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

theorem Th2:
   for f being Monomorphism of R,S, a being Element of rng f
   holds f".(a) = 0.R iff a = 0.S
   proof
     let f be Monomorphism of R,S, a be Element of rng f;
A1:   now assume f".(a) = 0.R; then
       f.(0.R) = a by FUNCT_1:35;
       hence a = 0.S by RING_2:6;
     end;
A2:  dom f = [#]R by FUNCT_2:def 1;
     now assume a = 0.S; then
       f.(0.R) = a by RING_2:6;
       hence f".(a) = 0.R by A2,FUNCT_1:34;
     end;
     hence thesis by A1;
   end;
