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

theorem Th14:
  (for n holds s1.n<=s2.n) implies for n holds Partial_Sums(s1).n
  <=Partial_Sums(s2).n
proof
  defpred X[Nat] means
Partial_Sums(s1).$1 <= Partial_Sums(s2).$1;
  assume
A1: for n holds s1.n<=s2.n;
A2: for n st X[n] holds X[n+1]
  proof
    let n such that
A3: Partial_Sums(s1).n <= Partial_Sums(s2).n;
A4: s1.(n+1)<=s2.(n+1) by A1;
    Partial_Sums(s1).(n+1) = Partial_Sums(s1).n + s1.(n+1) & Partial_Sums(
    s2).(n +1) = Partial_Sums(s2).n + s2.(n+1) by Def1;
    hence thesis by A3,A4,XREAL_1:7;
  end;
  Partial_Sums(s2).0 = s2.0 & Partial_Sums(s1).0 = s1.0 by Def1;
  then
A5: X[0] by A1;
  thus for n holds X[n] from NAT_1:sch 2(A5,A2);
end;
