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);

theorem Th15:
   for R,M,N,f,g,h 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 R,M,N,f,g,h;
     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 Lm15;
       thus h.x = ((the addF of N).:(f,g)).x by A1,Def15
       .= 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 Lm15;
         thus (ADD(M,N).(f,g)).x = ((the addF of N).:(f,g)).x by Def15
         .= f.x + g.x by A4,FUNCOP_1:22 .= h.x by A3;
       end;
       hence h = (ADD(M,N)).(f,g);
     end;
