reserve Z for open Subset of REAL;

theorem Th15:
  for x be Real holds x rExpSeq is absolutely_summable
proof
  let x be Real;
  for n be Nat holds |.x rExpSeq.n.| = (|.x.| rExpSeq).n
  proof
   let n be Nat;
    |.x rExpSeq.n.| = |.x |^ n / (n!).| by SIN_COS:def 5
      .= |.x |^ n.| / |.n!.| by COMPLEX1:67
      .= |.x |^ n.| / (n!) by ABSVALUE:def 1
      .= |.x.| |^ n / (n!) by Th1
      .= (|.x.| rExpSeq).n by SIN_COS:def 5;
    hence thesis;
  end;
  then |.x.| rExpSeq = |.x rExpSeq.| by SEQ_1:12;
  then |.x rExpSeq.| is summable by SIN_COS:45;
  hence thesis by SERIES_1:def 4;
end;
