 reserve n for Nat;

theorem Th12:
  for R being non degenerated Ring, p being non zero Polynomial of R holds
   LM p = anpoly(p.(deg p),deg p)
   proof
     let R be non degenerated Ring, p be non zero Polynomial of R;
     set q = anpoly(p.(deg p),deg p), r = LM p, n = deg p;
     reconsider degp = deg p as Element of NAT;
A1:   n = len p -' 1 by XREAL_0:def 2;
     now let i be Element of NAT;
     per cases;
       suppose A3: i <> n; then
         r.i = 0.R by A1,POLYNOM4:def 1
            .= q.i by A3,POLYDIFF:25;
         hence r.i = q.i;
       end;
       suppose A4: i = n; then
         r.i = p.n by A1,POLYNOM4:def 1
         .= q.i by A4,POLYDIFF:24;
         hence r.i = q.i;
       end;
     end;
     hence thesis;
   end;
