reserve T for TopSpace;

theorem
  for F,G being Subset-Family of T holds F c= G implies Int F c= Int G
proof
  let F,G be Subset-Family of T;
  reconsider F0 = Int F, G0 = Int G as set;
  assume
A1: F c= G;
  now
    let X be object;
    assume
A2: X in F0;
    then reconsider X0 = X as Subset of T;
    ex V being Subset of T st X0 = Int V & V in F by A2,Def1;
    hence X in G0 by A1,Def1;
  end;
  hence thesis;
end;
