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

theorem Th16:
  for x be Element of A, p be Polynomial of A st A is Subring of B holds
    Ext_eval(p,In(x,B)) = In(eval(p,x),B)
proof
   let x be Element of A, p be Polynomial of A;
    assume
A0:  A is Subring of B;
     consider F1 be FinSequence of B such that
A1:  Ext_eval(p,In(x,B)) = Sum F1 and
A2:  len F1 = len p and
A3:  for n be Element of NAT st n in dom F1 holds
     F1.n = In(p.(n-'1),B) * (power B).(In(x,B),n-'1) by Def1;
     consider F2 be FinSequence of A such that
A4:  eval(p,x) = Sum F2 and
A5:  len F2 = len p and
A6:  for n be Element of NAT st n in dom F2 holds
     F2.n = p.(n-'1) * (power A).(x,n-'1) by POLYNOM4:def 2;
     F1 = F2
     proof
A11:   rng F2 c= the carrier of A;
A8:    dom F1 = dom F2 by A2,A5,FINSEQ_3:29;
       for k be Nat st k in dom F1 holds F1.k = F2.k
       proof
         let k be Nat;
         assume
A10:     k in dom F1; then
         F2.k is Element of A by A8,FUNCT_1:3,A11; then
A13:      F2.k is Element of B by A0,Lm6;
          F1.k = In(p.(k-'1),B) * (power B).(In(x,B),k-'1) by A3,A10
              .= In(p.(k-'1) * (power A).(x,k-'1),B) by A0,Th15
              .= In(F2.k,B) by A6,A10,A8
              .= F2.k by A13,SUBSET_1:def 8;
          hence thesis;
       end;
       hence thesis by A2,A5,FINSEQ_3:29;
     end;
     hence thesis by A1,A4,A0,Th14;
   end;
