Foreign national has the burden of proving they are only married to the petitioner, and to the extent they were married before, there is a valid divorce certificate in the application package. Yes, they may independently do a background check and if they catch lies....it's worse than if you never applied
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There are various checks that are done. In one check they can do there is a list of associates and past spouses and other relationships will come up. Consult an attorney who can advise and represent you in your area. Be advised to always be truthful honest and complete in all answers to USCIS and consult an attorney before filing any forms.
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If the sponsor has been married previously, the sponsor will be required to provide evidence of a divorce in order to sponsor someone based on marriage. If you plan on marrying someone who is already married, then even if a green card were issued, it would eventually be revoked because it would have been issued in error. Yes, USCIS does run various checks with various systems inside and outside DHS as soon as a case is accepted for processing.
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