
theorem Th18:
  for L being complete LATTICE, x,y being Element of L st x << y
  for X being Subset of L st y <= sup X
  ex A being finite Subset of L st A c= X & x <= sup A
proof
  let L be complete LATTICE, x,y be Element of L;
  assume
A1: x << y;
  let X be Subset of L;
  assume
A2: y <= sup X;
  defpred P[set] means
  ex Y being finite Subset of X st ex_sup_of Y,L & $1 = "\/"(Y,L);
  consider F being Subset of L such that
A3: for a being Element of L holds a in F iff P[a] from SUBSET_1:sch 3;
A4: for Y being finite Subset of X st Y <> {} holds "\/"(Y,L) in F
          by A3,YELLOW_0:17;
A5: for Y being finite Subset of X st Y <> {} holds ex_sup_of Y,L
  by YELLOW_0:17;
A6: {} c= X;
  ex_sup_of {},L by YELLOW_0:17;
  then "\/"({},L) in F by A3,A6;
  then reconsider F as directed non empty Subset of L by A3,A4,A5,WAYBEL_0:51;
  ex_sup_of X,L by YELLOW_0:17;
  then sup X = sup F by A3,A4,A5,WAYBEL_0:54;
  then consider d being Element of L such that
A7: d in F and
A8: x <= d by A1,A2;
  consider Y being finite Subset of X such that
  ex_sup_of Y,L and
A9: d = "\/"(Y,L) by A3,A7;
  reconsider Y as finite Subset of L by XBOOLE_1:1;
  take Y;
  thus thesis by A8,A9;
end;
