reserve f for Function;
reserve n,k,n1 for Nat;
reserve r,p for Real;
reserve x,y,z for object;
reserve seq,seq1,seq2,seq3,seq9,seq19 for Real_Sequence;

theorem Th44:
  (r(#)seq)"=r"(#)seq"
proof
  now
    let n be Element of NAT;
    thus (r(#)seq)".n=((r(#)seq).n)" by VALUED_1:10
      .=(r*(seq.n))" by Th9
      .=r"*(seq.n)" by XCMPLX_1:204
      .=r"*seq".n by VALUED_1:10
      .=(r"(#)seq").n by Th9;
  end;
  hence thesis by FUNCT_2:63;
end;
