
theorem lem23a:
for F being Field
for p,q being Polynomial of F st LC p + LC q <> 0.F
holds deg(p + q) = max(deg p, deg q)
proof
let F be Field, p1,q1 be Polynomial of F;
assume AS: LC p1 + LC q1 <> 0.F;
per cases;
suppose p1 <> 0_.(F) & q1 <> 0_.(F); then
  reconsider p = p1, q = q1 as non zero Polynomial of F by UPROOTS:def 5;
  per cases;
  suppose A: deg p = deg q;
    (p + q).(deg p) = (p.(deg p)) + (q.(deg p)) by NORMSP_1:def 2
                   .= (LC p) + (q.(deg q)) by A,FIELD_6:2
                   .= LC p + LC q by FIELD_6:2; then
    len(p+q) >= deg p + 1 by AS,ALGSEQ_1:8,INT_1:7; then
    len(p+q) - 1 >= deg p + 1 - 1 by XREAL_1:9; then
    B: deg(p + q) >= deg p by HURWITZ:def 2;
    deg(p + q) <= max(deg p,deg q) by HURWITZ:22;
    hence thesis by A,B,XXREAL_0:1;
    end;
  suppose deg p <> deg q;
    hence thesis by HURWITZ:21;
    end;
  end;
suppose A: p1 = 0_.(F); then
  C: deg p1 = -1 by HURWITZ:20;
  len q1 - 1 >= 0 - 1 by XREAL_1:9; then
  deg q1 >= deg p1 by C,HURWITZ:def 2;
  hence thesis by A,XXREAL_0:def 10;
  end;
suppose A: q1 = 0_.(F); then
  C: deg q1 = -1 by HURWITZ:20;
  len p1 - 1 >= 0 - 1 by XREAL_1:9; then
  deg p1 >= deg q1 by C,HURWITZ:def 2;
  hence thesis by A,XXREAL_0:def 10;
  end;
end;
