
theorem RE1:
for S being Ring,
    R1,R2 being strict Subring of S
holds R1 = R2 iff the carrier of R1 = the carrier of R2
  proof
    let K be Ring;
    let SK1, SK2 be strict Subring of K;
    thus SK1 = SK2 implies
    the carrier of SK1 = the carrier of SK2;
    assume A1: the carrier of SK1 = the carrier of SK2;
    then A2: SK2 is strict Subring of SK1 by Th6;
    SK1 is strict Subring of SK2 by A1,Th6;
    hence thesis by A2,RE;
  end;
