reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;

theorem Th10:
  FX is finite implies FX is locally_finite
proof
  assume
A1: FX is finite;
  for x ex W being Subset of T st x in W & W is open & { V : V in FX & V
  meets W } is finite
  proof
    let x;
    take [#]T;
    thus x in [#]T;
    thus [#]T is open;
    thus thesis by A1,Th8,FINSET_1:1;
  end;
  hence thesis;
end;
