let X be non empty set ; :: thesis: for S being SigmaField of X
for f being with_the_same_dom Functional_Sequence of X,ExtREAL
for F being SetSequence of S
for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom (f . n),(R_EAL r)) ) holds
for n being natural number holds (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r))
let S be SigmaField of X; :: thesis: for f being with_the_same_dom Functional_Sequence of X,ExtREAL
for F being SetSequence of S
for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom (f . n),(R_EAL r)) ) holds
for n being natural number holds (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r))
let f be with_the_same_dom Functional_Sequence of X,ExtREAL ; :: thesis: for F being SetSequence of S
for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom (f . n),(R_EAL r)) ) holds
for n being natural number holds (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r))
let F be SetSequence of S; :: thesis: for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom (f . n),(R_EAL r)) ) holds
for n being natural number holds (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r))
let r be real number ; :: thesis: ( ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom (f . n),(R_EAL r)) ) implies for n being natural number holds (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r)) )
set E = dom (f . 0 );
assume A1: for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom (f . n),(R_EAL r)) ; :: thesis: for n being natural number holds (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r))
let n be natural number ; :: thesis: (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r))
reconsider n9 = n as Element of NAT by ORDINAL1:def 13;
set f1 = f ^\ n9;
set F1 = F ^\ n9;

A2: now

let k be natural number ; :: thesis: (F ^\ n9) . k = (dom (f . 0 )) /\ (great_eq_dom ((f ^\ n9) . k),(R_EAL r))
reconsider k9 = k as Element of NAT by ORDINAL1:def 13;
(F ^\ n9) . k = F . (n + k9) by NAT_1:def 3;
then (F ^\ n9) . k = (dom (f . 0 )) /\ (great_eq_dom (f . (n + k9)),(R_EAL r)) by A1;
hence (F ^\ n9) . k = (dom (f . 0 )) /\ (great_eq_dom ((f ^\ n9) . k),(R_EAL r)) by NAT_1:def 3; :: thesis: verum

end;

A3: meet (rng (F ^\ n9)) = (inferior_setsequence F) . n by Th2;

now

let k be Nat; :: thesis: (F ^\ n9) . k in S
A4: F . (n + k) in rng F by NAT_1:52;
A5: (F ^\ n9) . k = F . (n + k) by NAT_1:def 3;
rng F c= S by RELAT_1:def 19;
hence (F ^\ n9) . k in S by A4, A5; :: thesis: verum

end;

then rng (F ^\ n9) c= S by NAT_1:53;
then A6: F ^\ n9 is SetSequence of S by RELAT_1:def 19;
consider g being Function of NAT ,(PFuncs X,ExtREAL ) such that
A7: f = g and
f ^\ n9 = g ^\ n9 ;
(f ^\ n9) . 0 = g . (n + 0 ) by A7, NAT_1:def 3;
then dom ((f ^\ n9) . 0 ) = dom (f . 0 ) by A7, Def2;
then meet (rng (F ^\ n9)) = (dom (f . 0 )) /\ (great_eq_dom (inf (f ^\ n9)),(R_EAL r)) by A6, A2, Th16;
hence (inferior_setsequence F) . n = (dom (f . 0 )) /\ (great_eq_dom ((inferior_realsequence f) . n),(R_EAL r)) by A3, Th8; :: thesis: verum