let X be non empty set ; :: thesis: for S being SigmaField of X
for f, g being PartFunc of X,COMPLEX st f is_simple_func_in S & g is_simple_func_in S holds
f + g is_simple_func_in S
let S be SigmaField of X; :: thesis: for f, g being PartFunc of X,COMPLEX st f is_simple_func_in S & g is_simple_func_in S holds
f + g is_simple_func_in S
let f, g be PartFunc of X,COMPLEX; :: thesis: ( f is_simple_func_in S & g is_simple_func_in S implies f + g is_simple_func_in S )
assume A1: ( f is_simple_func_in S & g is_simple_func_in S ) ; :: thesis: f + g is_simple_func_in S
then ( Im f is_simple_func_in S & Im g is_simple_func_in S ) by MESFUN7C:43;
then (Im f) + (Im g) is_simple_func_in S by MESFUNC6:72;
then A2: Im (f + g) is_simple_func_in S by MESFUN6C:5;
( Re f is_simple_func_in S & Re g is_simple_func_in S ) by A1, MESFUN7C:43;
then (Re f) + (Re g) is_simple_func_in S by MESFUNC6:72;
then Re (f + g) is_simple_func_in S by MESFUN6C:5;
hence f + g is_simple_func_in S by A2, MESFUN7C:43; :: thesis: verum