
theorem Th13:
  for S1, S2 being RelStr, D being Subset of S1 for f being
  Function of S1,S2 st f is monotone holds f.:(downarrow D) c= downarrow (f.:D)
proof
  let S1, S2 be RelStr, D be Subset of S1, f be Function of S1,S2 such that
A1: f is monotone;
  let q be object;
  assume
A2: q in f.:(downarrow D);
  then consider x being object such that
A3: x in dom f and
A4: x in downarrow D and
A5: q = f.x by FUNCT_1:def 6;
  reconsider s1 = S1, s2 = S2 as non empty RelStr by A2;
  reconsider x as Element of s1 by A3;
  consider y being Element of s1 such that
A6: x <= y and
A7: y in D by A4,WAYBEL_0:def 15;
  reconsider f1 = f as Function of s1,s2;
  f1.x is Element of s2;
  then reconsider q1 = q, fy = f1.y as Element of s2 by A5;
  the carrier of s2 <> {};
  then dom f = the carrier of s1 by FUNCT_2:def 1;
  then
A8: f.y in f.:D by A7,FUNCT_1:def 6;
  q1 <= fy by A1,A5,A6;
  hence thesis by A8,WAYBEL_0:def 15;
end;
