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Further in the W-9 instructions ...  

When you got the EIN for the single member LLC that was not incorporated you had to report your SS#  so the EIN is automatically linked to your SS# and both numbers are reported on the Sch C  thus if you get a 1099 in either number they are both recognised as income to the same Sch C on your personal return.   So once again, it you cannot use it on the W-9 then why did you bother to get the EIN to begin with.