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;

theorem Th25:
  L is 0_Lattice & Bottom L in D implies <.D.) = <.L.) & <.D.) =
  the carrier of L
proof
  assume that
A1: L is 0_Lattice and
A2: Bottom L in D;
A3: <.Bottom L.) = <.L.) by A1,Th17;
  hence <.D.) c= <.L.) & <.L.) c= <.D.) by A2,Th23;
  thus <.D.) c= the carrier of L & the carrier of L c= <.D.) by A2,A3,Th23;
end;
