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 Th41:
  i => j in FI & j => k in FI implies i => k in FI
proof
  assume that
A1: i => j in FI and
A2: j => k in FI;
A3: FI \/ {i} c= <.FI \/ {i}.) by Def4;
  {i} c= FI \/ {i} by XBOOLE_1:7;
  then
A4: {i} c= <.FI \/ {i}.) by A3;
  FI c= FI \/ {i} by XBOOLE_1:7;
  then
A5: FI c= <.FI \/ {i}.) by A3;
  i in {i} by TARSKI:def 1;
  then j in <.FI \/ {i}.) by A1,A5,A4,Th29;
  then
A6: k in <.FI \/ {i}.) by A2,A5,Th29;
  <.FI.) = FI by Th21;
  hence thesis by A6,Th40;
end;
