reserve C for Simple_closed_curve,
  i, j, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  p in BDD C implies ex n,i,j being Nat st 1 < i & i < len
Gauge(C,n) & 1 < j & j < width Gauge(C,n) & p`1 <> (Gauge(C,n)*(i,j))`1 & p in
  cell(Gauge(C,n),i,j) & cell(Gauge(C,n),i,j) c= BDD C
proof
  reconsider P = p as Point of Euclid 2 by Lm3;
  set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C;
  set EW = E-W, NS = N-S;
  assume
A1: p in BDD C;
  then consider r being Real such that
A2: r > 0 and
A3: Ball(P,r) c= BDD C by Th17;
  set l = r/4;
  l > 0 by A2,XREAL_1:139;
  then consider n being Nat such that
  1 < n and
A4: dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < l and
A5: dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < l by GOBRD14:11;
  set I = [\ ((p`1 - W) / EW * 2|^n) + 2 /], J = [\ ((p`2 - S) / NS * 2|^n) +
  2 /];
A6: 1 < J by A1,Th12;
  set G = Gauge(C,n);
A7: I+1 <= len G by A1,Th11;
A8: J+1 <= width G by A1,Th12;
  take n;
A9: 1 < I by A1,Th10;
  then reconsider I, J as Element of NAT by A6,INT_1:3;
A10: I < I + 1 by XREAL_1:29;
  then
A11: I <= len G by A7,XXREAL_0:2;
  1 <= J + 1 by NAT_1:11;
  then [I,J+1] in Indices G by A9,A8,A11,MATRIX_0:30;
  then G*(I,J+1) = |[W+(EW/(2|^n))*(I-2), S+(NS/(2|^n))*(J+1-2)]| by
JORDAN8:def 1;
  then
A12: G*(I,J+1)`2 = S+(NS/(2|^n))*(J-1) by EUCLID:52;
  then
A13: p`2 < G*(I,J+1)`2 by Th16;
A14: J < J + 1 by XREAL_1:29;
  then
A15: J <= width G by A8,XXREAL_0:2;
  then [I,J] in Indices G by A9,A6,A11,MATRIX_0:30;
  then
A16: G*(I,J) = |[W+(EW/(2|^n))*(I-2), S+(NS/(2|^n))*(J-2)]| by JORDAN8:def 1;
  then G*(I,J)`1 = W+(EW/(2|^n))*(I-2) by EUCLID:52;
  then
A17: G*(I,J)`1 <= p`1 by Th13;
  1 <= I + 1 by NAT_1:11;
  then [I+1,J] in Indices G by A7,A6,A15,MATRIX_0:30;
  then G*(I+1,J) = |[W+(EW/(2|^n))*(I+1-2), S+(NS/(2|^n))*(J-2)]| by
JORDAN8:def 1;
  then G*(I+1,J)`1 = W+(EW/(2|^n))*(I-1) by EUCLID:52;
  then
A18: p`1 < G*(I+1,J)`1 by Th14;
  G*(I,J)`2 = S+(NS/(2|^n))*(J-2) by A16,EUCLID:52;
  then
A19: G*(I,J)`2 <= p`2 by Th15;
A20: S + (NS/(2|^n))*(J-1) > p`2 by Th16;
  then
A21: p in cell(G,I,J) by A9,A7,A6,A8,A17,A19,A18,A12,JORDAN9:17;
  per cases;
  suppose
A22: p`1 <> G*(I,J)`1;
    take I, J;
    thus 1 < I & I < len G & 1 < J & J < width G by A1,A7,A8,A10,A14,Th10,Th12,
XXREAL_0:2;
    cell(G,I,J) c= Ball(P,r) by A2,A4,A5,A9,A7,A6,A8,A21,Lm6;
    hence thesis by A3,A9,A7,A6,A8,A20,A17,A19,A18,A12,A22,JORDAN9:17;
  end;
  suppose
A23: p`1 = G*(I,J)`1;
    then
A24: p`1 <= G*(I-'1+1,J)`1 by A9,XREAL_1:235;
A25: I-'1+1 <= len G by A9,A11,XREAL_1:235;
A26: 1 <= J by A1,Th12;
A27: 1 <= I-'1 by A1,Th10,NAT_D:49;
    then I-'1 < I by NAT_D:51;
    then
A28: p`1 > G*(I-'1,J)`1 by A11,A15,A23,A27,A26,GOBOARD5:3;
    take I-'1, J;
A29: J + 1 <= width G by A1,Th12;
A30: 1 <= I-'1 by A1,Th10,NAT_D:49;
    then
A31: I-'1 < I by NAT_D:51;
    len G = width G by JORDAN8:def 1;
    then
A32: J <= len G by A8,A14,XXREAL_0:2;
    I-'1 <> 1
    proof
      assume I-'1 = 1;
      then 1 = I - 1 by NAT_D:39;
      then G*(I,J)`1 = W-bound C by A6,A32,JORDAN8:11;
      then p`1 <= W-bound BDD C by A1,A23,Th6;
      then
A33:  p`1 < W-bound BDD C by A1,Th22,XXREAL_0:1;
      BDD C is bounded by JORDAN2C:106;
      hence thesis by A1,A33,Th5;
    end;
    hence 1 < I-'1 & I-'1 < len G & 1 < J & J < width G by A1,A14,A11,A30,A29
,A31,Th12,XXREAL_0:1,2;
A34: I-'1+1 <= len G by A9,A11,XREAL_1:235;
A35: J + 1 <= width G by A1,Th12;
A36: p`1 <= G*(I-'1+1,J)`1 by A9,A23,XREAL_1:235;
A37: 1 <= J by A1,Th12;
A38: I-'1+1 = I by A9,XREAL_1:235;
    then
A39: G*(I-'1,J)`2 = G*(I,J)`2 by A11,A30,A37,A29,JORDAN9:16;
A40: G*(I-'1,J+1)`2 = G*(I,J+1)`2 by A11,A38,A30,A37,A29,JORDAN9:16;
    p`1 > G*(I-'1,J)`1 by A11,A15,A23,A30,A37,A31,GOBOARD5:3;
    then p in cell (G,I-'1,J) by A19,A13,A30,A34,A37,A29,A36,A39,A40,JORDAN9:17
;
    then cell(G,I-'1,J) c= Ball(P,r) by A2,A4,A5,A27,A25,A26,A35,Lm6;
    hence thesis by A3,A19,A13,A39,A40,A27,A25,A26,A35,A24,A28,JORDAN9:17;
  end;
end;
