Gauges and Cages . Part II

D denotes a non empty subset of E 2 T , and C denotes a compact connected non vertical non horizontal subset of E 2 T. We now state the proposition (1) For all sets A, B such that for every set x such that x ∈ A there exists a set K such that K ⊆ B and x ⊆ K holds A ⊆ B. Let m be an even integer. Note that m + 2 is even. Let m be an odd integer. One can verify that m + 2 is odd. Let m be a non empty natural number. One can check that 2 m is even. Let n be an even natural number and let m be a non empty natural number. One can verify that n m is even. One can prove the following propositions: (2) If r = 0, then