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);

theorem Th16:
  for R,M,N holds
  for a be Element of the carrier of R,
  f,h be Element of Funcs(the carrier of M, the carrier of N) holds
  h = (LMULT(M,N)).[a,f] iff for x being Element of M holds h.x = a*(f.x)
  proof
    let R,M,N;
    let a be Element of the carrier of R,
    f,h be Element of Funcs(the carrier of M, the carrier of N);
    thus
    h = (LMULT(M,N)).[a,f] implies for x being Element of the carrier of M
    holds h.x = a*(f.x) by Def16;
    now
      assume
A1:   for x being Element of the carrier of M holds h.x = a*(f.x);
      for x being Element of the carrier of M holds h.x = (LMULT(M,N).[a,f]).x
      proof
        let x be Element of the carrier of M;
        thus h.x = a*(f.x) by A1 .= ((LMULT(M,N)).[a,f]).x by Def16;
      end;
      hence h = (LMULT(M,N)).[a,f];
    end;
    hence thesis;
  end;
