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 Th45:
  |.seq(#)seq9.|=|.seq.|(#)|.seq9.|
proof
  now
    let n;
    thus (|.seq(#)seq9.|).n=|.((seq(#)seq9).n).| by VALUED_1:18
      .=|.(seq.n)*(seq9.n).| by VALUED_1:5
      .=|.(seq.n).|*|.(seq9.n).| by COMPLEX1:65
      .=((|.seq.|).n)*|.(seq9.n).| by VALUED_1:18
      .=((|.seq.|).n)*(|.seq9.|).n by VALUED_1:18
      .=(|.seq.|(#)|.seq9.|).n by VALUED_1:5;
  end;
  hence thesis by FUNCT_2:63;
