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;
