In this lecture we compare three types of analogies based on generalizations and their instantiations:

1. Generalizations w.r.t. invariant parts of proofs (e.g. graphs of rule applications etc);
2. Generalizations w.r.t. to an underlying meaning.(Here proofs and calculations are considered as trees of formal expressions. We analyze the well known calculation of Euler demonstrating that the 5th fermat number is compound.)
3. 3. Generalizations w.r.t. the premises of a proof (This type of analogies is especially important for juridical reasoning.).