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;

theorem Th40:
  j in <.DI \/ {i}.) implies i => j in <.DI.)
proof
  assume
A1: j in <.DI \/ {i}.);
  <.DI \/ {i}.) = <.<.DI.) \/ {i}.) by Th34
    .= <.<.DI.) \/ <.{i}.).) by Th34
    .= <.<.DI.) \/ <.i.).) by Th24
    .= <.DI.) "/\" <.i.) by Th38;
  then consider i1,i2 being Element of I such that
A2: j = i1"/\"i2 and
A3: i1 in <.DI.) and
A4: i2 in <.i.) by A1;
  i [= i2 by A4,Th15;
  then i1"/\"i [= i1"/\"i2 by LATTICES:9;
  then i1 [= i => j by A2,Def7;
  hence thesis by A3,Th9;
end;
