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
  FX is locally_finite & FX is closed implies union FX is closed
proof
  assume that
A1: FX is locally_finite and
A2: FX is closed;
  union (clf FX) c= union FX
  proof
    set UFX = union FX, UCFX = union(clf FX);
    for x st x in UCFX holds x in UFX
    proof
      let x;
      assume x in UCFX;
      then consider X be set such that
A3:   x in X and
A4:   X in clf FX by TARSKI:def 4;
      reconsider X as Subset of T by A4;
      consider W being Subset of T such that
A5:   X = Cl W and
A6:   W in FX by A4,Def2;
      reconsider W as Subset of T;
      W is closed by A2,A6,TOPS_2:def 2;
      then x in W by A3,A5,PRE_TOPC:22;
      hence thesis by A6,TARSKI:def 4;
    end;
    hence thesis;
  end;
  then
A7: Cl(union FX) c= union FX by A1,Th20;
  union FX c= union(clf FX) by Th17,SETFAM_1:13;
  then union FX c= Cl(union FX) by A1,Th20;
  hence thesis by A7,XBOOLE_0:def 10;
end;
