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
  B/\/FB is B_Lattice
proof
  set L = B/\/FB;
  set R = equivalence_wrt FB;
A1: L is 0_Lattice by Th18;
A2: Bottom L = (Bottom B)/\/FB by Th18;
A3: Top L = (Top B)/\/FB by Th19;
  reconsider L as 01_Lattice by A1;
A4: L is complemented
  proof
    let x be Element of L;
    L = LattStr (#Class R, (the L_join of B)/\/R, (the L_meet of B)/\/R #)
    by Def5;
    then consider a being Element of B such that
A5: x = Class(R,a) by EQREL_1:36;
    reconsider y = a`/\/FB as Element of L;
    take y;
A6: x = a/\/FB by A5,Def6;
    hence y"\/"x = (a`"\/"a)/\/FB by Th15
      .= (Top B)/\/FB by LATTICES:21
      .= Top L by A3;
    hence x"\/"y = Top L;
    thus y"/\"x = (a`"/\"a)/\/FB by A6,Th15
      .= Bottom L by A2,LATTICES:20;
    hence x"/\"y = Bottom L;
  end;
  thus thesis by A4;
end;
