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 j1,i2 being Nat st front_right_cell(f,(
  len f)-'1,Gauge(C,n)) meets C & [i2+1,j1] in Indices Gauge(C,n) & f/.((len f)
-'1) = Gauge(C,n)*(i2+1,j1) & [i2,j1] in Indices Gauge(C,n) & f/.len f = Gauge(
  C,n)*(i2,j1) holds [i2,j1+1] 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);
  let f be FinSequence of TOP-REAL 2 such that
A1: f is_sequence_on G and
A2: len f > 1;
A3: 1 <= (len f)-'1 & (len f) -'1 +1 = len f by A2,NAT_D:49,XREAL_1:235;
A4: len G = width G by JORDAN8:def 1;
  let j1,i2 being Nat such that
A5: front_right_cell(f,(len f)-'1,G) meets C & [i2+1,j1] in Indices G &
  f/.( (len f) -'1) = G*(i2+1,j1) and
A6: [i2,j1] in Indices G and
A7: f/.len f = G*(i2,j1);
A8: i2 <= len G by A6,MATRIX_0:32;
A9: j1 <= width G by A6,MATRIX_0:32;
A10: now
    assume j1+1 > len G;
    then
A11: (len G)+1 <= j1+1 by NAT_1:13;
    j1+1 <= (len G)+1 by A9,A4,XREAL_1:6;
    then j1+1 = (len G)+1 by A11,XXREAL_0:1;
    then cell(G,i2-'1,len G) meets C by A1,A5,A6,A7,A3,GOBRD13:39;
    hence contradiction by A8,JORDAN8:15,NAT_D:44;
  end;
A12: 1 <= j1+1 by NAT_1:11;
  1 <= i2 by A6,MATRIX_0:32;
  hence thesis by A8,A12,A4,A10,MATRIX_0:30;
end;
