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 Th30:
  for S being Segmentation of C
  holds Segm(S,len S) /\ Segm(S,len S -' 1) = {S/.len S}
proof
  let S be Segmentation of C;
A1: Segm(S,len S) = Segment(S/.len S,S/.1,C) by Def4;
A2: len S >= 8 by Def3;
  then len S >= 1+1 by XXREAL_0:2;
  then
A3: 1 <= len S -' 1 by NAT_D:55;
A4: len S -' 1 + 1 = len S by A2,XREAL_1:235,XXREAL_0:2;
  then len S -' 1 < len S by NAT_1:13;
  then Segm(S,len S -' 1) = Segment(S/.(len S -' 1),S/.len S,C) by A3,A4,Def4;
  hence thesis by A1,Def3;
end;
