let X be non empty set ; :: thesis: for Y, Z being non empty Subset of ExtREAL
for F being Function of X,Y
for G being Function of X,Z holds sup (F + G) <= (sup F) + (sup G)
let Y, Z be non empty Subset of ExtREAL; :: thesis: for F being Function of X,Y
for G being Function of X,Z holds sup (F + G) <= (sup F) + (sup G)
let F be Function of X,Y; :: thesis: for G being Function of X,Z holds sup (F + G) <= (sup F) + (sup G)
let G be Function of X,Z; :: thesis: sup (F + G) <= (sup F) + (sup G)
A1: sup ((rng F) + (rng G)) <= (sup (rng F)) + (sup (rng G)) by Th7;
sup (F + G) <= sup ((rng F) + (rng G)) by Th15, XXREAL_2:59;
hence sup (F + G) <= (sup F) + (sup G) by A1, XXREAL_0:2; :: thesis: verum