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
  for C being compact non vertical non horizontal non empty Subset of
  TOP-REAL 2 for n for f being FinSequence of TOP-REAL 2 st f is_sequence_on
  Gauge(C,n) & len f > 1 for i1,j2 being Nat st front_right_cell(f,(
  len f)-'1,Gauge(C,n)) meets C & [i1,j2+1] in Indices Gauge(C,n) & f/.((len f)
-'1) = Gauge(C,n)*(i1,j2+1) & [i1,j2] in Indices Gauge(C,n) & f/.len f = Gauge(
  C,n)*(i1,j2) holds [i1-'1,j2] in Indices Gauge(C,n)
proof
  let C be compact non vertical non horizontal non empty Subset of TOP-REAL 2;
  let n;
  set G = Gauge(C,n);
A1: len G = width G by JORDAN8:def 1;
  let f be FinSequence of TOP-REAL 2 such that
A2: f is_sequence_on G and
A3: len f > 1;
A4: 1 <= (len f)-'1 & (len f) -'1 +1 = len f by A3,NAT_D:49,XREAL_1:235;
  let i1,j2 being Nat such that
A5: front_right_cell(f,(len f)-'1,G) meets C & [i1,j2+1] in Indices G &
  f/.( (len f) -'1) = G*(i1,j2+1) and
A6: [i1,j2] in Indices G and
A7: f/.len f = G*(i1,j2);
A8: j2 <= width G by A6,MATRIX_0:32;
A9: 1 <= i1 by A6,MATRIX_0:32;
A10: now
    assume i1-'1 < 1;
    then i1 <= 1 by NAT_1:14,NAT_D:36;
    then i1 = 1 by A9,XXREAL_0:1;
    then cell(G,1-'1,j2-'1) meets C by A2,A5,A6,A7,A4,GOBRD13:41;
    then cell(G,0,j2-'1) meets C by XREAL_1:232;
    hence contradiction by A1,A8,JORDAN8:18,NAT_D:44;
  end;
  i1 <= len G by A6,MATRIX_0:32;
  then
A11: i1-'1 <= len G by NAT_D:44;
  1 <= j2 by A6,MATRIX_0:32;
  hence thesis by A8,A11,A10,MATRIX_0:30;
end;
