reserve n,k for Element of NAT;
reserve x,y,X for set;
reserve g,r,p for Real;
reserve S for RealNormSpace;
reserve rseq for Real_Sequence;
reserve seq,seq1 for sequence of S;
reserve x0 for Point of S;
reserve Y for Subset of S;

theorem Th9:
  for rseq be Real_Sequence for seq1, seq2 be sequence of S holds
  rseq(#)(seq1+seq2)=rseq(#)seq1+rseq(#)seq2
proof
  let rseq be Real_Sequence;
  let seq1, seq2 be sequence of S;
  now
    let n;
    thus (rseq(#)(seq1+seq2)).n =rseq.n*(seq1+seq2).n by Def2
      .=rseq.n*(seq1.n+seq2.n ) by NORMSP_1:def 2
      .=rseq.n*seq1.n+rseq.n*seq2.n by RLVECT_1:def 5
      .=(rseq(#)seq1).n+rseq.n*seq2.n by Def2
      .=(rseq(#)seq1).n+(rseq(#)seq2).n by Def2
      .=((rseq(#)seq1)+(rseq(#)seq2)).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
