Has Network violated Systems’s trademark by purchasing it as a Google keyword?

Network Automation and Advanced Systems Concepts both sold job scheduling and management software, and both advertised on the Internet. Network sold its software under the trademarked name Auto-Mate, while Systems used the trademark ActiveBatch. Customers paid between $995 and $10,995 to use these software programs.
Google AdWords is a program that sells “keywords,” which are search terms that trigger the display of a sponsor’s advertisement. When a user enters a keyword, Google displays the links generated by its own algorithm in the main part of the page, along with advertisements in a separate “Sponsored Links” section next to or above the objective results. Multiple advertisers can purchase the same keyword.
Although ActiveBatch was Systems’s trademark, Network purchased it as a keyword. This purchase meant that anyone who googled “ActiveBatch” would see a web page where the top results were links to Systems’ own website and various articles about the product. But in the “Sponsored Sites” section of the page, users would see the following ad:
Job Scheduler Windows Job Scheduling + Much More. Easy to Deploy, Scalable. D/L Trial www.NetworkAutomation.com
Sometimes, they would also see an equivalent ad for Systems’ software—the real ActiveBatch.
Systems alleged that this use of ActiveBatch was a violation of its trademark on the word. The trial court issued an injunction prohibiting Network’s purchase of the Google keyword. Network appealed.