Section Subtype.
Comments "In Coq type theory, there is no primitive notion of subtype".
Comments "Then we have to define such a notion".
Variable E : Setoid.
Variable F : Type.
Variable i : F -> E.
Comments "We have implicitely defined a subset of" E "which is the image of"
i ".".
Comments "As a setoid, this subset has" F
" as carrier, and we identify two elements of" F
"which have the same image by" i ":".
Definition subtype_image_equal (x y : F) : Prop := Equal (i x) (i y).

Lemma subtype_image_equiv : equivalence subtype_image_equal.