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,j1+1] in Indices Gauge(C,n) & f/.len f = Gauge(C,n)*(i1,j1+
  1) 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: left_cell(f,(len f)-'1,G) meets C and
A5: [i1,j1] in Indices G and
A6: f/.((len f) -'1) = G*(i1,j1) and
A7: [i1,j1+1] in Indices G and
A8: f/.len f = G*(i1,j1+1);
A9: 1 <= i1 by A7,MATRIX_0:32;
A10: len G = width G & j1 <= width G by A5,JORDAN8:def 1,MATRIX_0:32;
A11: 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,j1) meets C by A1,A4,A5,A6,A7,A8,A3,GOBRD13:21;
    then cell(G,0,j1) meets C by XREAL_1:232;
    hence contradiction by A10,JORDAN8:18;
  end;
A12: i1-'1 <= i1 by NAT_D:35;
  i1 <= len G by A5,MATRIX_0:32;
  then
A13: i1-'1 <= len G by A12,XXREAL_0:2;
  1 <= j1+1 & j1+1 <= width G by A7,MATRIX_0:32;
  hence thesis by A13,A11,MATRIX_0:30;
end;
