reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th44:
  for L being lower-bounded meet-continuous LATTICE, x being Element of L holds
  meet {AR-below x where AR is auxiliary Relation of L : AR in App L}
  = waybelow x
proof
  let L be lower-bounded meet-continuous LATTICE, x be Element of L;
  set A = {AR-below x where AR is auxiliary Relation of L : AR in App L};
  set AA = the approximating auxiliary Relation of L;
  AA in App L by Def19;
  then
A1: AA-below x in A;
A2: meet { I where I is Ideal of L : x <= sup I } = waybelow x by Th34;
A3: meet the set of all  (DownMap I).x where I is Ideal of L  =
  waybelow x by Th43;
  set I1 = the Ideal of L;
A4: (DownMap I1).x in the set of all  (DownMap I).x where I is Ideal of L ;
  the set of all  (DownMap I).x where I is Ideal of L  c= A
  proof
    let a be object;
    assume a in the set of all  (DownMap I).x where I is Ideal of L ;
    then consider I be Ideal of L such that
A5: a = (DownMap I).x;
A6: DownMap I is approximating by Th37;
    DownMap I in the carrier of MonSet L by Th35;
    then consider AR be auxiliary Relation of L such that
A7: AR = (Map2Rel L).(DownMap I) and
A8: for x,y be object holds [x,y] in AR iff ex x9,y9 be Element of L, s9
be Function of L, InclPoset Ids L st x9 = x & y9 = y & s9 = DownMap I & x9 in
    s9.y9
    by Def15;
A9: ex AR1 be approximating auxiliary Relation of L st ( AR1 =
    (Map2Rel L).(DownMap I)) by A6,Th35,Th42;
A10: ex ii be Subset of L st ( ii = (DownMap I).x)&( x = sup ii) by A6;
A11: AR-below x c= (DownMap I).x
    proof
      let a be object;
      assume a in AR-below x;
      then [a,x] in AR by Th13;
      then ex x9,y9 be Element of L, s9 be Function of L, InclPoset
      Ids L st ( x9 = a)&( y9 = x)&( s9 = DownMap I)&( x9 in s9.y9) by A8;
      hence thesis;
    end;
    (DownMap I).x c= AR-below x
    proof
      let a be object;
      assume
A12:  a in (DownMap I).x;
      then reconsider a9 = a as Element of L by A10;
      [a9,x] in AR by A8,A12;
      hence thesis;
    end;
    then
A13: AR-below x = (DownMap I).x by A11;
    AR in App L by A7,A9,Def19;
    hence thesis by A5,A13;
  end;
  hence meet A c= waybelow x by A3,A4,SETFAM_1:6;
  A c= { I where I is Ideal of L : x <= sup I }
  proof
    let a be object;
    assume a in A;
    then consider AR be auxiliary Relation of L such that
A14: a = AR-below x and
A15: AR in App L;
    reconsider AR as approximating auxiliary Relation of L by A15,Def19;
    sup (AR-below x) = x by Def17;
    hence thesis by A14;
  end;
  hence waybelow x c= meet A by A1,A2,SETFAM_1:6;
end;
