reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem Th3:
  (g(#)seq)*Ns = g(#)(seq*Ns)
proof
  now
    let n be Element of NAT;
    thus ((g(#)seq)*Ns).n = (g(#)seq).(Ns.n) by FUNCT_2:15
      .= g*(seq.(Ns.n)) by VALUED_1:6
      .= g*((seq*Ns).n) by FUNCT_2:15
      .= (g(#)(seq*Ns)).n by VALUED_1:6;
  end;
  hence thesis by FUNCT_2:63;
end;
