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 Th39:
  (r(#)seq)"=r"(#)seq"
proof
  now
    let n;
    thus (r(#)seq)".n=((r(#)seq).n)" by VALUED_1:10
      .=(r*(seq.n))" by VALUED_1:6
      .=r"*(seq.n)" by XCMPLX_1:204
      .=r"*seq".n by VALUED_1:10
      .=(r"(#)seq").n by VALUED_1:6;
  end;
  hence thesis by FUNCT_2:63;
end;
