The efficiency that you post all the time is only valid for an ideal gas. Therefore the approximation is the ideal gas law, not the second law. Using a real gas will yield a lower efficiency which is not a violation of the law.

Find an English version of a thermodynamics textbook and take a screenshot to prove your argument.

I'll just take German Wikipedia (and my textbooks and lecture notes here are German, too, but they also state clearly it's in ideal gas approximation):

https://de.wikipedia.org/wiki/Carnot-Prozess#Thermodynamik

In this paragraph it's clearly stated that it's "ein Zylinder mit Kolben und idealem Gas als Arbeitsmedium" which translates to "a cylinder with a piston and an ideal gas as the working medium".

I don't believe the information you crawled from the internet. I want a screenshot of the English version of the thermodynamics textbook for college.

The screenshot below is the Chinese version. The Carnot's Law states that Carnot efficiency is related to T1 and T2, and is independent of the type of working fluid.

It is not emphasized that it must be an ideal gas, and the Carnot efficiency of an ideal gas is the same as that of a real gas.

You can take a screenshot of the German version of the textbook, translate it into English, and display it.

If you insist: here is a screenshot from Max Planck's "Treatise on Thermodynamics" which I hope doesn't constitute copyright infringement. It was translated as "perfect gas", which, as a German I can guarantee you, is identical to an ideal gas.

So, Max Planck says it needs to be an ideal gas.

This is a calculation case, you may have read it too carelessly.

You can look at other chapters or textbooks. You are the birthplace of the second law of thermodynamics, don't make fun of yourself.

All Chinese textbooks have my own discourse.

No, this is the derivation of the n=1-TA/TB Carnot efficiency. By the way, if you read the full paragraph (§90 in the free upload of the Gutenberg project, the efficiency is implicated in equation 44), you will find that actually this is a calculation of the Carnot cycle by using the first law of thermodynamics upon whose validity even you insist.

But, as always, you provide great entertainment.

Please take another look at this picture.

Additionally, I have published a simple yet comprehensive article.

Article Breaking away from the second law of thermodynamics, the fir...

Your "comprehensive" article is based on the same false assumption as this one. Read Planck's original book, it's free at Project Gutenberg and, as you said yourself, Germany in general and Planck in particular are the original sources, so he knows what assumptions he made. If you base your life on bad resources and don't want to see that, that's not the fault of my countrymen...