reserve L,L1,L2 for Lattice,
  F1,F2 for Filter of L,
  p,q,r,s for Element of L,
  p1,q1,r1,s1 for Element of L1,
  p2,q2,r2,s2 for Element of L2,
  X,x,x1,x2,y,y1,y2 for set,
  D,D1,D2 for non empty set,
  R for Relation,
  RD for Equivalence_Relation of D,
  a,b,d for Element of D,
  a1,b1,c1 for Element of D1,
  a2,b2,c2 for Element of D2,
  B for B_Lattice,
  FB for Filter of B,
  I for I_Lattice,
  FI for Filter of I ,
  i,i1,i2,j,j1,j2,k for Element of I,
  f1,g1 for BinOp of D1,
  f2,g2 for BinOp of D2;
reserve F,G for BinOp of D,RD;

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;
