reserve x for set;

theorem
  for L1, L2 being complete LATTICE st the RelStr of L1 = the RelStr of
L2 for X1 being non empty Subset of L1 for X2 being non empty Subset of L2 for
  F1 being Filter of BoolePoset X1, F2 being Filter of BoolePoset X2 st F1 = F2
  holds lim_inf F1 = lim_inf F2
proof
  let L1,L2 be complete LATTICE such that
A1: the RelStr of L1 = the RelStr of L2;
  let X1 be non empty Subset of L1;
  let X2 be non empty Subset of L2;
  let F1 be Filter of BoolePoset X1, F2 be Filter of BoolePoset X2 such that
A2: F1 = F2;
  set Y2 = {inf B2 where B2 is Subset of L2: B2 in F2};
  set Y1 = {inf B1 where B1 is Subset of L1: B1 in F1};
A3: Y2 c= Y1
  proof
    let x be object;
    assume x in Y2;
    then consider B2 being Subset of L2 such that
A4: x = inf B2 and
A5: B2 in F2;
    F1 c= the carrier of BoolePoset X1;
    then F1 c= bool X1 by WAYBEL_7:2;
    then reconsider B1=B2 as Subset of L1 by A2,A5,XBOOLE_1:1;
    inf B1 = inf B2 by A1,YELLOW_0:17,27;
    hence thesis by A2,A4,A5;
  end;
  Y1 c= Y2
  proof
    let x be object;
    assume x in Y1;
    then consider B1 being Subset of L1 such that
A6: x = inf B1 and
A7: B1 in F1;
    F2 c= the carrier of BoolePoset X2;
    then F2 c= bool X2 by WAYBEL_7:2;
    then reconsider B2=B1 as Subset of L2 by A2,A7,XBOOLE_1:1;
    inf B1 = inf B2 by A1,YELLOW_0:17,27;
    hence thesis by A2,A6,A7;
  end;
  then Y1 = Y2 by A3;
  hence thesis by A1,YELLOW_0:17,26;
end;
