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Prerequisites: Algebraic General Topology.

Conjecture. L[f][f]idomAatomsLi for every pre-multifuncoid f of the form whose elements are atomic posets.

A weaker conjecture: It is true for forms whose elements are powersets.

The following is an attempted proof:

If arityf=0 our theorem is trivial, so let arityf0. Let is a well-ordering of arityf with greatest element m.

Let Φ is a function which maps non-least elements of posets into atoms under these elements and least elements into themselves. (Note that Φ is defined on least elements only for completeness, Φ is never taken on a least element in the proof below.) {\color{brown} [TODO: Fix the universal set paradox here.]}

Define a transfinite sequence a by transfinite induction with the formula Failed to parse (syntax error): {\displaystyle a_c = \Phi \left\langle f \right\rangle_c  \left( a|_{X \left( c \right) \setminus \left\{ c \right\}} \cup L|_{\left( \operatorname{arity} f \right) \setminus X \left( c \right)} \right)} .

Let bc=a|X(c){c}L|(arityf)X(c). Then ac=Φfcbc.

Let us prove by transfinite induction acatomsLc. ac=ΦfcL|(arityf){c}fcL|(arityf){c}. Thus acLc. [TODO: Is it true for pre-multifuncoids?]

The only thing remained to prove is that fcbc0

that is Failed to parse (syntax error): {\displaystyle \langle f \rangle _ c  ( a|_{ X ( c ) \setminus \{ c \} } \cup L|_{( \operatorname{arity} f ) \setminus X ( c )} ) \neq 0} that is y≭fcbc.