reserve n,m,k for Nat;
reserve r,r1 for Real;
reserve f,seq,seq1 for Real_Sequence;
reserve x,y for set;
reserve e1,e2 for ExtReal;
reserve Nseq for increasing sequence of NAT;

theorem
  (r(#)seq) ^\k=r(#)(seq ^\k)
proof
  now
    let n be Element of NAT;
    thus ((r(#)seq) ^\k).n=(r(#)seq).(n+k) by NAT_1:def 3
      .=r*(seq.(n+k)) by SEQ_1:9
      .=r*((seq ^\k).n) by NAT_1:def 3
      .=(r(#)(seq ^\k)).n by SEQ_1:9;
  end;
  hence thesis by FUNCT_2:63;
end;
