Yes, if I understand your question.
The confusion about the 23% is the way the bill is written. I believe it is normal for federal tax bills to be written in language that thinks about the taxation as taking a portion of what is there, not adding an additional portion to it, (the FairTax is no diferent) hence the wording being 23% of gross. It is the vendor who is required to remit the correct portion of his proceeds. If $129.87 is collected by a vendor, they would be expected to remit 23% (or $29.87) to the state on behalf of the federal government. The consumer would see this as a sales tax of 29.87% on what he would pay without the tax.
see #836