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

theorem
  (for n holds 0<=s1.n & s1.n<=s2.n) & s2 is summable implies s1 is
  summable & Sum(s1)<=Sum(s2)
proof
  assume that
A1: for n holds 0<=s1.n & s1.n<=s2.n and
A2: s2 is summable;
  for n holds 0<=n implies s1.n<=s2.n by A1;
  hence s1 is summable by A1,A2,Th19;
  then
A3: Partial_Sums(s1) is convergent;
  Partial_Sums(s2) is convergent & for n holds Partial_Sums(s1).n<=
  Partial_Sums(s2).n by A1,A2,Th14;
  hence thesis by A3,SEQ_2:18;
end;
