let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable & A c= dom f holds
(A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S
let S be SigmaField of X; :: thesis: for f being PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable & A c= dom f holds
(A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S
let f be PartFunc of X,ExtREAL; :: thesis: for A being Element of S st f is A -measurable & A c= dom f holds
(A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S
let A be Element of S; :: thesis: ( f is A -measurable & A c= dom f implies (A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S )
assume that
A1: f is A -measurable and
A2: A c= dom f ; :: thesis: (A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S
A3: A /\ (great_dom (f,-infty)) in S

proof

defpred S₁[ Element of NAT , set ] means A /\ (great_dom (f,(- $1))) = $2;
A4: for n being Element of NAT ex y being Element of S st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of S st S₁[n,y]
reconsider y = A /\ (great_dom (f,(- n))) as Element of S by A1, A2, Th29;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being sequence of S such that
A5: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A4);
A /\ (great_dom (f,-infty)) = union (rng F) by A5, Th26;
hence A /\ (great_dom (f,-infty)) in S ; :: thesis: verum

end;

A6: A /\ (less_dom (f,+infty)) in S

proof

defpred S₁[ Element of NAT , set ] means A /\ (less_dom (f,$1)) = $2;
A7: for n being Element of NAT ex y being Element of S st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of S st S₁[n,y]
reconsider y = A /\ (less_dom (f,n)) as Element of S by A1;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being sequence of S such that
A8: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A7);
A /\ (less_dom (f,+infty)) = union (rng F) by A8, Th24;
hence A /\ (less_dom (f,+infty)) in S ; :: thesis: verum

end;

(A /\ (great_dom (f,-infty))) /\ (A /\ (less_dom (f,+infty))) = ((A /\ (great_dom (f,-infty))) /\ A) /\ (less_dom (f,+infty)) by XBOOLE_1:16
.= ((great_dom (f,-infty)) /\ (A /\ A)) /\ (less_dom (f,+infty)) by XBOOLE_1:16 ;
hence (A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S by A3, A6, FINSUB_1:def 2; :: thesis: verum