  The probability distribution of a function of a subsystem conditioned on the
value of the function of the whole, in the limit when the ratio of their values
goes to zero, has a limit law: It equals the unconditioned marginal probability
distribution weighted by an exponential factor whose exponent is uniquely
determined by the condition. We apply this theorem to explain the canonical
equilibrium ensemble of a system in contact with a heat reservoir. Since the
theorem only requires analysis at the level of the function of the subsystem
and reservoir, it is applicable even without the knowledge of the composition
of the reservoir itself, which extends the applicability of the canonical
ensemble. Furthermore, we generalize our theorem to a model with strong
interaction that contributes an additional term to the exponent, which is
beyond the typical case of approximately additive functions. This result is new
in both physics and mathematics, as a theory for the Gibbs conditioning
principle for strongly correlated systems. A corollary provides a precise
formulation of what a temperature bath is in probabilistic term
