 reserve n for Nat;

theorem Th10:
  for R being non degenerated comRing,
    a,b being non zero Element of R holds b * anpoly(a,n) = anpoly(a*b,n)
 proof
   let R be non degenerated comRing, a,b be non zero Element of R;
   now let i be Element of NAT;
     set p = anpoly(a,n), q = anpoly(a*b,n);
     per cases;
       suppose
A1:    i = n;
       (b*p).i = b * (p.i) by POLYNOM5:def 4
        .= b * a by A1,POLYDIFF:24
        .= q.i by A1,POLYDIFF:24;
       hence (b*p).i = q.i;
       end;
       suppose
A2:    i <> n;
       (b*p).i = b * (p.i) by POLYNOM5:def 4
        .= b * 0.R by A2,POLYDIFF:25
        .= q.i by A2,POLYDIFF:25;
       hence (b*p).i = q.i;
       end;
    end;
    hence thesis;
   end;
