reserve C for Simple_closed_curve,
  p,q,p1 for Point of TOP-REAL 2,
  i,j,k,n for Nat,
  r,s for Real;

theorem Th31:
  for S being Segmentation of C holds Segm(S,len S) meets Segm(S,len S -' 1)
proof
  let S be Segmentation of C;
  Segm(S,len S) /\ Segm(S,len S -' 1) = {S/.len S} by Th30;
  hence thesis;
end;
