reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;
reserve a,b for Object of Ens(V);
reserve f,g,f1,f2 for Morphism of Ens(V);
reserve C for Category,
  a,b,a9,b9,c for Object of C,
  f,g,h,f9,g9 for Morphism of C;

theorem Th51:
  [[Hom(cod f,dom g),Hom(dom f,cod g)],hom(f,g)] is Element of Maps(Hom(C))
proof
A1: Hom(dom f,cod g) in Hom(C) & Hom(cod f,dom g) in Hom(C);
  Hom(dom f,cod g) = {} implies Hom(cod f,dom g) = {} by Th50;
  hence thesis by A1,Th5;
end;
