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 Th38:
  m <= n & 1 <= i & i+1 <= len Gauge(C,n) & 1 <= j & j+1 <= width
Gauge(C,n) implies ex i1,j1 st 1 <= i1 & i1+1 <= len Gauge(C,m) & 1 <= j1 & j1+
  1 <= width Gauge(C,m) & cell(Gauge(C,n),i,j) c= cell(Gauge(C,m),i1,j1)
proof
  assume that
A1: m <= n and
A2: 1 <= i & i+1 <= len Gauge(C,n) and
A3: 1 <= j & j+1 <= width Gauge(C,n);
  consider i1,j1 such that
A4: i1 = [\ (i-2)/2|^(n-'m)+2 /] and
A5: j1 = [\ (j-2)/2|^(n-'m)+2 /] and
A6: cell(Gauge(C,n),i,j) c= cell(Gauge(C,m),i1,j1) by A1,A2,A3,Th37;
  take i1,j1;
  thus 1 <= i1 & i1+1 <= len Gauge(C,m) by A1,A2,A4,Th36;
  len Gauge(C,m) = width Gauge(C,m) & len Gauge(C,n) = width Gauge(C,n) by
JORDAN8:def 1;
  hence 1 <= j1 & j1+1 <= width Gauge(C,m) by A1,A3,A5,Th36;
  thus thesis by A6;
end;
