# Shoenfield's Mathematical Logic Chapter 1 Problem 3

Edit for context:

An expression in this context is any finite sequence of symbols of the language in question.

"A designator is an expression which is either a term or a formula. Every designator has the form $$uv$$1...$$v$$n where $$u$$ is a symbol and the $$v$$i are designators, and n is a natural number determined by $$u$$. For example, if $$u$$ is a variable, $$n = 0$$; if $$u$$ is a $$k$$-ary function symbol, $$n = k$$; if $$u$$ is the existential quantifier, $$n = 2$$. We call $$n$$ the index of $$u$$. We say two expressions are compatible if one can be obtained from the other by adding an expression to the right of the other (possibly the empty expression). If $$uv$$ and $$u'v'$$ are compatible, then $$u$$ and $$u'$$ are compatible. If $$uv$$ and $$uv'$$ are compatible, then $$v$$ and $$v'$$ are compatible."

Show that if $$uv$$ and $$vv'$$ are designators, then either $$v$$ or $$v'$$ is the empty expression.