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 e being Real st e > 0 ex S being Segmentation of C st diameter S < e
proof
  let e be Real;
  assume e > 0;
  then consider h being FinSequence of the carrier of TOP-REAL 2 such that
A1: h.1=W-min C and
A2: h is one-to-one and
A3: 8<=len h and
A4: rng h c= C and
A5: for i being Nat st 1<=i & i<len h holds LE h/.i,h/.(i+1),C and
A6: for i being Nat,W being Subset of Euclid 2
  st 1<=i & i<len h & W=Segment(h/.i,h/.(i+1),C) holds diameter(W)<e and
A7: for W being Subset of Euclid 2 st
  W=Segment(h/.len h,h/.1,C) holds diameter(W)<e and
A8: for i being Nat st 1<=i & i+1<len h holds
  Segment(h/.i,h/.(i+1),C)/\ Segment(h/.(i+1),h/.(i+2),C)={h/.(i+1)} and
A9: Segment(h/.len h,h/.1,C)/\ Segment(h/.1,h/.2,C)={h/.1} and
A10: Segment(h/.(len h-' 1),h/.len h,C)/\ Segment(h/.len h,h/.1,C)={h/.len h}
  and
A11: Segment(h/.(len h-'1),h/.len h,C) misses Segment(h/.1,h/.2,C) and
A12: for i,j being Nat st 1<=i & i < j & j < len h &
   i,j aren't_adjacent
  holds Segment(h/.i,h/.(i+1),C) misses Segment(h/.j,h/.(j+1),C) and
A13: for i being Nat st 1 < i & i+1 < len h holds
  Segment(h/.len h,h/.1,C) misses Segment(h/.i,h/.(i+1),C) by JORDAN7:20;
  h <> {} by A3,CARD_1:27;
  then 1 in dom h by FINSEQ_5:6;
  then h/.1 = W-min C by A1,PARTFUN1:def 6;
  then reconsider h as Segmentation of C
  by A2,A3,A4,A5,A8,A9,A10,A11,A12,A13,Def3;
  take h;
  diameter Segm(h,i) < e
  proof
A14: ex W being Subset of Euclid 2 st ( W = Segm(h,i))&(
    diameter Segm(h,i) = diameter W) by Def5;
    per cases;
    suppose
A15:  1<=i & i<len h;
      then Segm(h,i) = Segment(h/.i,h/.(i+1),C) by Def4;
      hence thesis by A6,A14,A15;
    end;
    suppose not(1<=i & i<len h);
      then Segm(h,i) = Segment(h/.len h,h/.1,C) by Def4;
      hence thesis by A7,A14;
    end;
  end;
  hence thesis by Th33;
end;
