 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 Th43:
  for E1,E2 be non empty Subset of A holds
    ex E3 be non empty Subset of A st
      PrimeIdeals(A,E1) \/ PrimeIdeals(A,E2) = PrimeIdeals(A,E3)
  proof
    let E1,E2 be non empty Subset of A;
    set F1 = PrimeIdeals(A,E1);
    set F2 = PrimeIdeals(A,E2);
    set I1 = E1-Ideal;
    set I2 = E2-Ideal;
    reconsider I3 = I1 *' I2 as Ideal of A;
A4: PrimeIdeals(A,E1) = PrimeIdeals(A,I1) by Th34;
A5: PrimeIdeals(A,I3) c= PrimeIdeals(A,I1) \/ PrimeIdeals(A,I2)
    proof
      let x be object;
      assume x in PrimeIdeals(A,I3); then
      consider x1 be prime Ideal of A such that
A7:   x1 = x and
A8:   I3 c= x1;
A9:   I1 c= x1 or I2 c= x1 by A8,Th40;
      x1 in {p where p is prime Ideal of A: I1 c= p } or
      x1 in {p where p is prime Ideal of A: I2 c= p } by A9;
      hence thesis by A7,XBOOLE_0:def 3;
    end;
A11:PrimeIdeals(A,I1) \/ PrimeIdeals(A,I2) c= PrimeIdeals(A,I3)
    proof
A12:  I1*'I2 c= I1 /\ I2 by IDEAL_1:79;
      I1 /\ I2 c= I1 by XBOOLE_1:17; then
A13:  I1*'I2 c= I1 by A12;
      I1 /\ I2 c= I2 by XBOOLE_1:17; then
      I1*'I2 c= I2 by A12; then
A14:  PrimeIdeals(A,I2) c= PrimeIdeals(A,I1*'I2) by Th38;
      PrimeIdeals(A,I1) c= PrimeIdeals(A,I1*'I2) by A13,Th38;
      hence thesis by A14,XBOOLE_1:8;
    end;
    PrimeIdeals(A,I3) =
      PrimeIdeals(A,E1) \/ PrimeIdeals(A,E2) by A4,A5,A11,Th34;
    hence thesis;
  end;
