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 Th34:
    for f, g be Element of the carrier of Polynom-Ring R
    st f is constant holds (Der1(R)).(f*g) = ((Der1(R)).f)*g + f*((Der1(R)).g)
    proof
      let f,g be Element of the carrier of Polynom-Ring R;
      assume f is constant; then
      consider a being Element of R such that
A2:   f = a|R by RING_4:20;
      reconsider p = f, q = g as Polynomial of R;
      reconsider dp = (Der1(R)).f, dq = (Der1(R)).g as Polynomial of R;
      f = anpoly(a,0) by A2; then
A4:   dp = 0_.R by Th29;
      f*((Der1(R)).g) = p*'dq & ((Der1(R)).f)*g = dp*'q
        by POLYNOM3:def 10; then
      ((Der1(R)).f)*g + f*((Der1(R)).g) = 0_.R + p*'dq by A4,POLYNOM3:def 10
      .= (Der1(R)).(f*g) by A2,Th28;
      hence thesis;
    end;
