Why bitcoin private key WIF format always be started in K or L?

Question

According to https://en.bitcoin.it/wiki/Wallet_import_format, the bitcoin private wif format key is create by below step. And it says that the compressed wif key always start with 'K' or 'L', this mean 80+private-key+01/ mod 58' last result is always 18 or 19. I know the big number mod 58 will result in 0-57, why the last remainder of wif key mod 58 always is 18 or 19 Would someone give me Mathematical Proof of this?

 #1 take a private key
 priv='d7c9e36a4d31039d4b94a43246a4a5f0767f2f55ef9a359901c82c41fb1e4bff'
 compressed='01'

 #2 add version
 version='80' #mainnet
 priv=version+priv+compressed

 #3 perform double hash-256 of priv in step#2
 checksum=dhash256(priv.decode('hex'))

 #4 get first 4 bytes for checksum
 checksum=checksum.encode('hex')[0:8]

 #5 add checksum to priv in step#2
 wifhash=priv+checksum

 #6 final bash58encode
 wif= b58encode(wifhash.decode('hex'))

 # compressed --> K or L
 print wif

 #compressed: L4TB9Z3dXeKW4tyF7EVqbczrEA5hjSdjG1QvBd1CeYTQHm2YincA

Answer 1

Base58 check encoding is really just a base conversion. It follows the standard base conversion algorithm in mathematics.

Right before we perform the base conversion, we will have a sequence of 38 bytes. It's 0x80 + 32 byte private key + 1 byte compression indicator + 4 bytes checksum. For the encoding itself, these 38 bytes are interpreted as a very large number, and then a base 58 version of that number is computed.

Let's look at our number in hexadecimal (base 16). The constraints we have mean that the number will be at least 0x0x8000000000000000000000000000000000000000000000000000000000000000000000000000 and at most 0x80ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff. Now in base 58, these two numbers are KwDiBf89QgGbjEhKnhXJuH7LrciVrZi3qYjgd9M7rFU73NTmk4WB and L5oLkpV3aqBjhki6LmvChTCq73v9gyymzzMpBbhDLjDpKhfJuTe6.

Since every possible value that can be encoded falls between those two base 58 numbers, all possible values that can be encoded must begin with a K or L.