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 st ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom (((inferior_realsequence f) . n),r)) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),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 st ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom (((inferior_realsequence f) . n),r)) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),r))
let f be 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 (((inferior_realsequence f) . n),r)) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),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 (((inferior_realsequence f) . n),r)) ) holds
union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),r))
let r be Real; :: thesis: ( ( for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom (((inferior_realsequence f) . n),r)) ) implies union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),r)) )
set E = dom (f . 0);
set g = inferior_realsequence f;
assume A1: for n being Nat holds F . n = (dom (f . 0)) /\ (great_dom (((inferior_realsequence f) . n),r)) ; :: thesis: union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),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)) by A1, Th15;
hence union (rng F) = (dom (f . 0)) /\ (great_dom ((lim_inf f),r)) by Th11; :: thesis: verum