 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;
 reserve p for prime Ideal of A;
 reserve k for non zero Nat;

theorem Th41A:
  PrimeIdeals(A,{1.A}) = {}
  proof
    assume PrimeIdeals(A,{1.A}) <> {}; then
    consider p be object such that
A2: p in PrimeIdeals(A,{1.A}) by XBOOLE_0:def 1;
    consider p1 be prime Ideal of A such that
    p1 = p and
A3: {1.A} c= p1 by A2;
    1.A in {1.A} by TARSKI:def 1;
    hence contradiction by A3,IDEAL_1:19;
  end;
