theorem Th21:
  for f being real-valued FinSequence holds |.Sum f.| <= Sum abs f
  proof
    let f be real-valued FinSequence;
    defpred P[set] means ex F being real-valued FinSequence st F = $1 &
    |.Sum F.| <= Sum abs F;
A1: P[<*>REAL]
    proof
      take <*>REAL;
      thus thesis by ABSVALUE:2,RVSUM_1:72;
    end;
A2: for p being FinSequence of REAL, x being Element of REAL st P[p] holds
    P[p^<*x*>]
    proof
      let p be FinSequence of REAL, x be Element of REAL;
      given F being real-valued FinSequence such that
A3:   F = p and
A4:   |.Sum F.| <= Sum abs F;
A5:   |.Sum F.| + |.x.| <= Sum abs F + |.x.| by A4,XREAL_1:6;
      take G = p^<*x*>;
A6:   abs <*x*> = <*|.x.|*> by JORDAN2B:19;
A7:   Sum <*|.x.|*> = |.x.| by RVSUM_1:73;
      abs G = (abs p)^abs<*x*> by TOPREAL7:11;
      then
A8:   Sum abs G = Sum abs p + |.x.| by A6,A7,RVSUM_1:75;
      Sum G = Sum p + x by RVSUM_1:74;
      then |.Sum G.| <= |.Sum p.| + |.x.| by COMPLEX1:56;
      hence thesis by A8,A3,A5,XXREAL_0:2;
    end;
A9: for p being FinSequence of REAL holds P[p] from FINSEQ_2:sch 2(A1,A2);
    f is FinSequence of REAL by RVSUM_1:145;
    then P[f] by A9;
    hence thesis;
  end;
