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 holds S-Gap S > 0
proof
  let S be Segmentation of C;
  consider F being finite non empty Subset of REAL such that
A1: F = { dist_min(Segm(S,i),Segm(S,j)):
  1<=i & i<j & j<=len S & Segm(S,i) misses Segm(S,j)} and
A2: S-Gap S = min F by Th35;
  S-Gap S in F by A2,XXREAL_2:def 7;
  then ex i,j st S-Gap S = dist_min(Segm(S,i),Segm(S,j)) &
  1<=i & i<j & j<=len S & Segm(S,i) misses Segm(S,j) by A1;
  hence thesis by Th7;
end;
