let X, Y, Z be non empty TopSpace; :: thesis: for f being continuous Function of Y,Z holds oContMaps (f,X) is monotone
let f be continuous Function of Y,Z; :: thesis: oContMaps (f,X) is monotone
let a, b be Element of (oContMaps (Z,X)); :: according to WAYBEL_1:def 2 :: thesis: ( not a <= b or (oContMaps (f,X)) . a <= (oContMaps (f,X)) . b )
reconsider g1 = a, g2 = b as continuous Function of Z,(Omega X) by Th1;
set Xf = oContMaps (f,X);
( TopStruct(# the carrier of Y, the topology of Y #) = TopStruct(# the carrier of (Omega Y), the topology of (Omega Y) #) & TopStruct(# the carrier of Z, the topology of Z #) = TopStruct(# the carrier of (Omega Z), the topology 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: (oContMaps (f,X)) . b = g2 * f9 by Def3;
g1 is continuous Function of Z,X by Th2;
then A2: (oContMaps (f,X)) . a = g1 * f9 by Def3;
then reconsider fg1 = g1 * f9, fg2 = g2 * f9 as Function of Y,(Omega X) by A1, Th1;
assume a <= b ; :: thesis: (oContMaps (f,X)) . a <= (oContMaps (f,X)) . b
then A3: g1 <= g2 by Th3;

now :: thesis: for x being set st x in the carrier of Y holds
ex a, b being Element of (Omega X) st
( a = (g1 * f) . x & b = (g2 * f) . x & a <= b )

let x be set ; :: thesis: ( x in the carrier of Y implies ex a, b being Element of (Omega X) st
( a = (g1 * f) . x & b = (g2 * f) . x & a <= b ) )
assume x in the carrier of Y ; :: thesis: ex a, b being Element of (Omega X) st
( a = (g1 * f) . x & b = (g2 * f) . x & a <= b )
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; :: thesis: verum

end;

then fg1 <= fg2 ;
hence (oContMaps (f,X)) . a <= (oContMaps (f,X)) . b by A2, A1, Th3; :: thesis: verum