
theorem lemphi1:
for n being Nat
for F,E being Field
for x being Function of n,E
holds rng hom_Ext_eval(x,F)
    = the set of all Ext_eval(p,x) where p is Polynomial of n,F
proof
let n be Nat, F,E be Field, x be Function of n,E;
set g = hom_Ext_eval(x,F);
A: now let o be object;
   assume o in rng g; then
   consider p being object such that A1: p in dom g & o = g.p
     by FUNCT_1:def 3;
   reconsider p as Element of the carrier of Polynom-Ring(n,F) by A1;
   reconsider p as Polynomial of n,F by POLYNOM1:def 11;
   o = Ext_eval(p,x) by A1,dh;
   hence o in the set of all Ext_eval(p,x) where p is Polynomial of n,F;
   end;
now let o be object;
   assume o in the set of all Ext_eval(p,x)
                                 where p is Polynomial of n,F;
   then consider p being Polynomial of n,F such that
   A2: o = Ext_eval(p,x);
   A3: g.p = o by A2,dh;
   A4: dom g = the carrier of Polynom-Ring(n,F) by FUNCT_2:def 1;
   p is Element of the carrier of Polynom-Ring(n,F)
      by POLYNOM1:def 11;
   hence o in rng g by A3,A4,FUNCT_1:3;
   end;
hence thesis by A,TARSKI:2;
end;
