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

theorem Th18:
  for S being non empty 1-sorted, N being net of S, M being subnet
  of N, X st M is_often_in X holds N is_often_in X
proof
  let S be non empty 1-sorted, N be net of S, M be subnet of N, X such that
A1: M is_often_in X;
  let i be Element of N;
  consider f being Function of M, N such that
A2: the mapping of M = (the mapping of N)*f and
A3: for m being Element of N ex n being Element of M st for p being
  Element of M st n <= p holds m <= f.p by Def9;
  consider n being Element of M such that
A4: for p being Element of M st n <= p holds i <= f.p by A3;
  consider m being Element of M such that
A5: n <= m and
A6: M.m in X by A1;
  take f.m;
  thus i <= f.m by A4,A5;
  thus thesis by A2,A6,FUNCT_2:15;
end;
