
theorem Th10:
  for X,Y,Z being non empty TopSpace for f being continuous
  Function of Y,Z holds oContMaps(f, X) is monotone
proof
  let X,Y,Z be non empty TopSpace;
  let f be continuous Function of Y,Z;
  let a,b be Element of oContMaps(Z, X);
  reconsider g1 = a, g2 = b as continuous Function of Z, Omega X by Th1;
  set Xf = oContMaps(f, X);
  the TopStruct of Y = the TopStruct of Omega Y & the TopStruct of Z = the
  TopStruct of Omega Z by WAYBEL25:def 2;
  then reconsider f9 = f as continuous Function of Omega Y, Omega Z by
YELLOW12:36;
  g2 is continuous Function of Z,X by Th2;
  then
A1: Xf.b = g2 qua Function*f9 by Def3;
  g1 is continuous Function of Z,X by Th2;
  then
A2: Xf.a = g1 qua Function*f9 by Def3;
  then reconsider
  fg1 = g1 qua Function*f9, fg2 = g2 qua Function*f9 as Function of
  Y, Omega X by A1,Th1;
  assume a <= b;
  then
A3: g1 <= g2 by Th3;
  now
    let x be set;
    assume x in the carrier of Y;
    then reconsider x9 = x as Element of Y;
    (g1*f).x9 = g1.(f.x9) & (g2*f).x9 = g2.(f.x9) by FUNCT_2:15;
    hence ex a, b being Element of Omega X st a = (g1*f).x & b = (g2*f).x & a
    <= b by A3;
  end;
  then fg1 <= fg2;
  hence Xf.a <= Xf.b by A2,A1,Th3;
end;
