
theorem  :: Remark 1.5 (ii)
  for L being lower-bounded LATTICE st L is meet-continuous
  for x,y being Element of L holds
  x << y iff for I being Ideal of L st y = sup I holds x in I
proof
  let L be lower-bounded LATTICE;
  assume
A1: L is up-complete satisfying_MC;
  let x,y be Element of L;
  hereby
    assume
A2: x << y;
    let I be Ideal of L;
    assume y = sup I;
    then ex d being Element of L st d in I & x <= d by A2;
    hence x in I by WAYBEL_0:def 19;
  end;
  assume
A3: for I being Ideal of L st y = sup I holds x in I;
  now
    let I be Ideal of L;
A4: ex_sup_of finsups ({y}"/\"I), L by A1,WAYBEL_0:75;
    assume y <= sup I;
    then y "/\" sup I = y by YELLOW_0:25;
    then y = sup ({y} "/\" I) by A1
      .= sup finsups ({y}"/\"I) by A1,WAYBEL_0:55
      .= sup downarrow finsups ({y}"/\"I) by A4,WAYBEL_0:33;
    then x in downarrow finsups ({y} "/\" I) by A3;
    then consider z being Element of L such that
A5: x <= z and
A6: z in finsups ({y}"/\"I) by WAYBEL_0:def 15;
    consider Y being finite Subset of {y}"/\"I such that
A7: z = "\/"(Y,L) and ex_sup_of Y, L by A6;
    set i = the Element of I;
    reconsider i as Element of L;
A8: ex_sup_of {i}, L by YELLOW_0:38;
A9: sup {i} = i by YELLOW_0:39;
    ex_sup_of {},L by A1,YELLOW_0:17;
    then "\/"({},L) <= sup {i} by A8,XBOOLE_1:2,YELLOW_0:34;
    then
A10: "\/"({},L) in I by A9,WAYBEL_0:def 19;
    Y c= I
    proof
      let a be object;
      assume a in Y;
      then a in {y}"/\"I;
      then a in {y"/\"j where j is Element of L: j in I} by YELLOW_4:42;
      then consider j being Element of L such that
A11:  a = y"/\"j and
A12:  j in I;
      y"/\"j <= j by YELLOW_0:23;
      hence thesis by A11,A12,WAYBEL_0:def 19;
    end;
    then z in I by A7,A10,WAYBEL_0:42;
    hence x in I by A5,WAYBEL_0:def 19;
  end;
  hence thesis by A1,Th21;
end;
