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You would use $.ajax(). You wouldn't need the silly code to check which browser.
But do you need to support IE5 and IE6? The latest jQuery (3.2.1) no longer supports them, so could not replace your code. 2.x and 3.x don't support old IE. Late 1.x and 2.x versions are parallel with same functionality but different code. The 1.x series supports old IE.
But you could use jQuery 1.1.12, the last version that supports old IE.
Where did you get that code? An old site that you are updating? A musty book that should be burned? A blog that Google should bury?