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 Th18:
  for G1,G2,G3 being AddGroup, G being Morphism of G2,G3, F being
Morphism of G1,G2, g being Function of G2,G3, f being Function of G1,G2 st G =
  GroupMorphismStr(# G2,G3,g#) & F = GroupMorphismStr(# G1,G2,f#) holds G*F =
  GroupMorphismStr(# G1,G3,g*f#)
proof
  let G1,G2,G3 be AddGroup, G be Morphism of G2,G3, F be Morphism of G1,G2, g
  be Function of G2,G3, f be Function of G1,G2 such that
A1: G = GroupMorphismStr(# G2,G3,g#) & F = GroupMorphismStr(# G1,G2,f#);
  dom(G) = G2 by Def12
    .= cod(F) by Def12;
  hence thesis by A1,Def14;
end;
