reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem Th56:
  Bottom latt <.p.) = p
proof
  consider q9 being Element of latt <.p.) such that
A1: for r9 being Element of latt <.p.) holds q9"/\"r9 = q9 & r9"/\"q9 =
  q9 by LATTICES:def 13;
  the carrier of latt <.p.) = <.p.) by Th49;
  then reconsider p9 = p as Element of latt <.p.) by Th16;
  reconsider q = q9 as Element of <.p.) by Th49;
  q9"/\"p9 = q9 by A1;
  then q9 [= p9 by LATTICES:4;
  then
A2: q [= p by Th51;
A3: p [= q by Th15;
  q9 = Bottom latt <.p.) by A1,RLSUB_2:64;
  hence thesis by A2,A3,LATTICES:8;
end;
