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 Th15:
  (seq1 + seq2) (#) seq3 = seq1 (#) seq3 + seq2 (#) seq3
proof
  now
    let n be Element of NAT;
    thus ((seq1+seq2)(#)seq3).n=(seq1+seq2).n*seq3.n by Th8
      .=(seq1.n+seq2.n)*seq3.n by Th7
      .=seq1.n*seq3.n+seq2.n*seq3.n
      .=(seq1(#)seq3).n+seq2.n*seq3.n by Th8
      .=(seq1(#)seq3).n+(seq2(#)seq3).n by Th8
      .=((seq1(#)seq3)+(seq2(#)seq3)).n by Th7;
  end;
  hence thesis by FUNCT_2:63;
end;
