theorem Th16:
  (for m1,m2 be object st m1 in m-Spectrum A & m2 in m-Spectrum A holds
    m1 = m2)
     implies A is local
  proof
    assume
A1: for m1,m2 be object st m1 in m-Spectrum A & m2 in m-Spectrum A holds
      m1 = m2;
    reconsider m = the maximal Ideal of A as maximal Ideal of A;
A3: o = m implies o in m-Spectrum A;
    m in m-Spectrum A; then
    o in m-Spectrum A implies o = m by A1; then
    m-Spectrum A = {m} by A3,TARSKI:def 1;
    hence thesis;
  end;
