reserve n for Nat;

theorem Th53:
  for G be Go-board for f be FinSequence of TOP-REAL 2 for p be
  Point of TOP-REAL 2 for k be Nat st 1 <= k & k < p..f & f
is_sequence_on G holds left_cell(f-:p,k,G) = left_cell(f,k,G) & right_cell(f-:p
  ,k,G) = right_cell(f,k,G)
proof
  let G be Go-board;
  let f be FinSequence of TOP-REAL 2;
  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;
A4: k+1 <= p..f by A2,NAT_1:13;
A5: f|(p..f) = f-:p by FINSEQ_5:def 1;
  per cases by TOPREAL8:4;
  suppose
    p in rng f;
    then p..f <= len f by FINSEQ_4:21;
    then k+1 <= len f by A4,XXREAL_0:2;
    hence thesis by A1,A3,A5,A4,GOBRD13:31;
  end;
  suppose
    p..f = 0;
    hence thesis by A2;
  end;
end;
