Symmetry properties of the products of symmetrical functions

The product of two functions with axial symmetry is a function with axial symmetry.

The product of two functions symmetrical to the origin is a function with axial symmetry.

The product of a function symmetrical to the origin and a function with axial symmetry is a function symmetrical to the origin.

The product of two functions with the property of half wave symmetry of the 1st degree is a function with the property of half wave symmetry of the 2nd degree.

The product of a function with the property of half wave symmetry of the 1st degree and a function with the property of half wave symmetry of the 2nd degree is a function with the property of half wave symmetry of the 1st degree.