First, repair disk permissions and trash the FCP preference files. You can use Preference Manager for that.
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FCP6 is not officially supported from Lion, from what I remember. FCP7 is the version that is supported from Lion. FCP6 does not install on Lion because it isn't supported, also because some parts ran on older code. I remember specifically the install module was still in old code, which required Rosetta to run. Rosetta was an invisible app running in the background to run applications from the old PPC days. Rosetta was dead in Lion, but there was a workaround to install it. But FCP6 is not an application I would suggest running in ML. Stick to Snow Leopard if you must. If you want to run Mountain Lion, you can get an FCP7 upgrade, but it's pointless because FCP7 is dead software and is also no longer supported. FCP7 works pretty okay in ML, but who knows which future cat will kill it.
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