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

theorem Th46:
  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;
    case
      i1<i2;
      hence thesis by A1,Th44;
    end;
    case
      i1>=i2;
      then i1>i2 by A2,XXREAL_0:1;
      hence thesis by A1,Lm1;
    end;
  end;
  hence thesis;
end;
