\(RM1,\aleph_{\alpha }\): The representation theorem for multi-algebras with \(\aleph_{\alpha }\) unary operations:  Assume \((A,F)\) is  a  multi-algebra  with \(\aleph_{\alpha }\) unary operations (and no other operations). Then  there  is  an  algebra \((B,G)\)  with \(\aleph_{\alpha }\) unary operations and an equivalence relation \(E\) on \(B\) such that \((B/E,G/E)\) and \((A,F)\) are isomorphic multi-algebras.

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

The Closed Graph Theorem for operations between Fréchet Spaces: Suppose \(X\) and \(Y\) are Fréchet spaces, \(T:X\to Y\) is linear and \(G=\{(x,Tx): x \in X \}\) is closed in \(X\times Y\). Then \(T\) is continuous. Rudin [1991] p. 51.