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;

theorem
  1 <= i & i <= len Gauge(T,n) implies
  Gauge(T,n)*(i,len Gauge(T,n)-'1)`2 = N-bound T
proof
  set G = Gauge(T,n);
  set W = W-bound T, S = S-bound T, E = E-bound T, N = N-bound T;
  assume that
A1: 1 <= i and
A2: i <= len Gauge(T,n);
A3: len G = 2|^n + 3 by Def1;
A4: len G = width G by Def1;
A5: len G >= 4 by Th10;
  then 1 < len G by XXREAL_0:2;
  then
A6: 1 <= (len G)-'1 by NAT_D:49;
A7: (len G)-'1-2 = (len G)-1-2 by A5,XREAL_1:233,XXREAL_0:2
    .= 2|^n by A3;
A8: 2|^n > 0 by NEWTON:83;
  (len G)-'1 <= len G by NAT_D:35;
  then [i,(len G)-'1] in Indices G by A1,A2,A4,A6,MATRIX_0:30;
  then G*(i,(len G)-'1)
  = |[W+((E-W)/(2|^n))*(i-2), S+((N-S)/(2|^n))*((len G)-'1-2)]| by Def1;
  hence G*(i,(len G)-'1)`2 = S+((N-S)/(2|^n))*(2|^n) by A7,EUCLID:52
    .= S+(N-S) by A8,XCMPLX_1:87
    .= N-bound T;
end;
