reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;
reserve N for normal Subgroup of G;
reserve S,T1,T2 for Element of G./.N;
reserve g,h for Homomorphism of G,H;
reserve h1 for Homomorphism of H,I;

theorem Th39:
  for G,H,I being unital non empty multMagma,
      h being multiplicative Function of G,H,
      h1 being multiplicative Function of H,I holds
  h1 * h is multiplicative
proof
  let G,H,I be unital non empty multMagma,
  h be multiplicative Function of G,H,
  h1 be multiplicative Function of H,I;
  set f = h1 * h;
  let a,b be Element of G;
  thus f.(a * b) = h1.(h.(a * b)) by FUNCT_2:15
  .= h1.(h.a * h.b) by Def6
  .= (h1.(h.a)) * (h1.(h.b)) by Def6
  .= f.a * (h1.(h.b)) by FUNCT_2:15
  .= f.a * f.b by FUNCT_2:15;
end;
