 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 Th15:
  x in I & I is proper Ideal of A implies x is NonUnit of A
   proof
     assume that
A1:  x in I and
A2:  I is proper Ideal of A;
     assume
A3:  not x is NonUnit of A;
     reconsider x as Element of A by A1;
     {x}-Ideal = [#]A by A3,RING_2:20; then
     [#]A = I by A1,RING_2:4;
     hence contradiction by A2;
   end;
