 reserve K,F,E for Field,
         R,S for Ring;

theorem
   for S being RingExtension of R,
   p being Element of the carrier of (Polynom-Ring R),
   q being Element of the carrier of Polynom-Ring S,
   a being Element of R, b being Element of S st q = p & b = a
   holds eval(q,b) = eval(p,a)
   proof
     let S be RingExtension of R,
     p be Element of the carrier of (Polynom-Ring R),
     q be Element of the carrier of Polynom-Ring S,
     a being Element of R, b being Element of S;
     assume that
A1:  p = q and
A2:  a = b;
     thus eval(p,a) = Ext_eval(p,b) by A2,Th14 .= eval(q,b) by A1,Th15;
   end;
