reserve x for set;

theorem Th17:
  for L being non empty 1-sorted for N being net of L for A being
set st N is_often_in A ex N9 being strict subnet of N st rng the mapping of N9
  c= A & N9 is SubNetStr of N
proof
  let L be non empty 1-sorted;
  let N be net of L;
  let A be set;
  assume N is_often_in A;
  then reconsider N9 = N"A as strict subnet of N by YELLOW_6:22;
  take N9;
  rng the mapping of N9 c= A
  proof
    let y be object;
    assume y in rng the mapping of N9;
    then consider x being object such that
A1: x in dom the mapping of N9 and
A2: y = (the mapping of N9).x by FUNCT_1:def 3;
A3: x in dom ((the mapping of N)|the carrier of N9) by A1,YELLOW_6:def 6;
    then x in (dom the mapping of N) /\ the carrier of N9 by RELAT_1:61;
    then x in the carrier of N9 by XBOOLE_0:def 4;
    then
A4: x in (the mapping of N)"A by YELLOW_6:def 10;
    y = ((the mapping of N)|the carrier of N9).x by A2,YELLOW_6:def 6;
    then y = (the mapping of N).x by A3,FUNCT_1:47;
    hence thesis by A4,FUNCT_1:def 7;
  end;
  hence thesis;
end;
