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;
reserve p,q for Element of StoneLatt(L);
reserve H for non trivial H_Lattice;
reserve p9,q9 for Element of H;

theorem Th33:
  StoneH(H).(p9 => q9) = (StoneH(H).p9) => (StoneH(H).q9)
proof
A1: the carrier of Open_setLatt(HTopSpace H) = the set of all
union A where A is Subset of
  StoneS(H) by Def12;
A2: now
    let r be Element of Open_setLatt(HTopSpace H);
    r in the carrier of Open_setLatt(HTopSpace H);
    then consider A being Subset of StoneS(H) such that
A3: r = union A by A1;
    assume StoneH(H).p9 "/\" r [= StoneH(H).q9;
    then StoneH(H).p9 "/\" r c= StoneH(H).q9 by Th6;
    then StoneH(H).p9 /\ union A c= StoneH(H).q9 by A3,Def3;
    then
A4: union INTERSECTION ({StoneH(H).p9}, A) c= StoneH(H).q9 by SETFAM_1:25;
    now
      let x;
      assume
A5:   x in A;
      then consider x9 being Element of H such that
A6:   x=StoneH(H).x9 by Th13;
      StoneH(H).p9 in {StoneH(H).p9} by TARSKI:def 1;
      then StoneH(H).p9 /\ x in INTERSECTION ({StoneH(H).p9}, A) by A5,
SETFAM_1:def 5;
      then StoneH(H).p9 /\ StoneH(H).x9 c= StoneH(H).q9 by A4,A6,SETFAM_1:41;
      then StoneH(H).(p9 "/\" x9) c= StoneH(H).q9 by Th15;
      then StoneH(H).(p9 "/\" x9) [= StoneH(H).q9 by Th6;
      then (p9 "/\" x9) [= q9 by LATTICE4:5;
      then x9 [= p9 => q9 by FILTER_0:def 7;
      then StoneH(H).x9 [= StoneH(H).(p9 => q9) by LATTICE4:4;
      hence x c= StoneH(H).(p9 => q9) by A6,Th6;
    end;
    then union A c= StoneH(H).(p9 => q9) by ZFMISC_1:76;
    hence r [= StoneH(H).(p9 => q9) by A3,Th6;
  end;
  p9 "/\" (p9 => q9) [= q9 by FILTER_0:def 7;
  then StoneH(H).(p9 "/\" (p9 => q9)) [= StoneH(H).q9 by LATTICE4:4;
  then StoneH(H).p9"/\" StoneH(H).(p9 => q9) [= StoneH(H).q9
  by LATTICE4:def 2;
  hence thesis by A2,FILTER_0:def 7;
end;
