
theorem exevalminus:
for R being Ring,
    S being RingExtension of R
for a being Element of S,
    p being Polynomial of R holds Ext_eval(-p,a) = - Ext_eval(p,a)
proof
let R be Ring, S be RingExtension of R;
let a be Element of S, p be Polynomial of R;
consider G being FinSequence of S such that
A: Ext_eval(p,a) = Sum G & len G = len p &
   for n being Element of NAT st n in dom G
   holds G.n = In(p.(n-'1),S) * (power S).(a,n-'1) by ALGNUM_1:def 1;
consider H being FinSequence of S such that
B: Ext_eval(-p,a) = Sum H & len H = len -p &
   for n being Element of NAT st n in dom H
   holds H.n = In((-p).(n-'1),S) * (power S).(a,n-'1) by ALGNUM_1:def 1;
K: R is Subring of S by FIELD_4:def 1; then
H: the carrier of R c= the carrier of S by C0SP1:def 3;
C: len G = len H by A,B,POLYNOM4:8;
now let n be Nat, b be Element of S;
  assume D1: n in dom H & b = G.n;
  D2: dom H = Seg(len G) by C,FINSEQ_1:def 3 .= dom G by FINSEQ_1:def 3;
  reconsider pn1 = p.(n-'1), mpn1 = -p.(n-'1) as Element of S by H;
  thus H.n = In((-p).(n-'1),S) * (power S).(a,n-'1) by B,D1
     .= In(-(p.(n-'1)),S) * (power S).(a,n-'1) by BHSP_1:44
     .= mpn1 * (power S).(a,n-'1)
     .= (- pn1) * (power S).(a,n-'1) by K,Th19
     .= -(In(p.(n-'1),S) * (power S).(a,n-'1)) by VECTSP_1:9
     .= - b by D1,D2,A;
  end;
hence thesis by A,B,POLYNOM4:8,RLVECT_1:40;
end;
