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 Th18:
  I is lower-bounded implies I/\/FI is 0_Lattice & Bottom (I/\/FI)
  = (Bottom I)/\/FI
proof
  set L = I/\/FI;
  set R = equivalence_wrt FI;
  assume
A1: I is lower-bounded;
  then consider i such that
A2: i"/\"j = i & j"/\"i = i;
  set x = i/\/FI;
A3: now
    let y be Element of L;
    L = LattStr (#Class R, (the L_join of I)/\/R, (the L_meet of I)/\/R #)
    by Def5;
    then consider j such that
A4: y = Class(R,j) by EQREL_1:36;
A5: i"/\"j = i by A2;
A6: y = j/\/FI by A4,Def6;
    hence x"/\"y = x by A5,Th15;
    thus y"/\"x = x by A5,A6,Th15;
  end;
  hence
A7: I/\/FI is 0_Lattice by LATTICES:def 13;
  Bottom I = i by A1,A2,LATTICES:def 16;
  hence thesis by A3,A7,LATTICES:def 16;
end;
