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;
