reserve L for Abelian left_zeroed add-associative associative right_zeroed
              right_complementable distributive non empty doubleLoopStr;
reserve a,b,c for Element of L;
reserve R for non degenerated comRing;
reserve n,m,i,j,k for Nat;
 reserve D for Function of R, R;
 reserve x,y,z for Element of R;
reserve D for Derivation of R;
reserve s for FinSequence of the carrier of R;
reserve h for Function of R,R;
 reserve R for domRing;
 reserve f,g for Element of the carrier of Polynom-Ring R;
reserve a for Element of R;

theorem Th28:
    for f,g being Element of the carrier of Polynom-Ring R, a be Element of R
    st f = a|R holds (Der1(R)).(f*g) = (a|R)*'((Der1(R)).g)
    proof
      let f,g be Element of the carrier of Polynom-Ring R,
      a be Element of R;
      assume
A1:   f = a|R;
      reconsider f1 = f, g1 = g as sequence of R;
      reconsider f9 = f, g9 = g as Polynomial of R;
      reconsider fg0 = f * g as Element of Polynom-Ring R;
      reconsider fg9 = f9 *' g9 as Element of Polynom-Ring R
      by POLYNOM3:def 10;
      reconsider fg1 = f1 *' g1 as sequence of R;
A2:   f * g = (a|R) *' g9 by A1,POLYNOM3:def 10;
      for n be Nat holds ((Der1(R)).(f*g)).n = ((a|R)*'((Der1(R)).g)).n
      proof
        let n be Nat;
        reconsider b = g9.(n+1) as Element of R;
        reconsider Dg = (Der1(R)).g as Polynomial of R;
        ((Der1(R)).(f*g)).n = (n+1)*((a|R)*'g9).(n+1) by A2,Def8
        .= (n+1)*(a*g9.(n+1)) by Th27
        .= a*((n+1)*(g9.(n+1))) by BINOM:19
        .= a*((Der1(R)).g).n by Def8
         .= ((a|R)*'((Der1(R)).g)).n by Th27;
        hence thesis;
      end;
      hence thesis;
    end;
