reserve n for Nat;

theorem Th52:
  for G be Go-board for f be S-Sequence_in_R2 for p be Point of
TOP-REAL 2 for k be Nat st 1 <= k & k < p..f & f is_sequence_on G &
  p in rng f holds left_cell(R_Cut(f,p),k,G) = left_cell(f,k,G) & right_cell(
  R_Cut(f,p),k,G) = right_cell(f,k,G)
proof
  let G be Go-board;
  let f be S-Sequence_in_R2;
  let p be Point of TOP-REAL 2;
  let k be Nat;
  assume that
A1: 1 <= k and
A2: k < p..f and
A3: f is_sequence_on G and
A4: p in rng f;
A5: f|(p..f) = mid(f,1,p..f) by A1,A2,FINSEQ_6:116,XXREAL_0:2
    .= R_Cut(f,p) by A4,JORDAN1G:49;
A6: k+1 <= p..f by A2,NAT_1:13;
  p..f <= len f by A4,FINSEQ_4:21;
  then k+1 <= len f by A6,XXREAL_0:2;
  hence thesis by A1,A3,A5,A6,GOBRD13:31;
end;
