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 Th32:
  "\/"({a"/\"b: b in X}, C) [= a"/\""\/"(X,C)
proof
  set Y = {a"/\"b: b in X};
  Y is_less_than a"/\""\/"(X,C)
  proof
    let c;
    assume c in Y;
    then consider b such that
A1: c = a"/\"b and
A2: b in X;
    X is_less_than "\/"(X,C) by Def21;
    then b [= "\/"(X,C) by A2;
    hence thesis by A1,LATTICES:9;
  end;
  hence thesis by Def21;
end;
