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

theorem Th25:
  for x be Element of B, z0 be Element of A
  st A is Subring of B holds Ext_eval(<%z0%>,x) = In(z0,B)
proof
  let x be Element of B, z0 be Element of A;
    assume
A0:   A is Subring of B;
   consider F be FinSequence of B such that
A1: Ext_eval(<%z0%>,x) = Sum F and
A2: len F = len <%z0%> and
A3: for n be Element of NAT st n in dom F holds
    F.n = In(<%z0%>.(n-'1),B)*(power B).(x,n-'1) by Def1;
  per cases by A2,ALGSEQ_1:def 5,NAT_1:25;
  suppose
A4: len F = 0;
A5: z0 = <%z0%>.0 by POLYNOM5:32 .= (0_.A).0 by A4,A2,POLYNOM4:5
      .=0.A by FUNCOP_1:7;
    Ext_eval(<%z0%>,x) = Ext_eval(0_.A,x) by A4,A2,POLYNOM4:5
     .= 0.B by Th17
     .= In(z0,B) by A5,A0,Lm5;
    hence thesis;
  end;
  suppose
A6: len F = 1; then
A7: F.1 = In(<%z0%>.(1-'1),B)*(power B).(x,1-'1) by A3,FINSEQ_3:25
      .=In( <%z0%>.0,B)*(power B).(x,1-'1) by XREAL_1:232
      .= In( <%z0%>.0,B)*(power B).(x,0) by XREAL_1:232
      .= In( z0,B) * (power B).(x,0) by POLYNOM5:32
      .= In(z0,B) * 1_B by GROUP_1:def 7
      .= In(z0,B);
      Ext_eval(<%z0%>,x) = Sum <*In(z0,B)*> by A6,A7,FINSEQ_1:40,A1
      .= In(z0,B) by RLVECT_1:44;
    hence thesis;
  end;
end;
