reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem Th33:
  m <= n & 1 < i & i < len Gauge(D,m) & 1 < j & j < width Gauge(D,
m) implies for i1,j1 being Nat st i1 = 2|^(n-'m)*(i-2)+2 & j1 = 2|^(
  n-'m)*(j-2)+2 holds Gauge(D,m)*(i,j) = Gauge(D,n)*(i1,j1)
proof
  assume that
A1: m <= n and
A2: 1 < i & i < len Gauge(D,m) and
A3: 1 < j & j < width Gauge(D,m);
  let i1,j1 be Nat such that
A4: i1 = 2|^(n-'m)*(i-2)+2 and
A5: j1 = 2|^(n-'m)*(j-2)+2;
A6: 1 < i1 & i1 < len Gauge(D,n) by A1,A2,A4,Th31;
  (j-2)/(2|^m) = (j-2)/(2|^(n-'(n-'m))) by A1,NAT_D:58
    .= (j-2)/((2|^n)/(2|^(n-'m))) by NAT_D:50,TOPREAL6:10
    .= (j1-2)/(2|^n) by A5,XCMPLX_1:77;
  then
A7: (((N-bound D)-(S-bound D))/(2|^m))*(j-2) = ((N-bound D)-(S-bound D))*((
  j1-2)/(2|^n)) by XCMPLX_1:75
    .= (((N-bound D)-(S-bound D))/(2|^n))*(j1-2) by XCMPLX_1:75;
  (i-2)/(2|^m) = (i-2)/(2|^(n-'(n-'m))) by A1,NAT_D:58
    .= (i-2)/((2|^n)/(2|^(n-'m))) by NAT_D:50,TOPREAL6:10
    .= (i1-2)/(2|^n) by A4,XCMPLX_1:77;
  then
A8: (((E-bound D)-(W-bound D))/(2|^m))*(i-2) = ((E-bound D)-(W-bound D))*((
  i1-2)/(2|^n)) by XCMPLX_1:75
    .= (((E-bound D)-(W-bound D))/(2|^n))*(i1-2) by XCMPLX_1:75;
  1 < j1 & j1 < width Gauge(D,n) by A1,A3,A5,Th32;
  then
A9: [i1,j1] in Indices Gauge(D,n) by A6,MATRIX_0:30;
  [i,j] in Indices Gauge(D,m) by A2,A3,MATRIX_0:30;
  hence
  Gauge(D,m)*(i,j) = |[(W-bound D)+(((E-bound D)-(W-bound D))/(2|^m))*(i-
  2), (S-bound D)+(((N-bound D)-(S-bound D))/(2|^m))*(j-2)]| by JORDAN8:def 1
    .= Gauge(D,n)*(i1,j1) by A9,A8,A7,JORDAN8:def 1;
end;
