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 Th22:
  for a be Element of the carrier of R,
  f be Homomorphism of R,M,N holds
  (lmult_Hom(M,N)).[a,f] = (LMULT(M,N)).[a,f] &
  (LMULT(M,N)).[a,f] is Homomorphism of R,M,N
  proof
    let a be Element of the carrier of R,
    f be Homomorphism of R,M,N;
    a in the carrier of R & f in set_Hom(M,N);
    hence thesis by ZFMISC_1:87,FUNCT_1:49,Th20;
  end;
