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"We managed to feed the twins continuously without any formula for 14 months".
Full breast feeding was defined as predominant breast feeding without any formula or solids, but allowing water and vitamins.
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I resubmitted a lengthy response (without any formulae), and I prevailed--because of the right words and not just the right formula!
We've gone out and done things our own way without subscribing to any formula of how things should be done.
Hume's moral system aims at the happiness of others (without any such formula as "the greatest happiness of the greatest number") and at the happiness of self.
Some of the analytical methods need transformation, and some of them deal with the equation without any transformation formula or other restrictive assumption.
Thus, the hybrid logic with the binder ∀ has the same expressive power as first-order logic with equality and the hybrid logic without the binder ∀ (but with the binder ↓) has the same expressive power as the bounded fragment of first-order logic (note that the translation STa of any formula φ without the binder ∀ is in the bounded fragment).
Classical predicate logic has the desirable feature that co-extensive open formulas may be substituted for one another in any formula salva veritate i.e., without changing that formula's truth value.
Results of tests on different alternative berm breakwater designs, which may be considered as practical/economic designs, have also been included, but without any comparison with any formulae simply because the formulae do not cover the specific designs.
You can perform calculations without having to input any formulas or copy any cells.
"Like everyone else in Brazil, we started off late in this business, without a formula for exporting," said Hugo Campos, the industrial production director.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com