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Theorems 3.2, 3.4 and 3.5 suggest a reduction procedure when determining the fixed point property for a set: Find a suitable retraction on the set and then decide if the retract has the fixed point property instead. Iterate this procedure until no suitable retraction can be found. In finite sets this procedure is well understood (cf. e.g., [7], [31], [39], [95]). In infinite sets one might be able to iterate the above idea infinitely often, thus being faced with the problem how to get past the limit ordinal.gif This problem has been addressed successfully by Li and Milner (cf. [70]-[77]) for large classes of ordered sets. We will not pursue their work in full generality here, but rather present a very simple-minded approach to the same problem. Our approach bypasses the problems at the limit ordinal in a brute force fashion, namely by demanding that there are no problems at the limit ordinal. While this might be very simplistic, the exposition is freed of many technical details and we can also ``mix" several different kinds of retractions in the dismantling process.