reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th19:
  (for n holds 0<=s2.n) & s1 is summable & (ex m st for n st m<=n
  holds s2.n<=s1.n) implies s2 is summable
proof
  assume that
A1: for n holds 0<=s2.n and
A2: s1 is summable;
  given m such that
A3: for n st m<=n holds s2.n<=s1.n;
  s1^\m is summable by A2,Th12;
  then Partial_Sums(s1^\m) is bounded_above;
  then consider r being Real such that
A4: for n holds Partial_Sums(s1^\m).n<r by SEQ_2:def 3;
A5: now
    let n;
    s2.(n+m)<=s1.(n+m) by A3,NAT_1:12;
    then (s2^\m).n<=s1.(n+m) by NAT_1:def 3;
    hence (s2^\m).n<=(s1^\m).n by NAT_1:def 3;
  end;
  now
    let n;
    Partial_Sums(s2^\m).n <= Partial_Sums(s1^\m).n by A5,Th14;
    hence Partial_Sums(s2^\m).n<r by A4,XXREAL_0:2;
  end;
  then
A6: Partial_Sums(s2^\m) is bounded_above by SEQ_2:def 3;
  now
    let n;
    (s2^\m).n = s2.(n+m) by NAT_1:def 3;
    hence 0<=(s2^\m).n by A1;
  end;
  then s2^\m is summable by A6,Th17;
  hence thesis by Th13;
end;
