theorem Th13:
  the L_join 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 (the carrier 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 "/\" (x1 => y1) [= y1 by FILTER_0:def 7;
  x1 "/\" ((x1 => y1) "/\" (x2 => y2)) = x1 "/\" (x1 => y1) "/\" (x2 =>
  y2 ) by LATTICES:def 7;
  then
A6: x1 "/\" ((x1 => y1) "/\" (x2 => y2)) [= y1 by A5,FILTER_0:2;
A7: x2 "/\" ((x1 => y1) "/\" (x2 => y2)) = x2 "/\" (x1 => y1) "/\" (x2 =>
  y2 ) by LATTICES:def 7;
A8: x2 "/\" (x2 => y2) [= y2 by FILTER_0:def 7;
  (x1 => y1) "/\" (x2 "/\" (x2 => y2)) = (x1 => y1) "/\" x2 "/\" (x2 => y2
  ) by LATTICES:def 7;
  then x2 "/\" ((x1 => y1) "/\" (x2 => y2)) [= y2 by A7,A8,FILTER_0:2;
  then
  x1 "/\" ((x1 => y1) "/\" (x2 => y2)) "\/" (x2 "/\" ((x1 => y1) "/\" (x2
  => y2))) [= y1 "\/" y2 by A6,FILTER_0:4;
  then (x1 "\/" x2) "/\" ((x1 => y1) "/\" (x2 => y2)) [= y1 "\/" y2 by
LATTICES:def 11;
  then
A9: (x1 => y1) "/\" (x2 => y2) [= (x1 "\/" x2) => (y1 "\/" y2) by
FILTER_0:def 7;
A10: y1 "/\" (y1 => x1) [= x1 by FILTER_0:def 7;
  y1 "/\" ((y1 => x1) "/\" (y2 => x2)) = y1 "/\" (y1 => x1) "/\" (y2 =>
  x2) by LATTICES:def 7;
  then
A11: y1 "/\" ((y1 => x1) "/\" (y2 => x2)) [= x1 by A10,FILTER_0:2;
A12: y2 "/\" ((y1 => x1) "/\" (y2 => x2)) = y2 "/\" (y1 => x1) "/\" (y2 =>
  x2) by LATTICES:def 7;
A13: y2 => x2 in FI by A3,FILTER_0:8;
A14: y2 "/\" (y2 => x2) [= x2 by FILTER_0:def 7;
  (y1 => x1) "/\" (y2 "/\" (y2 => x2)) = (y1 => x1) "/\" y2 "/\" (y2 =>
  x2) by LATTICES:def 7;
  then y2 "/\" ((y1 => x1) "/\" (y2 => x2)) [= x2 by A12,A14,FILTER_0:2;
  then
  y1 "/\" ((y1 => x1) "/\" (y2 => x2)) "\/" (y2 "/\" ((y1 => x1) "/\" (y2
  => x2))) [= x1 "\/" x2 by A11,FILTER_0:4;
  then (y1 "\/" y2) "/\" ((y1 => x1) "/\" (y2 => x2)) [= x1 "\/" x2 by
LATTICES:def 11;
  then
A15: (y1 => x1) "/\" (y2 => x2) [= (y1 "\/" y2) => (x1 "\/" x2) by
FILTER_0:def 7;
A16: x1 <=> y1 in FI by A1,FILTER_0:def 11;
  then y1 => x1 in FI by FILTER_0:8;
  then (y1 => x1) "/\" (y2 => x2) in FI by A13,FILTER_0:8;
  then
A17: (y1 "\/" y2) => (x1 "\/" x2) in FI by A15,FILTER_0:9;
  x1 => y1 in FI by A16,FILTER_0:8;
  then (x1 => y1) "/\" (x2 => y2) in FI by A4,FILTER_0:8;
  then (x1 "\/" x2) => (y1 "\/" y2) in FI by A9,FILTER_0:9;
  then (x1 "\/" x2) <=> (y1 "\/" y2) in FI by A17,FILTER_0:8;
  hence thesis by FILTER_0:def 11;
end;
