
theorem
  for S being reflexive RelStr, T being reflexive transitive RelStr, f
being Function of S, T, X being Subset of S st f is monotone holds downarrow (f
  .:X) = downarrow (f.:downarrow X)
proof
  let S be reflexive RelStr, T be reflexive transitive RelStr, f be Function
  of S, T, X be Subset of S such that
A1: f is monotone;
  thus downarrow f.:X c= downarrow f.:downarrow X by Th6;
  let a be object;
  assume
A2: a in downarrow (f.:downarrow X);
  then reconsider T1 = T as non empty reflexive transitive RelStr;
  reconsider b = a as Element of T1 by A2;
  consider fx being Element of T1 such that
A3: fx >= b and
A4: fx in f.:downarrow X by A2,WAYBEL_0:def 15;
  consider x being object such that
A5: x in dom f and
A6: x in downarrow X and
A7: fx = f.x by A4,FUNCT_1:def 6;
  reconsider S1 = S as non empty reflexive RelStr by A5;
  reconsider x as Element of S1 by A5;
  consider y being Element of S1 such that
A8: y >= x and
A9: y in X by A6,WAYBEL_0:def 15;
  reconsider f1 = f as Function of S1, T1;
  f1.x <= f1.y by A1,A8,ORDERS_3:def 5;
  then
A10: b <= f1.y by A3,A7,ORDERS_2:3;
  the carrier of T1 <> {};
  then dom f = the carrier of S by FUNCT_2:def 1;
  then f.y in f.:X by A9,FUNCT_1:def 6;
  hence thesis by A10,WAYBEL_0:def 15;
end;
