Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures.In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy.As applications of this technique it is shown that • — Graph Connectivity is not expressible in existential monadic second-order (...) logic , even in the presence of a built-in linear order,• — Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n0, and• — the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation. (shrink)