reserve x,y,X for set;

theorem Th19:
  for T being non empty 1-sorted, A being set for N being net of T
  st N is_eventually_in A for S being subnet of N holds S is_eventually_in A
proof
  let T be non empty 1-sorted, A be set;
  let N be net of T;
  given i being Element of N such that
A1: for j being Element of N st i <= j holds N.j in A;
  let S be subnet of N;
  consider f being Function of S, N such that
A2: the mapping of S = (the mapping of N)*f and
A3: for m being Element of N ex n being Element of S st for p being
  Element of S st n <= p holds m <= f.p by YELLOW_6:def 9;
  consider n being Element of S such that
A4: for p being Element of S st n <= p holds i <= f.p by A3;
  take n;
  let p be Element of S;
  assume n <= p;
  then N.(f.p) in A by A1,A4;
  hence thesis by A2,FUNCT_2:15;
end;
