The country was England, and "proved son" in this context means "legitimate son."
Leofric probably had many sons, but only one by his lawful spouse, Lady Godiva. More to the point, he had affairs with a bunch of other women, who produced sons that may or may not have been his.
The more important issue (absent today's DNA tests) is: was Lady Godiva's son actually his? English law strongly presumes this to be the case, except when it was basically impossible (e.g. when the man had been away from home for a whole year before the child's birth). Given that Leofric cohabited with his wife, and the child at least somewhat resembled him, English law presumed, and therefore "proved" that the child was his.