Final answer:
To negate the mathematical statement 'for every x, for every y, P(x, y)', one would assert 'there exists an x, there exists a y, such that not P(x, y)', which flips the inequality and changes the quantifiers.
Step-by-step explanation:
The given statement seems to be incomplete. However, based on the context provided, to negate a mathematical statement that asserts 'for every x, for every y, P(x, y)', you would assert 'there exists an x, there exists a y, such that not P(x, y)'. For example, if the original statement is 'for every x, for every y, x² - 2x - y < 0', the negation would be 'there exists an x, there exists a y, such that x² - 2x - y ≥ 0'. The process of negation essentially involves flipping the inequality and changing universal quantifiers 'for every' to existential quantifiers 'there exists', while also inserting a 'not' for the condition.