How can I get VMWare Fusion 4.0 to work on Mountain Lion?

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Upgrade to the latest version of VMWare Fusion, in your situation you will need manually download it as indicated below for the situation you describe in your question:

I upgraded to Mountain Lion and now VMware Fusion 4 won’t start. What happened?

Mountain Lion appears to detect some older versions of VMware Fusion 4 and prevent them from launching. To resolve this issue, login to http://my.vmware.com/ (free product registration required) and download VMware Fusion 4.1.3. It can be found under “Products & Downloads > All Downloads > VMware Fusion”

According to VMware Fusion and OS X Mountain Lion FAQ

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