theorem Th14:
  the L_meet of I is BinOp of the carrier of I, equivalence_wrt FI
proof
  set R = equivalence_wrt FI;
  let x1,y1, x2,y2 be Element of I;
  assume that
A1: [x1,y1] in R and
A2: [x2,y2] in R;
A3: x2 <=> y2 in FI by A2,FILTER_0:def 11;
  then
A4: x2 => y2 in FI by FILTER_0:8;
A5: x1 <=> y1 in FI by A1,FILTER_0:def 11;
  then x1 => y1 in FI by FILTER_0:8;
  then
A6: (x1 => y1) "/\" (x2 => y2) in FI by A4,FILTER_0:8;
A7: y2 "/\" (y2 => x2) [= x2 by FILTER_0:def 7;
  y1 "/\" (y1 => x1) [= x1 by FILTER_0:def 7;
  then
A8: y1 "/\" (y1 => x1) "/\" (y2 "/\" (y2 => x2)) [= x1 "/\" x2 by A7,FILTER_0:5
;
A9: x1 "/\" x2 "/\" (x1 => y1) "/\" (x2 => y2) = x1 "/\" x2 "/\" ((x1 => y1
  ) "/\" (x2 => y2)) by LATTICES:def 7;
A10: x2 "/\" (x2 => y2) [= y2 by FILTER_0:def 7;
  x1 "/\" (x1 => y1) [= y1 by FILTER_0:def 7;
  then
A11: x1 "/\" (x1 => y1) "/\" (x2 "/\" (x2 => y2)) [= y1 "/\" y2 by A10,
FILTER_0:5;
A12: x2 "/\" x1 "/\" (x1 => y1) = x2 "/\" (x1 "/\" (x1 => y1)) by
LATTICES:def 7;
A13: y2 => x2 in FI by A3,FILTER_0:8;
A14: y2 "/\" y1 "/\" (y1 => x1) = y2 "/\" (y1 "/\" (y1 => x1)) by
LATTICES:def 7;
  y1 => x1 in FI by A5,FILTER_0:8;
  then
A15: (y1 => x1) "/\" (y2 => x2) in FI by A13,FILTER_0:8;
A16: y1 "/\" y2 "/\" (y1 => x1) "/\" (y2 => x2) = y1 "/\" y2 "/\" ((y1 => x1
  ) "/\" (y2 => x2)) by LATTICES:def 7;
  y1 "/\" (y1 => x1) "/\" (y2 "/\" (y2 => x2)) = y1 "/\" (y1 => x1) "/\"
  y2 "/\" (y2 => x2) by LATTICES:def 7;
  then (y1 => x1) "/\" (y2 => x2) [= (y1 "/\" y2) => (x1 "/\" x2) by A14,A16,A8
,FILTER_0:def 7;
  then
A17: (y1 "/\" y2) => (x1 "/\" x2) in FI by A15,FILTER_0:9;
  x1 "/\" (x1 => y1) "/\" (x2 "/\" (x2 => y2)) = x1 "/\" (x1 => y1) "/\"
  x2 "/\" (x2 => y2) by LATTICES:def 7;
  then (x1 => y1) "/\" (x2 => y2) [= (x1 "/\" x2) => (y1 "/\" y2) by A12,A9,A11
,FILTER_0:def 7;
  then (x1 "/\" x2) => (y1 "/\" y2) in FI by A6,FILTER_0:9;
  then (x1 "/\" x2) <=> (y1 "/\" y2) in FI by A17,FILTER_0:8;
  hence thesis by FILTER_0:def 11;
end;
