
theorem
for F being Field,
    R being Subring of F holds R is Subfield of F iff R is Field
proof
let E be Field, R be Subring of E;
now assume B: R is Field;
    the carrier of R c= the carrier of E
  & the addF of R = (the addF of E)||the carrier of R
  & the multF of R = (the multF of E)||the carrier of R
  & 1.R = 1.E & 0.R = 0.E by C0SP1:def 3;
  hence R is Subfield of E by EC_PF_1:def 1,B;
  end;
hence thesis;
end;
