Multiplicative decomposition of arithmetic progressions in prime fields

We prove that there exists an absolute constant c>0c>0 such that if an arithmetic progression PP modulo a prime number p does not contain zero and has the cardinality less than cp  , then it cannot be represented as a product of two subsets of cardinality greater than 1, unless P=−PP=−P or P={−2r,r,4r}P={−2r,r,4r} for some residue r modulo p.