Superfluous thinking may be extremely valuable in facilitating grasp of the proof text and in enabling one to understand the idea underlying the proof steps; but it is not necessary for thinking through the proof.
This iterative method is the same underlying the proof of the First Recursion Theorem, and it has a long history.
The first axiomatization was due to Zermelo (1908) and it came as a result of the need to spell out the basic set-theoretic principles underlying his proof of Cantor's Well-Ordering Principle.
The approximation method underlying the proof of Proposition 29 essentially bridges the gap between the autonomous linear transport equation with "more regular" coefficients and its counterpart with "less regular" coefficients.
The method is based on the mathematical arguments underlying the proof that positive feedback loops are necessary for multistationarity to occur.
Creating ways for multi-party transactions to engage in queries that prove certain facts about a business, without ever accessing the data underlying those proofs.
The concept of a proof underlying Sophistical Refutations is similar to what is demanded of demonstrative knowledge in Posterior Analytics (I ii 71b20), viz., that the premises must be "true, primary, immediate, better known than, prior to, and causative of the conclusion," except that the first three conditions do not apply to deductions in which the premises are obtained through questioning.
A very conservative attitude would be to conclude that observed inter-group differences only provide a strong indication (rather than proof) of underlying differences in scanner calibration, signaling the need for additional quality assurance measures such as further phantom measurements to control or correct the current scanner calibration.
In cases of clinically suspected glomerulonephritis or interstitial nephropathy but missing biopsy proof, the underlying kidney pathology was grouped as 'unknown'unknown
