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 Th10:
  for rseq be Real_Sequence holds r*(rseq(#)seq) =rseq(#)(r*seq)
proof
  let rseq be Real_Sequence;
  now
    let n;
    thus (r*(rseq(#)seq)).n =r*(rseq(#)seq).n by NORMSP_1:def 5
      .=r*(rseq.n*seq.n) by Def2
      .=(r*rseq.n)*seq.n by RLVECT_1:def 7
      .=rseq.n*(r*seq.n) by RLVECT_1:def 7
      .=rseq.n*(r*seq).n by NORMSP_1:def 5
      .=(rseq(#)(r*seq)).n by Def2;
  end;
  hence thesis by FUNCT_2:63;
end;
