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 Th8:
  rng seq c= dom h implies (g(#)h)/*seq = g(#)(h/*seq)
proof
  assume
A1: rng seq c= dom h;
  then
A2: rng seq c= dom (g(#)h) by VALUED_1:def 5;
  now
    let n be Element of NAT;
A3: seq.n in rng seq by VALUED_0:28;
    thus ((g(#)h)/*seq).n = (g(#)h)/.(seq.n) by A2,FUNCT_2:109
      .= g * (h/.(seq.n)) by A2,A3,CFUNCT_1:4
      .= g *((h/*seq).n) by A1,FUNCT_2:109
      .= (g(#)(h/*seq)).n by VALUED_1:6;
  end;
  hence thesis by FUNCT_2:63;
end;
