 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 Th44:
  for G being Subset-Family of Spectrum A st
  for S being set st S in G holds
  (ex E be non empty Subset of A st S = PrimeIdeals(A,E))
  holds ex F be non empty Subset of A st Intersect G = PrimeIdeals(A,F)
  proof
    let G be Subset-Family of Spectrum A;
    assume
A1: for S being set st S in G holds
    (ex E be non empty Subset of A st S = PrimeIdeals(A,E));
    per cases;
      suppose
A2:     G <> {}; then
        consider o such that
A3:     o in G by XBOOLE_0:def 1;
        reconsider o as set by A3;
        consider E be non empty Subset of A such that
A4:     o = PrimeIdeals(A,E) by A1,A3;
A5:     meet G = Intersect G by A2,SETFAM_1:def 9;
        defpred X[set] means ex Z being non empty Subset of A
        st $1 = Z & ex D be set st D in G & D = PrimeIdeals(A,Z);
        consider SFE being set such that
A6:     for x being set holds x in SFE iff x in bool the carrier of A &
        X[x] from XFAMILY:sch 1;
        SFE c= bool the carrier of A by A6; then
        reconsider SFE as non empty Subset-Family of the carrier of A
          by A3,A4,A6;
        E c= union SFE by ZFMISC_1:74,A3,A4,A6; then
A8:     union SFE is non empty Subset of A;
A9:     x in Intersect G implies x in PrimeIdeals(A,union SFE)
        proof
          assume
A10:      x in Intersect G; then
A11:      x in meet G by A2,SETFAM_1:def 9;
          reconsider x as prime Ideal of A by A10,Th22;
          union SFE c= x
          proof
            let o1;
            assume o1 in union SFE; then
            consider Y be set such that
A13:        o1 in Y and
A14:        Y in SFE by TARSKI:def 4;
            consider Z being non empty Subset of A such that
A15:        Z = Y and
A16:        ex D be set st D in G & D = PrimeIdeals(A,Z) by A6,A14;
            consider D be set such that
A17:        D in G and
A18:        D = PrimeIdeals(A,Z) by A16;
            x in D by A17,A11,SETFAM_1:def 1; then
            consider x1 be prime Ideal of A such that
A19:        x1 = x and
A20:        Z c= x1 by A18;
            thus thesis by A13,A15,A20,A19;
          end;
          hence thesis;
        end;
        x in PrimeIdeals(A,union SFE) implies x in Intersect G
        proof
          assume
A23:      x in PrimeIdeals(A,union SFE);
          for Y be set holds Y in G implies x in Y
          proof
            let Y be set;
            assume
A24:        Y in G; then
            consider E be non empty Subset of A such that
A25:        Y = PrimeIdeals(A,E) by A1;
            E c= union SFE by A6,A24,A25,ZFMISC_1:74; then
            PrimeIdeals(A,union SFE) c= PrimeIdeals(A,E) by Th38;
            hence thesis by A25,A23;
          end;
          hence thesis by A2,A5,SETFAM_1:def 1;
        end;
        hence thesis by A8,A9,TARSKI:2;
      end;
      suppose G = {}; then
        Intersect G = Spectrum A by SETFAM_1:def 9;
        hence thesis by Th42;
      end;
    end;
