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

theorem Th1:
   for f being Monomorphism of R,S, a,b being Element of rng f
   holds f".(a+b) = f".a + f".b & f".(a*b) = f".a * f".b
   proof
      let f be Monomorphism of R,S, a,b be Element of rng f;
      consider x being object such that
A1:   x in [#]R & f.x = a by FUNCT_2:11;
      reconsider x as Element of R by A1;
      consider y being object such that
A2:   y in [#]R & f.y = b by FUNCT_2:11;
      reconsider y as Element of R by A2;
A3:   [#]R = dom f by FUNCT_2:def 1; then
A4:   f".b = y by A2,FUNCT_1:34;
A5:   [#]R = dom f by FUNCT_2:def 1;
      f.(x + y) = a + b by A1,A2,VECTSP_1:def 20;
      hence f".(a+b)=x+y by A5,FUNCT_1:34 .= f".a+f".b by A1,A3,A4,FUNCT_1:34;
      f.(x*y) = a*b by A1,A2, GROUP_6:def 6;
      hence f".(a*b)=x*y by A5,FUNCT_1:34 .= f".a*f".b by A1,A3,A4,FUNCT_1:34;
   end;
