reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem Th32:
  for g,f being Element of Morphs(V) st dom(g) = cod(f) holds g*f in Morphs(V)
proof
  set X = Morphs(V);
  defpred P[Element of X,Element of X] means dom($1) = cod($2);
  let g,f be Element of X;
  assume P[g,f];
  then consider G1,G2,G3 being strict Element of V such that
A1: g is Morphism of G2,G3 and
A2: f is Morphism of G1,G2 by Th31;
  reconsider f9 = f as Morphism of G1,G2 by A2;
  reconsider g9 = g as Morphism of G2,G3 by A1;
  g9*f9 is Morphism of G1,G3;
  hence thesis by Def23;
end;
