rule incorporates a simplification over a rule where are the free variables of the branch, not of alone. This rule can be further simplified by the reuse of a function symbol if it has already been used in a formula that is identical to up to variable renaming. The formula represented by a tableau is obtained in a way that is similar to the propositional case, with the additional assumption that free variables are considered universally quantified. As for the propositional case, formulae in each branch are conjoined and the resulting formulae are disjoined. In addition, all free variables of the resulting formula are universally quantified. All these quantifiers have the whole formula in their scope. In other words, if is the formula obtained by disjoining the conjunction of the formulae in each branch, and are the free variables in it, then is the formula represented by the tableau. The following considerations apply: The assumption that free variables are universally quantified is what makes the application of a most general unifier a sound rule: since means that is true for every possible value of , then is true for the term that the most general unifier replaces with. Free variables in a tableau are rigid: all occurrences of the same variable have to be replaced all with the same term. Every variable can be considered a symbol representing a term that is yet to be decided. This is a consequence of free variables being assumed universally quantified over the whole formula represented by the tableau: if the same variable occurs free in two different nodes, both occurrences are in the scope of the same quantifier. As an example, if the formulae in two nodes are and , where is free in both, the formula represented by the tableau is something in the form . This formula implies that is true for any value of , but does not in general imply for two different terms and , as these two terms may in general take different values. This means that cannot be replaced by two different terms in and . Free variables in a formula to check for validity are also considered universally quantified. However, these variables cannot be left free when building a tableau, because tableau rules works on the converse of the formula but still treats free variables as universally quantified. For example, is not valid (it is