reserve R for Ring, S for R-monomorphic Ring,
        K for Field, F for K-monomorphic Field,
        T for K-monomorphic comRing;

theorem Th3:
   for f being Monomorphism of R,S holds f".(1.S) = 1.R & f".(0.S) = 0.R
   proof
     let f be Monomorphism of R,S;
A1:  [#]R = dom f by FUNCT_2:def 1;
     f.(1_R) = 1_S by GROUP_1:def 13;
     hence f".(1.S) = 1.R by A1,FUNCT_1:34;
     f.(0.R) = 0.S by RING_2:6; then
     reconsider o = 0.S as Element of (rng f) by FUNCT_2:4;
     f".(0.S) = f".(o + o) = f".o + f".o by Th1;
     hence thesis by RLVECT_1:9;
   end;
