reserve M,N for AbGroup;

theorem Th4:
   for M,N be AbGroup
   for f,g be Element of Funcs(the carrier of M, the carrier of N)
   holds (ADD(M,N)).(f,g) = (ADD(M,N)).(g,f)
   proof
     let M,N be AbGroup;
     let f,g be Element of Funcs(the carrier of M, the carrier of N);
     for x being Element of M holds ((ADD(M,N)).(f,g)).x = ((ADD(M,N)).(g,f)).x
     proof
       let x be Element of M;
       ((ADD(M,N)).(f,g)).x = f.x + g.x by Th1 .= ((ADD(M,N)).(g,f)).x by Th1;
       hence thesis;
     end;
     hence thesis;
   end;
