
theorem
  for r be Real_Sequence st (for n be Nat holds 0 <= r.n)
  holds ( for n be Nat holds 0 <= (Partial_Sums r).n ) & ( for n be
  Nat holds r.n <= (Partial_Sums r).n ) & ( r is summable implies (
  for n be Nat holds (Partial_Sums r).n <= Sum r) &
   for n be Nat holds r.n <= Sum r ) by Lm1;
