reserve i,j for Nat;
reserve A,B for Ring;

theorem Th17:
  for x be Element of B holds Ext_eval(0_.A,x) = 0.B
proof
  let x be Element of B;
  consider F be FinSequence of B such that
A1: Ext_eval(0_.A,x) = Sum F and
A2: len F = len 0_.A and
  for n be Element of NAT st n in dom F holds
  F.n = In((0_.A).(n-'1),B) * (power B).(x,n-'1) by Def1;
  F = <*>the carrier of B by A2,POLYNOM4:3;
  hence thesis by A1,RLVECT_1:43;
end;
