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It has been four years since Herman Cain rode his “9-9-9″ tax plan to a short-lived lead in the race for the Republican Party’s presidential nomination, and the former chief executive of Godfather Pizza does not like being used to exemplify a flash-in-the-pan candidate.

Especially by Jeb Bush.

The former Florida governor has referred to Mr. Cain when facing questions about his lackluster poll numbers, telling supporters that it is too soon to place much stock in such indicators because hot candidates can fizzle fast. Mr. Cain, whose campaign was undone by accusations of sexual misconduct, said on Monday that Mr. Bush should stop trying to use him to comfort himself.

“At least I was once winning,” Mr. Cain said of Mr. Bush in an essay on his website. “Jeb Bush has been doing nothing but losing throughout this entire campaign.”

Mr. Cain, who is attending a campaign rally in Georgia with Donald J. Trump on Monday, criticized Mr. Bush for underestimating political outsiders and said that he should be doing far better considering his financial advantages and name recognition. Mr. Bush has been stuck in single digits in many national and state polls.

Analysts have pointed to the 2012 race, when Michele Bachmann and Mr. Cain both briefly surged to the top of the polls, months before any state caucuses or primaries took place, as precedents to show why candidates such as Mr. Trump and Ben Carson are likely to fade once voters get serious. But Mr. Trump’s personal wealth and the crowded Republican field have so far made this cycle a different story.

“Trump is very smart, has done his homework and has learned a lot from what happened in many previous campaigns — including mine,” Mr. Cain said.