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Source. “Main Theorem on the Equivalence,” PDF p. 2 of the author survey. The survey identifies it as Theorem 4 of Sun's 1989 paper Systems of congruences with multipliers.

Conventions

Two weighted systems

$$A=\{(\lambda_s,a_s,n_s)\}_{s=1}^k, \qquad B=\{(\mu_t,b_t,m_t)\}_{t=1}^{\ell}$$

are equivalent, written $A\sim B$, when their covering maps agree. Let $P$ be a set of primes, let $M$ be a left $R$-module, and let $F$ map into $M$. Assume that whenever $p\in P$, $(x,y)\in\operatorname{Dom}(F)$, and $r\in R(p)=\{0,\ldots,p-1\}$, one has

$$\left(\frac{x+r}{p},py\right)\in\operatorname{Dom}(F).$$

Statement

The following two conditions are equivalent.

1. Whenever $A\sim B$, all weights lie in $R$, and every prime divisor of every modulus belongs to $P$, one has

$$\sum_{s=1}^k\lambda_s F\left(\frac{x+a_s}{n_s},n_sy\right) = \sum_{t=1}^{\ell}\mu_t F\left(\frac{x+b_t}{m_t},m_ty\right)$$

for every $(x,y)\in\operatorname{Dom}(F)$.

2. For every $p\in P$ and $(x,y)\in\operatorname{Dom}(F)$,

$$\sum_{r=0}^{p-1}F\left(\frac{x+r}{p},py\right)=F(x,y).$$

The survey prints a lowercase $f$ in the theorem's opening phrase and uses $F$ throughout the hypotheses and formulas; this transcription uses $F$ consistently.

Proof scope. The survey says this follows by induction from the Generating Theorem. The original 1989 proof was not acquired or independently checked here.