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 Th29:
    for f,g,h be Endomorphism of R,M holds
    AbGr(h) = (FuncComp(AbGr(M))).(AbGr(f),AbGr(g)) iff
    for x being Element of the carrier of AbGr(M)
    holds (AbGr(h)).x = (AbGr(f) * (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) = (FuncComp(AbGr(M))).(AbGr(f),AbGr(g)) implies
      for x being Element of the carrier of AbGr(M)
      holds (AbGr(h)).x = (AbGr(f) * (AbGr(g))).x
      proof
        assume AbGr(h) = (FuncComp(AbGr(M))).(AbGr(f),AbGr(g)); then
        for x being Element of the carrier of AbGr(M)
        holds h1.x = (f1 * g1).x by Def4;
        hence thesis;
      end;
      assume
A3:   for x being Element of the carrier of AbGr(M) holds
      (AbGr(h)).x = ((AbGr(f))*(AbGr(g))).x;
      AbGr(h) = (FuncComp(AbGr(M))).(AbGr(f),AbGr(g))
      proof
A4:     for x be Element of the carrier of AbGr(M) holds
        (AbGr(h)).x = ((FuncComp(AbGr(M))).(AbGr(f),AbGr(g))).x
        proof
          let x be Element of the carrier of AbGr(M);
          ((FuncComp(AbGr(M))).(f1,g1)).x = (f1*g1).x by Def4 .= h1.x by A3;
          hence thesis;
        end;
        reconsider h2 = AbGr(h), addfg2 = (FuncComp(AbGr(M))).(f1,g1) as
          Function of the carrier of AbGr(M), the carrier of AbGr(M);
        h2 = addfg2 by A4;
        hence thesis;
      end;
      hence thesis;
    end;
