Section well_def_1.
Variable F : Ens.
Hypothesis w : well_definedness F.
Variable R : Ens.
Hypothesis h : SUB R F.
Variable X Y Z : Ens.
Hypothesis i : IN (PAIR X Y) R.
Hypothesis j : IN (PAIR X Z) R.
Let step1 : IN (PAIR X Y) F.
Proof h (PAIR X Y) i.
Let step2 : IN (PAIR X Z) F.
Proof h (PAIR X Z) j.
Proof w X Y Z step1 step2.
End well_def_1.
Proof well_def_1_Out.
Record is_a_function (F : Ens) : Prop :=
{function_is_a_relation : is_a_relation F;
function_well_definedness : well_definedness F}.

Theorem function_sub :
forall (F : Ens) (f : is_a_function F) (G : Ens) (s : SUB G F),
is_a_function G.