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;

theorem Th17:
  for G1,G2,G3 being AddGroup, G being Morphism of G2,G3, F being
  Morphism of G1,G2 holds G*F is Morphism of G1,G3
proof
  let G1,G2,G3 be AddGroup, G be Morphism of G2,G3, F be Morphism of G1,G2;
  consider g being Function of G2,G3 such that
A1: the GroupMorphismStr of G = GroupMorphismStr(# G2,G3,g#) and
  g is additive by Th12;
  consider f being Function of G1,G2 such that
A2: the GroupMorphismStr of F = GroupMorphismStr(# G1,G2,f#) and
  f is additive by Th12;
  dom(G) = G2 by Def12
    .= cod(F) by Def12;
  then G*F = GroupMorphismStr(# G1,G3,g*f#) by A1,A2,Def14;
  then dom(G*F) = G1 & cod(G*F) = G3;
  hence thesis by Def12;
end;
