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 Th23:
   for R,M,N
   for f1,g1 be Element of Func_Mod(R,M,N),
   f,g be Element of Hom(R,M,N) st f1=f & g1= g holds f + g = f1 + g1
   proof
     let R,M,N;
     let f1,g1 be Element of Func_Mod(R,M,N), f,g be Element of Hom(R,M,N);
     assume
A1:  f1=f & g1= g;
     reconsider f0 = f as Homomorphism of R,M,N by Lm29;
     reconsider g0 = g as Homomorphism of R,M,N by Lm29;
     f+g = (ADD(M,N)).(f0,g0) by Th21 .= f1+g1 by A1;
     hence thesis;
   end;
