As [x.sup.n] > 0, we deduce by assumption with m = 0 that f (s, [x.sup.n.sub.s]) [greater than or equal to] -[gamma][x.sup.n.sub.s] for all s [member of] [0,[T.sub.0]] hence

By induction, we then deduce that [mathematical expression not reproducible] for all t [member of] [0,[T.sub.0]].

By induction, we deduce that [[parallel]x - y[parallel].sub.[0,t]] [less than or equal to] 2C[k.sup.n][t.sup.n]/n!

Then, we also deduce that the sequence [([x.sup.m]).sub.m[greater than or equal to]1] uniformly converges to some function [z.sup.n] on [[t.sub.n], [t.sub.n] + [T.sup.*.sub.0]] such that

Therefore, by definition of [T.sup.*], we deduce that [t.sub.n] + [T.sup.*.sub.0] [less than or equal to] [T.sup.*] for all n.

remarking via the hypotheses that 10[x.sub.n,1] = n + 71, clearly 10[x.sub.n,1] < n + 71, now using the previous inequality and (4), then it becomes trivial to deduce that statement Z([x.sub.n,1], [gamma](n), [gamma]'(n)) is true, otherwise, Z(xni1,7(n),7'(n)) is false, and using (4), then we clearly deduce that it is false that 10xn1 < n + 71, therefore, 10xn1 > n + 71.

In this case, using the definition of Y([x.sub.n,1], [gamma](n), [gamma]'(n)) (see Definitions 2.1), then we immediately deduce that statement Y([x.sub.n,1], [gamma](n), [gamma]'(n)) is of the form

Indeed, remarking by (5) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and recalling that we are in the case where Z([x.sub.n,1], [gamma](n), [gamma]'(n)) is true, then, using the previous, it becomes trivial to deduce that Z([x.sub.n,1], [gamma](n), [gamma]'(n)) and Y([x.sub.n,1], [gamma](n), [gamma]'(n)), are simultaneously true.

Otherwise, using the definition of statement Z([x.sub.n,1], [gamma](n), [gamma]'(n)) (see Definitions 2.1), then we immediately deduce that statement Z([x.sub.n,1], [gamma](n), [gamma]'(n)) is of the form

Here, using only the immediate part of the generalized Fermat induction, simple definitions, elementary arithmetic congruences, elementary complex analysis, elementary arithmetic calculus, reasoning by reduction to absurd and properties (2.4) and (2.3) of Remark 2, we prove a Theorem which implies the Mersenne primes conjecture; moreover, from our Theorem, we immediately deduce that the Mersenne primes conjecture that we solved, was only an elementary consequence of the Goldbach conjecture.

For that, let n + 11; recalling that in particular n is of type 37, clearly (by using the definition of type 37), n [equivalent to] 0 mod(37) and using the previous congruence, we immediately deduce that

That being so, if [m.sub.n,1] <n + 11, then, using congruences (R.1.5) and (R.1.4), it becomes trivial to deduce that the previous inequality immediately implies that

Now consider the quantity R[Z(n)] + I[Z(n)]; then, using (R.1.7), it becomes immediate to deduce that

Cullen

deduces what he can from prefatory references suggesting associations with the Pierrepont family of Dorchester and with James Howell, author of Dodona's Grove; other scholars will no doubt discover additional facts in due course.

He insists on a dark power as the cause and iconography of men in black: "Alone or in ranks, the man in black is the agent of a serious power; and of a power claimed over women and the feminine." Accordingly, he

deduces gravely, "We live now in the aftertow of the black wave's latest rise and breaking....