 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 Th18:
  a in [#]A \ m implies {a}-Ideal + m = [#]A
  proof
    assume
A1: a in [#]A \ m;
A2: a in {a}-Ideal by IDEAL_1:66;
    0.A in m by IDEAL_1:3; then
A4: a+0.A in {x+y where x,y is Element of A :x in {a}-Ideal & y in m} by A2;
    reconsider a as Element of A;
A5: a in {a}-Ideal + m by A4,IDEAL_1:def 19;
    {a}-Ideal + m=m or {a}-Ideal+m is non proper by IDEAL_1:74,RING_1:def 3;
    hence thesis by A1,A5,XBOOLE_0:def 5;
  end;
