reserve n for Nat;

theorem Th51:
  for G be Go-board for f,g be FinSequence of TOP-REAL 2 for k be
  Nat holds 1 <= k & k < len f & f^'g is_sequence_on G implies
left_cell(f^'g,k,G) = left_cell(f,k,G) & right_cell(f^'g,k,G) = right_cell(f,k,
  G)
proof
  let G be Go-board;
  let f,g be FinSequence of TOP-REAL 2;
  let k be Nat;
  assume that
A1: 1 <= k and
A2: k < len f and
A3: f^'g is_sequence_on G;
A4: k+1 <= len f by A2,NAT_1:13;
A5: (f^'g)|len f = f by Th50;
  len f <= len (f^'g) by TOPREAL8:7;
  then k+1 <= len (f^'g) by A4,XXREAL_0:2;
  hence thesis by A1,A3,A5,A4,GOBRD13:31;
end;
