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 Th23:
  for f being standard special_circular_sequence st 1 <= k & k+1
  <= len f holds right_cell(f,k,GoB f) = right_cell(f,k)
proof
  let f be standard special_circular_sequence such that
A1: 1 <= k and
A2: k+1 <= len f;
  set G = GoB f;
A3: f is_sequence_on GoB f by GOBOARD5:def 5;
  for i1,j1,i2,j2 being Nat st [i1,j1] in Indices G & [i2,j2]
  in Indices G & f/.k = G*(i1,j1) & f/.(k+1) = G*(i2,j2) holds i1 = i2 & j1+1 =
j2 & right_cell(f,k) = cell(G,i1,j1) or i1+1 = i2 & j1 = j2 & right_cell(f,k) =
cell(G,i1,j1-'1) or i1 = i2+1 & j1 = j2 & right_cell(f,k) = cell(G,i2,j2) or i1
  = i2 & j1 = j2+1 & right_cell(f,k) = cell(G,i1-'1,j2)
  proof
    1 <= k+1 by NAT_1:11;
    then
A4: k+1 in dom f by A2,FINSEQ_3:25;
    set IT = right_cell(f,k);
    let i1,j1,i2,j2 be Nat such that
A5: [i1,j1] in Indices G & [i2,j2] in Indices G & f/.k = G*(i1,j1) & f
    /.( k+1) = G*(i2,j2);
    k < len f by A2,NAT_1:13;
    then k in dom f by A1,FINSEQ_3:25;
    then |.i1-i2.|+|.j1-j2.| = 1 by A3,A5,A4,GOBOARD1:def 9;
    then
A6: |.i1-i2.|=1 & j1=j2 or |.j1-j2.|=1 & i1=i2 by SEQM_3:42;
    right_cell(f,k) = right_cell(f,k);
    then
A7: i1 = i2 & j1+1 = j2 & IT = cell(GoB f,i1,j1) or i1+1 = i2 & j1 = j2 &
IT = cell(GoB f,i1,j1-'1) or i1 = i2+1 & j1 = j2 & IT = cell(GoB f,i2,j2) or i1
    = i2 & j1 = j2+1 & IT = cell(GoB f,i1-'1,j2) by A1,A2,A5,GOBOARD5:def 6;
    per cases by A6,SEQM_3:41;
    case
      i1 = i2 & j1+1 = j2;
      hence thesis by A7;
    end;
    case
      i1+1 = i2 & j1 = j2;
      hence thesis by A7;
    end;
    case
      i1 = i2+1 & j1 = j2;
      hence thesis by A7;
    end;
    case
      i1 = i2 & j1 = j2+1;
      hence thesis by A7;
    end;
  end;
  hence thesis by A1,A2,A3,GOBRD13:def 2;
end;
