reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

theorem Th16:
  i/\/FI [= j/\/FI iff i => j in FI
proof
  set R = equivalence_wrt FI;
  set a = i"\/"j;
  set b = a => j;
A1: j"/\"(i => j) [= j by FILTER_0:2;
A2: j"\/"j = j;
  thus i/\/FI [= j/\/FI implies i => j in FI
  proof
    assume (i/\/FI) "\/" (j/\/FI) = j/\/FI;
    then
A3: (i"\/"j)/\/FI = j/\/FI by Th15;
A4: i"\/"j in Class(R,i"\/"j) by EQREL_1:20;
A5: i"/\"b [= (i"/\"b)"\/"(j"/\"b) by LATTICES:5;
A6: j/\/FI = Class(R,j) by Def6;
A7: j in Class(R,j) by EQREL_1:20;
    Class(R,i"\/"j) = (i"\/"j)/\/FI by Def6;
    then [i"\/"j,j] in R by A3,A6,A4,A7,EQREL_1:22;
    then (i"\/"j) <=> j in FI by FILTER_0:def 11;
    then
A8: (i"\/"j) => j in FI by FILTER_0:8;
A9: a"/\"b [= j by FILTER_0:def 7;
    a"/\"b = (i"/\"b)"\/"(j"/\"b) by LATTICES:def 11;
    then i"/\"b [= j by A9,A5,LATTICES:7;
    then (i"\/"j) => j [= i => j by FILTER_0:def 7;
    hence thesis by A8,FILTER_0:9;
  end;
  j [= i"\/"j by FILTER_0:3;
  then j"/\"Top I [= i"\/"j;
  then
A10: Top I [= j => (i"\/"j) by FILTER_0:def 7;
  Top I in FI by FILTER_0:11;
  then
A11: j => (i"\/"j) in FI by A10;
A12: (i"/\"(i => j))"\/"(j"/\"(i => j)) = (i"\/"j)"/\"(i => j) by
LATTICES:def 11;
  i"/\"(i => j) [= j by FILTER_0:def 7;
  then (i"\/"j)"/\"(i => j) [= j by A1,A2,A12,FILTER_0:4;
  then
A13: i => j [= (i"\/"j) => j by FILTER_0:def 7;
  assume i => j in FI;
  then (i"\/"j) => j in FI by A13,FILTER_0:9;
  then (i"\/"j) <=> j in FI by A11,FILTER_0:8;
  then
A14: [i"\/"j,j] in R by FILTER_0:def 11;
  thus (i/\/FI) "\/" (j/\/FI) = (i"\/"j)/\/FI by Th15
    .= Class(R,i"\/"j) by Def6
    .= Class(R,j) by A14,EQREL_1:35
    .= j/\/FI by Def6;
end;
