The best replacement for assumption is presumption.

The best alternatives to assumption are rated as following:
★★★★★  presumption
★★  hypothesis
★★  opinion
★★  inference
★★  constraint
★  refinance
★  expectation
★  dependency
★  ascension
★  perception
★  novation
★  poveda
  guess
  implication
  premise
  assignment
  prediction
  hypothetical condition
  presupposition
  supposition
  deduction
  stereotype
  educated guess
  miriam
  dormition
  on the assumption
  stonehill
  subject to
  merrimack
  assuming
  sacred heart
  bettendorf

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Speaking about assumption, one probably shouldn't forget about release of liability.

Other search terms related to assumption and their ratings are:
★★★★★ release of liability
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