reserve x,y,z,X for set,
  T for Universe;

theorem Th32:
  for T being non empty TopSpace, N being net of T, Y being subnet
  of N holds Lim N c= Lim Y
proof
  let T be non empty TopSpace, N be net of T, Y be subnet of N;
  let x be object;
  consider f being Function of Y, N such that
A1: the mapping of Y = (the mapping of N)*f and
A2: for m being Element of N ex n being Element of Y st for p being
  Element of Y st n <= p holds m <= f.p by Def9;
  assume
A3: x in Lim N;
  then reconsider p = x as Point of T;
  for V being a_neighborhood of p holds Y is_eventually_in V
  proof
    let V be a_neighborhood of p;
    N is_eventually_in V by A3,Def15;
    then consider ii being Element of N such that
A4: for j being Element of N st ii <= j holds N.j in V;
    consider n being Element of Y such that
A5: for p being Element of Y st n <= p holds ii <= f.p by A2;
    take n;
    let j be Element of Y;
    assume
A6: n <= j;
    N.(f.j) = Y.j by A1,FUNCT_2:15;
    hence thesis by A4,A5,A6;
  end;
  hence thesis by Def15;
end;
