reserve A for non degenerated comRing;
reserve R for non degenerated domRing;
reserve n for non empty Ordinal;
reserve o,o1,o2 for object;
reserve X,Y for Subset of Funcs(n,[#]R);
reserve S,T for Subset of Polynom-Ring(n,R);
reserve F,G for FinSequence of the carrier of Polynom-Ring(n,R);
reserve x for Function of n,R;

theorem Th3:
    for n being Ordinal, L being right_zeroed add-associative
    right_complementable Abelian well-unital distributive
    non trivial commutative associative non empty doubleLoopStr,
    f,g being Element of Polynom-Ring(n,L),
    x being Function of n, L holds
    E_eval(f*g,x) = E_eval(f,x) * E_eval(g,x)
    proof
      let n be Ordinal, L be right_zeroed add-associative
      right_complementable Abelian well-unital distributive
      non trivial commutative associative non empty doubleLoopStr,
      f,g be Element of Polynom-Ring(n,L),
      x be Function of n, L;
      f in [#]Polynom-Ring(n,L) & g in [#]Polynom-Ring(n,L) by SUBSET_1:def 1;
      then
      f is Polynomial of n,L & g is Polynomial of n,L by POLYNOM1:def 11; then
      consider p,q be Polynomial of n,L such that
A1:   p = f & q = g;
A2:   f*g = p*'q by A1,POLYNOM1:def 11;
A3:   E_eval(f,x) = eval(p,x) & E_eval(g,x) = eval(q,x) by A1,Def1;
      E_eval(f*g,x) = eval(p*'q,x) by A2,Def1
      .= E_eval(f,x) * E_eval(g,x) by A3,POLYNOM2:25;
      hence thesis;
    end;
