 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;
 reserve m for maximal Ideal of A;

theorem
  (for a holds a in m implies 1.A + a is Unit of A) implies A is local
  proof
    assume
A1: (for a holds a in m implies 1.A + a is Unit of A);
             for x holds x in [#]A \ m implies x is Unit of A
     proof
       let x;
       assume
A2:    x in [#]A \ m; then
       reconsider a0=x as Element of A;
       {a0}-Ideal + m = [#]A by A2,Th18; then
       1.A in {a0}-Ideal + m; then
       1.A in {p+q where p,q is Element of A:
       p in {a0}-Ideal & q in m } by IDEAL_1:def 19; then
       consider p,q be Element of A such that
A3:    1.A = p+q and
A4:    p in {a0}-Ideal and
A5:    q in m;
A6:    {a0}-Ideal = the set of all a0*s where s is Element of A by IDEAL_1:64;
       consider s be Element of A such that
A7:    p = a0 * s by A4,A6;
       1.A+(-q) = a0*s+(-q+q) by RLVECT_1:def 3,A3,A7
       .= a0*s+0.A by RLVECT_1:5 .= a0*s; then
       a0*s is Unit of A by A1,A5,IDEAL_1:13; then
       {a0*s}-Ideal = [#]A by RING_2:20; then
A9:    1.A in {a0*s}-Ideal;
       {a0*s}-Ideal = the set of all (a0*s)*t where t is Element of A
         by IDEAL_1:64; then
       consider t1 be Element of A such that
A11:   1.A = (a0*s)*t1 by A9;
A12:   a0*(s*t1) = 1.A by A11,GROUP_1:def 3;
       reconsider t = s*t1 as Element of A;
       1.A in {a0}-Ideal by A6,A12; then
       not {a0}-Ideal is proper by IDEAL_1:19; then
       {a0}-Ideal = [#]A;
       hence thesis by RING_2:20;
     end;
     hence thesis by Th17;
   end;
