reserve M,N for AbGroup;

theorem Th7:
   for M be AbGroup
   for f be Element of set_End(M) holds (ADD(M,M)).(f,Inv(f)) = ZeroMap(M,M)
   proof
     let M be AbGroup;
     let f be Element of set_End(M);
     f in set_End(M); then
     consider f1 be Function of the carrier of M, the carrier of M such that
A1:  f1 = f & f1 is Endomorphism of M;
     Inv(f) in set_End(M); then
     consider g1 be Function of the carrier of M, the carrier of M such that
A2:  g1 = Inv(f) &
     g1 is Endomorphism of M;
     reconsider f1, g1 as Element of Funcs(the carrier of M, the carrier of M)
        by FUNCT_2:8;
     for x being Element of the carrier of M holds ((ADD(M,M)).(f1,g1)).x
     = (ZeroMap(M,M)).x
     proof
       let x be Element of the carrier of M;
A3:    g1.x = f1.(-x) by A1,A2,Def8;
A4:    f1.0.M + f1.0.M = f1.(0.M + 0.M) by A1,VECTSP_1:def 20
       .= f1.0.M by ALGSTR_1:7 .= f1.0.M + 0.M by ALGSTR_1:7;
       ((ADD(M,M)).(f1,g1)).x
       = f1.x + f1.(-x) by Th1,A3 .= f1.(x + (-x)) by A1,VECTSP_1:def 20
       .= f1.(0.M) by VECTSP_1:19
       .=((the carrier of M) --> 0.M).x by A4,RLVECT_1:8
       .= (ZeroMap(M,M)).x by GRCAT_1:def 7;
       hence thesis;
     end;
     hence thesis by A1,A2;
   end;
