reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for set,
  f for FinSequence of D,
  G for Matrix of D;
reserve G for Go-board,
  p for Point of TOP-REAL 2;
reserve T for non empty Subset of TOP-REAL 2;
reserve C for
  compact non vertical non horizontal non empty Subset of TOP-REAL 2;
reserve i, j, n for Nat;

theorem
  j <= len Gauge(C,n) implies cell(Gauge(C,n),len Gauge(C,n),j) misses C
proof
  set G = Gauge(C,n);
  assume
A1: j <= len G;
A2: len G = 2|^n + (1+2) by Def1;
A3: len G = width G by Def1;
  assume cell(G,len G,j) /\ C <> {};
  then consider p such that
A4: p in cell(G,len G,j) /\ C by SUBSET_1:4;
A5: p in cell(G,len G,j) by A4,XBOOLE_0:def 4;
A6: p in C by A4,XBOOLE_0:def 4;
  4 <= len G by Th10;
  then
A7: 1 <= len G by XXREAL_0:2;
  set W = W-bound C, S = S-bound C, E = E-bound C, N = N-bound C;
  set EW = (E-W)/(2|^n);
  [len G,1] in Indices G by A3,A7,MATRIX_0:30;
  then
A8: G*(len G,1) = |[W+EW*((len G)-2), S+((N-S)/(2|^n))*(1-2)]| by Def1;
A9: 2|^n > 0 by NEWTON:83;
  E > W by Th8;
  then
A10: E-W > 0 by XREAL_1:50;
  EW*((len G)-2) = EW*(2|^n) + EW*1 by A2
    .= E-W+EW by A9,XCMPLX_1:87;
  then G*(len G,1)`1 = E + EW by A8,EUCLID:52;
  then
A11: G*(len G,1)`1 > E by A9,A10,XREAL_1:29,139;
A12: j = 0 or j >= 1+0 by NAT_1:9;
  per cases by A1,A12,XXREAL_0:1;
  suppose j = 0;
    then cell(G,len G,j) = {|[r,s]| : G*(len G,1)`1 <= r & s <= G*(1,1)`2}
    by GOBRD11:27;
    then consider r,s such that
A13: p = |[r,s]| and
A14: G*(len G,1)`1 <= r and s <= G*(1,1)`2 by A5;
    p`1 = r by A13,EUCLID:52;
    then E < p`1 by A11,A14,XXREAL_0:2;
    hence contradiction by A6,PSCOMP_1:24;
  end;
  suppose j = len G;
    then cell(G,len G,j)={|[r,s]| : G*(len G,1)`1 <= r & G*(1,width G)`2 <= s}
    by A3,GOBRD11:28;
    then consider r,s such that
A15: p = |[r,s]| and
A16: G*(len G,1)`1 <= r and G*(1,width G)`2 <= s by A5;
    p`1 = r by A15,EUCLID:52;
    then E < p`1 by A11,A16,XXREAL_0:2;
    hence contradiction by A6,PSCOMP_1:24;
  end;
  suppose 1 <= j & j < len G;
    then cell(G,len G,j)
    = { |[r,s]| : G*(len G,1)`1 <= r & G*(1,j)`2 <= s & s <= G*(1,j+1)`2 }
    by A3,GOBRD11:29;
    then consider r,s such that
A17: p = |[r,s]| and
A18: G*(len G,1)`1 <= r and G*(1,j)`2 <= s
    and s <= G*(1,j+1)`2 by A5;
    p`1 = r by A17,EUCLID:52;
    then E < p`1 by A11,A18,XXREAL_0:2;
    hence contradiction by A6,PSCOMP_1:24;
  end;
end;
