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 st ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom ((f . n),r)) ) holds
for n being Nat holds (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),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 st ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom ((f . n),r)) ) holds
for n being Nat holds (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r))
let f be with_the_same_dom Functional_Sequence of X,ExtREAL; :: thesis: for F being SetSequence of S
for r being Real st ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom ((f . n),r)) ) holds
for n being Nat holds (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r))
let F be SetSequence of S; :: thesis: for r being Real st ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom ((f . n),r)) ) holds
for n being Nat holds (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r))
let r be Real; :: thesis: ( ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom ((f . n),r)) ) implies for n being Nat holds (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r)) )
set E = dom (f . 0);
assume A1: for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom ((f . n),r)) ; :: thesis: for n being Nat holds (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r))
let n be Nat; :: thesis: (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r))
reconsider n9 = n as Element of NAT by ORDINAL1:def 12;
set f1 = f ^\ n9;
set F1 = F ^\ n9;

A2: now :: thesis: for k being Nat holds (F ^\ n9) . k = (dom (f . 0)) /\ (great_dom (((f ^\ n9) . k),r))

let k be Nat; :: thesis: (F ^\ n9) . k = (dom (f . 0)) /\ (great_dom (((f ^\ n9) . k),r))
reconsider k9 = k as Element of NAT by ORDINAL1:def 12;
(F ^\ n9) . k = F . (n + k9) by NAT_1:def 3;
then (F ^\ n9) . k = (dom (f . 0)) /\ (great_dom ((f . (n + k9)),r)) by A1;
hence (F ^\ n9) . k = (dom (f . 0)) /\ (great_dom (((f ^\ n9) . k),r)) by NAT_1:def 3; :: thesis: verum

end;

A3: union (rng (F ^\ n9)) = (superior_setsequence F) . n by Th2;
consider g being sequence of (PFuncs (X,ExtREAL)) such that
A4: f = g and
f ^\ n9 = g ^\ n9 ;
(f ^\ n9) . 0 = g . (n + 0) by A4, NAT_1:def 3;
then dom ((f ^\ n9) . 0) = dom (f . 0) by A4, Def2;
then union (rng (F ^\ n9)) = (dom (f . 0)) /\ (great_dom ((sup (f ^\ n9)),r)) by A2, Th15;
hence (superior_setsequence F) . n = (dom (f . 0)) /\ (great_dom (((superior_realsequence f) . n),r)) by A3, Th9; :: thesis: verum