reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem Th05:
  for X be set,B1,B2 be Subset-Family of X st B1,B2 are_equivalent_generators
  holds <.B1.] c= <.B2.]
  proof
    let X be set,B1,B2 be Subset-Family of X;
    assume
A1: B1,B2 are_equivalent_generators;
    let x be object;
    assume
A2: x in <.B1.];
    then reconsider x0=x as Subset of X;
    consider b1 be Element of B1 such that
A3: b1 c= x0 by A2,def3;
    consider b2 be Element of B2 such that
A4: b2 c= b1 by A1;
    b2 c= x0 by A3,A4;
    hence thesis by def3;
  end;
