theorem
  (i"\/"j) => i = j => i & i => (i"/\"j) = i => j
proof
  j"/\"(j=>i) [= i by FILTER_0:def 7;
  then
A1: i"\/"(j"/\"(j=>i)) [= i;
A2: (i"\/"j)"/\"(j=>i) [= (i"\/"j)"/\"(i"\/" (j=>i)) by LATTICES:5,9;
A3: j"/\"((i"\/"j)=>i) [= (i"\/"j)"/\"((i"\/" j)=>i) by LATTICES:5,9;
  (i"\/"j)"/\"((i"\/"j)=>i) [= i by FILTER_0:def 7;
  then j"/\"((i"\/"j)=>i) [= i by A3,LATTICES:7;
  then
A4: (i"\/"j)=>i [= j=>i by FILTER_0:def 7;
  i"\/"(j"/\"(j=>i)) = (i"\/"j)"/\"(i"\/"(j=> i)) by LATTICES:11;
  then (i"\/"j)"/\"(j=>i) [= i by A1,A2,LATTICES:7;
  then j=>i [= (i"\/"j)=>i by FILTER_0:def 7;
  hence (i"\/"j) => i = j => i by A4,LATTICES:8;
A5: j"/\"i [= j by LATTICES:6;
  i"/\"(i=>(i"/\"j)) [= i"/\"j by FILTER_0:def 7;
  then i"/\"(i=>(i"/\"j)) [= j by A5,LATTICES:7;
  then
A6: i=>(i"/\"j) [= i=>j by FILTER_0:def 7;
  i"/\"(i=>j) [= j by FILTER_0:def 7;
  then
A7: i"/\"(i"/\"(i=>j)) [= i"/\"j by LATTICES:9;
  i"/\"(i"/\"(i=>j)) = i"/\"i"/\"(i=>j) by LATTICES:def 7;
  then i=>j [= i=>(i"/\"j) by A7,FILTER_0:def 7;
  hence thesis by A6,LATTICES:8;
end;
