
theorem RE:
for R1,R2 being strict Ring
st R1 is Subring of R2 & R2 is Subring of R1 holds R1 = R2
proof
let K1,K2 be strict Ring;
assume A1: K1 is Subring of K2 & K2 is Subring of K1;
A2: the carrier of K1 c= the carrier of K2 &
    the carrier of K2 c= the carrier of K1 by A1,C0SP1:def 3; then
A3: the carrier of K1 = the carrier of K2 by XBOOLE_0:def 10;
A4: the addF of K2 = (the addF of K2) || the carrier of K1 by A3
                  .= the addF of K1 by A1,C0SP1:def 3;
A5: the multF of K2 = (the multF of K2)||the carrier of K1 by A3
                   .= the multF of K1 by A1,C0SP1:def 3;
1.K1 = 1.K2 & 0.K1 = 0.K2 by A1,C0SP1:def 3;
hence thesis by A4,A5,A2,XBOOLE_0:def 10;
end;
