
theorem Th1:
   for M,N be AbGroup,
   f,g,h be Element of Funcs(the carrier of M, the carrier of N) holds
   h = ADD(M,N).(f,g) iff
   for x being Element of the carrier of M holds h.x = f.x + g.x
   proof
     let M,N be AbGroup,
     f,g,h be Element of Funcs(the carrier of M, the carrier of N);
     hereby
       assume
A1:    h = ADD(M,N).(f,g);
       let x be Element of the carrier of M;
A2:    x in dom ((the addF of N).:(f,g)) by Lm1;
       thus h.x = ((the addF of N).:(f,g)).x by A1,Def1
       .= f.x + g.x by A2,FUNCOP_1:22;
     end;
     assume
A3:  for x being Element of M holds h.x = f.x + g.x;
     now
       let x be Element of M;
A4:    x in dom ((the addF of N).:(f,g)) by Lm1;
       thus (ADD(M,N).(f,g)).x = ((the addF of N).:(f,g)).x by Def1
       .= f.x + g.x by A4,FUNCOP_1:22 .= h.x by A3;
     end;
     hence h = (ADD(M,N)).(f,g);
   end;
