reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;
reserve p9,q9 for Element of LattPOSet L;
reserve C for complete Lattice,
  a,a9,b,b9,c,d for Element of C,
  X,Y for set;

theorem
  C is 0_Lattice & Bottom C = "\/"({},C)
proof
A1: now
    let a;
    {} is_less_than ("\/"({},C))"/\"a;
    then
A2: "\/"({},C) [= ("\/"({},C))"/\"a by Def21;
A3: ("\/"({},C))"/\"a [= "\/"({},C ) by LATTICES:6;
    hence ("\/"({},C))"/\"a = "\/"({},C) by A2,LATTICES:8;
    thus a"/\"("\/"({},C)) = "\/"({},C) by A2,A3,LATTICES:8;
  end;
  then C is lower-bounded;
  hence thesis by A1,LATTICES:def 16;
end;
