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
  for S being Segmentation of C st i in dom S holds Segm(S,i) c= C
proof
  let S be Segmentation of C;
  assume i in dom S;
  then
A1: 1 <= i by FINSEQ_3:25;
  i < len S or i >= len S;
  then Segm(S,i) = Segment(S/.i,S/.(i+1),C) or
  Segm(S,i) = Segment(S/.len S,S/.1,C) by A1,Def4;
  hence thesis by JORDAN16:6;
end;
