let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,COMPLEX
for A, B being Element of S st dom f = A holds
( f is_measurable_on B iff f is_measurable_on A /\ B )
let S be SigmaField of X; :: thesis: for f being PartFunc of X,COMPLEX
for A, B being Element of S st dom f = A holds
( f is_measurable_on B iff f is_measurable_on A /\ B )
let f be PartFunc of X,COMPLEX ; :: thesis: for A, B being Element of S st dom f = A holds
( f is_measurable_on B iff f is_measurable_on A /\ B )
let A, B be Element of S; :: thesis: ( dom f = A implies ( f is_measurable_on B iff f is_measurable_on A /\ B ) )
assume dom f = A ; :: thesis: ( f is_measurable_on B iff f is_measurable_on A /\ B )
then A1: ( dom (Re f) = A & dom (Im f) = A ) by Def1, Def2;

hereby :: thesis: ( f is_measurable_on A /\ B implies f is_measurable_on B )

assume f is_measurable_on B ; :: thesis: f is_measurable_on A /\ B
then ( Re f is_measurable_on B & Im f is_measurable_on B ) by Def3;
then ( Re f is_measurable_on A /\ B & Im f is_measurable_on A /\ B ) by A1, MESFUNC6:80;
hence f is_measurable_on A /\ B by Def3; :: thesis: verum

end;

hereby :: thesis: verum

assume f is_measurable_on A /\ B ; :: thesis: f is_measurable_on B
then ( Re f is_measurable_on A /\ B & Im f is_measurable_on A /\ B ) by Def3;
then ( Re f is_measurable_on B & Im f is_measurable_on B ) by A1, MESFUNC6:80;
hence f is_measurable_on B by Def3; :: thesis: verum

end;