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 Th52:
  L is upper-bounded implies latt F is upper-bounded
proof
  given p such that
A1: p"\/"q = p & q"\/"p = p;
  consider q such that
A2: q in F by SUBSET_1:4;
A3: ex o1,o2 st o1 = (the L_join of L)||F & o2 = (the L_meet of L)||F & latt
  F = LattStr (#F, o1, o2#) by Def9;
  p"\/"q = p by A1;
  then reconsider p9 = p as Element of latt F by A2,A3,Th10;
  take p9;
  let r be Element of latt F;
  reconsider r9 = r as Element of F by A3;
  thus p9"\/"r = p"\/"r9 by Th50
    .= p9 by A1;
  hence thesis;
end;
