"Provided that" and "on the condition that" are common synonyms for "with the stipulation that." These phrases indicate that there is a requirement or condition that must be met for something to occur. "Subject to the understanding that" and "contingent upon" can also be used interchangeably with "with the stipulation that," while "assuming that" implies a hypothetical situation that is conditional upon a particular agreement. In contract law, "under the terms of" and "pursuant to" can be used instead of "with the stipulation that" to specify the precise conditions that must be fulfilled. Overall, these synonyms allow for greater flexibility in wording when expressing that certain requirements must be met before a particular action can be taken.