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;
