
theorem
  for G be Go-board for f be FinSequence of TOP-REAL 2
  for k be Nat holds 1 <= k & k+1 <= len f & f is_sequence_on G
   implies left_cell(f,k,G) is closed
proof
  let G be Go-board;
  let f be FinSequence of TOP-REAL 2;
  let k be Nat;
  assume that
A1: 1 <= k and
A2: k+1 <= len f and
A3: f is_sequence_on G;
  consider i1,j1,i2,j2 be Nat such that
A4: [i1,j1] in Indices G and
A5: f/.k = G*(i1,j1) and
A6: [i2,j2] in Indices G and
A7: f/.(k+1) = G*(i2,j2) and
A8: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or
  i1 = i2 & j1 = j2+1 by A1,A2,A3,JORDAN8:3;
  i1 = i2 & j1+1 = j2 & left_cell(f,k,G) = cell(G,i1-'1,j1) or i1+1 = i2 &
j1 = j2 & left_cell(f,k,G) = cell(G,i1,j1) or i1 = i2+1 & j1 = j2 & left_cell(f
,k,G) = cell(G,i2,j2-'1) or i1 = i2 & j1 = j2+1 & left_cell(f,k,G) = cell(G,i1,
  j2) by A1,A2,A3,A4,A5,A6,A7,A8,GOBRD13:def 3;
  hence thesis by GOBRD11:33;
end;
