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
  FB <> <.B.) & FB is prime iff FB is being_ultrafilter
proof
  thus FB <> <.B.) & FB is prime implies FB is being_ultrafilter
  proof
    assume that
A1: FB <> <.B.) and
A2: for a,b holds a"\/"b in FB iff a in FB or b in FB;
    now
      let a such that
A3:   not a in FB;
      a"\/"a` = Top B by LATTICES:21;
      then a"\/"a` in FB by Th11;
      hence a` in FB by A2,A3;
    end;
    hence thesis by A1,Th44;
  end;
  assume
A4: FB is being_ultrafilter;
  hence FB <> <.B.);
  let a,b;
A5: FB <> the carrier of B by A4;
  thus a"\/"b in FB implies a in FB or b in FB
  proof
    assume that
A6: a"\/"b in FB and
A7: ( not a in FB)& not b in FB;
    a` in FB & b` in FB by A4,A7,Th44;
    then a`"/\"b` in FB by Th8;
    then (a"\/"b)` in FB by LATTICES:24;
    then a"\/"b"/\"(a"\/"b)` in FB by A6,Th8;
    then
A8: Bottom B in FB by LATTICES:20;
    FB = <.FB.) by Th21;
    hence contradiction by A5,A8,Th25;
  end;
  thus thesis by Th10;
end;
