Complex sentence: Chirac was a member of the RPR and a political opponent of Mitterrand's, and despite the fact the Mitterrand's own Socialist Party was still the largest party in the Assembly, the RPR had an ally in the UDF, which gave them a majority.
Split into simpler sentences: Chirac was a member of the RPR and a political opponent of Mitterrand's. The RPR had an ally in the UDF, which gave them a majority.