reserve F for RealNormSpace;
reserve G for RealNormSpace;
reserve X for set;
reserve x,x0,g,r,s,p for Real;
reserve n,m,k for Element of NAT;
reserve Y for Subset of REAL;
reserve Z for open Subset of REAL;
reserve s1,s3 for Real_Sequence;
reserve seq for sequence of G;
reserve f,f1,f2 for PartFunc of REAL,the carrier of F;
reserve h for 0-convergent non-zero Real_Sequence;
reserve c for constant Real_Sequence;

theorem Th2:
  (s1^\k)(#)(seq^\k)= (s1(#)seq) ^\k
  proof
    now let n;
      thus ((s1(#)seq) ^\k).n=(s1(#)seq).(n+k) by NAT_1:def 3
      .=s1.(n+k)*seq.(n+k) by NDIFF_1:def 2
      .=(s1 ^\k).n*seq.(n+k) by NAT_1:def 3
      .=(s1 ^\k).n*(seq ^\k).n by NAT_1:def 3
      .=((s1 ^\k)(#)(seq ^\k)).n by NDIFF_1:def 2;
    end;
    hence thesis by FUNCT_2:63;
  end;
