reserve R,R1 for commutative Ring;
reserve A,B for non degenerated commutative Ring;
reserve o,o1,o2 for object;
reserve r,r1,r2 for Element of R;
reserve a,a1,a2,b,b1 for Element of A;
reserve f for Function of R, R1;
reserve p for Element of Spectrum A;
reserve S for non empty multiplicatively-closed Subset of R;
reserve u,v,w,x,y,z for Element of Frac(S);
reserve a, b, c for Element of Frac(S);
reserve x, y, z for Element of S~R;
reserve S for without_zero non empty multiplicatively-closed Subset of A;
reserve p for Element of Spectrum A;
reserve a,m,n for Element of A~p;

theorem
  A~p is local & Loc-Ideal(p) is maximal Ideal of A~p
  proof
    reconsider J = Loc-Ideal(p) as proper Ideal of A~p by Th53;
A1: A~p is local
    proof
      (for x be object holds x in [#](A~p) \ J
        implies x is Unit of A~p) implies A~p is local by TOPZARI1:13;
      hence thesis by Th54;
    end;
    J is maximal Ideal of A~p
    proof
      consider m be maximal Ideal of A~p such that
A3:   J c= m by TOPZARI1:8;
      o in m implies o in J
      proof
        assume
A4:     o in m; then
A5:     o is NonUnit of A~p by TOPZARI1:11;
        per cases;
        suppose o in m \ J; then
A7:       o in m & not o in J by XBOOLE_0:def 5;
          o in [#](A~p) \ J by A7,XBOOLE_0:def 5;
          hence thesis by A5,Th54;
        end;
        suppose not o in m \ J;
          hence thesis by A4,XBOOLE_0:def 5;
        end;
      end; then
      m c= J;
      hence thesis by A3,XBOOLE_0:def 10;
    end;
    hence thesis by A1;
  end;
