The Finite Development theorem (FD) is a fundamental theorem in the theory of the syntax of the lambda-calculus. It gives sense to parallel reductions by stating that one can contract any given set of (possibly nested) redexes in any lambda term without looping and caring about the order in which these redexes are contracted. This theorem can be used to prove the Church-Rosser property, thus insuring determinism of reductions and uniqueness of normal forms. This paper explains how to extend the

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... FD theorem to a finite number of creations of new redexes, i.e. redexes which do not exist in the initial term. This generalized theorem (gFD) also provides a proof technique to show the completeness of various reduction strategies. Finally it gives a natural intuition to the strong normalization property of the standard first-order typed lambda-calculus. The results in this article are not new, but were often mixed with other arguments; the aim of this paper is to stress on this sole gFD theorem.