Text: sometimes it can be shown mathematically that a cryptosystem is strong (if x can break this system, than x can solve the well-known difficult problem of factoring integers)

Correct capitalization: Sometimes it can be shown mathematically that a cryptosystem is strong (if x can break this system, than x can solve the well-known difficult problem of factoring integers)

Text: several scientific research have verified this theory, proving that consumers wouldn't have to be concerned about future health issues with continued application of the product.

Correct capitalization: Several scientific research have verified this theory, proving that consumers wouldn't have to be concerned about future health issues with continued application of the product.

Text: at least some of them are continuing her legacy.

Correct capitalization: At least some of them are continuing her legacy.

Text: both themed and "free" sessions are available.

Correct capitalization:
Both themed and "free" sessions are available.