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 Th25:
  for S being Segmentation of C for j st 1<j & j<len S -' 1
  holds Segm(S,len S) misses Segm(S,j)
proof
  let S be Segmentation of C;
  let j such that
A1: 1<j and
A2: j<len S -' 1;
A3: Segm(S,len S) = Segment(S/.len S,S/.1,C) by Def4;
A4: j+1 < len S by A2,NAT_D:53;
  j < len S by A2,NAT_D:44;
  then Segm(S,j) = Segment(S/.j,S/.(j+1),C) by A1,Def4;
  hence thesis by A1,A3,A4,Def3;
end;
