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 Th2:
   for n being Ordinal, L being right_zeroed add-associative
   right_complementable Abelian well-unital distributive non trivial
   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
     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,POLYNOM2:23;
     hence thesis;
   end;
