-----Description-----  
This task requires writing a Lean 4 method that computes the product of the first even and the first odd number encountered in a list of integers. The method should search the list for the earliest even number and the earliest odd number, then return the product of these two numbers.

-----Input-----  
The input consists of:  
lst: A list of integers.

-----Output-----  
The output is an integer:  
Returns the product resulting from multiplying the first even number and the first odd number found in the list.

-----Note-----  
The input list is assumed to contain at least one even number and one odd number.