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 Th18:
  (seq") ^\k=(seq ^\k)"
proof
  now
    let n be Element of NAT;
    thus ((seq") ^\k).n=(seq").(n+k) by NAT_1:def 3
      .=(seq.(n+k))" by VALUED_1:10
      .=((seq ^\k).n)" by NAT_1:def 3
      .=((seq ^\k)").n by VALUED_1:10;
  end;
  hence thesis by FUNCT_2:63;
end;
