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 Th14:
  FX c= GX implies clf FX c= clf GX
proof
  reconsider CFX = clf FX,CGX = clf GX as set;
  assume
A1: FX c= GX;
  for X be object st X in CFX holds X in CGX
  proof
    let X be object;
    assume
A2: X in CFX;
    then reconsider X as Subset of T;
    ex V st X = Cl V & V in FX by A2,Def2;
    hence thesis by A1,Def2;
  end;
  hence thesis;
end;
