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
f - g is_integrable_on M
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
f - g is_integrable_on M
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
f - g is_integrable_on M
let f, g be PartFunc of X,COMPLEX; :: thesis: ( f is_integrable_on M & g is_integrable_on M implies f - g is_integrable_on M )
assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M ; :: thesis: f - g is_integrable_on M
A3: Im g is_integrable_on M by A2;
Im f is_integrable_on M by A1;
then (Im f) - (Im g) is_integrable_on M by A3, Th34;
then A4: Im (f - g) is_integrable_on M by Th6;
A5: Re g is_integrable_on M by A2;
Re f is_integrable_on M by A1;
then (Re f) - (Re g) is_integrable_on M by A5, Th34;
then Re (f - g) is_integrable_on M by Th6;
hence f - g is_integrable_on M by A4; :: thesis: verum