let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,COMPLEX st f is_integrable_on M & g is_integrable_on M holds
dom (f + g) in S
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,COMPLEX st f is_integrable_on M & g is_integrable_on M holds
dom (f + g) in S
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,COMPLEX st f is_integrable_on M & g is_integrable_on M holds
dom (f + g) in S
let f, g be PartFunc of X,COMPLEX; :: thesis: ( f is_integrable_on M & g is_integrable_on M implies dom (f + g) in S )
assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M ; :: thesis: dom (f + g) in S
A3: Re g is_integrable_on M by A2;
A4: Im g is_integrable_on M by A2;
Im f is_integrable_on M by A1;
then dom ((Im f) + (Im g)) in S by A4, MESFUNC6:99;
then A5: dom (Im (f + g)) in S by Th5;
Re f is_integrable_on M by A1;
then dom ((Re f) + (Re g)) in S by A3, MESFUNC6:99;
then A6: dom (Re (f + g)) in S by Th5;
dom (<i> (#) (Im (f + g))) = dom (Im (f + g)) by VALUED_1:def 5;
then dom ((Re (f + g)) + (<i> (#) (Im (f + g)))) = (dom (Re (f + g))) /\ (dom (Im (f + g))) by VALUED_1:def 1;
then dom ((Re (f + g)) + (<i> (#) (Im (f + g)))) in S by A6, A5, FINSUB_1:def 2;
hence dom (f + g) in S by Th8; :: thesis: verum