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 Th30:
   for f,g,h be Endomorphism of R,M holds
   AbGr(h) = ADD(AbGr(M),AbGr(M)).(AbGr(f),AbGr(g)) iff
   for x being Element of the carrier of AbGr(M) holds
   (AbGr(h)).x = (AbGr(f)).x + (AbGr(g)).x
   proof
     let f,g,h be Endomorphism of R,M;
     reconsider f1 = AbGr(f), g1 = AbGr(g), h1 = AbGr(h) as Element of
       Funcs(the carrier of AbGr(M),the carrier of AbGr(M)) by FUNCT_2:8;
     thus AbGr(h) = ADD(AbGr(M),AbGr(M)).(AbGr(f),AbGr(g)) implies
     for x being Element of the carrier of AbGr(M) holds
     (AbGr(h)).x = (AbGr(f)).x + (AbGr(g)).x
     proof
       assume
A2:    AbGr(h) = ADD(AbGr(M),AbGr(M)).(AbGr(f),AbGr(g));
       for x being Element of AbGr(M) holds
       (AbGr(h)).x = (AbGr(f)).x + (AbGr(g)).x
       proof
         let x be Element of AbGr(M);
A3:      x in dom ((the addF of AbGr(M)).:(f1,g1)) by Lm1;
         h1.x = ((the addF of AbGr(M)).:(f1,g1)).x by A2,Def1
         .= f1.x + g1.x by A3,FUNCOP_1:22;
         hence thesis;
       end;
       hence thesis;
     end;
     assume
A4:  for x being Element of the carrier of AbGr(M) holds
     (AbGr(h)).x = (AbGr(f)).x + (AbGr(g)).x;
     AbGr(h) = ADD(AbGr(M),AbGr(M)).(AbGr(f),AbGr(g))
     proof
A5:    for x be Element of the carrier of AbGr(M) holds
       (AbGr(h)).x = (ADD(AbGr(M),AbGr(M)).(AbGr(f),AbGr(g))).x
       proof
         let x be Element of the carrier of AbGr(M);
A6:      x in dom ((the addF of AbGr(M)).:(f1,g1)) by Lm1;
         (ADD(AbGr(M),AbGr(M)).(f1,g1)).x
         = ((the addF of AbGr(M)).:(f1,g1)).x by Def1
         .= f1.x + g1.x by A6,FUNCOP_1:22 .= h1.x by A4;
         hence thesis;
       end;
  reconsider h2 = AbGr(h), addfg2 = (ADD(AbGr(M),AbGr(M)).(f1,g1)) as
        Function of the carrier of AbGr(M), the carrier of AbGr(M);
        h2 = addfg2 by A5;
        hence thesis;
      end;
      hence thesis;
    end;
