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 Th47:
  m <= n & 1 < i & i+1 < len Gauge(E,m) & 1 < j & j+1 < width
Gauge(E,m) implies for i1,j1 being Nat st 2|^(n-'m)*(i-2)+2 <= i1 &
i1 < 2|^(n-'m)*(i-1)+2 & 2|^(n-'m)*(j-2)+2 <= j1 & j1 < 2|^(n-'m)*(j-1)+2 holds
  cell(Gauge(E,n),i1,j1) c= cell(Gauge(E,m),i,j)
proof
  set G = Gauge(E,m), G1 = Gauge(E,n);
  assume that
A1: m <= n and
A2: 1 < i and
A3: i+1 < len G and
A4: 1 < j and
A5: j+1 < width G;
  set i2 = 2|^(n-'m)*(i-'2)+2, j2 = 2|^(n-'m)*(j-'2)+2, i3 = 2|^(n-'m)*(i-'1)+
  2, j3 = 2|^(n-'m)*(j-'1)+2;
  let i1,j1 be Nat such that
A6: 2|^(n-'m)*(i-2)+2 <= i1 and
A7: i1 < 2|^(n-'m)*(i-1)+2 and
A8: 2|^(n-'m)*(j-2)+2 <= j1 and
A9: j1 < 2|^(n-'m)*(j-1)+2;
A10: j-1 = j-'1 by A4,XREAL_1:233;
  then
A11: j3 <= width G1 by A1,A4,A5,Th35;
A12: 1+1 <= i by A2,NAT_1:13;
  then
A13: i2 = 2|^(n-'m)*(i-2)+2 by XREAL_1:233;
  i < i+1 by XREAL_1:29;
  then
A14: i < len G by A3,XXREAL_0:2;
  then
A15: 1 <= 2|^(n-'m)*(i-2)+2 by A1,A2,Th31;
  then
A16: 1 <= i1 by A6,XXREAL_0:2;
  j1+1 <= 2|^(n-'m)*(j-'1)+2 by A9,A10,NAT_1:13;
  then
A17: j1+1 < j3 or j1+1 = j3 by XXREAL_0:1;
  let e be object;
  assume
A18: e in cell(G1,i1,j1);
  then reconsider p = e as Point of TOP-REAL 2;
  2|^(n-'m)*(i-1)+2 <= len G1 by A1,A2,A3,Th34;
  then
A19: i1 < len G1 by A7,XXREAL_0:2;
  then
A20: i1+1 <= len G1 by NAT_1:13;
A21: j+1-(1+1) = j-1 .= j-'1 by A4,XREAL_1:233;
  1 < j+1 by A4,XREAL_1:29;
  then
A22: G*(i,j+1) = G1*(i2,j3) by A1,A2,A5,A14,A21,A13,Th33;
A23: i-1 = i-'1 by A2,XREAL_1:233;
  then
A24: i3 <= len G1 by A1,A2,A3,Th34;
  i1+1 <= 2|^(n-'m)*(i-'1)+2 by A7,A23,NAT_1:13;
  then
A25: i1+1 < i3 or i1+1 = i3 by XXREAL_0:1;
A26: i2 = 2|^(n-'m)*(i-2)+2 by A12,XREAL_1:233;
A27: i+1-(1+1) = i-1 .= i-'1 by A2,XREAL_1:233;
A28: i2 = 2|^(n-'m)*(i-2)+2 by A12,XREAL_1:233;
  then
A29: i2 <= len G1 by A6,A19,XXREAL_0:2;
  j < j+1 by XREAL_1:29;
  then
A30: j < width G by A5,XXREAL_0:2;
  then
A31: 1 <= 2|^(n-'m)*(j-2)+2 by A1,A4,Th32;
  then
A32: 1 <= j1 by A8,XXREAL_0:2;
  then 1 < j1+1 by NAT_1:13;
  then
A33: G1*(i1,j1+1)`2 <= G1*(i1,j3)`2 by A19,A16,A11,A17,GOBOARD5:4;
  2|^(n-'m)*(j-1)+2 <= width G1 by A1,A4,A5,Th35;
  then
A34: j1 < width G1 by A9,XXREAL_0:2;
  then
A35: j1+1 <= width G1 by NAT_1:13;
  then
A36: G1*(i1,j1)`1 <= p`1 by A18,A20,A16,A32,JORDAN9:17;
A37: 1+1 <= j by A4,NAT_1:13;
  then
A38: j2 = 2|^(n-'m)*(j-2)+2 by XREAL_1:233;
  then j2 < j1 or j2 = j1 by A8,XXREAL_0:1;
  then
A39: G1*(i1,j2)`2 <= G1*(i1,j1)`2 by A19,A34,A16,A31,A38,GOBOARD5:4;
A40: j2 = 2|^(n-'m)*(j-2)+2 by A37,XREAL_1:233;
  then
A41: G*(i,j) = Gauge(E,n)*(i2,j2) by A1,A2,A4,A14,A30,A28,Th33;
  1 < i+1 by A2,XREAL_1:29;
  then
A42: G*(i+1,j) = G1*(i3,j2) by A1,A3,A4,A30,A27,A38,Th33;
A43: p`1 <= G1*(i1+1,j1)`1 by A18,A20,A35,A16,A32,JORDAN9:17;
  1 < i1+1 by A16,NAT_1:13;
  then
A44: G1*(i1+1,j1)`1 <= G1*(i3,j1)`1 by A34,A32,A24,A25,GOBOARD5:3;
A45: G1*(i1,j1)`2 <= p`2 by A18,A20,A35,A16,A32,JORDAN9:17;
A46: j2 <= width G1 by A8,A34,A40,XXREAL_0:2;
  then G1*(i1,j2)`2 = G1*(1,j2)`2 by A19,A16,A31,A38,GOBOARD5:1
    .= G1*(i2,j2)`2 by A15,A31,A29,A46,A26,A38,GOBOARD5:1;
  then
A47: G*(i,j)`2 <= p`2 by A45,A41,A39,XXREAL_0:2;
A48: p`2 <= G1* (i1,j1+1)`2 by A18,A20,A35,A16,A32,JORDAN9:17;
A49: 1 < j3 by A9,A32,A10,XXREAL_0:2;
  then G1*(i1,j3)`2 = G1*(1,j3)`2 by A19,A16,A11,GOBOARD5:1
    .= G1*(i2,j3)`2 by A15,A29,A13,A11,A49,GOBOARD5:1;
  then
A50: p`2 <= G*(i,j+1)`2 by A48,A22,A33,XXREAL_0:2;
  i2 < i1 or i2 = i1 by A6,A28,XXREAL_0:1;
  then
A51: G1*(i2,j1)`1 <= G1*(i1,j1)`1 by A19,A34,A15,A32,A28,GOBOARD5:3;
A52: 1 < i3 by A7,A16,A23,XXREAL_0:2;
  then G1*(i3,j1)`1 = G1*(i3,1)`1 by A34,A32,A24,GOBOARD5:2
    .= G1*(i3,j2)`1 by A31,A46,A38,A24,A52,GOBOARD5:2;
  then
A53: p`1 <= G*(i+1,j)`1 by A43,A42,A44,XXREAL_0:2;
  G1*(i2,j1)`1 = G1*(i2,1)`1 by A34,A15,A32,A28,A29,GOBOARD5:2
    .= G1*(i2,j2)`1 by A15,A31,A28,A40,A29,A46,GOBOARD5:2;
  then G*(i,j)`1 <= p`1 by A36,A41,A51,XXREAL_0:2;
  hence thesis by A2,A3,A4,A5,A53,A47,A50,JORDAN9:17;
end;
