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
  for rseq1,rseq2 be Real_Sequence holds (rseq1+rseq2)(#)seq=rseq1(#)seq
  +rseq2(#)seq
proof
  let rseq1,rseq2 be Real_Sequence;
  now
    let n;
    thus ((rseq1+rseq2)(#)seq).n =(rseq1+rseq2).n*seq.n by Def2
      .=(rseq1.n+rseq2.n)*seq.n by SEQ_1:7
      .=rseq1.n*seq.n+rseq2.n*seq.n by RLVECT_1:def 6
      .=(rseq1(#)seq).n+rseq2.n*seq.n by Def2
      .=(rseq1(#)seq).n+(rseq2(#)seq).n by Def2
      .=((rseq1(#)seq)+(rseq2(#)seq)).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
