# Article

**Keywords:**

isomorphic representation; extensional relation; well-founded relation; set-pro\-lon\-gation

**Summary:**

By an $\in $-representation of a relation we mean its isomorphic embedding to $\Bbb E = \{\langle x,y\rangle;\,x\in y\}$. Some theorems on such a representation are presented. Especially, we prove a version of the well-known theorem on isomorphic representation of extensional and well-founded relations in $\Bbb E$, which holds in Zermelo-Fraenkel set theory. This our version is in Zermelo-Fraenkel set theory false. A general theorem on a set-prolongation is proved; it enables us to solve the task of the representation in question.

References:

[V] Vopěnka P.:

**Mathematics in the Alternative Set Theory**. TEUBNER TEXTE Leipzig (1979).

MR 0581368