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As far as I'm aware, Vintager, the only updating that has been done is with Kaspersky. I really don't want to change antivirus as, up to now, this has been the least intrusive one and the most effective that I've found.
PinkyBoyid, thanks for your continued efforts. I've followed your instructions, but the troubleshooter said that it could not detect a problem. Tried the ping thing again, with (possibly predictably) the same result as before. Can I just check - all the spaces in the command thread you told me to type in should be there, should they? Just a thought that maybe there's a problem there? I'm not sure why this would cause a problem with Firefox and not Internet Explorer.