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
  m <= n & 3 <= i & i < len Gauge(E,m) & 1 < j & j+1 < width Gauge(E,m)
implies for i1,j1 being Nat st i1 = 2|^(n-'m)*(i-2)+2 & j1 = 2|^(n-'
  m)*(j-2)+2 holds cell(Gauge(E,n),i1-'1,j1) c= cell(Gauge(E,m),i-'1,j)
proof
  assume that
A1: m <= n and
A2: 3 <= i and
A3: i < len Gauge(E,m) and
A4: 1 < j & j+1 < width Gauge(E,m);
A5: i-2 = i-'2 by A2,XREAL_1:233,XXREAL_0:2;
A6: 2+1 <= i by A2;
  then 1+1 < i by NAT_1:13;
  then
A7: 1 < i-1 by XREAL_1:20;
A8: 2|^(n-'m) > 0 by NEWTON:83;
A9: i-3 = i-'3 by A2,XREAL_1:233;
  then i -' 3 < i -' 2 by A5,XREAL_1:10;
  then 2|^(n-'m)*(i-'3) < 2|^(n-'m)*(i-'2) by A8,XREAL_1:68;
  then 2|^(n-'m)*(i-'3) + 1 <= 2|^(n-'m)*(i-'2) by NAT_1:13;
  then 2|^(n-'m)*(i-'3) <= 2|^(n-'m)*(i-'2)-'1 by NAT_D:55;
  then
A10: 2|^(n-'m)*(i-'3)+2 <= 2|^(n-'m)*(i-'2)-'1+2 by XREAL_1:6;
A11: i-'1 = i-1 by A2,XREAL_1:233,XXREAL_0:2;
  then
A12: i -'1 -1 = i-(1+1);
  i > 2+0 by A6,NAT_1:13;
  then i-2 > 0 by XREAL_1:20;
  then
A13: 2|^(n-'m)*(i-'2) > 0 by A8,A5,XREAL_1:129;
  then 2|^(n-'m)*(i-'2) >= 0+1 by NAT_1:13;
  then
A14: 2|^(n-'m)*(i-'1-2)+2 <= 2|^(n-'m)*(i-'2)+2-'1 by A9,A11,A10,NAT_D:38;
A15: i-'1+1 < len Gauge(E,m) by A2,A3,XREAL_1:235,XXREAL_0:2;
  let i1,j1 be Nat such that
A16: i1 = 2|^(n-'m)*(i-2)+2 and
A17: j1 = 2|^(n-'m)*(j-2)+2;
  i1 < i1+1 by XREAL_1:29;
  then
A18: i1-1 < i1 by XREAL_1:19;
  i1 > 0+2 by A16,A5,A13,XREAL_1:6;
  then
A19: i1-'1 < i1 by A18,XREAL_1:233,XXREAL_0:2;
  j-2 < j-1 by XREAL_1:10;
  then 2|^(n-'m)*(j-2) < 2|^(n-'m)*(j-1) by A8,XREAL_1:68;
  then j1 < 2|^(n-'m)*(j-1)+2 by A17,XREAL_1:6;
  hence thesis by A1,A4,A16,A17,A7,A15,A5,A14,A12,A19,Th47;
end;
