theorem Th32:
  a is not_quadratic_residue iff Lege_p(a) = -1
  proof
    hereby assume a is not_quadratic_residue;
      then a <> 0 & not ex x being Element
      of GF(p) st x|^2= a;
      then a <> 0 & not a is quadratic_residue;
      hence Lege_p(a) = -1 by Def5;
    end;
    assume Lege_p(a) = -1;
    then a <> 0 & not a is quadratic_residue by Def5;
    then a <> 0 & not ex x being Element of GF(p)
    st x|^2 = a;
    hence thesis;
  end;
