 reserve R for commutative Ring;
 reserve A,B for non degenerated commutative Ring;
 reserve h for Function of A,B;
 reserve I0,I,I1,I2 for Ideal of A;
 reserve J,J1,J2 for proper Ideal of A;
 reserve p for prime Ideal of A;
 reserve S,S1 for non empty Subset of A;
 reserve E,E1,E2 for Subset of A;
 reserve a,b,f for Element of A;
 reserve n for Nat;
 reserve x,o,o1 for object;

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;
