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

theorem Th23:
  for a,b being Real_Sequence st for i being Nat holds
  a.i <= b.i & a.i <= a.(i+1) & b.(i+1) <= b.i holds
  IntervalSequence(a,b) is non-ascending
  proof
    let a,b be Real_Sequence;
    assume
A1: for i being Nat holds a.i <= b.i & a.i <= a.(i+1) & b.(i+1) <= b.i;
    now
      let n,m be Nat;
      assume
A2:   n <= m;
      product <* [.a.m,b.m.] *> c= product <*[.a.n,b.n.]*>
      proof
        let x be object;
        assume x in product <*[.a.m,b.m.]*>;
        then consider f be Function such that
A3:     x = f and
A4:     dom f = dom <* [.a.m,b.m.] *> and
A5:     for i be object st i in dom <* [.a.m,b.m.] *> holds
        f.i in (<* [.a.m,b.m.] *>).i by CARD_3:def 5;
A6:     dom <* [.a.m,b.m.] *> = Seg 1 by FINSEQ_1:def 8;
        now
          thus x = f by A3;
          thus dom f = dom <* [.a.n,b.n.] *> by A6,A4,FINSEQ_1:def 8;
          hereby
            let i be object;
            assume
A7:         i in dom <*[.a.n,b.n.]*>; then
A8:         i in dom <*[.a.m,b.m.]*> by A6,FINSEQ_1:def 8;
            i in Seg 1 by A7,FINSEQ_1:def 8; then
A9:         i = 1 by TARSKI:def 1,FINSEQ_1:2; then
A10:        (<*[.a.m,b.m.]*>).i = [.a.m,b.m.];
A11:        (<*[.a.n,b.n.]*>).i = [.a.n,b.n.] by A9;
A12:        a is non-decreasing by A1;
            dom a = NAT by SEQ_1:1;
            then n in dom a & m in dom a by ORDINAL1:def 12; then
A13:        a.n <= a.m by A12,A2;
A14:        b is non-increasing by A1;
            dom b = NAT by SEQ_1:1;then
            n in dom b & m in dom b by ORDINAL1:def 12;
            then b.m <= b.n by A14,A2;
            then [.a.m,b.m.] c= [.a.n,b.n.] by A13,XXREAL_1:34;
            hence f.i in (<* [.a.n,b.n.] *>).i by A8,A5,A10,A11;
          end;
        end;
        hence thesis by CARD_3:def 5;
      end;
      then IntervalSequence(a,b).m c= product <*[.a.n,b.n.]*> by Def1;
      hence IntervalSequence(a,b).m c= IntervalSequence(a,b).n by Def1;
    end;
    hence thesis by PROB_1:def 4;
  end;
