reserve f for Function;
reserve n,k,n1 for Element of NAT;
reserve r,p for Complex;
reserve x,y for set;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Complex_Sequence;

theorem Th46:
  |. seq .| " = |. seq" .|
proof
  now
    let n;
    thus (|.seq".|).n=|.seq".n.| by VALUED_1:18
      .=|.(seq.n)".| by VALUED_1:10
      .=|.1r/(seq.n).| by XCMPLX_1:215
      .=1/|.seq.n.| by COMPLEX1:48,67
      .=|.seq.n.|" by XCMPLX_1:215
      .=(|.seq.|.n)" by VALUED_1:18
      .=(|.seq.|)".n by VALUED_1:10;
  end;
  hence thesis by FUNCT_2:63;
end;
