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 st A c= dom f holds
( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,r) in S )
let S be SigmaField of X; :: thesis: for f being PartFunc of X,REAL
for A being Element of S st A c= dom f holds
( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,r) in S )
let f be PartFunc of X,REAL ; :: thesis: for A being Element of S st A c= dom f holds
( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,r) in S )
let A be Element of S; :: thesis: ( A c= dom f implies ( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,r) in S ) )
assume A1: A c= dom f ; :: thesis: ( f is_measurable_on A iff for r being real number holds A /\ (great_eq_dom f,r) in S )
A2: ( f is_measurable_on A iff R_EAL f is_measurable_on A ) by Def6;
thus ( f is_measurable_on A implies for r being real number holds A /\ (great_eq_dom f,r) in S ) :: thesis: ( ( for r being real number holds A /\ (great_eq_dom f,r) in S ) implies f is_measurable_on A )

proof

assume A3: f is_measurable_on A ; :: thesis: for r being real number holds A /\ (great_eq_dom f,r) in S
let r be real number ; :: thesis: A /\ (great_eq_dom f,r) in S
A /\ (great_eq_dom f,(R_EAL r)) in S by A1, A2, A3, MESFUNC1:31;
hence A /\ (great_eq_dom f,r) in S ; :: thesis: verum

end;

assume for r being real number holds A /\ (great_eq_dom f,r) in S ; :: thesis: f is_measurable_on A
then for r being real number holds A /\ (great_eq_dom f,(R_EAL r)) in S ;
hence f is_measurable_on A by A1, A2, MESFUNC1:31; :: thesis: verum