A family of functions f(), g(),... is defined on integer pairs in |Z^2 . They are nonzero in general `on land' but have zeros which congregate in n x n lattice square regions `in ponds'. I want to establish that they satisfy a certain system of linear relations `bridges' between quartets of values f(w), f(x), f(y), f(z) at arguments w, x, y, z , lying one on each bank of a pond. An inductive argument exists which constructs a bridge over an n-pond of f() from one over an (n-2)-pond of a related family member g() ; normally all that should then be necessary is progressively shrinking the pond until it dries up when n passes zero. But there is a snag. The construction of g() from f() proceeds stepwise: starting from (say) x on the bank, it fans out point-by-point around the pond and thence over the entire land (essentially common to both functions); however it breaks down inside ponds, which must be filled separately once their banks are built. As a result, after a full circuit skirting the pond, g(x) might conceivably become inconsistent with its previous value; at this stage g() may be assumed single-valued only within simply-connected regions of land. Now if only I could somehow prove g() globally single-valued, I could go ahead and build the bridge; or if only I already had the bridge, I could deduce g() must be single-valued. One way to break into this cycle might be to assume that g() branches around ponds, then formulate the induction on n to avoid reliance on consistency by cutting the plane from pond centre away from currently selected w, x, y, z , in order finally to deduce that the entire flapdoodle was unnecessary! I would be happier about resorting to this bizarre stratagem if I knew of a similar situation elsewhere. For example, is there an instance of an identity involving some function of a complex variable, constructed via iterated analytic extension around a singularity, such the proof requires one initially to assume a branch point; despite which the identity finally implies that the function is single-valued? Fred Lunnon