Let's denote the number of cans already on the shelf as \( x \).
According to the problem, the total capacity of the shelf is 58 cans, and Alexei can still add 39 cans to the shelf. This means that the total number of cans on the shelf plus the cans he can still add must equal the shelf's capacity.
We can set up the equation:
\[ x + 39 = 58 \]
Now, we will solve for \( x \):
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Subtract 39 from both sides of the equation: \[ x = 58 - 39 \]
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Calculate the right side: \[ x = 19 \]
So, the number of cans already on the shelf is \( \boxed{19} \).