Section two_arg_convenience_results.
Variable X Y Z:TopologicalSpace.
Variable f:point_set X -> point_set Y -> point_set Z.
Definition continuous_2arg :=
continuous (fun p:point_set X * point_set Y =>
let (x,y):=p in f x y)
(X:=ProductTopology2 X Y).
Definition continuous_at_2arg (x:point_set X) (y:point_set Y) :=
continuous_at (fun p:point_set X * point_set Y =>
let (x,y):=p in f x y)  (x, y)
(X:=ProductTopology2 X Y).
End two_arg_convenience_results.
Arguments continuous_2arg {X} {Y} {Z}.
Arguments continuous_at_2arg {X} {Y} {Z}.

Lemma continuous_composition_at_2arg:
forall (W X Y Z:TopologicalSpace)
(f:point_set X -> point_set Y -> point_set Z)
(g:point_set W -> point_set X) (h:point_set W -> point_set Y)
(w:point_set W),
continuous_at_2arg f (g w) (h w) ->
continuous_at g w -> continuous_at h w ->
continuous_at (fun w:point_set W => f (g w) (h w)) w.