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
  |.r(#)seq.|=|.r.|(#)|.seq.|
proof
  now
    let n;
    thus |.r(#)seq.|.n=|.(r(#)seq).n.| by VALUED_1:18
      .=|.r*(seq.n).| by VALUED_1:6
      .=|.r.|*|.seq.n.| by COMPLEX1:65
      .=|.r.|*(|.seq.|).n by VALUED_1:18
      .=(|.r.|(#)|.seq.|).n by VALUED_1:6;
  end;
  hence thesis by FUNCT_2:63;
end;
