Proposition 10 can be proved in a way analogous to the proof of Proposition 3. The statement of Proposition 10 remains true if we replace (27) with mu_{t,t+1}(m,V =phi_{t}(m+V_{t}),quad minbar{L}^{0}_{t+1}, Vinmathbb{V}^{p}.
Since by Proposition 1, we have shown that BPeMRF behaves as BP, it suffices to prove the statement of Proposition 2 for the BP algorithm.
As I establish in the second statement of Proposition 1, women's human capital and the degree of polygyny are inversely related.
The following example shows that the finite-≤ property of the extended quasi-metric space cannot be deleted in the statement of Proposition 19.
See Appendix A. The intuition behind the first statement of Proposition 1 is as follows.14 Holding the income of poor men constant, an increase in the income level of rich men enlarges their choice sets.
The proof of the theorem follows from the statements of Propositions 1 and 2 given below.
(ii) Since from (3.56), we have that (3.15) and (3.37) are fulfilled, we get from (3.16) and statement (ii) of Propositions 3.1 and 3.2 that the fuzzy difference equation (1.6) has unique positive solution such that (3.27) holds, and a nonnegative equilibrium, such that (3.38) and (3.39) hold.
Then from the statement (ii) of Proposition 3.1 and statement (ii) of Proposition 3.3 we have that the unique positive solution of (3.61) with initial values nearly converges to the nonnegative equilibrium with respect to as and converges to with respect to as.
(i) If (3.36) holds then from (3.10), (3.41), (3.42), and statement (i) of Proposition B, we get that (3.43) This completes the proof of statement (i).
Thus, in view of the statement (2) of Proposition 4.2, we only need to consider the cases (2a)–(2d) in this statement.
