The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Replace In Subquery With Correlated Exists (modified query below): In many cases, an EXISTS subquery with a correlated condition will perform better than a non correlated IN subquery.

Optimal indexes for this query:

ALTER TABLE `FactInternetSales` ADD INDEX `factinternetsales_idx_salesordernumbe_salesorderlinen` (`SalesOrderNumber`,`SalesOrderLineNumber`);
ALTER TABLE `FactInternetSalesReason` ADD INDEX `factinternetsalesr_idx_salesreasonkey` (`SalesReasonKey`);
ALTER TABLE `FactInternetSalesReason` ADD INDEX `factinternetsalesr_idx_salesordernumbe_salesreasonkey` (`SalesOrderNumber`,`SalesReasonKey`);

The optimized query:

SELECT
        count(DISTINCT f.SalesOrderNumber) 
    FROM
        FactInternetSales f 
    JOIN
        FactInternetSalesReason fs 
            ON f.SalesOrderNumber = fs.SalesOrderNumber 
            AND f.SalesOrderLineNumber = fs.SalesOrderLineNumber 
    WHERE
        fs.SalesReasonKey = 1 
        AND EXISTS (
            SELECT
                1 
            FROM
                FactInternetSalesReason AS fs11 
            WHERE
                (
                    fs11.SalesReasonKey = 2
                ) 
                AND (
                    fs.SalesOrderNumber = fs11.SalesOrderNumber
                )
        )