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 Th19:
  (seq(#)seq1) ^\k=(seq ^\k)(#)(seq1 ^\k)
proof
  now
    let n be Element of NAT;
    thus ((seq(#)seq1) ^\k).n=(seq(#)seq1).(n+k) by NAT_1:def 3
      .=seq.(n+k)*seq1.(n+k) by SEQ_1:8
      .=(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 SEQ_1:8;
  end;
  hence thesis by FUNCT_2:63;
end;
