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
    for f, g be constant Element of the carrier of Polynom-Ring R holds
    (Der1(R)).(f*g) = ((Der1(R)).f)*g + f*((Der1(R)).g)
    proof
      let f,g be constant Element of the carrier of Polynom-Ring R;
      consider a being Element of R such that
A1:   f = (a|R) by RING_4:20;
      consider b being Element of R such that
A2:   g = b|R by RING_4:20;
A3:   f = anpoly(a,0) by A1;
A4:   g = anpoly(b,0) by A2;
      reconsider p = f, q = g as Polynomial of R;
      reconsider dp = (Der1(R)).f, dq = (Der1(R)).g as Polynomial of R;
A5:   f*g = (a|R)*'(b|R) by A1,A2,POLYNOM3:def 10
      .= (a * b)|R by RING_4:18;
      reconsider h = f*g as Element of Polynom-Ring R;
      consider c be Element of R such that
A6:   h = c|R by A5;
A7:   h = anpoly(c,0) by A6;
A8:   dp = 0_.R by A3,Th29 .= (0.R)|R by RING_4:13;
A9:   dq = 0_.R by A4,Th29 .= (0.R)|R by RING_4:13;
A10:  dp*'q = (0.R*b)|R by A8,A2,RING_4:18 .= 0.R|R;
      p*'dq = (a*0.R)|R by A9,A1,RING_4:18 .= 0.R|R; then
A11:  dp*'q + p*'dq = 0_.R by A10,RING_4:13;
      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 by A11,POLYNOM3:def 10
       .= (Der1(R)).(f*g) by A7,Th29;
      hence thesis;
    end;
