The Truth: This is accurate. Charles Thompson said it denoted that the United State of America ought to rely on their own virtue.
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Heraldically speaking, the eagle is the supporter of the shield. While it is more common to have two supporters, one on either side, it is perfectly correct, and not unheard of, to have a single supporter behind the shield.
The founding fathers were, for the most part, armigerous gentlemen and well-versed in the laws of heraldry. The seal they brought forth violated none of these laws.