We must decide whether the “purchase” element of the crime of equity skimming requires proof of the exchange of adequate consideration.

I

Appellants Sandra Weaver and Richard Buschman, together with several co-defendants not involved in this appeal, were indicted, charged with multiple counts of mail fraud and with one count of equity skimming, and subsequently convicted after a bench trial. Weaver was a licensed real estate agent, and Buschman was a licensed broker. The charges against them stemmed from their involvement in a business called “Foreclosure Intervention Services.” Under that name, Weaver and Buschman contacted homeowners in the Sacramento area who were in or near default on their mortgages, and claimed to have the ability to delay or to prevent foreclosure. Weaver and Buschman worked with co-defendant Raymond Hall, operator of “Financial Management Services,” in soliciting clients and selling their services.

In most instances, the defendants told the homeowner that they could register a “common law lien” that would take priority over the security interest in his home and prevent the secured party from foreclosing. Each homeowner was told that, to effectuate this strategy, he had to convey his home to a trust, with one of the defendants as trustee. Most of the homeowners continued to live in their homes after the transfer of title, paying rent to the trust. In some cases, the homeowners testified that they thought the payments would go toward legal expenses in the fight against foreclosure or would be held in trust for their benefit. However, the payments to the trusts were distributed among the defendants according to various fee-splitting agreements; they were neither used to make payments on the homeowners’ mortgages, nor used to pay legal expenses (indeed, none of the defendants is an attorney), nor held for the homeowners’ benefit.

Weaver, Buschman, and Hall waived trial by jury and proceeded to a bench trial. Both Weaver and Buschman faced eight counts of mail fraud, in violation of 18 U.S.C. § 1341 , and one count of equity skimming, in violation of 12 U.S.C. § 1709-2 . “Equity skimming is defined as the practice of purchasing one-to-four family dwellings subject to a loan insured by either the [Federal Housing Administration] or the [Department of Veterans Affairs (“VA”) ] and applying the rent receipts for personal gain rather than towards payment of the mortgages.” United States v. Laykin, 886 F.2d 1534, 1537 (9th Cir.1989).

The evidence of equity skimming related to the defendants’ obtaining title to two properties, 4809 Hinchman Way in Sacramento (“Hinchman Way”) and 6944 Alvilde Court in Rio Linda (“Alvilde Court”). Both properties were also the subject of mail fraud counts; the count pertaining to the Hinchman Way transaction named Weaver, Buschman, and Hall, while the count pertaining to the Alvilde Court transfer named only Hall.

Dhruwa and Veena Sharma owned Hinchman Way subject to a loan insured by the VA, which was in default when the Sharmas received a solicitation from Foreclosure Intervention Services. Weaver and Buschman persuaded the Sharmas to deed their home to a trust, with Hall as trustee, and to sign a “Rental Agreement with Option to Purchase,” in which the Sharmas agreed to make monthly payments to the trust in the amount of $600. (Pursuant to an agreement among the defendants, Weaver and Buschman together received one-third of the monthly rental payments.) Dhruwa Sharma testified that the defendants said that they would pay off the mortgage and that he was not to contact the mortgage lender or the VA. However, after he and his wife had paid the defendants about $2000 in rent, the lender foreclosed, and they lost the house. The foreclosure sale left the Sharmas owing about $18,000 on the loan.

Terrance and Tina Biggers owned Al-vilde Court subject to a mortgage guaranteed by the VA. They were having difficulty keeping up with the mortgage payments when a mutual friend introduced Terrance Biggers to Hall. According to Biggers, Hall told him that he could prevent foreclosure and thereby protect the Biggers’ credit rating. The Biggers agreed to the strategy and deeded Alvilde Court to a trust with Hall as trustee. However, unlike the Sharmas, they did not continue to live in the house and pay rent; instead, they moved out of state. Terrance Big-gers understood that Hall had arranged to rent the house to a man named Johnson. In fact, the tenant was Karen Markell, who had previously deeded her own property on Posada Way — the subject of one of the mail fraud counts — to a trust controlled by Hall. Markell had continued to live in the Posada Way house, paying rent to the trust, but the foreclosing lender evicted her. Hall told her that the eviction “was a mistake” and that she could rent Alvilde Court in the meantime. Unbeknownst to her, the Biggers’ mortgagee had sold Alvilde Court at a public foreclosure sale two days before, and two months after Markell moved in, she received a notice to quit from the VA. She had paid three months’ rent on that property; under the defendants’ fee — splitting agreement, Weaver and Buschman together received one-fifth of her rental payments. The foreclosure sale of Alvilde Court left the Biggers owing about $33,000 on the loan.

No direct evidence connected Weaver or Buschman with the transaction by which Hall obtained title to Alvilde Court. (Weaver and Buschman participated in resettling Markell into Alvilde Court, and they received a portion of her rent payments, but they never met or dealt with the Biggers.) However, the indictment also alleged aiding and abetting liability under 18 U.S.C. § 2 (a), and it was on this theory that the government relied in building its equity skimming case against Weaver and Buschman.

All of the above facts were presented at trial, and the district court convicted Weaver, Buschman, and Hall on all charges, including equity skimming, mail fraud with respect to Hinchman Way, and (in Hall’s case) mail fraud with respect to Alvilde Court. Before sentencing, Hall reached a plea agreement and waived his right to appeal. The district court sentenced Weaver and Buschman to ten months’ imprisonment on each count, to run concurrently. Weaver and Buschman then filed these timely appeals.

II

Weaver and Buschman argue first that, to convict them of both mail fraud and equity skimming simultaneously, the government necessarily proved two mutually exclusive sets of facts, a legal impossibility that requires reversal. They reason as follows: The defendants obtained title to the properties in question (and, consequently, rental payments from the former owners who continued to live on the premises) in return for using the common law lien strategy to delay or prevent foreclosure. In order to make out the “scheme to defraud” element of the mail fraud charge, the government would have to prove that the common law lien strategy was merit-less, and therefore that the defendants’ services were worthless. Yet in order to make out the “purchase” element of equity skimming, the government would have to prove that the transaction involved valuable consideration, and therefore that the defendants’ services were not worthless. Weaver and Buschman also assert that this supposed inconsistency also carries over to the mental states required for each crime, ie., that the government would have to prove simultaneously that the defendants knew that their services were worthless and that they knew that those services were valuable.

In pressing this argument, Weaver and Buschman invoke the classic philosophical conundrum posed by Epimenides of Crete, who reportedly said, “All Cretans are bars.” “This statement presents a paradox because Epimenides himself was a Cretan and the statement was taken to mean that every statement by a Cretan is false.” John M. Rogers & Robert E. Molzon, Some Lessons About the Law from Self Referential Problems in Mathematics, 90 MICH. L. REV. 992, 994 (1992). “Taken together,” two sentences may “have the same effect as the original Epi-menides paradox,” even when the same sentences are “separately ... harmless and even potentially useful.” DOUGLAS R. HOFSTADTER, GODEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID 21 (20th anniversary ed.1999). For example:

The following sentence is false.

The preceding sentence is true.

Id. Weaver and Buschman assert that, in conjunction with each other and with a statement by the government that “the mail fraud proves the equity skimming,” the mail fraud and equity skimming counts create the same sort of “Strange Loop,” id. at 10, 21 , as the two cross-referential statements quoted above.

We need not tackle this argument head-on, however, because even the Epimenides paradox breaks down once the Cretan liar turns out to hail from Sacramento. A similarly basic flaw afflicts Weaver and Buschman’s contention about the equity skimming statute: it simply does not require a transfer for value.

The elements of equity skimming include a pattern of “purchasing one- to four-family dwellings.” 12 U.S.C. § 1709-2 (1) (2000) (emphasis added). The statute does not define “purchasing.” Weaver and Buschman refer us to a section of the Internal Revenue Code that defines “purchaser” to mean “a person who, for adequate and full consideration in money or money’s worth, acquires an interest ... in property which is valid under local law against subsequent purchasers without actual notice.” 26 U.S.C. § 6328 (h)(6) (2000) (emphasis added). Weaver and Buschman reason that the same requirement of “adequate and full consideration” inheres in the “purchasing” element of equity skimming.

We disagree. “Where Congress uses terms that have accumulated a settled meaning under either equity or the common law, a court must infer, unless the statute otherwise dictates, that Congress means to incorporate the established meaning of these terms.” Black v. Comm’r, 765 F.2d 862, 864-65 (9th Cir.1985) (quoting NLRB v. Amax Coal Co., 453 U.S. 322, 329 , 101 S.Ct. 2789 , 69 L.Ed.2d 672 (1981)) (internal quotation marks omitted). And the term “purchase” has, and has long had, a settled meaning in the context of real estate transfers: “the acquisition of real estate by any means whatever except descent.” BLACK’S LAW DICTIONARY 1399 (4th ed.1951) (emphasis added); accord, e.g., BLACK’S LAW DICTIONARY 1248 (7th ed.1999); WEBSTER’S THIRD NEW INTERNATIONAL DICTIONARY 1844 (1986). As the crime of equity skimming by definition involves real property, specifically “one- to four-family dwellings,” and as the congressional enactment that created the offense maintained a similar focus, see Housing and Urban Development Act of 1970, Pub.L. No. 91-609, 84 Stat. 1770, we conclude that Congress used the term “purchasing” in accord with its settled meaning in the real estate context.

Indeed, the provision of the Internal Revenue Code to which Weaver and Buschman refer us for the contrary definition seems particularly inappropriate for importation into this context. The tax code requires the purchaser of a property to have paid full value before he may avoid any tax liens on the property at the time of purchase, see 26 U.S.C. §§ 6321 , 6323(a), (h)(6) (2000), and the rationale is readily apparent: to prevent a tax deadbeat from lifting the IRS’s lien simply by transferring his property for nominal consideration. No analogous reason supports an adequacy-of-consideration requirement in the equity skimming context; to the contrary, we believe that Congress intended to punish those who cause or exacerbate default on a government-backed loan irrespective of whether they acquire title to the mortgaged property with gold or with fool’s gold.

Because the government could establish the “purchase” element of the equity skimming offense simply by showing the transfer of title from the homeowners to Hall as trustee, the value attributable to the services the homeowners received in exchange is immaterial. Where a Cretan liar obtains title to mortgaged property through his sheer mendacity, the Epimen-ides paradox is no bar to his simultaneous conviction of both equity skimming and mail fraud.

Ill

Weaver and Buschman also challenge the sufficiency of the evidence on the equity skimming count. Because they were charged as aiders and abettors, the government was obliged to establish not only that “someone committed the underlying substantive offense,” but also “(1) that the accused had the specific intent to facilitate the commission of a crime by another, (2) that the accused had the requisite intent of the underlying substantive offense”-in this case, specific intent to defraud, and “(3) that the accused assisted or participated in the commission of the underlying substantive offense.” United States v. Gaskins, 849 F.2d 454, 459 (9th Cir.1988).

Construing all reasonable inferences in the government’s favor, as we must, we conclude that a reasonable fact-finder could decide beyond a reasonable doubt that Hall, the principal, had committed each and every element of equity skimming. Jackson v. Virginia, 443 U.S. 307, 319 , 99 S.Ct. 2781 , 61 L.Ed.2d 560 (1979). As we discuss in Part II, above, the adequacy of the consideration Hall received in exchange for control over both Hinchman Way and Alvilde Court is immaterial to the satisfaction of the “purchasing” element of the offense. In both cases, Hall acquired the homeowner’s interest in the property by inter vivos transfer, not by descent or inheritance. The conduct proved at trial was also sufficient to prove a “pattern or practice”; even assuming arguendo that the jurisdictional element of federal insurance is part of the pattern to be established (meaning that similar conduct with respect to homes not subject to a federally backed loan is not probative of the practice of equity skimming), the evidence pertaining directly to Hinchman Way and Alvilde Court suffices to prove the practice element. By carving out an exception for the purchase of only a single qualifying home (even if made with intent to defraud), the statute itself makes clear that two such purchases may be sufficient to establish the pattern element of the offense, and we conclude that the two proven purchases satisfy that element here.

That leaves intent to defraud. Weaver and Buschman do not seriously dispute that the government presented adequate proof of Hall’s intent to defraud; in particular, documents found in Hall’s files indicated that he had been told by several sources that his documents purporting to claim common law liens could not be recorded and had no legal effect. Weaver and Buschman point to evidence of their own sincerity, seeking to establish that they were Hall’s unwitting dupes. Indeed, there was support for that contention; for example, Markell testified that she thought Weaver and Buschman were surprised when Biggers’s mortgagee foreclosed on Alvilde Court, because they were under the impression that Hall owned that property. However, Weaver and Buschman were not involved in soliciting Biggers to convey his home to a trust; their surprise that Alvilde Court was subject to foreclosure is not conclusive proof that they did not act fraudulently, and other evidence cuts in the opposite direction. In particular, they apparently made false representations to Sharma about past success in stopping foreclosure using common law liens, and they continued to market the common law lien strategy even after it had repeatedly failed to work. Cf. United States v. Peters, 962 F.2d 1410, 1414 (9th Cir.1992) (holding that where a co-defendant continued mailing out credit card solicitations to potential customers, even though she knew that numerous past recipients had sent in their “membership fees” but never received a card, the jury could infer that she acted with the requisite fraudulent intent). Further, the government presented evidence that both Weaver and Buschman would have known from their continuing education courses in real estate and property law that the supposed services they were selling could not stop foreclosure or protect homeowners’ credit ratings. Under the deferential standard by which we review sufficiency-of-the-evidence challenges, this evidence is enough to support a reasonable finding of specific intent. For the same reasons, we conclude that the government adequately established that Weaver and Buschman possessed “the specific intent to facilitate the commission of a crime by another.” Gaskins, 849 F.2d at 459 .

Finally, we examine whether there was adequate evidence of the accomplices’ participation in the underlying offense. Indisputably, Weaver and Busch-man assisted Hall in the Hinchman Way transaction. However, there is no evidence that they aided Hall in obtaining title to Alvilde Court; their only connection with that property was in resettling Markell into it once she moved, and in receiving a portion of the rent she paid. Weaver and Buschman assert that if they were not involved in the second transaction, aiding and abetting liability is foreclosed by the equity skimming statute’s exemption of those who purchase only one qualifying dwelling. We disagree. The two-dwelling requirement simply operates “[f]or the protection of those purchasing in good faith,” H.R. REP. 91-1556 (1970), reprinted in 1970 U.S.C.C.A.N. 5582, 5650, by establishing a threshold for federal involvement slightly higher than the bare constitutional minimum; it is thus more akin to a jurisdictional requirement than to a grading element or to a fact that establishes the activity as criminal. See generally 1 SARAH N. WELLING, SARA SUN BEALE & PAMELA H. BUCEY, FEDERAL CRIMINAL LAW AND RELATED ACTIONS: CRIMES, FORFEITURE, THE FALSE CLAIMS ACT AND RICO § 4.2(c)(vi), at 119-22 (1998) (distinguishing among elements that go “to the definition of the criminal activity,” “to the grading of the offense,” or to “which sovereign has jurisdiction”). Although a jurisdictional element must be charged in the indictment and proved beyond a reasonable doubt, that requirement applies only to establishing the underlying offense; an aider and abettor of equity skimming need not specifically involve herself in at least two transactions, so long as she participates in a part of the principal’s “pattern or practice.” Cf. e.g., United States v. Sigalow, 812 F.2d 783, 785 (2d Cir.1987) (holding that involvement in the conduct satisfying a jurisdictional element is not necessary to establish aiding and abetting liability). Although a § 2(a) defendant “must have knowingly and intentionally aided and abetted the principals in each essential element of the crime,” United States v. Dinkane, 17 F.3d 1192, 1196 (9th Cir.1994), the type of “essential element” for which we require discrete proof of the accomplice’s intentional abetment is one that aggravates the crime from a less serious to a more serious offense, e.g., id. at 1196-97 (reducing an aiding and abetting conviction from armed bank robbery to unai’med bank robbery), or one that by itself imposes an additional criminal offense, e.g., United States v. Bancalari, 110 F.3d 1425, 1429-30 (9th Cir.1997) (reversing a conviction for aiding and abetting a violation of 18 U.S.C. § 924 (c)(1)(A), which creates an additional offense for using a firearm in the course of a federal crime). The two-dwelling requirement simply lacks this essential character. Therefore, the evidence of Weaver and Buschman’s involvement in the Hinchman Way transaction, which formed a key part of the “pattern or practice” element, suffices to establish their participation in the equity skimming offense.

IV

For the foregoing reasons, we conclude that Weaver and Buschman’s equity skimming convictions were not precluded by their mail fraud convictions and were supported by sufficient evidence to convince a rational factfinder. We therefore affirm the judgment of the district court on the equity skimming counts.

AFFIRMED.

. In this opinion, we consider only those claims of error relating to the equity skimming count. We address the appellants’ challenges to their mail fraud convictions in a separate, concurrently filed memorandum disposition. . Each of the documents purporting to claim a common law lien stated: Common Law Liens are superior to and supersede mortgages and equity liens. DRUMMOND CARRIAGE CO. v. MILL and others. The U.S. Supreme Court in RICH v. BRAXTON [ 158 U.S. 375 , 15 S.Ct. 1006 , 39 L.Ed. 1022 (1895)] specifically forbids judges from invoking equity jurisdiction to interfere with or remove common law liens or similar “clouds of title”. Even if a preponderance of evidence displays the lien to be void or voidable. This Common Law Lien shall be valid notwithstanding any other provision of statute or rule regarding the form or content, nor shall it be dischargeable for 100 years, nor extinguishable due to claimants death whether accidental or purposely, nor dis-chargeable by plaintiff's heirs, assigns or executors. Anyone wishing to challenge this Common Law Lien must do so with an action at Common Law within thirty (30) days from date of recording, otherwise they shall have been deemed to have waived any and all rights to such challenge. The first case citation may refer to Drummond Carriage Co. v. Mills, 54 Neb. 417 , 74 N.W. 966 (1898). . The full text of the equity skimming statute is as follows: Whoever, with intent to defraud, willfully engages in a pattern or practice of— (1) purchasing one- to four-family dwellings (including condominiums and cooperatives) which are subject to a loan in default at time of purchase or in default within one year subsequent to the purchase and the loan is secured by a mortgage or deed of trust insured or held by the Secretary of Housing and Urban Development or guaranteed by the Department of Veterans Affairs, or the loan is made by the Department of Veterans Affairs, (2) failing to make payments under the mortgage or deed of trust as the payments become due, regardless of whether the purchaser is obligated on the loan, and (3)applying or authorizing the application of rents from such dwellings for his own use, shall be fined not more than $ 250,000 or imprisoned not more than 5 years, or both. This section shall apply to a purchaser of such a dwelling, or a beneficial owner under any business organization or trust purchasing such dwelling, or to an officer, director, or agent of any such purchaser. Nothing in this section shall apply to the purchaser of only one such dwelling. 12 U.S.C. § 1709-2 (2000). . The statement is recounted in St. Paul's Letter to Titus. See Titus 1:12. . "The problem is presented more purely by the simple statement, 'This statement is a lie.' ” Rogers & Molzon, supra, at 994 n. 5. .Cf. James Boyle, A Theory of Law and Information: Copyright, Spleens, Blackmail, and Insider Trading, 80 C AL. L. REV. 1413, 1479 (1992) ("[T]his is similar to ‘solving’ Epini-mides’ paradox ... by suggesting that Epini-mides was actually from los.”). . Two elements' — that Hinchman Way and Al-vilde Court were qualifying one- to four-family dwellings subject to federally insured mortgages, and that Hall failed to apply rent receipts from those properties to pay the mortgages — were satisfied by stipulation at trial. Two additional elements — that the mortgages on the two properties went into default within one year of the purchases, and that Hall had applied rents from those properties to his own use — were adequately proven by the live testimony of Sharma, Biggers, and Markell and the stipulated testimony of an FBI financial analyst; Weaver and Busch-man do not dispute the sufficiency of the evidence on these points. With respect to the failure to make mortgage payments, Weaver and Buschman assert that neither Hall's conduct nor their own was criminal, simply because no payment less than the full amount due would have staved off foreclosure. However, on its face the equity skimming statute applies to properties "subject to a loan in default at time of purchase or in default within one year subsequent to the purchase,” and it criminalizes failure to make payments on that loan “regardless of whether the purchaser is obligated on the loan.” 12 U.S.C. § 1709-2 (1), (2). It is true that, following a default, once a lender invokes an acceleration clause and demands payment of the entire amount of the debt, a partial payment will not avert foreclosure. However, if the auction value of the real estate is less than the amount of the debt, as was the case with both Hinchman Way and Alvilde Court, any dollar in rent receipts devoted to the purchaser's "own use” is a dollar not available to reduce the amount of the deficiency judgment following foreclosure. Thus, we think the statute contemplates liability even if the purchaser keeps for himself rental payments less than the amount of a mortgage payment due. . To be sure, two purchases may not suffice to make out a "pattern or practice” on the facts of every individual case.