let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,REAL
for A being Element of S
for r, s being Real st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for A being Element of S
for r, s being Real st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S
let f be PartFunc of X,REAL; :: thesis: for A being Element of S
for r, s being Real st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S
let A be Element of S; :: thesis: for r, s being Real st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S
let r, s be Real; :: thesis: ( f is_measurable_on A & A c= dom f implies (A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S )
A1: ( R_EAL r = r & R_EAL s = s ) ;
assume that
A2: f is_measurable_on A and
A3: A c= dom f ; :: thesis: (A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S
R_EAL f is_measurable_on A by A2, Def6;
hence (A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S by A1, A3, MESFUNC1:32; :: thesis: verum