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
for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom ((superior_realsequence f) . n),(R_EAL r)) ) holds
meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r))
let S be SigmaField of X; :: thesis: for f being Functional_Sequence of X,ExtREAL
for F being SetSequence of
for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom ((superior_realsequence f) . n),(R_EAL r)) ) holds
meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r))
let f be Functional_Sequence of X,ExtREAL ; :: thesis: for F being SetSequence of
for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom ((superior_realsequence f) . n),(R_EAL r)) ) holds
meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r))
let F be SetSequence of ; :: thesis: for r being real number st ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom ((superior_realsequence f) . n),(R_EAL r)) ) holds
meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r))
let r be real number ; :: thesis: ( ( for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom ((superior_realsequence f) . n),(R_EAL r)) ) implies meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r)) )
set E = dom (f . 0 );
set g = superior_realsequence f;
assume A1: for n being natural number holds F . n = (dom (f . 0 )) /\ (great_eq_dom ((superior_realsequence f) . n),(R_EAL r)) ; :: thesis: meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r))
dom ((superior_realsequence f) . 0 ) = dom (f . 0 ) by Def6;
then meet (rng F) = (dom (f . 0 )) /\ (great_eq_dom (inf (superior_realsequence f)),(R_EAL r)) by A1, Th16;
hence meet F = (dom (f . 0 )) /\ (great_eq_dom (lim_sup f),(R_EAL r)) by Th12; :: thesis: verum