
theorem exevalconst:
for R being Ring,
    S being RingExtension of R
for a being Element of S,
    p being constant Polynomial of R holds Ext_eval(p,a) = LC p
proof
let R be Ring, S be RingExtension of R;
let a be Element of S, p be constant Polynomial of R;
A: R is Subring of S by FIELD_4:def 1;
per cases;
suppose A0: p = 0_.(R);
hence Ext_eval(p,a) = 0.S by ALGNUM_1:13
                   .= p.(len p-'1) by A0,A,C0SP1:def 3
                   .= LC p by RATFUNC1:def 6;
end;
suppose A0: p <> 0_.(R);
the carrier of R c= the carrier of S by A,C0SP1:def 3; then
reconsider p0 = p.0 as Element of S;
consider F being FinSequence of S such that
  A1: Ext_eval(p,a) = Sum F and
  A2: len F = len p and
  A3: for n be Element of NAT st n in dom F holds
      F.n = In(p.(n-'1),S)*(power S).(a,n-'1) by ALGNUM_1:def 1;
reconsider degp = deg p as Element of NAT by A0,FIELD_1:1;
A7: degp = 0 by RATFUNC1:def 2;
A6: deg p = len p - 1 by HURWITZ:def 2; then
A4: F.1 = In(p.(1-'1),S) * (power S).(a,1-'1) by A7,A2,A3,FINSEQ_3:25
       .= In(p.0,S) * (power S).(a,1-'1) by XREAL_1:232
       .= p0 * (power S).(a,0) by XREAL_1:232
       .= p0 * 1_S by GROUP_1:def 7
       .= p0;
Sum F = Sum <*p0*> by A6,A7,A2,FINSEQ_1:40,A4
     .= p.(1-1) by RLVECT_1:44
     .= p.(1-'1) by XREAL_0:def 2
     .= LC p by A6,A7,RATFUNC1:def 6;
hence thesis by A1;
end;
end;
