reserve L for Lattice,
  p,p1,q,q1,r,r1 for Element of L;
reserve x,y,z,X,Y,Z,X1,X2 for set;
reserve H,F for Filter of L;
reserve D for non empty Subset of L;
reserve D1,D2 for non empty Subset of L;
reserve I for I_Lattice,
  i,j,k for Element of I;
reserve B for B_Lattice,
  FB,HB for Filter of B;
reserve I for I_Lattice,
  i,j,k for Element of I,
  DI for non empty Subset of I,
  FI for Filter of I;
reserve F1,F2 for Filter of I;
reserve a,b,c for Element of B;
reserve o1,o2 for BinOp of F;

theorem Th57:
  L is upper-bounded implies Top latt <.p.) = Top L
proof
  given q such that
A1: for r holds q"\/"r = q & r"\/"q = q;
  L is 1_Lattice by A1,LATTICES:def 14;
  then Top L in <.p.) by Th11;
  then reconsider q9 = Top L as Element of latt <.p.) by Th49;
A2: q = Top L by A1,RLSUB_2:65;
  now
    let r9 be Element of latt <.p.);
    reconsider r = r9 as Element of <.p.) by Th49;
    thus r9"\/"q9 = q"\/"r by A2,Th50
      .= q9 by A1,A2;
  end;
  hence thesis by RLSUB_2:65;
end;
