reserve T for TopSpace;
reserve A,B for Subset of T;
reserve T for non empty TopSpace;
reserve P,Q for Element of Topology_of T;
reserve p,q for Element of Open_setLatt(T);
reserve L for D_Lattice;
reserve F for Filter of L;
reserve a,b for Element of L;
reserve x,X,X1,X2,Y,Z for set;

theorem Th20:
  not b [= a implies ex F st F in F_primeSet(L) & not a in F & b in F
proof
  set A = SF_have b \ SF_have a;
  assume not b [= a;
  then
A1: A <> {} by Th19;
  for Z st Z <> {} & Z c= SF_have b \ SF_have a & Z is c=-linear ex Y st Y
  in SF_have b\ SF_have a & for X1 st X1 in Z holds X1 c= Y by Th18;
  then consider Y such that
A2: Y in A and
A3: for Z st Z in A & Z <> Y holds not Y c= Z by A1,LATTICE4:1;
  reconsider Y as Filter of L by A2,Th17;
A4: b in Y by A2,Th17;
A5: not a in Y by A2,Th17;
A6: Y is prime
  proof
    let a1,a2 be Element of L;
    thus a1"\/"a2 in Y implies a1 in Y or a2 in Y
    proof
      set F2=<.<.a2.) \/ Y.);
      set F1=<.<.a1.) \/ Y.);
      assume that
A7:   a1"\/"a2 in Y and
A8:   not a1 in Y and
A9:   not a2 in Y;
A10:  <.a1.) c= F1 by LATTICE4:2;
      a1 in <.a1.);
      then
A11:  a1 in F1 by A10;
A12:  Y c= F1 by LATTICE4:2;
A13:  <.a2.) c= F2 by LATTICE4:2;
      a2 in <.a2.);
      then
A14:  a2 in F2 by A13;
A15:  Y c= F2 by LATTICE4:2;
      not a in F1 or not a in F2
      proof
        assume that
A16:    a in F1 and
A17:    a in F2;
        consider c1 being Element of L such that
A18:    c1 in Y and
A19:    a1 "/\" c1 [= a by A16,LATTICE4:3;
        consider c2 being Element of L such that
A20:    c2 in Y and
A21:    a2 "/\" c2 [= a by A17,LATTICE4:3;
        set c = c1 "/\" c2;
        a2 "/\" c [= a2 "/\" c2 by LATTICES:6,9;
        then
A22:    a2 "/\" c [= a by A21,LATTICES:7;
        a1 "/\" c [= a1 "/\" c1 by LATTICES:6,9;
        then a1 "/\" c [= a by A19,LATTICES:7;
        then (a1 "/\" c) "\/"( a2 "/\" c) [= a by A22,FILTER_0:6;
        then
A23:    (a1 "\/" a2) "/\" c [= a by LATTICES:def 11;
        c in Y by A18,A20,FILTER_0:8;
        then (a1 "\/" a2) "/\" c in Y by A7,FILTER_0:8;
        hence contradiction by A5,A23,FILTER_0:9;
      end;
      then F1 in A or F2 in A by A4,A12,A15,Lm1;
      hence contradiction by A3,A8,A9,A11,A14,A12,A15;
    end;
    thus thesis by FILTER_0:10;
  end;
  Y <> the carrier of L by A2,Th17;
  then Y in F_primeSet(L) by A6;
  hence thesis by A5,A4;
end;
