let X be non empty set ; :: thesis: for S being SigmaField of X
for f being 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_dom (((inferior_realsequence f) . n),(R_EAL r))) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(R_EAL r)))
let S be SigmaField of X; :: thesis: for f being 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_dom (((inferior_realsequence f) . n),(R_EAL r))) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(R_EAL r)))
let f be 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_dom (((inferior_realsequence f) . n),(R_EAL r))) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(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_dom (((inferior_realsequence f) . n),(R_EAL r))) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(R_EAL r)))
let r be real number ; :: thesis: ( ( for n being natural number holds F . n = (dom (f . 0)) /\ (great_dom (((inferior_realsequence f) . n),(R_EAL r))) ) implies union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(R_EAL r))) )
set E = dom (f . 0);
set g = inferior_realsequence f;
assume A1: for n being natural number holds F . n = (dom (f . 0)) /\ (great_dom (((inferior_realsequence f) . n),(R_EAL r))) ; :: thesis: union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(R_EAL r)))
dom ((inferior_realsequence f) . 0) = dom (f . 0) by Def5;
then union (rng F) = (dom (f . 0)) /\ (great_dom ((sup (inferior_realsequence f)),(R_EAL r))) by A1, Th15;
hence union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),(R_EAL r))) by Th11; :: thesis: verum