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 Th9:
  (seq1+seq2)(#)seq3=seq1(#)seq3+seq2(#)seq3
proof
  now
    let n;
    thus ((seq1+seq2)(#)seq3).n=(seq1+seq2).n*seq3.n by VALUED_1:5
      .=(seq1.n+seq2.n)*seq3.n by VALUED_1:1
      .=seq1.n*seq3.n+seq2.n*seq3.n
      .=(seq1(#)seq3).n+seq2.n*seq3.n by VALUED_1:5
      .=(seq1(#)seq3).n+(seq2(#)seq3).n by VALUED_1:5
      .=((seq1(#)seq3)+(seq2(#)seq3)).n by VALUED_1:1;
  end;
  hence thesis by FUNCT_2:63;
end;
