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 Th15:
 for k being Nat
  holds (seq+seq1) ^\k=(seq ^\k) +(seq1 ^\k)
proof let k be Nat;
  now
    let n;
    thus ((seq+seq1) ^\k).n=(seq+seq1).(n+k) by NAT_1:def 3
      .=seq.(n+k) + seq1.(n+k) by NORMSP_1:def 2
      .=(seq ^\k).n +seq1.(n+k) by NAT_1:def 3
      .=(seq ^\k).n +(seq1 ^\k).n by NAT_1:def 3
      .=((seq ^\k) +(seq1 ^\k)).n by NORMSP_1:def 2;
  end;
  hence thesis by FUNCT_2:63;
end;
