
theorem
  for T being non empty 1-sorted, V,W being Subset of T for S being
  TopAugmentation of BoolePoset{0}
    for f, g being Function of T, S st f = chi(V,
  the carrier of T) & g = chi(W, the carrier of T) holds V c= W iff f <= g
proof
  let T be non empty 1-sorted, V,W be Subset of T;
  let S be TopAugmentation of BoolePoset{0};
  let c1, c2 be Function of T, S such that
A1: c1 = chi(V, the carrier of T) and
A2: c2 = chi(W, the carrier of T);
A3: the RelStr of S = BoolePoset{0} by YELLOW_9:def 4;
  hereby
    assume
A4: V c= W;
    now
      let z be set;
      assume z in the carrier of T;
      then reconsider x = z as Element of T;
      reconsider a = c1.x, b = c2.x as Element of BoolePoset{0} by A3;
      x in V & x in W or not x in V by A4;
      then c1.x = 1 & c2.x = 1 or c1.x = 0 by A1,A2,FUNCT_3:def 3;
      then c1.x c= c2.x;
      then a <= b by YELLOW_1:2;
      hence ex a,b being Element of S st a = c1.z & b = c2.z & a <= b by A3,
YELLOW_0:1;
    end;
    hence c1 <= c2;
  end;
  assume
A5: c1 <= c2;
  let x be object;
  assume that
A6: x in V and
A7: not x in W;
  reconsider x as Element of T by A6;
A8: c2.x = 0 by A2,A7,FUNCT_3:def 3;
A9: 0 c= {0};
  reconsider a = c1.x, b = c2.x as Element of BoolePoset{0} by A3;
  ex a,b being Element of S st a = c1.x & b = c2.x & a <= b by A5;
  then
A10: a <= b by A3,YELLOW_0:1;
  c1.x = 1 by A1,A6,FUNCT_3:def 3;
  then {0} c= 0 by A8,A10,YELLOW_1:2,CARD_1:49;
  hence thesis by A9,XBOOLE_0:def 10;
end;
