
theorem
for R being comRing,
    p being Polynomial of R
for a being Element of R holds eval(<%0.R,1.R%> *' p,a) = a * eval(p,a)
proof
let R be comRing, p be Polynomial of R; let a be Element of R;
per cases;
suppose R is degenerated; then
A: the carrier of R = {0.R} by degen;
hence eval(<%0.R,1.R%> *' p,a) = 0.R by TARSKI:def 1
  .= a * eval(p,a) by A,TARSKI:def 1;
end;
suppose R is non degenerated;
hence eval(<%0.R,1.R%> *' p,a)
   = eval(<%0.R,1.R%>,a) * eval(p,a) by POLYNOM4:24
  .= (0.R + 1.R * a) * eval(p,a) by POLYNOM5:44
  .= a * eval(p,a);
end;
end;
