
theorem Th14:
 for p be FinSequence of the carrier of F_Complex holds |.Sum p.| <= Sum|.p.|
proof
  set D = the carrier of F_Complex;
  defpred P[FinSequence of D] means |.Sum $1.| <= Sum|.$1.|;
A1: now
    let p be FinSequence of D;
    let x be Element of D;
    assume P[p];
    then
A2: |.Sum p.| + |.x.| <= Sum|.p.| + |.x.| by XREAL_1:6;
    Sum (p^<*x*>) = Sum p + x by FVSUM_1:71;
    then
A3: |.Sum (p^<*x*>).| <= |.Sum p.| + |.x.| by COMPLFLD:62;
    reconsider xx = |.x.| as Element of REAL by XREAL_0:def 1;
    Sum|.p.| + |.x.| = Sum|.p.| + Sum <*xx*> by FINSOP_1:11
      .= Sum|.p.| + Sum |.<*x*>.| by Th9
      .= Sum(|.p.|^|.<*x*>.|) by RVSUM_1:75
      .= Sum|.p^<*x*>.| by Th12;
    hence P[p^<*x*>] by A2,A3,XXREAL_0:2;
  end;
A4: P[<*>D] by Th8,COMPLFLD:57,RLVECT_1:43,RVSUM_1:72;
  thus for p be FinSequence of D holds P[p] from FINSEQ_2:sch 2(A4,A1 );
end;
