reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  x for set;

theorem Th26:
  F c= G implies Der F c= Der G
proof
  assume
A1: F c= G;
  Der F c= Der G
  proof
    let x be object;
    assume
A2: x in Der F;
    then reconsider A = x as Subset of T;
    ex B being Subset of T st A = Der B & B in F by A2,Def6;
    hence thesis by A1,Def6;
  end;
  hence thesis;
end;
