theorem Th91:
  abs(c*z) = |.c.|*(abs z)
proof
  now
    let j be Nat;
    reconsider w = j as Element of NAT by ORDINAL1:def 12;
    assume
A1: j in Seg n;
    then reconsider c9 = z.j, cc = (c*z).j as Element of COMPLEX by Th57;
    reconsider ac = (abs z).w as Real;
    thus (abs(c*z)).j = |.cc.| by A1,Th88
      .= |.c*c9.| by A1,Th81
      .= |.c.|*|.c9.| by COMPLEX1:65
      .= |.c.|*ac by A1,Th88
      .= (|.c.|*(abs z)).j by RVSUM_1:45;
  end;
  hence thesis by FINSEQ_2:119;
end;
