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 Th51:
  C is I_Lattice implies a => b = "\/"({c: a"/\"c [= b}, C)
proof
  set X = {a9: a"/\"a9 [= b};
  assume
A1: C is I_Lattice;
  then a"/\"(a=>b) [= b by FILTER_0:def 7;
  then
A2: a=>b in X;
  X is_less_than a=>b
  proof
    let c;
    assume c in X;
    then ex a9 st c = a9 & a"/\"a9 [= b;
    hence c [= a=>b by A1,FILTER_0:def 7;
  end;
  hence thesis by A2,Th40;
end;
