>>562の続き

一方、
(dg/ds)(s, μ + h) = f(s, g(s, μ + h), μ + h)
(dg/ds)(s, μ) = f(s, g(s, μ), μ)

よって、
g(t, μ + h) - g(t, μ)
= ∫[t_0, t] (f(s, g(s, μ + h), μ + h) - f(s, g(s, μ), μ)) ds ← (1)

同様に
∫[t_0, t] (dψ/ds)(s, μ)h ds = ψ(t, μ)h - ψ(t_0, μ)h = ψ(t, μ)h

(dψ/ds)(s, μ) = (d_2)f(s, g(s, μ), μ)ψ(s, μ) + (d_3)f(s, g(s, μ), μ)

よって、
ψ(t, μ)h = ∫[t_0, t] (d_2)f(s, g(s, μ), μ)ψ(s, μ)h ds
+ ∫[t_0, t] (d_3)f(s, g(s, μ), μ)h ds ← (2)

(1) と (2) より

g(t, μ + h) - g(t, μ) - ψ(t, μ)h
= ∫[t_0, t] (f(s, g(s, μ + h), μ + h) - f(s, g(s, μ), μ)) ds
- ∫[t_0, t] (d_2)f(s, g(s, μ), μ)ψ(s, μ)h ds
- ∫[t_0, t] (d_3)f(s, g(s, μ), μ)h ds ← (3)

(続く)