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

theorem Th32:
  for a,b being Real_Sequence st a.0 <= b.0 &
  (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 meet IntervalSequence(a,b) is non empty
  proof
    let a,b be Real_Sequence;
    assume that
A1: a.0 <= b.0 and
A2: 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));
    IntervalSequence(a,b) is SetSequence of Euclid 1 by Th17; then
A3: for i being Nat holds
    a.i <= b.i & a.i <= a.(i+1) & b.(i+1) <= b.i by A1,A2,Th28;
    reconsider S = IntervalSequence(a,b) as non-empty pointwise_bounded closed
    SetSequence of (Euclid 1) by A3,Th22;
A4: S is non-ascending by A3,Th23;
    lim diameter S = 0 by A1,A2,Th31;
    then meet S is non empty by A4,COMPL_SP:10;
    hence thesis;
  end;
