reserve m,n for Nat;
reserve r for Real;
reserve c for Element of F_Complex;

theorem Th15:
  for R,T being Ring, S being Subring of R
  for f being Polynomial of S
  for g being Polynomial of R st f = g
  for a being Element of R
  holds Ext_eval(f,In(a,T)) = Ext_eval(g,In(a,T))
  proof
    let R,T be Ring;
    let S be Subring of R;
    let f be Polynomial of S;
    let g be Polynomial of R;
    assume
A1: f = g;
    let a be Element of R;
    consider F being FinSequence of T such that
A2: Ext_eval(f,In(a,T)) = Sum F and
A3: len F = len f and
A4: for n being Element of NAT st n in dom F holds
    F.n = In(f.(n-'1),T) * (power T).(In(a,T),n-'1) by ALGNUM_1:def 1;
    consider G being FinSequence of T such that
A5: Ext_eval(g,In(a,T)) = Sum G and
A6: len G = len g and
A7: for n being Element of NAT st n in dom G holds
    G.n = In(g.(n-'1),T) * (power T).(In(a,T),n-'1) by ALGNUM_1:def 1;
    consider Z being sequence of T such that
A8: Sum G = Z.(len G) and
A9: Z.0 = 0.T and
A10: for j being Nat, v being Element of T st j < len G & v = G.(j+1)
    holds Z.(j+1) = Z.j + v by RLVECT_1:def 12;
A11: Sum G = Z.(len F) by A1,A3,A6,A8,Th9;
    now
      let j be Nat, v be Element of T such that
A12:  j < len F and
A13:  v = F.(j+1);
A14:  len F = len G by A1,A3,A6,Th9;
A15:  dom F = dom G by A1,A3,A6,Th9,FINSEQ_3:29;
      j+1 <= len F by A12,NAT_1:13;
      then
A16:  j+1 in dom F by NAT_1:11,FINSEQ_3:25;
      then F.(j+1) = In(f.(j+1-'1),T) * (power T).(In(a,T),j+1-'1) by A4
      .= G.(j+1) by A1,A7,A15,A16;
      hence Z.(j+1) = Z.j + v by A10,A12,A13,A14;
    end;
    hence thesis by A2,A5,A9,A11,RLVECT_1:def 12;
  end;
