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

theorem Th16:
  for R being Ring, S being Subring of R
  for f being Polynomial of S
  for r being Element of R, s being Element of S st r = s
  holds Ext_eval(f,r) = Ext_eval(f,s)
  proof
    let R be Ring;
    let S be Subring of R;
    let f be Polynomial of S;
    let r be Element of R;
    let s be Element of S;
    assume
A1: r = s;
    consider F being FinSequence of R such that
A2: Ext_eval(f,r) = 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),R) * (power R).(r,n-'1) by ALGNUM_1:def 1;
    consider G being FinSequence of S such that
A5: Ext_eval(f,s) = Sum G and
A6: len G = len f and
A7: for n being Element of NAT st n in dom G holds
    G.n = In(f.(n-'1),S) * (power S).(s,n-'1) by ALGNUM_1:def 1;
    now
      let n such that
A8:   n in dom F;
A9:   dom F = dom G by A3,A6,FINSEQ_3:29;
A10:  r = In(s,R) by A1;
A11:  f.(n-'1)*(power S).(s,n-'1) is Element of R by Th7;
      thus F.n = In(f.(n-'1),R)*(power R).(r,n-'1) by A4,A8
      .= In(f.(n-'1)*(power S).(s,n-'1),R) by A10,ALGNUM_1:11
      .= In(f.(n-'1),S) * (power S).(s,n-'1) by A11
      .= G.n by A7,A8,A9;
    end;
    then F = G by A3,A6,FINSEQ_2:9; then
A12: In(Sum G,R) = Sum F by ALGNUM_1:10;
    Sum G is Element of R by Th7;
    hence Ext_eval(f,r) = Ext_eval(f,s) by A2,A5,A12;
  end;
