reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem
  ex M be non empty MetrSpace st M is complete & ex S be non-empty
pointwise_bounded SetSequence of M st S is closed & S is non-ascending &
meet S is empty
proof
  reconsider D=DiscreteSpace 2 as non empty MetrSpace;
  0 in Segm 2 & 1 in Segm 2 by NAT_1:44;
  then reconsider a=0,b=1 as Point of D;
  TopSpaceMetr D is compact;
  then
A1: D is complete by TBSP_1:8;
A2: 1=dist(a,b) by METRIC_1:def 10;
  then
A3: ex S be non-empty pointwise_bounded SetSequence of WellSpace(a,NAT) st
    S is closed
  & S is non-ascending & meet S is empty by A1,Th42;
  WellSpace(a,NAT) is complete by A2,A1,Th42;
  hence thesis by A3;
end;
