for p being natural prime number
for a being integer number holds
( ( Leg (a,p) = 1 implies ( a gcd p = 1 & a is_quadratic_residue_mod p ) ) & ( a gcd p = 1 & a is_quadratic_residue_mod p implies Leg (a,p) = 1 ) & ( Leg (a,p) = 0 implies p divides a ) & ( p divides a implies Leg (a,p) = 0 ) & ( Leg (a,p) = - 1 implies ( a gcd p = 1 & a is_quadratic_non_residue_mod p ) ) & ( a gcd p = 1 & a is_quadratic_non_residue_mod p implies Leg (a,p) = - 1 ) )