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;
reserve o1,o2 for BinOp of F;

theorem
  L is I_Lattice implies latt F is implicative
proof
  assume
A1: L is I_Lattice;
  then reconsider I = L as I_Lattice;
  reconsider FI = F as Filter of I;
  let p9,q9 be Element of latt F;
  reconsider p = p9, q = q9 as Element of F by Th49;
  consider r such that
A2: p"/\"r [= q and
A3: for r1 st p"/\"r1 [= q holds r1 [= r by A1,Def6;
  reconsider i = p, j = q as Element of I;
A4: i => j in FI by Th30;
  i => j = r by A2,A3,Def7;
  then reconsider r9 = r as Element of latt F by A4,Th49;
  take r9;
  p"/\"r = p9"/\"r9 by Th50;
  hence p9"/\"r9 [= q9 by A2,Th51;
  let s9 be Element of latt F such that
A5: p9"/\"s9 [= q9;
  reconsider s = s9 as Element of F by Th49;
  p"/\"s = p9"/\"s9 by Th50;
  then p"/\"s [= q by A5,Th51;
  then s [= r by A3;
  hence thesis by Th51;
end;
