
theorem exevalLM:
for R being non degenerated Ring,
    S being RingExtension of R
for a,b being Element of S,
    p being non zero Polynomial of R st b = LC p holds
Ext_eval(Leading-Monomial p,a) = b * a|^(deg p)
proof
let R be non degenerated Ring, S be RingExtension of R;
let a,b be Element of S, p be non zero Polynomial of R;
assume AS: b = LC p;
H0: R is Subring of S by FIELD_4:def 1;
H2: p.(len p-'1) = LC p by RATFUNC1:def 6;
deg p = len p - 1 by HURWITZ:def 2; then
H3: len p -' 1 = deg p by XREAL_0:def 2;
thus Ext_eval(Leading-Monomial p,a)
   = In(p.(len p-'1),S) * (power S).(a,len p-'1) by H0,ALGNUM_1:17
  .= b * a|^(deg p) by AS,H2,H3,BINOM:def 2;
end;
