 reserve n for Nat;
 reserve s1 for sequence of Euclid n,
         s2 for sequence of REAL-NS n;

theorem Th28:
  for a,b being Real_Sequence, S being SetSequence of Euclid 1 st
  a.0 <= b.0 & S = IntervalSequence(a,b) & (for i being Nat holds
  ((a.(i+1) = a.i & b.(i+1) = (a.i+b.i)/2 ) or
  (a.(i+1) = (a.i + b.i)/2 & b.(i+1) = b.i)))
  holds for i being Nat holds
  a.i <= b.i & a.i <= a.(i+1) & b.(i+1) <= b.i &
  (diameter S).i = b.i - a.i
  proof
    let a,b be Real_Sequence, S be SetSequence of Euclid 1;
    assume that
A1: a.0 <= b.0 and
A2: S = IntervalSequence(a,b) and
A3: for i being Nat holds
    (a.(i+1) = a.i & b.(i+1) = (a.i+b.i)/2 ) or
    (a.(i+1) = (a.i + b.i)/2 & b.(i+1) = b.i);
A4: for i being Nat holds
    a.i <= b.i & a.i <= a.(i+1) & b.(i+1) <= b.i
    proof
      let i be Nat;
        thus
A6:     a.i <= b.i by Th27,A1,A3;
        thus a.i <= a.(i+1)
        proof
          a.(i+1) = a.i or a.(i+1) = (a.i+b.i)/2 by A3;
          hence thesis by A6,Th8;
        end;
        thus b.(i+1) <= b.i
        proof
          b.(i+1) = b.i or b.(i+1) = (a.i+b.i)/2 by A3;
          hence thesis by A6,Th8;
        end;
    end;
    now
      let i be Nat;
A7:   IntervalSequence(a,b) is SetSequence of Euclid 1 by Th17;
      reconsider IntervalSequence1 = IntervalSequence(a,b).i as
        Subset of Euclid 1 by ORDINAL1:def 12,A7,FUNCT_2:5;
      S.i = product <* [.a.i,b.i.] *> by A2,Def1;
      then reconsider IntervalSequence2 = product <* [.a.i,b.i.] *> as
        Subset of Euclid 1;
      diameter(S.i) = diameter IntervalSequence2 by A2,Def1
                   .= b.i - a.i by A4,Th25;
      hence (diameter S).i = b.i - a.i by COMPL_SP:def 2;
    end;
    hence thesis by A4;
  end;
