
theorem Th21:
  for L being non empty 1-sorted, C1,C2 being Convergence-Class of
L holds C1 c= C2 implies the topology of ConvergenceSpace C2 c= the topology of
  ConvergenceSpace C1
proof
  let L be non empty 1-sorted, C1,C2 be Convergence-Class of L;
  assume
A1: C1 c= C2;
  let A be object;
  assume A in the topology of ConvergenceSpace C2;
  then
  A in { V where V is Subset of L: for p being Element of L st p in V for
  N being net of L st [N,p] in C2 holds N is_eventually_in V} by
YELLOW_6:def 24;
  then consider V1 being Subset of L such that
A2: A=V1 and
A3: for p being Element of L st p in V1 for N being net of L st [N,p] in
  C2 holds N is_eventually_in V1;
  ex V being Subset of L st A=V & for p being Element of L st p in V for N
  being net of L st [N,p] in C1 holds N is_eventually_in V
  proof
    take V1;
    thus A=V1 by A2;
    let p be Element of L;
    assume
A4: p in V1;
    let N be net of L;
    assume [N,p] in C1;
    hence thesis by A1,A3,A4;
  end;
  then
  A in { V where V is Subset of L: for p being Element of L st p in V for
  N being net of L st [N,p] in C1 holds N is_eventually_in V};
  hence thesis by YELLOW_6:def 24;
end;
