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
A5: A /\ (less_dom f,(R_EAL r1)) in S by A1, Def17;
A /\ (great_eq_dom f,(R_EAL r1)) = A \ (A /\ (less_dom f,(R_EAL r1))) by A2, Th18;
hence A /\ (great_eq_dom f,(R_EAL r1)) in S by A5, MEASURE1:20; :: 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;
A6: 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
A7: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A6);
A /\ (great_dom f,(R_EAL r1)) = union (rng F) by A7, Th26;
hence A /\ (great_dom f,(R_EAL r1)) in S ; :: thesis: verum

end;

then A8: 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, A8, MEASURE1:19; :: thesis: verum