reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem
  (r(#)s)*' = (r*')(#)(s*')
proof
  now
    let n be Element of NAT;
    thus (r(#)s)*'.n = ((r(#)s).n)*' by Def2
      .= (r*s.n)*' by VALUED_1:6
      .= (r*')*(s.n)*' by COMPLEX1:35
      .= (r*')*(s*'.n) by Def2
      .= ((r*')(#)(s*')).n by VALUED_1:6;
  end;
  hence thesis by FUNCT_2:63;
end;
