let X, Y be non empty TopSpace; :: thesis: for a, b being Element of (oContMaps (X,Y))
for f, g being Function of X,(Omega Y) st a = f & b = g holds
( a <= b iff f <= g )
let a, b be Element of (oContMaps (X,Y)); :: thesis: for f, g being Function of X,(Omega Y) st a = f & b = g holds
( a <= b iff f <= g )
A1: oContMaps (X,Y) is full SubRelStr of (Omega Y) |^ the carrier of X by WAYBEL24:def 3;
then reconsider x = a, y = b as Element of ((Omega Y) |^ the carrier of X) by YELLOW_0:58;
A2: ( a <= b iff x <= y ) by A1, YELLOW_0:59, YELLOW_0:60;
let f, g be Function of X,(Omega Y); :: thesis: ( a = f & b = g implies ( a <= b iff f <= g ) )
assume ( a = f & b = g ) ; :: thesis: ( a <= b iff f <= g )
hence ( a <= b iff f <= g ) by A2, WAYBEL10:11; :: thesis: verum