2;
    then
A18: len G-'1 = len G - 1 by XREAL_0:def 2;
A19: len G - 1 < len G - 0 by XREAL_1:15;
    then Int cell(G,k,len G-'1) <> {} by A5,A11,A18,GOBOARD9:14;
    then consider p being object such that
A20: p in Int cell(G,k,len G-'1) by XBOOLE_0:def 1;
    reconsider p as Point of TOP-REAL 2 by A20;
    per cases;
    suppose
      q in L~f;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
A21:  not q in L~f;
A22:  LSeg(p,s) c= (L~f)`
      proof
        3 <= len G-'1 by GOBRD14:7;
        then
A23:    1 <= len G-'1 by XXREAL_0:2;
        then
A24:    Int cell(G,k,len G-'1) = { |[x,y]| where x, y is Real:
    G*(k,1)`1
< x & x < G*(k+1,1)`1 & G*(1,len G-'1)`2 < y & y < G*(1,len G-'1+1)`2 } by A1
,A5,A11,A18,A19,GOBOARD6:26;
A25:    cell(G,k,len G-'1) = { |[m,o]| where m, o is Real:
     G*(k,1)`1 <= m
& m <= G*(k+1,1)`1 & G*(1,len G-'1)`2 <= o & o <= G*(1,len G-'1+1)`2 } by A1,A5
,A11,A18,A19,A23,GOBRD11:32;
        Fr cell(G,k,len G-'1) c= cell(G,k,len G-'1) by A9,A12,GOBRD13:30
,TOPS_1:35;
        then q in cell(G,k,len G-'1) by A17;
        then consider m, o being Real such that
A26:    s = |[m,o]| and
A27:    G*(k,1)`1 <= m and
A28:    m <= G*(k+1,1)`1 and
A29:    G*(1,len G-'1)`2 <= o and
A30:    o <= G*(1,len G-'1+1)`2 by A25;
A31:    s`2 = o by A26,EUCLID:52;
        consider x, y being Real such that
A32:    p = |[x,y]| and
A33:    G*(k,1)`1 < x and
A34:    x < G*(k+1,1)`1 and
A35:    G*(1,len G-'1)`2 < y and
A36:    y < G*(1,len G-'1+1)`2 by A20,A24;
A37:    p`1 = x by A32,EUCLID:52;
        let a be object;
        assume
A38:    a in LSeg(p,s);
        then reconsider b = a as Point of TOP-REAL 2;
A39:    b = |[b`1,b`2]| by EUCLID:53;
A40:    p`2 = y by A32,EUCLID:52;
A41:    s`1 = m by A26,EUCLID:52;
        now
          per cases;
          case
            b = s;
            hence thesis by A21,SUBSET_1:29;
          end;
          case
A42:        b <> s;
            now
              per cases by XXREAL_0:1;
              case
A43:            s`1 < p`1 & s`2 < p`2;
                then s`2 < b`2 by A38,A42,TOPREAL6:30;
                then
A44:            G*(1,len G-'1)`2 < b`2 by A29,A31,XXREAL_0:2;
                b`1 <= p`1 by A38,A43,TOPREAL6:29;
                then
A45:            b`1 < G*(k+1,1)`1 by A34,A37,XXREAL_0:2;
                b`2 <= p`2 by A38,A43,TOPREAL6:30;
                then
A46:            b`2 < G*(1,len G-'1+1)`2 by A36,A40,XXREAL_0:2;
                s`1 < b`1 by A38,A42,A43,TOPREAL6:29;
                then G*(k,1)`1 < b`1 by A27,A41,XXREAL_0:2;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A45,A44,A46;
              end;
              case
A47:            s`1 < p`1 & s`2 > p`2;
                then b`2 < s`2 by A38,A42,TOPREAL6:30;
                then
A48:            b`2 < G*(1,len G-'1+1)`2 by A30,A31,XXREAL_0:2;
                b`1 <= p`1 by A38,A47,TOPREAL6:29;
                then
A49:            b`1 < G*(k+1,1)`1 by A34,A37,XXREAL_0:2;
                p`2 <= b`2 by A38,A47,TOPREAL6:30;
                then
A50:            G*(1,len G-'1)`2 < b`2 by A35,A40,XXREAL_0:2;
                s`1 < b`1 by A38,A42,A47,TOPREAL6:29;
                then G*(k,1)`1 < b`1 by A27,A41,XXREAL_0:2;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A49,A50,A48;
              end;
              case
A51:            s`1 < p`1 & s`2 = p`2;
                then b`1 <= p`1 by A38,TOPREAL6:29;
                then
A52:            b`1 < G*(k+1,1)`1 by A34,A37,XXREAL_0:2;
                s`1 < b`1 by A38,A42,A51,TOPREAL6:29;
                then
A53:            G*(k,1)`1 < b`1 by A27,A41,XXREAL_0:2;
                p`2 = b`2 by A38,A51,GOBOARD7:6;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A35,A36,A40,A53
,A52;
              end;
              case
A54:            s`1 > p`1 & s`2 < p`2;
                then s`2 < b`2 by A38,A42,TOPREAL6:30;
                then
A55:            G*(1,len G-'1)`2 < b`2 by A29,A31,XXREAL_0:2;
                b`1 >= p`1 by A38,A54,TOPREAL6:29;
                then
A56:            G*(k,1)`1 < b`1 by A33,A37,XXREAL_0:2;
                b`2 <= p`2 by A38,A54,TOPREAL6:30;
                then
A57:            b`2 < G*(1,len G-'1+1)`2 by A36,A40,XXREAL_0:2;
                s`1 > b`1 by A38,A42,A54,TOPREAL6:29;
                then b`1 < G*(k+1,1)`1 by A28,A41,XXREAL_0:2;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A56,A55,A57;
              end;
              case
A58:            s`1 > p`1 & s`2 > p`2;
                then s`2 > b`2 by A38,A42,TOPREAL6:30;
                then
A59:            b`2 < G*(1,len G-'1+1)`2 by A30,A31,XXREAL_0:2;
                b`1 >= p`1 by A38,A58,TOPREAL6:29;
                then
A60:            G*(k,1)`1 < b`1 by A33,A37,XXREAL_0:2;
                b`2 >= p`2 by A38,A58,TOPREAL6:30;
                then
A61:            G*(1,len G-'1)`2 < b`2 by A35,A40,XXREAL_0:2;
                s`1 > b`1 by A38,A42,A58,TOPREAL6:29;
                then b`1 < G*(k+1,1)`1 by A28,A41,XXREAL_0:2;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A60,A61,A59;
              end;
              case
A62:            s`1 > p`1 & s`2 = p`2;
                then b`1 >= p`1 by A38,TOPREAL6:29;
                then
A63:            G*(k,1)`1 < b`1 by A33,A37,XXREAL_0:2;
                s`1 > b`1 by A38,A42,A62,TOPREAL6:29;
                then
A64:            b`1 < G*(k+1,1)`1 by A28,A41,XXREAL_0:2;
                b`2 = p`2 by A38,A62,GOBOARD7:6;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A35,A36,A40,A63
,A64;
              end;
              case
A65:            s`1 = p`1 & s`2 > p`2;
                then b`2 >= p`2 by A38,TOPREAL6:30;
                then
A66:            G*(1,len G-'1)`2 < b`2 by A35,A40,XXREAL_0:2;
                s`2 > b`2 by A38,A42,A65,TOPREAL6:30;
                then
A67:            b`2 < G*(1,len G-'1+1)`2 by A30,A31,XXREAL_0:2;
                b`1 = p`1 by A38,A65,GOBOARD7:5;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A33,A34,A37,A66
,A67;
              end;
              case
A68:            s`1 = p`1 & s`2 < p`2;
                then b`2 <= p`2 by A38,TOPREAL6:30;
                then
A69:            b`2 < G*(1,len G-'1+1)`2 by A36,A40,XXREAL_0:2;
                s`2 < b`2 by A38,A42,A68,TOPREAL6:30;
                then
A70:            G*(1,len G-'1)`2 < b`2 by A29,A31,XXREAL_0:2;
                b`1 = p`1 by A38,A68,GOBOARD7:5;
                hence b in Int cell(G,k,len G-'1) by A24,A39,A33,A34,A37,A70
,A69;
              end;
              case
                s`1 = p`1 & s`2 = p`2;
                then p = s by TOPREAL3:6;
                then LSeg(p,s) = {p} by RLTOPSP1:70;
                hence b in Int cell(G,k,len G-'1) by A20,A38,TARSKI:def 1;
              end;
            end;
            then not b in L~f by A15,A14,XBOOLE_0:def 4;
            hence thesis by SUBSET_1:29;
          end;
        end;
        hence thesis;
      end;
A71:  s in LSeg(p,s) by RLTOPSP1:68;
      now
        take a = p;
        thus a in {p} & a in LSeg(p,s) by RLTOPSP1:68,TARSKI:def 1;
      end;
      then
A72:  {p} meets LSeg(p,s) by XBOOLE_0:3;
      RightComp f is_a_component_of (L~f)` & {p} c= RightComp f by A14,A20,
GOBOARD9:def 2,ZFMISC_1:31;
      then LSeg(p,s) c= RightComp f by A22,A72,GOBOARD9:4;
      hence thesis by A71,XBOOLE_0:def 3;
    end;
  end;
  C misses L~f by Th5;
  then N-min C in C & C /\ L~f = {} by SPRECT_1:11;
  then
A73: not N-min C in L~f by XBOOLE_0:def 4;
  RightComp f c= RightComp f \/ L~f by XBOOLE_1:7;
  then Int cell(G,k,len G-'1) c= RightComp f \/ L~f by A14;
  then right_cell(f,1,G) c= RightComp f \/ L~f by A12,A16,A10,XBOOLE_1:8;
  hence thesis by A73,A4,A5,A12,XBOOLE_0:def 3;
end;
