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

theorem Th4:
  s is summable implies s is convergent & lim s = 0
proof
  assume s is summable;
  then
A1: Partial_Sums(s) is convergent;
  now
    let n;
    (Partial_Sums(s)).(n+1) = (Partial_Sums(s)).n + s.(n+1) by Def1;
    then (Partial_Sums(s)).(n+1) = (Partial_Sums(s)).n + (s ^\1).n by
NAT_1:def 3;
    hence (Partial_Sums(s) ^\1).n = (Partial_Sums(s)).n + (s ^\1).n by
NAT_1:def 3;
  end;
  then
A2: (Partial_Sums(s) ^\1) = Partial_Sums(s) + s ^\1 by SEQ_1:7;
  now
    let n be Element of NAT;
    thus (s ^\1 + (Partial_Sums(s) - Partial_Sums(s))).n = (s ^\1).n + (
    Partial_Sums(s) +- Partial_Sums(s)).n by SEQ_1:7
      .= (s ^\1).n + ((Partial_Sums(s)).n + (-Partial_Sums(s)).n) by SEQ_1:7
      .= (s ^\1).n + ((Partial_Sums(s)).n +- (Partial_Sums(s)).n) by SEQ_1:10
      .= (s ^\1).n;
  end;
  then s ^\1 + (Partial_Sums(s) - Partial_Sums(s)) = s ^\1;
  then
A3: s ^\1 = Partial_Sums(s) ^\1 - Partial_Sums(s) by A2,SEQ_1:31;
  then
A4: s ^\1 qua Real_Sequence is convergent by A1;
  hence s is convergent by SEQ_4:21;
  lim(Partial_Sums(s) ^\1) = lim(Partial_Sums(s)) by A1,SEQ_4:20;
  then
  lim(Partial_Sums(s) ^\1 - Partial_Sums(s)) = lim(Partial_Sums(s)) - lim(
  Partial_Sums(s)) by A1,SEQ_2:12
    .= 0;
  hence thesis by A3,A4,SEQ_4:22;
end;
