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
for r, s being Real st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom (f,(R_EAL r)))) /\ (less_dom (f,(R_EAL s))) in S
let S be SigmaField of X; :: thesis: for f being PartFunc of X,ExtREAL
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_EAL r)))) /\ (less_dom (f,(R_EAL s))) in S
let f be PartFunc of X,ExtREAL; :: 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_EAL r)))) /\ (less_dom (f,(R_EAL 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_EAL r)))) /\ (less_dom (f,(R_EAL s))) in S
let r, s be Real; :: thesis: ( f is_measurable_on A & A c= dom f implies (A /\ (great_dom (f,(R_EAL r)))) /\ (less_dom (f,(R_EAL s))) in S )
assume that
A1: f is_measurable_on A and
A2: A c= dom f ; :: thesis: (A /\ (great_dom (f,(R_EAL r)))) /\ (less_dom (f,(R_EAL s))) in S
A3: A /\ (less_dom (f,(R_EAL s))) in S by A1, Def17;
A4: for r1 being Real holds A /\ (great_eq_dom (f,(R_EAL r1))) in S

proof

let r1 be Real; :: thesis: A /\ (great_eq_dom (f,(R_EAL r1))) in S
( A /\ (less_dom (f,(R_EAL r1))) in S & A /\ (great_eq_dom (f,(R_EAL r1))) = A \ (A /\ (less_dom (f,(R_EAL r1)))) ) by A1, A2, Def17, Th18;
hence A /\ (great_eq_dom (f,(R_EAL r1))) in S by MEASURE1:6; :: thesis: verum

end;

for r1 being Real holds A /\ (great_dom (f,(R_EAL r1))) in S

proof

let r1 be Real; :: thesis: A /\ (great_dom (f,(R_EAL r1))) in S
defpred S₁[ Element of NAT , set ] means A /\ (great_eq_dom (f,(R_EAL (r1 + (1 / ($1 + 1)))))) = $2;
A5: 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_eq_dom (f,(R_EAL (r1 + (1 / (n + 1)))))) as Element of S by A4;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being Function of NAT,S such that
A6: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A5);
A /\ (great_dom (f,(R_EAL r1))) = union (rng F) by A6, Th26;
hence A /\ (great_dom (f,(R_EAL r1))) in S ; :: thesis: verum

end;

then A7: A /\ (great_dom (f,(R_EAL r))) in S ;
(A /\ (great_dom (f,(R_EAL r)))) /\ (A /\ (less_dom (f,(R_EAL s)))) = ((A /\ (great_dom (f,(R_EAL r)))) /\ A) /\ (less_dom (f,(R_EAL s))) by XBOOLE_1:16
.= ((great_dom (f,(R_EAL r))) /\ (A /\ A)) /\ (less_dom (f,(R_EAL s))) by XBOOLE_1:16 ;
hence (A /\ (great_dom (f,(R_EAL r)))) /\ (less_dom (f,(R_EAL s))) in S by A3, A7, FINSUB_1:def 2; :: thesis: verum