reserve i, i1, i2, j, k for Nat,
  r, s for Real;
reserve D for non empty set,
  f1 for FinSequence of D;
reserve g, g1, g2 for FinSequence of TOP-REAL 2;

theorem Th52:
  for f being non constant standard special_circular_sequence,
      g1,g2 st g1 is_a_part>_of f,i1,i2 & g2 is_a_part>_of f,i1,i2 holds
        g1=g2
proof
  let f be non constant standard special_circular_sequence, g1,g2;
  assume that
A1: g1 is_a_part>_of f,i1,i2 and
A2: g2 is_a_part>_of f,i1,i2;
  now
    per cases;
    case
A3:   i1<=i2;
      then g1=mid(f,i1,i2) by A1,Th25;
      hence thesis by A2,A3,Th25;
    end;
    case
A4:   i1>i2;
      then g1=mid(f,i1,len f-'1)^mid(f,1,i2) by A1,Th26;
      hence thesis by A2,A4,Th26;
    end;
  end;
  hence thesis;
end;
