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

theorem Th37:
  m <= n & 1 <= i & i+1 <= len Gauge(C,n) & 1 <= j & j+1 <= width
Gauge(C,n) implies ex i1,j1 st i1 = [\ (i-2)/2|^(n-'m)+2 /] & j1 = [\ (j-2)/2|^
  (n-'m)+2 /] & cell(Gauge(C,n),i,j) c= cell(Gauge(C,m),i1,j1)
proof
  assume that
A1: m <= n and
A2: 1 <= i and
A3: i+1 <= len Gauge(C,n) and
A4: 1 <= j and
A5: j+1 <= width Gauge(C,n);
  reconsider i1 = [\ (i-2)/2|^(n-'m)+2 /], j1 = [\ (j-2)/2|^(n-'m)+2 /] as
  Nat by A2,A4,Th35;
  set Gm = Gauge(C,m), Gn = Gauge(C,n);
A6: 2|^(n-'m) >n-'m by NEWTON:86;
  take i1,j1;
  thus i1 = [\ (i-2)/2|^(n-'m)+2 /];
  thus j1 = [\ (j-2)/2|^(n-'m)+2 /];
  let e be object;
  assume
A7: e in cell(Gauge(C,n),i,j);
  then reconsider p = e as Point of TOP-REAL 2;
A8: p`2 <= Gn*(i,j+1)`2 by A2,A3,A4,A5,A7,JORDAN9:17;
  i <= len Gn & j <= width Gn by A3,A5,NAT_1:13;
  then [i,j] in Indices Gn by A2,A4,MATRIX_0:30;
  then
A9: Gn*(i,j) =|[(W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i-2), (
  S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2)]| by JORDAN8:def 1;
  then
A10: Gn*(i,j)`1 =(W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*((i-2) / 1)
    .= (W-bound C)+((E-bound C)-(W-bound C))/1 *((i-2)/(2|^n)) by XCMPLX_1:85;
A11: Gn*(i,j)`1 <= p`1 by A2,A3,A4,A5,A7,JORDAN9:17;
A12: Gn*(i,j)`2 =(S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*((j-2)/1) by A9

    .= (S-bound C)+(((N-bound C)-(S-bound C))/1)*((j-2)/(2|^n)) by XCMPLX_1:85;
A13: p`1 <= Gn*(i+1,j)`1 by A2,A3,A4,A5,A7,JORDAN9:17;
  (E-bound C) >= (W-bound C)+(0 qua Nat) by SPRECT_1:21;
  then
A14: (E-bound C)-(W-bound C) >= 0 by XREAL_1:19;
  1 <= j+1 & i <= len Gn by A3,NAT_1:11,13;
  then [i,j+1] in Indices Gn by A2,A5,MATRIX_0:30;
  then Gn*(i,j+1) =|[(W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i-2), (
  S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j+1-2)]| by JORDAN8:def 1;
  then
A15: Gn*(i,j+1)`2 =(S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*((j+1- 2)/
  1)
    .= (S-bound C)+(((N-bound C)-(S-bound C))/1)*((j+1-2)/(2|^n)) by
XCMPLX_1:85;
  n -' m + 1 >= 0 qua Nat + 1 & n-'m +1 <= 2|^(n-'m) by NEWTON:85,XREAL_1:6;
  then 0 qua Nat + 1 <= 2|^(n-'m) by XXREAL_0:2;
  then
A16: (-1)/1 <= (-1)/2|^(n-'m) by XREAL_1:120;
A17: Gn*(i,j)`2 <= p`2 by A2,A3,A4,A5,A7,JORDAN9:17;
  1 <= i+1 & j <= width Gn by A5,NAT_1:11,13;
  then [i+1,j] in Indices Gn by A3,A4,MATRIX_0:30;
  then
  Gn*(i+1,j) =|[(W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*(i+1-2), (
  S-bound C)+(((N-bound C)-(S-bound C))/(2|^n))*(j-2)]| by JORDAN8:def 1;
  then
A18: Gn*(i+1,j)`1 =(W-bound C)+(((E-bound C)-(W-bound C))/(2|^n))*((i+1- 2)/
  1)
    .= (W-bound C)+(((E-bound C)-(W-bound C))/1)*((i+1-2)/(2|^n)) by
XCMPLX_1:85;
  (i-2)/2|^(n-'m)+2-1 = (i-2)/2|^(n-'m)+(2-1);
  then
A19: (i-2)/2|^(n-'m)+1 < i1 by INT_1:def 6;
  1-2 <= i-2 by A2,XREAL_1:9;
  then (-1)/2|^(n-'m) <= (i-2)/2|^(n-'m) by XREAL_1:72;
  then -1 <= (i-2)/2|^(n-'m) by A16,XXREAL_0:2;
  then -1 + 1 <= (i-2)/2|^(n-'m) + 1 by XREAL_1:6;
  then
A20: i1 >= 1+(0 qua Nat) by A19,INT_1:7;
  (N-bound C) >= (S-bound C)+(0 qua Nat) by SPRECT_1:22;
  then
A21: (N-bound C)-(S-bound C) >= 0 by XREAL_1:19;
  (i-2)/2|^(n-'m)+2-1 < i1 by INT_1:def 6;
  then (i-2)/2|^(n-'m)+2 < i1+1 by XREAL_1:19;
  then (i-2)/2|^(n-'m) < i1+1-2 by XREAL_1:20;
  then (i-2) + 1 <= (i1+1-2)*2|^(n-'m) by A6,INT_1:7,XREAL_1:77;
  then (i+1-2)/2|^(n-'m) <= i1+1-2 by A6,XREAL_1:79;
  then (i+1-2)/2|^(n-'m)/2|^m <= (i1+1-2)/2|^m by XREAL_1:72;
  then (i+1-2)/(2|^(n-'m)*2|^m) <= (i1+1-2)/2|^m by XCMPLX_1:78;
  then (i+1-2)/2|^(n-'m+m) <= (i1+1-2)/2|^m by NEWTON:8;
  then (i+1-2)/2|^n <= (i1+1-2)/2|^m by A1,XREAL_1:235;
  then
A22: ((E-bound C)-(W-bound C)) *((i+1-2)/(2|^n)) <= (((E-bound C)-(W-bound C
  )))*((i1+1-2)/(2|^m)) by A14,XREAL_1:64;
  i1 <= (i-2)/2|^(n-'m)+2 by INT_1:def 6;
  then i1-2 <= (i-2)/2|^(n-'m) by XREAL_1:20;
  then (i1-2)/2|^m <= (i-2)/2|^(n-'m)/2|^m by XREAL_1:72;
  then (i1-2)/2|^m <= (i-2)/(2|^(n-'m)*2|^m) by XCMPLX_1:78;
  then (i1-2)/2|^m <= (i-2)/2|^(n-'m+m) by NEWTON:8;
  then (i1-2)/2|^m <= (i-2)/2|^n by A1,XREAL_1:235;
  then
A23: ((E-bound C)-(W-bound C)) *((i1-2)/(2|^m)) <= (((E-bound C)-(W-bound C)
  ))*((i-2)/(2|^n)) by A14,XREAL_1:64;
  j1 <= (j-2)/2|^(n-'m)+2 by INT_1:def 6;
  then j1-2 <= (j-2)/2|^(n-'m) by XREAL_1:20;
  then (j1-2)/2|^m <= (j-2)/2|^(n-'m)/2|^m by XREAL_1:72;
  then (j1-2)/2|^m <= (j-2)/(2|^(n-'m)*2|^m) by XCMPLX_1:78;
  then (j1-2)/2|^m <= (j-2)/2|^(n-'m+m) by NEWTON:8;
  then (j1-2)/2|^m <= (j-2)/2|^n by A1,XREAL_1:235;
  then
A24: ((N-bound C)-(S-bound C)) *((j1-2)/(2|^m)) <= (((N-bound C)-(S-bound C)
  ))*((j-2)/(2|^n)) by A21,XREAL_1:64;
  (j-2)/2|^(n-'m)+2-1 = (j-2)/2|^(n-'m)+(2-1);
  then
A25: (j-2)/2|^(n-'m)+1 < j1 by INT_1:def 6;
  1-2 <= j-2 by A4,XREAL_1:9;
  then (-1)/2|^(n-'m) <= (j-2)/2|^(n-'m) by XREAL_1:72;
  then -1 <= (j-2)/2|^(n-'m) by A16,XXREAL_0:2;
  then -1 + 1 <= (j-2)/2|^(n-'m) + 1 by XREAL_1:6;
  then
A26: j1 >= 1+(0 qua Nat) by A25,INT_1:7;
  (j-2)/2|^(n-'m)+2-1 < j1 by INT_1:def 6;
  then (j-2)/2|^(n-'m)+2 < j1+1 by XREAL_1:19;
  then (j-2)/2|^(n-'m) < j1+1-2 by XREAL_1:20;
  then (j-2) + 1 <= (j1+1-2)*2|^(n-'m) by A6,INT_1:7,XREAL_1:77;
  then (j+1-2)/2|^(n-'m) <= j1+1-2 by A6,XREAL_1:79;
  then (j+1-2)/2|^(n-'m)/2|^m <= (j1+1-2)/2|^m by XREAL_1:72;
  then (j+1-2)/(2|^(n-'m)*2|^m) <= (j1+1-2)/2|^m by XCMPLX_1:78;
  then (j+1-2)/2|^(n-'m+m) <= (j1+1-2)/2|^m by NEWTON:8;
  then (j+1-2)/2|^n <= (j1+1-2)/2|^m by A1,XREAL_1:235;
  then
A27: ((N-bound C)-(S-bound C)) *((j+1-2)/(2|^n)) <= (((N-bound C)-(S-bound
  C)))*((j1+1-2)/(2|^m)) by A21,XREAL_1:64;
  len Gauge(C,m) = width Gauge(C,m) & len Gauge(C,n) = width Gauge(C,n) by
JORDAN8:def 1;
  then
A28: j1+1 <= width Gauge(C,m) by A1,A4,A5,Th36;
  then
A29: j1 <= width Gauge(C,m) by NAT_1:13;
A30: i1+1 <= len Gauge(C,m) by A1,A2,A3,Th36;
  then 1 <= j1+1 & i1 <= len Gauge(C,m) by NAT_1:11,13;
  then [i1,j1+1] in Indices Gm by A20,A28,MATRIX_0:30;
  then Gm*(i1,j1+1) =|[(W-bound C)+(((E-bound C)-(W-bound C))/(2|^m))*(i1-2),
  (S-bound C)+(((N-bound C)-(S-bound C))/(2|^m))*(j1+1-2)]| by JORDAN8:def 1;
  then
  Gm*(i1,j1+1)`2 =(S-bound C)+(((N-bound C)-(S-bound C))/(2|^m))*((j1+1-2
  )/1)
    .= (S-bound C)+(((N-bound C)-(S-bound C))/1)*((j1+1-2)/(2|^m)) by
XCMPLX_1:85;
  then Gn*(i,j+1)`2 <= Gm*(i1,j1+1)`2 by A15,A27,XREAL_1:6;
  then
A31: p`2 <= Gm*(i1,j1+1)`2 by A8,XXREAL_0:2;
  i1 <= len Gauge(C,m) by A30,NAT_1:13;
  then [i1,j1] in Indices Gm by A20,A26,A29,MATRIX_0:30;
  then
A32: Gm*(i1,j1) =|[(W-bound C)+(((E-bound C)-(W-bound C))/(2|^m))*(i1- 2 ),
  (S-bound C)+(((N-bound C)-(S-bound C))/(2|^m))*(j1-2)]| by JORDAN8:def 1;
  then
  Gm*(i1,j1)`1 =(W-bound C)+(((E-bound C)-(W-bound C))/(2|^m))*((i1 - 2)/
  1)
    .= (W-bound C)+(((E-bound C)-(W-bound C))/1)*((i1-2)/(2|^m)) by XCMPLX_1:85
;
  then Gm*(i1,j1)`1 <= Gn*(i,j)`1 by A10,A23,XREAL_1:6;
  then
A33: Gm*(i1,j1)`1 <= p`1 by A11,XXREAL_0:2;
  Gm*(i1,j1)`2 =(S-bound C)+(((N-bound C)-(S-bound C))/(2|^m))*((j1-2)/1)
  by A32
    .= (S-bound C)+(((N-bound C)-(S-bound C))/1)*((j1-2)/(2|^m)) by XCMPLX_1:85
;
  then Gm*(i1,j1)`2 <= Gn*(i,j)`2 by A12,A24,XREAL_1:6;
  then
A34: Gm*(i1,j1)`2 <= p`2 by A17,XXREAL_0:2;
  1 <= i1+1 & j1 <= width Gauge(C,m) by A28,NAT_1:11,13;
  then [i1+1,j1] in Indices Gm by A26,A30,MATRIX_0:30;
  then
  Gm*(i1+1,j1) =|[(W-bound C)+(((E-bound C)-(W-bound C))/(2|^m))*(i1+1-2)
  , (S-bound C)+(((N-bound C)-(S-bound C))/(2|^m))*(j1-2)]| by JORDAN8:def 1;
  then
  Gm*(i1+1,j1)`1 =(W-bound C)+(((E-bound C)-(W-bound C))/(2|^m))*((i1+1-2
  )/1)
    .= (W-bound C)+(((E-bound C)-(W-bound C))/1)*((i1+1-2)/(2|^m)) by
XCMPLX_1:85;
  then Gn*(i+1,j)`1 <= Gm*(i1+1,j1)`1 by A18,A22,XREAL_1:6;
  then p`1 <= Gm*(i1+1,j1)`1 by A13,XXREAL_0:2;
  hence thesis by A20,A26,A30,A28,A33,A34,A31,JORDAN9:17;
end;
