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,2)`2 = S-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 = width G by Def1;
  len G >= 4 by Th10;
  then 2 <= len G by XXREAL_0:2;
  then [i,2] in Indices G by A1,A2,A3,MATRIX_0:30;
  then G*(i,2) = |[W+((E-W)/(2|^n))*(i-2), S+((N-S)/(2|^n))*(2-2)]| by Def1;
  hence thesis by EUCLID:52;
end;
