
theorem thdLM:
for R being non degenerated Ring
for p being non zero Polynomial of R holds deg(p - LM p) < deg p
proof
let R be non degenerated Ring, p be non zero Polynomial of R;
per cases;
suppose p - LM p = 0_.(R);
  hence thesis by HURWITZ:20;
  end;
suppose p - LM p <> 0_.(R);
  then reconsider q = p-LM p as non zero Polynomial of R by UPROOTS:def 5;
  A: now assume deg q = deg p; then
     LC q = q.(deg p) by thLC
         .= p.(deg p) - (LM p).(deg p) by POLYNOM3:27
         .= p.(deg p) - (LM p).(deg(LM p)) by thdegLM
         .= p.(deg p) - LC(LM p) by thLC
         .= p.(deg p) - LC p by thdegLC
         .= LC p - LC p by thLC
         .= 0.R by RLVECT_1:15;
     hence contradiction;
     end;
  now assume B0: deg q > deg p; then
     deg q >= deg p + 1 by INT_1:7; then
     B1: deg q >= (len p - 1) + 1 by HURWITZ:def 2;
     p <> 0_.(R); then
     len p <> 0 by POLYNOM4:5; then
     len p + 1 > 0 + 1 by XREAL_1:6; then
     D: len p >= 1 by NAT_1:13;
     deg q <> len p - 1 by B0,HURWITZ:def 2; then
     deg q <> len p -' 1 by D,XREAL_0:def 2; then
     B2: (LM p).(deg q) = 0.R by POLYNOM4:def 1;
     LC q = q.(deg q) by thLC
         .= p.(deg q) - (LM p).(deg q) by POLYNOM3:27
         .= 0.R - 0.R by ALGSEQ_1:8,B1,B2;
     hence contradiction;
     end;
  hence thesis by A,XXREAL_0:1;
  end;
end;
