
theorem Th25:
  for p being Prime,
      a being Integer holds
  (Leg(a,p) =  1 iff a gcd p = 1 & a is_quadratic_residue_mod p) &
  (Leg(a,p) =  0 iff p divides a) &
  (Leg(a,p) = -1 iff a gcd p = 1 & a is_quadratic_non_residue_mod p)
proof
let p be Prime,
    a be Integer;
A1:now assume A2: Leg(a,p) = 0;
  now assume not p divides a;
    then a gcd p = 1 by Th6;
    hence contradiction by A2,Def4;
    end;
  hence p divides a;
  end;
now assume A3: Leg(a,p) = 1;
   then a gcd p = 1 by Th6,Def4;
   hence a gcd p = 1 & a is_quadratic_residue_mod p by A3,Def4;
   end;
hence Leg(a,p) = 1 iff a gcd p = 1 & a is_quadratic_residue_mod p by Def4;
now assume A4: Leg(a,p) = -1;
   then a gcd p = 1 by Th6,Def4;
   hence a gcd p = 1 & a is_quadratic_non_residue_mod p by A4,Def4;
   end;
hence thesis by A1,Def4;
end;
