>>598の続き

よって、
∫[t_0, t_1] ((dg/dt)(t, p + h) - (dg/dt)(t, p)) dt = g(t_1, p + h) - g(t_1, p)

一方、
(dg/dt)(t, p + h) = f(t, g(t, p + h), p + h)
(dg/dt)(t, p) = f(t, g(t, p), p)

よって、
g(t_1, p + h) - g(t_1, p)
= ∫[t_0, t_1] (f(t, g(t, p + h), p + h) - f(t, g(t, p), p)) dt ← (1)

同様に
∫[t_0, t_1] (dψ/dt)(t, p)h dt = ψ(t_1, p)h - ψ(t_0, p)h = ψ(t_1, p)h

(dψ/dt)(t, p) = (d_2)f(t, g(t, p), p)ψ(t, p) + (d_3)f(t, g(t, p), p)

よって、
ψ(t_1, p)h = ∫[t_0, t_1] (d_2)f(t, g(t, p), p)ψ(t, p)h dt
+ ∫[t_0, t_1] (d_3)f(t, g(t, p), p)h dt ← (2)

(1) と (2) より

g(t_1, p + h) - g(t_1, p) - ψ(a, p)h
= ∫[t_0, t_1] (f(t, g(t, p + h), p + h) - f(t, g(t, p), p)) dt
- ∫[t_0, t_1] (d_2)f(t, g(t, p), p)ψ(t, p)h dt
- ∫[t_0, t_1] (d_3)f(t, g(t, p), p)h dt ← (3)

(続く)