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;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;

theorem Th43:
  a [= b iff a"/\"b` = Bottom B
proof
  reconsider B9 = B as 0_Lattice;
  reconsider B99 = B as 1_Lattice;
  reconsider D = B as D_Lattice;
  reconsider a9 = a, b9 = b, c9 = a"/\"b` as Element of B9;
  reconsider a99 = a, b99 = b as Element of B99;
  reconsider a1 = a, b1 = b as Element of D;
  thus a [= b implies a"/\"b` = Bottom B
  proof
    assume a [= b;
    then a = a"/\"b by LATTICES:4;
    hence a"/\"b` = a"/\"(b"/\"b`) by LATTICES:def 7
      .= a9"/\"Bottom B9 by LATTICES:20
      .= Bottom B;
  end;
  assume a"/\"b` = Bottom B;
  then b = b9"\/"c9
    .= (b1"\/"a1)"/\"(b1"\/"b1`) by LATTICES:11
    .= (b99"\/"a99)"/\"Top B99 by LATTICES:21
    .= a"\/"b;
  hence thesis;
end;
