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

theorem Th18:
  for A,B being non degenerated Ring
  for x be Element of B st A is Subring of B holds
  Ext_eval(1_.A,x) = 1.B
proof
  let A,B be non degenerated Ring;
  let x be Element of B;
    assume
A0: A is Subring of B;
    consider F be FinSequence of B such that
A1: Ext_eval(1_.A,x) = Sum F and
A2: len F = len 1_.A and
A3: for n be Element of NAT st n in dom F holds
    F.n = In((1_.A).(n-'1),B)*(power B).(x,n-'1) by Def1;
    len F = 1 by A2,POLYNOM4:4; then
A4: F.1 = In((1_.A).(1-'1),B) * (power B).(x,1-'1) by A3,FINSEQ_3:25
    .= In((1_.A).(0),B) * (power B).(x,1-'1) by XREAL_1:232
    .= In(1.A,B) * (power B).(x,1-'1) by POLYNOM3:30
    .= 1.B * (power B).(x,1-'1) by A0,Lm5
    .= (power B).(x,0) by XREAL_1:232 .= 1_B by GROUP_1:def 7 .= 1.B;
     Sum F = Sum <*1.B*> by A2,POLYNOM4:4,FINSEQ_1:40,A4 .= 1.B by RLVECT_1:44;
  hence thesis by A1;
end;
