Two programs are said to be mutually terminating if they terminate on exactly the same inputs. We suggest a proof rule that uses a mapping between the functions of the two programs for proving mutual termination of functions f, f'. The rule's premise requires proving that given the same arbitrary input in, f(in) and f'(in) call mapped functions with the same arguments. A variant of this rule with a weaker premise allows to prove termination of one of the programs if the other is known to terminate for all inputs. We present an algorithm for decomposing the verification problem of whole programs to that of proving mutual termination of individual functions, based on our suggested rules.
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