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

theorem Th9:
  Partial_Sums(r(#)s) = r(#)Partial_Sums(s)
proof
A1: now
    let n;
    thus (r(#)Partial_Sums(s)).(n+1) = r*Partial_Sums(s).(n+1) by SEQ_1:9
      .= r*(Partial_Sums(s).n + s.(n+1)) by Def1
      .= r*Partial_Sums(s).n + r*s.(n+1)
      .= (r(#)Partial_Sums(s)).n + r*s.(n+1) by SEQ_1:9
      .= (r(#)Partial_Sums(s)).n + (r(#)s).(n+1) by SEQ_1:9;
  end;
  (r(#)Partial_Sums(s)).0 = r*Partial_Sums(s).0 by SEQ_1:9
    .= r*s.0 by Def1
    .= (r(#)s).0 by SEQ_1:9;
  hence thesis by A1,Def1;
end;
