I'm trying to apply the continuity and differentiability theorems, shown in the book "Ordinary Differential Equations" from Philip Hartman pages 94 and 95 respectively,

Continuity Theorem. Letf(t, y, z) be continuous on an open (t, y, z)-set E with the property that for every (t0, y0, z) e E, the initial value problem (1.2), with z fixed, has a unique solution y(t) = n(t, t0, y0, z). Let w_ < t < w+ be the maximal interval of existence of y(t) = n(t, t0, y0, z). Then w+ =co+(t0, y0, z) [or w_ = w_(t0, y0, z)] is a lower [or upper] semicontinuous function of (f0, y0, z)e E and n(t, tQ, yQ, z) is continuous on the set w_ < t < w+, (t0, y0, z) e E.

Differentiability Theorem (Peano). Letf(t, y, z) be continuous on an open (t, y, z)-set E and possess continuous first order partials df/dy^k, df/dz^i with respect to the components of y and z: (i) Then the unique solution y = n(t, t0, y0, z) of (1.2) is of class C^1 on its open domain of definition w_ < t < w+, (t0, y0, z) e E, where w± = w±(t0, y0, z). (ii) Furthermore, if J(t) = J(t, t0, y0, z) is the Jacobian matrix (df/dy) of f(t, y, z) with respect to y at y = n(j, to, y0, z)...

to the Prey-Predator system whose differential equations are

x1'= x1 * (a-b*x2),

x2' = x2 * (d*x1-c),

where a, b, c and d are constants.

I would like to see step by step to test the theorems for those differential equations.