reserve i, i1, i2, j, k for Nat,
  r, s for Real;
reserve D for non empty set,
  f1 for FinSequence of D;

theorem Th48:
  for f being non constant standard special_circular_sequence,
      g being FinSequence of TOP-REAL 2,i1,i2 being Nat st
      g is_a_part_of f,i1,i2 & i1<>i2 holds
      L~g is_S-P_arc_joining f/.i1,f/.i2
proof
  let f be non constant standard special_circular_sequence, g be FinSequence
  of TOP-REAL 2,i1,i2 be Nat;
  assume that
A1: g is_a_part_of f,i1,i2 and
A2: i1<>i2;
  now
    per cases by A1;
    case
      g is_a_part>_of f,i1,i2;
      hence thesis by A2,Th46;
    end;
    case
      g is_a_part<_of f,i1,i2;
      hence thesis by A2,Th47;
    end;
  end;
  hence thesis;
end;
