reserve i,j,k,n for Nat;

theorem
  for f being non constant standard special_circular_sequence, p being
Point of TOP-REAL 2 st p in rng f holds left_cell(f,p..f) = left_cell(Rotate(f,
  p),1)
proof
  set n = 1;
  let f be non constant standard special_circular_sequence, p be Point of
  TOP-REAL 2 such that
A1: p in rng f;
  set k = p..f;
  len f > 1 by GOBOARD7:34,XXREAL_0:2;
  then k < len f by A1,SPRECT_5:7;
  then
A2: k+1 <= len f by NAT_1:13;
A3: 1 <= k by A1,FINSEQ_4:21;
A4: for i1,j1,i2,j2 being Nat st [i1,j1] in Indices GoB Rotate(f,
p) & [i2,j2] in Indices GoB Rotate(f,p) & Rotate(f,p)/.n = (GoB Rotate(f,p))*(
i1,j1) & Rotate(f,p)/.(n+1) = (GoB Rotate(f,p))*(i2,j2) holds i1 = i2 & j1+1 =
  j2 & left_cell(f,k) = cell(GoB Rotate(f,p),i1-'1,j1) or i1+1 = i2 & j1 = j2 &
left_cell(f,k) = cell(GoB Rotate(f,p),i1,j1) or i1 = i2+1 & j1 = j2 & left_cell
(f,k) = cell(GoB Rotate(f,p),i2,j2-'1) or i1 = i2 & j1 = j2+1 & left_cell(f,k)
  = cell(GoB Rotate(f,p),i1,j2)
  proof
    Rotate(f,p)/.(1 + 1 + k -' p..f) = Rotate(f,p)/.(n+1) by NAT_D:34;
    then
A5: Rotate(f,p)/.(k+1 + 1 -' p..f) = Rotate(f,p)/.(n+1);
A6: left_cell(f,k) = left_cell(f,k );
    let i1,j1,i2,j2 be Nat such that
A7: [i1,j1] in Indices GoB Rotate(f,p) & [i2,j2] in Indices GoB Rotate
    (f,p) and
A8: Rotate(f,p)/.n = (GoB Rotate(f,p))*(i1,j1) and
A9: Rotate(f,p)/.(n+1) = (GoB Rotate(f,p))*(i2,j2);
A10: GoB Rotate(f,p) = GoB f by REVROT_1:28;
    len Rotate(f,p) = len f by FINSEQ_6:179;
    then Rotate(f,p)/.(len f) = Rotate(f,p)/.1 by FINSEQ_6:def 1;
    then Rotate(f,p)/.(k + len f -' p..f) = Rotate(f,p)/.1 by NAT_D:34;
    then
A11: f/.k = (GoB f)*(i1,j1) by A1,A3,A8,A10,FINSEQ_6:183;
    k < k+1 by NAT_1:13;
    then
A12: f/.(k+1) = (GoB f)*(i2,j2) by A1,A2,A9,A10,A5,FINSEQ_6:175;
    then
A13: i1 = i2 & j1+1 = j2 & left_cell(f,k) = cell(GoB f,i1-'1,j1) or i1+1 =
    i2 & j1 = j2 & left_cell(f,k) = cell(GoB f,i1,j1) or i1 = i2+1 & j1 = j2 &
left_cell(f,k) = cell(GoB f,i2,j2-'1) or i1 = i2 & j1 = j2+1 & left_cell(f,k) =
    cell(GoB f,i1,j2) by A3,A2,A7,A10,A11,A6,GOBOARD5:def 7;
    per cases by A3,A2,A7,A10,A11,A12,A6,GOBOARD5:def 7;
    case
      i1 = i2 & j1+1 = j2;
      hence thesis by A13,REVROT_1:28;
    end;
    case
      i1+1 = i2 & j1 = j2;
      hence thesis by A13,REVROT_1:28;
    end;
    case
      i1 = i2+1 & j1 = j2;
      hence thesis by A13,REVROT_1:28;
    end;
    case
      i1 = i2 & j1 = j2+1;
      hence thesis by A13,REVROT_1:28;
    end;
  end;
  n+1 <= len Rotate(f,p) by GOBOARD7:34,XXREAL_0:2;
  hence thesis by A4,GOBOARD5:def 7;
end;
