reserve M,N for AbGroup;
 reserve R for Ring;
 reserve r for Element of R;
reserve M,N for LeftMod of R;
reserve f,g,h for Element of Funcs(the carrier of M, the carrier of N);
reserve a,b for Element of the carrier of R;
reserve R for comRing;
reserve M,M1,N,N1 for LeftMod of R;

theorem Th28:
    for R,M,N for f be Homomorphism of R,M,N holds
    AbGr(f) is Homomorphism of AbGr(M),AbGr(N)
    proof
      let R, M,N;
      let f be Homomorphism of R,M,N;
      for x,y being Element of AbGr(M) holds
      (AbGr(f)).(x+y) = (AbGr(f)).x + (AbGr(f)).y
      proof
        let x,y be Element of AbGr(M);
        reconsider x1 = x, y1 = y as Element of the carrier of M;
A2:     (AbGr(f)).x = f.x1 by Def26;
A4:     (AbGr(f)).y = f.y1 by Def26;
        reconsider z1 = x+y as Element of the carrier of M;
A6:     (AbGr(f)).z1 = f.z1 by Def26;
A7:     z1 = x1 + y1;
        (AbGr(f)).(x+y)
        = f.x1 + f.y1 by A6,A7,Def10,VECTSP_1:def 20
        .= (AbGr(f)).x + (AbGr(f)).y by A2,A4;
        hence thesis;
      end;
      hence thesis by VECTSP_1:def 20;
    end;
