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,j1 being Nat st front_right_cell(f,(
len f)-'1,Gauge(C,n)) meets C & [i1,j1] in Indices Gauge(C,n) & f/.((len f) -'1
) = Gauge(C,n)*(i1,j1) & [i1+1,j1] in Indices Gauge(C,n) & f/.len f = Gauge(C,n
  )*(i1+1,j1) holds [i1+1,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;
  let i1,j1 being Nat such that
A4: front_right_cell(f,(len f)-'1,G) meets C & [i1,j1] in Indices G & f
  /.(( len f) -'1) = G*(i1,j1) and
A5: [i1+1,j1] in Indices G and
A6: f/.len f = G*(i1+1,j1);
A7: i1+1 <= len G by A5,MATRIX_0:32;
A8: 1 <= j1 by A5,MATRIX_0:32;
A9: now
    assume j1-'1 < 1;
    then j1 <= 1 by NAT_1:14,NAT_D:36;
    then j1 = 1 by A8,XXREAL_0:1;
    then cell(G,i1+1,1-'1) meets C by A1,A4,A5,A6,A3,GOBRD13:37;
    then cell(G,i1+1,0) meets C by XREAL_1:232;
    hence contradiction by A7,JORDAN8:17;
  end;
A10: j1-'1 <= j1 by NAT_D:35;
  j1 <= width G by A5,MATRIX_0:32;
  then
A11: j1-'1 <= width G by A10,XXREAL_0:2;
  1 <= i1+1 by A5,MATRIX_0:32;
  hence thesis by A7,A11,A9,MATRIX_0:30;
end;
