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

theorem Th34:
  for L being complete LATTICE, x being Element of L holds
  meet { I where I is Ideal of L : x <= sup I } = waybelow x
proof
  let L be complete LATTICE, x be Element of L;
  set X = { I where I is Ideal of L : x <= sup I };
  X c= bool the carrier of L
  proof
    let a be object;
    assume a in X;
    then ex I be Ideal of L st ( a = I)&( x <= sup I);
    hence thesis;
  end;
  then reconsider X9 = X as Subset-Family of L;
  sup downarrow x = x by WAYBEL_0:34;
  then
A1: downarrow x in X;
  thus meet X c= waybelow x
  proof
    let a be object;
    assume
A2: a in meet X;
    then a in meet X9;
    then reconsider y = a as Element of L;
    now
      let I be Ideal of L;
      assume x <= sup I;
      then I in X;
      hence y in I by A2,SETFAM_1:def 1;
    end;
    then y << x by WAYBEL_3:21;
    hence thesis by WAYBEL_3:7;
  end;
  thus waybelow x c= meet X
  proof
    let a be object;
    assume a in waybelow x;
    then a in {y where y is Element of L: y << x} by WAYBEL_3:def 3;
    then
A3: ex a9 be Element of L st ( a9 = a)&( a9 << x);
    for Y be set holds Y in X implies a in Y
    proof
      let Y be set;
      assume Y in X;
      then ex I be Ideal of L st ( Y = I)&( x <= sup I);
      hence thesis by A3,WAYBEL_3:20;
    end;
    hence thesis by A1,SETFAM_1:def 1;
  end;
end;
