Lufthansa does not cancel the tickets per se in case of a missed first segment, they are obligated to honor the booking, but they have a right under the contract to recalculate the fare based on the new itinerary. They will not transport you, though, if you do not cough up the fare difference before your flight.
(As an aside, they are even permitted to recalculate the fare in case of a missing last leg. They just do not have any immediate leverage to collect the fare difference, short of going to court afterwards, which I have not heard happen, yet)