In Theorem 2.14, if we assume that the quasi-ordered metric space ( X, d, ⪯ ) is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed ⪯-monotone property can be dropped, since the proof is still valid in this case.
In Theorem 2.3, if we assume that the quasi-ordered metric space ( X, d, ⪯ ) is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed ≼ I -monotone can be dropped, since the proof is still valid in this case.
In Theorem 2.16, if we assume that the quasi-ordered metric space ( X, d, ⪯ ) is complete (not mixed-monotonically complete), then the assumption for f having the sequentially mixed ≼ I -monotone or ⪯-monotone property can be dropped, since the proof is still valid in this case.
The wording of a fairness opinion describes its scope, the work that was completed and the assumptions made.
Note also that whenever the cone metric space ( X, d ) is complete, then the assumption of the second part of Theorem 9.9 is satisfied automatically.
When he signed a new deal after his switch to the Bernabeu fell through in the summer, it was completed amid the assumption a buy-out clause was inserted in his contact.
"I think that's plenty of time to get an act through the UK parliament and for the preparatory work to be completed for the assumption of those powers by the people of Wales".
"To rescind a grant award after it was announced and after a project was completed under the assumption that those funds would be paid would be economically devastating to this small community".
The first optimization-based control analysis was completed with the assumption that sunitinib would be administered every day during the 28-day cycle.
The authors are advised to present the complete underlying assumptions and governing PDE equations, boundary conditions, initial conditions, and the parameters used for the modeling.
