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 Th16:
  r(#)(seq1+seq2)=r(#)seq1+r(#)seq2
proof
  now
    let n;
    thus (r(#)(seq1 + seq2)).n=r*(seq1+seq2).n by VALUED_1:6
      .=r*(seq1.n+seq2.n) by VALUED_1:1
      .=r*seq1.n+r*seq2.n
      .=(r(#)seq1).n+r*seq2.n by VALUED_1:6
      .=(r(#)seq1).n+(r(#)seq2).n by VALUED_1:6
      .=((r(#)seq1)+(r(#)seq2)).n by VALUED_1:1;
  end;
  hence thesis by FUNCT_2:63;
end;
