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 ( f is_integrable_on M & g is_integrable_on M ) ; :: thesis: f - g is_integrable_on M
then ( Re f is_integrable_on M & Im f is_integrable_on M & Re g is_integrable_on M & Im g is_integrable_on M ) by Def4;
then ( (Re f) - (Re g) is_integrable_on M & (Im f) - (Im g) is_integrable_on M ) by Th100x;
then ( Re (f - g) is_integrable_on M & Im (f - g) is_integrable_on M ) by COM48;
hence f - g is_integrable_on M by Def4; :: thesis: verum