
theorem field427:
for n being Nat
for F being Field, E being FieldExtension of F,
    a being Element of F, b being Element of E st a = b holds a|(n,F) = b|(n,E)
proof
let n be Nat, F be Field, E be FieldExtension of F,
    a be Element of F, b be Element of E;
assume AS: a = b;
set ap = a|(n,F), bp = b|(n,E);
H: F is Subring of E by FIELD_4:def 1;
now let u be bag of n;
  per cases;
  suppose A: u = EmptyBag n;
    hence ap.u = a by POLYNOM7:18 .= bp.u by AS,A,POLYNOM7:18;
    end;
  suppose A: u <> EmptyBag n;
    hence ap.u = 0.F by POLYNOM7:def 7 .= 0.E by H,C0SP1:def 3
              .= bp.u by A,POLYNOM7:def 7;
    end;
  end;
hence thesis;
end;
