as proved above

USAGE SUMMARY

The phrase "as proved above" is correct and usable in written English.
It is typically used to refer back to evidence or arguments that have been presented earlier in the text. Example: "The results of the experiment support our hypothesis, as proved above."

✓ Grammatically correct

Science

News & Media

Encyclopedias

Human-verified examples from authoritative sources

Exact Expressions

1 human-written examples

Obviously, in (19), the source transmit power is independent of X 1, and the minimum of the source power is achieved when ( mathbf {U}_{tilde {H}_{1}}^{H} = mathbf {U}_{1}.) Besides, because we also have X 1=I M as proved above, B can be obtained as mathbf{B} = mathbf{V}_{1} mathbf{Lambda}_{1}^{-frac{1}{2}} mathbf{Lambda}_{tilde{H}_{1}}^{frac{1}{2}}mathbf{R}_{x}^{-frac{1}{2}}.

EURASIP Journal on Wireless Communications and Networking

Human-verified similar examples from authoritative sources

Similar Expressions

59 human-written examples

For (beta=-frac{1}{p}), the cosine system is minimal since, as was proved above, the coefficients (B_{n}) are found by concrete formulas in the form of an integral.

Journal of Inequalities and Applications

If (betain 2-frac {1}{p},4-frac{1}{p})), then, using the subetain 2-fraceta-2=beta') and removing the first cosine, we obtain the cosine system ({1}{p}-frac{beta'}{2})theta^{infty }_{n=1},1}), which, as was proved above, composes a basis in ((W^{1}_{p}(0,pi))), and thenefore the initial cosine system is not minimal in ((W^{1}_{p}(0,pi ))).

Journal of Inequalities and Applications

As we proved above, u defined by (20) is an optimal control of Problem (LQ SOC).

Advances in Difference Equations

The inequality (n_k ge frac{N}{2}) was proved above as an immediate consequence of the fact that the Weyl group W effectively acts only on the (psi _j,).

Complex Analysis and its Synergies

In this section, as an application of our results proved above, we deduce certain Suzuki-Wardowski type fixed point theorems.

Fixed Point Theory and Applications

Although this value was achieved by H qNMR measurements, it can be employed in subsequent H and P qNMR measurements as proven above.

Analytical and Bioanalytical Chemistry

It remains to be shown though whether this reaction can also be catalysed by GGT1 isolated from apocrine sweat glands, as opposed to GGT1 isolates from human liver and recombinant GGT1 as proven above.

Experimental Dermatology

Using the same approach above, the solution can be obtained as, as proved in [11].

EURASIP Journal on Wireless Communications and Networking

The possibility that these A. barbouri individuals embrace A. barbouri and complete unisexual-like mtgenomes is minimal as proven by the experiments we conducted above.

BMC Evolutionary Biology

We reveal the regional locations where house prices are soaring above average as well as proving popular with renters.

Independent

Expert writing Tips

Best practice

Use "as proved above" to clearly signal that you're building upon a point already substantiated within your document. This improves the logical flow and strengthens your argument.

Common error

Avoid using "as proved above" if the 'proof' is merely asserted and not rigorously demonstrated. Ensure the preceding text genuinely establishes the point you're referencing; otherwise, your argument will lack credibility.

Linguistic Context

The phrase "as proved above" functions as an adverbial phrase, modifying a clause by indicating that the statement is based on evidence or reasoning presented earlier in the text. As demonstrated by Ludwig, it refers back to a point already substantiated.

Expression frequency: Rare

Frequent in

Science

55%

News & Media

23%

Encyclopedias

22%

Less common in

Formal & Business

0%

Wiki

0%

Reference

0%

Ludwig's WRAP-UP

In summary, the phrase "as proved above" is a grammatically correct adverbial phrase used to refer back to previously substantiated arguments. As confirmed by Ludwig AI, it strengthens claims by linking them to prior proofs. Though relatively rare, it's most frequently found in scientific and academic writing, reflecting its formal tone. When using this phrase, ensure that the referenced proof is genuinely established to maintain credibility. Alternatives such as "as demonstrated previously" can be considered for nuanced meaning.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

as demonstrated previously

Focuses on the act of demonstrating something earlier in the text.

as shown earlier

Emphasizes the visual or evidential aspect of the prior information.

as established before

Highlights the act of setting up or confirming a fact or idea.

as mentioned earlier

Refers to something that was previously brought up in the discussion.

as noted above

Indicates a reference to something observed or written in a preceding part of the text.

as indicated previously

Emphasizes that something was pointed out or suggested before.

as we have seen

Draws attention to a conclusion or observation the reader has already encountered.

as previously stated

Highlights that something was articulated or expressed in the past.

pursuant to the foregoing

Uses a more formal tone to refer to what has already been presented.

in light of the foregoing

Implies that the current statement is based on the previously mentioned information.

FAQs

How can I use "as proved above" in a sentence?

Use "as proved above" to refer back to an argument or point that you've already substantiated earlier in your text. For instance, "The hypothesis is valid, as proved above, and can now be used for further testing."

What's a good alternative to "as proved above"?

You can use alternatives like "as demonstrated previously", "as shown earlier", or "as established before" depending on the specific context.

Is "as proved above" formal or informal?

"As proved above" is generally considered suitable for formal and academic writing, but it can also be used in other contexts where you need to refer back to something previously demonstrated or proven.

Can I use "as proved above" if I only presented evidence and not a formal proof?

While "as proved above" implies a certain level of rigor, it can be used if you've presented convincing evidence. However, consider using alternatives like "as demonstrated previously" or "as shown earlier" if the evidence is not a formal proof.