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_eq_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_eq_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_eq_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_eq_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_eq_dom f,r)) /\ (less_dom f,s) in S )
assume that
A1: f is_measurable_on A and
A2: A c= dom f ; :: thesis: (A /\ (great_eq_dom f,r)) /\ (less_dom f,s) in S
R_EAL f is_measurable_on A by A1, Def6;
then A3: A /\ (less_dom (R_EAL f),(R_EAL s)) in S by MESFUNC1:def 17;
A4: (A /\ (great_eq_dom f,r)) /\ (A /\ (less_dom f,s)) = ((A /\ (great_eq_dom f,r)) /\ A) /\ (less_dom f,s) by XBOOLE_1:16
.= ((great_eq_dom f,r) /\ (A /\ A)) /\ (less_dom f,s) by XBOOLE_1:16 ;
A /\ (great_eq_dom f,r) in S by A1, A2, Th13;
hence (A /\ (great_eq_dom f,r)) /\ (less_dom f,s) in S by A3, A4, FINSUB_1:def 2; :: thesis: verum