reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;
reserve fr,f for FinSequence of INT;
reserve b,m for Nat;
reserve b for Integer;

theorem Th25:
  Lege (a,p) = 1 or Lege (a,p) = 0 or Lege (a,p) = -1
  proof
  per cases;
  suppose a is_quadratic_residue_mod p & a mod p <> 0;
    hence thesis by Def3;
  end;
  suppose a is_quadratic_residue_mod p & a mod p = 0;
    hence thesis by Def3;
  end;
  suppose not a is_quadratic_residue_mod p;
    hence thesis by Def3;
  end;
  end;
