
theorem ev0:
for R,S being non degenerated comRing
for n being Ordinal
for x being Function of n,S holds Ext_eval(0_(n,R),x) = 0.S
proof
let A,B be non degenerated comRing, n be Ordinal, x be Function of n,B;
set p = 0_(n,A);
consider y being FinSequence of the carrier of B such that
A: Ext_eval(p,x) = Sum y & len y = len SgmX(BagOrder n, Support p) &
   for i being Element of NAT st 1 <= i & i <= len y
   holds y.i = In( (p * SgmX(BagOrder n, Support p)).i, B) *
               eval(((SgmX(BagOrder n, Support p))/.i),x) by defeval;
   field(BagOrder n) = Bags n by ORDERS_1:12; then
B: BagOrder n linearly_orders (Support p) by ORDERS_1:37,ORDERS_1:38;
Support p = {} by YY;
then card(Support p) = 0;
then y = <*>the carrier of B by A,B,PRE_POLY:11;
hence thesis by A,RLVECT_1:43;
end;
