
theorem help3:
for R being domRing,
    S being domRingExtension of R
for n being non zero Element of NAT
for a being Element of S holds Ext_eval(<%0.R,1.R%>`^n,a) = a|^n
proof
let R be domRing, S be domRingExtension of R;
let n be non zero Element of NAT; let a be Element of S;
reconsider q = <%0.S,1.S%>`^n as Element of the carrier of Polynom-Ring S
  by POLYNOM3:def 10;
reconsider p = <%0.R,1.R%>`^n as Element of the carrier of Polynom-Ring R
  by POLYNOM3:def 10;
R is Subring of S by FIELD_4:def 1; then
0.R = 0.S & 1.R = 1.S by C0SP1:def 3; then
<%0.R,1.R%>`^n = <%0.S,1.S%>`^n by helpp; then
Ext_eval(p,a) = eval(q,a) by FIELD_4:26 .= a|^n by help3a;
hence thesis;
end;
