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

theorem Th27:
  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 for i being Nat holds a.i <= b.i
  proof
    let a,b be Real_Sequence;
    assume that
A1: a.0 <= b.0 and
A2: for i be 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));
    defpred P[object] means ex i be Nat st $1 = i & a.i <= b.i;
A3: P[0] by A1;
A4: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P[k];
      then consider i be Nat such that
      k = i and
A5:   a.k <= b.k;
      ((a.(k+1) = a.k & b.(k+1) = (a.k+b.k)/2 ) or
      (a.(k+1) = (a.k + b.k)/2 & b.(k+1) = b.k)) by A2;
      hence thesis by A5,Th8;
    end;
A6: for k be Nat holds P[k] from NAT_1:sch 2(A3,A4);
    let k be Nat;
    ex i be Nat st i = k & a.i <= b.i by A6;
    hence a.k <= b.k;
  end;
