
theorem FIELD427:
for R being Ring,
    S being RingExtension of R
for p being Element of the carrier of Polynom-Ring R,
    a being Element of R,
    b being Element of S st b = a holds Ext_eval(p,b) = eval(p,a)
   proof
     let R be Ring, S be RingExtension of R;
     let p be Element of the carrier of Polynom-Ring R;
     let a be Element of R, b be Element of S;
     assume
A1:  b = a;
A2:  R is Subring of S by FIELD_4:def 1; then
A3:  the carrier of R c= the carrier of S by C0SP1:def 3;
     Ext_eval(p,In(a,S)) = In(eval(p,a),S) by A2,ALGNUM_1:12
     .= eval(p,a) by A3,SUBSET_1:def 8;
     hence thesis by A1;
   end;
