Christmas Streak 11/88: Mathematicians, Coffee, and Theorems

Logic Level 3

Hungarian mathematician Erdős Paul once asserted: "A mathematician is a device for turning coffee into theorems."

Why don't we prove it mathematically?

Solve the cryptogram below, and submit your answer as the 7-digit integer $$\overline{\color{blue}T\color{purple}H\color{green}E\color{orange}O\color{pink}R\color{green}E\color{brown}M}.$$

$\begin{array}{ccccccccc} &&&\color{red}C&\color{orange}O&\color{yellow}F&\color{yellow}F&\color{green}E&\color{green}E \\ &&&\color{red}C&\color{orange}O&\color{yellow}F&\color{yellow}F&\color{green}E&\color{green}E \\ +&&&\color{red}C&\color{orange}O&\color{yellow}F&\color{yellow}F&\color{green}E&\color{green}E \\ \hline &&\color{blue}T&\color{purple}H&\color{green}E&\color{orange}O&\color{pink}R&\color{green}E&\color{brown}M \end{array}$

In a cryptogram, each symbol represents a distinct, non-negative, single digit, and all leading digits are non-zero.

This problem is a part of <Christmas Streak 2017> series.

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