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

theorem
  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;
  reconsider P=L~g as Subset of TOP-REAL 2;
  reconsider p2=f/.i2,p1=f/.i1 as Point of TOP-REAL 2;
  L~Rev g is_S-P_arc_joining f/.i2,f/.i1 by A1,A2,Th30,Th44;
  then P is_S-P_arc_joining p2,p1 by SPPOL_2:22;
  hence thesis by SPPOL_2:49;
end;
