reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;

theorem Th37:
  <.F \/ H.) = <.F "/\" H.)
proof
  F c= F "/\" H & H c= F "/\" H by Th36;
  then F \/ H c= F "/\" H by XBOOLE_1:8;
  hence <.F \/ H.) c= <.F "/\" H.) by Th22;
  F"/\"H c= <.F \/ H.)
  proof
    let x be object;
    assume x in F"/\"H;
    then consider p,q such that
A1: x = p"/\"q and
A2: p in F and
A3: q in H;
    H c= F \/ H by XBOOLE_1:7;
    then
A4: q in F \/ H by A3;
A5: F \/ H c= <.F \/ H.) by Def4;
    F c= F \/ H by XBOOLE_1:7;
    then p in F \/ H by A2;
    hence thesis by A1,A4,A5,Th9;
  end;
  then <.F"/\"H.) c= <.<.F \/ H.).) by Th22;
  hence thesis by Th21;
end;
