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 left_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;
A4: len G = width G by JORDAN8:def 1;
  let i1,j1 being Nat such that
A5: left_cell(f,(len f)-'1,G) meets C and
A6: [i1,j1] in Indices G and
A7: f/.((len f) -'1) = G*(i1,j1) and
A8: [i1+1,j1] in Indices G and
A9: f/.len f = G*(i1+1,j1);
A10: j1 <= width G by A8,MATRIX_0:32;
A11: i1 <= len G by A6,MATRIX_0:32;
A12: now
    assume j1+1 > len G;
    then
A13: (len G)+1 <= j1+1 by NAT_1:13;
    j1+1 <= (len G)+1 by A4,A10,XREAL_1:6;
    then j1+1 = (len G)+1 by A13,XXREAL_0:1;
    then cell(G,i1,len G) meets C by A1,A5,A6,A7,A8,A9,A3,GOBRD13:23;
    hence contradiction by A11,JORDAN8:15;
  end;
A14: 1 <= j1+1 by NAT_1:11;
  1 <= i1+1 & i1+1 <= len G by A8,MATRIX_0:32;
  hence thesis by A4,A14,A12,MATRIX_0:30;
end;
