My problem:
I have 3 polygons (a, b, c) on layer 1 and two other polygons (1 and 2) on layer 2. Now I want to "crop and melt" (forgive me I don't know the terminus technicus) my three polygons (a,b,c) with the polygons (1 and 2) that (on a new layer) I have two new areas: area I (marked in red in my picture) and area II (marked in blue in my picture). Each of my new areas has two parts.

Thank you very much!