Suppose R is the set of real numbers and all integrals are of the subdivisionrefinement
type. Suppose each of G and H is a function from R ×R to R and each of f and h is
a function from R to R such that f(a) = h(a), dh is of bounded variation on
[a,x], and ∫
_{a}^{x}H^{2} = ∫
_{a}^{x}G^{2} = 0 for x > a. The following two statements are
equivalent:
(1) If x > a, then f is bounded on [a,x], ∫
_{a}^{x}H exists, ∫
_{a}^{x}G exists,
(RL)∫
_{a}^{x}(fG + fH) exists, and
(2) If a ≦ p < q ≦ x, then each of ⋅_{p} Π^{q}(1 + H) and ⋅_{p} Π^{q}(1 − G)^{−1} exists and
neither is zero,
exists, and
f(x)  = f(a)⋅_{a} Π^{x}(1 + H)(1 − G)^{−1}  
  + (R)∫
_{a}^{x}[⋅_{
t} Π^{x}(1 + H)(1 + G)][(1 − G)^{−1}]dh.   
