theorem Th57:
  (i => j)"/\"(i => k) [= i => (j"/\"k)
proof
A1: i"/\"(i=>j) [= j by FILTER_0:def 7;
  i"/\"(i=>k) [= k by FILTER_0:def 7;
  then
A2: (i"/\"(i=>j))"/\"(i"/\"(i=>k)) [= j"/\"k by A1,FILTER_0:5;
A3: (i"/\"(i=>j))"/\"(i"/\"(i=>k)) = ((i"/\"(i=>j))"/\"i)"/\"(i=>k) by
LATTICES:def 7;
A4: i"/\"((i=>j)"/\"(i=>k)) = i"/\"(i=>j)"/\" (i=>k) by LATTICES:def 7;
A5: i"/\"(i"/\"(i=>j)) = i"/\"i"/\" (i=>j) by LATTICES:def 7;
  thus thesis by A2,A3,A5,A4,FILTER_0:def 7;
end;
