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 Th31:
    for f,g,h be Endomorphism of R,M st
    h = ADD(M,M).(f,g) holds AbGr(h) = ADD(AbGr(M),AbGr(M)).(AbGr(f),AbGr(g))
    proof
      let f,g,h be Endomorphism of R,M;
      assume
A1:   h = ADD(M,M).(f,g);
      reconsider f1 = AbGr(f), g1 = AbGr(g) as
Element of Funcs(the carrier of AbGr(M),the carrier of AbGr(M)) by FUNCT_2:8;
      reconsider f0 = f, g0 = g, h0 = h as
        Element of Funcs(the carrier of M,the carrier of M) by FUNCT_2:8;
      for x being Element of the carrier of AbGr(M) holds
      (AbGr(h)).x = (AbGr(f)).x + (AbGr(g)).x
      proof
        let x be Element of the carrier of AbGr(M);
        reconsider x0 = x as Element of the carrier of M;
A2:     (AbGr(f)).x = f.x0 by Def26;
A3:     (AbGr(g)).x = g.x0 by Def26;
A4:     (AbGr(h)).x = h.x0 by Def26;
        (AbGr(h)).x = h0.x0 by A4
        .= f0.x0 + g0.x0 by A1,Th15
        .= (AbGr(f)).x + (AbGr(g)).x by A2,A3;
        hence thesis;
      end;
      hence thesis by Th30;
    end;
