Let's solve the expression step by step:
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Start with the fractional part: \(-13 \frac{5}{7}\).
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Convert the mixed number to an improper fraction: \[ -13 \frac{5}{7} = -\frac{13 \times 7 + 5}{7} = -\frac{91 + 5}{7} = -\frac{96}{7} \]
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Next, we incorporate the integers and fractions: \[ -\frac{96}{7} + 6 - \frac{2}{7} - 4 \]
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Convert \(6\) and \(4\) to fractions with a denominator of \(7\): \[ 6 = \frac{6 \times 7}{7} = \frac{42}{7} \] \[ 4 = \frac{4 \times 7}{7} = \frac{28}{7} \]
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Now replace the integers in the expression: \[ -\frac{96}{7} + \frac{42}{7} - \frac{2}{7} - \frac{28}{7} \]
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Combine all the fractions: \[ -\frac{96 + 2 + 28 - 42}{7} = -\frac{96 + 2 - 42 + 28}{7} = -\frac{96 - 12}{7} = -\frac{84}{7} \]
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Simplify the final result: \[ -\frac{84}{7} = -12 \]
So, the final answer is: \[ \boxed{-12} \]