I can't understand, a native bitcoin client for calculations - will he be able to take part in mining? If not, what kind of native mining software should use? Does he even exist? I only found third party products like cgminer, cudaminer, etc.
Is the native Bitcoin client intended for mining?
Long long A decade or so ago, back in the mists of time, in prehistoric days when dinosaurs roamed the earth, early versions of the Bitcoin software did indeed include a mining capability that used the pitiful capabilities of CPUs. But this function has long been rendered defunct by the advance of mining hardware such that one small ASIC is now more powerful at mining than a billion 4.6 million† modern high-end decentish desktop computers even if each were crammed with dozens of overclocked multi-core CPUs and GPUs. So the vestigial and useless appendix was removed from Bitcoin-core's DNA mainnet phenotype.
Your choice of mining hardware determines what specialised control software you use.
(Health warning: answer may contain microscopic traces of hyperbole)
† Roughly. Give or take an order of magnitude, or two.
I almost hate to provide a second answer to pair against RedGrittyBrick's beautifully worded answer, but there is a consideration that matters here beyond that of Mhash/s: profitability.
Most people run a miner for profit. Not many people are out there running miners just for the philosophical principle behind bitcoin.
The ASIC miners are extraordinarily efficient in terms of power. The Antminer 19 linked in RedGrittyBrick's answer mentions that it has an efficiency of 29.5 J/TH. That means for every 29.5J of energy which goes into the Antminer, it can do 1 trillion hashes.
For the poor i7 in the example, the numbers are dismal. An i7's TDP is about 95W. At 20GH/s (a number that does appear roughly in bitcoins wiki) it will take 50s to do trillion hashes, which is 4,750 J/TH.
The economics of bitcoin are such that you literally cannot pay for the power it takes to mine with the profits from the mining.