
theorem Th30:
  for S,T being non empty TopPoset for X being non empty filtered Subset of S
  for f being monotone Function of S,T
  for Y being non empty filtered Subset of T st Y = f.:X
  holds f*(X opp+id) is subnet of Y opp+id
proof
  let S,T be non empty TopPoset;
  let X be non empty filtered Subset of S;
  let f be monotone Function of S,T;
  let Y be non empty filtered Subset of T such that
A1: Y = f.:X;
  set N = f*(X opp+id), M = Y opp+id;
A2: the RelStr of N = the RelStr of X opp+id by WAYBEL_9:def 8;
A3: the mapping of N = f*the mapping of X opp+id by WAYBEL_9:def 8;
A4: the carrier of M = Y by YELLOW_9:7;
A5: the mapping of M = id Y by WAYBEL19:27;
A6: the carrier of X opp+id = X by YELLOW_9:7;
  the mapping of X opp+id = id X by WAYBEL19:27;
  then
A7: the mapping of N = f|X by A3,RELAT_1:65;
  then
A8: rng the mapping of N = f.:X by RELAT_1:115;
  dom the mapping of N = X by A2,A6,FUNCT_2:def 1;
  then reconsider g = f|X as Function of N,M by A1,A2,A4,A6,A7,A8,FUNCT_2:def 1
,RELSET_1:4;
  take g;
  thus the mapping of N = (the mapping of M)*g by A1,A5,A7,A8,RELAT_1:53;
  let m be Element of M;
  consider n being object such that
A9: n in the carrier of S and
A10: n in X and
A11: m = f.n by A1,A4,FUNCT_2:64;
  reconsider n as Element of N by A2,A10,YELLOW_9:7;
  take n;
  let p be Element of N;
  p in X by A2,A6;
  then reconsider n9 = n, p9 = p as Element of S by A9;
  reconsider fp = f.p9 as Element of M by A1,A2,A4,A6,FUNCT_2:35;
  X opp+id is SubRelStr of S opp by YELLOW_9:7;
  then
A12: N is SubRelStr of S opp by A2,Th12;
A13: M is full SubRelStr of T opp by YELLOW_9:7;
  assume n <= p;
  then n9~ <= p9~ by A12,YELLOW_0:59;
  then n9 >= p9 by LATTICE3:9;
  then f.n9 >= f.p9 by WAYBEL_1:def 2;
  then (f.n9)~ <= (f.p9)~ by LATTICE3:9;
  then fp >= m by A11,A13,YELLOW_0:60;
  hence m <= g.p by A2,A6,FUNCT_1:49;
end;
