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 Th21:
   for f,g be Homomorphism of R,M,N holds
   f in Funcs(the carrier of M,the carrier of N) &
   g in Funcs(the carrier of M,the carrier of N) &
   (add_Hom(M,N)).[f,g] = (ADD(M,N)).(f,g) &
   (ADD(M,N)).(f,g) is Homomorphism of R,M,N
   proof
     let f,g be Homomorphism of R,M,N;
     f in set_Hom(M,N) & g in set_Hom(M,N);
     hence thesis by ZFMISC_1:87,FUNCT_1:49,Th18;
  end;
