let X be set ; :: thesis: for Si being SigmaField of X
for S1, S2, S3 being SetSequence of Si st ( for n being Nat holds S3 . n = (S1 . n) /\ (S2 . n) ) holds
lim_sup S3 c= (lim_sup S1) /\ (lim_sup S2)
let Si be SigmaField of X; :: thesis: for S1, S2, S3 being SetSequence of Si st ( for n being Nat holds S3 . n = (S1 . n) /\ (S2 . n) ) holds
lim_sup S3 c= (lim_sup S1) /\ (lim_sup S2)
let S1, S2, S3 be SetSequence of Si; :: thesis: ( ( for n being Nat holds S3 . n = (S1 . n) /\ (S2 . n) ) implies lim_sup S3 c= (lim_sup S1) /\ (lim_sup S2) )
A1: for B, A2, A3 being SetSequence of X st ( for n being Nat holds A3 . n = (B . n) /\ (A2 . n) ) holds
Intersection (superior_setsequence A3) c= (Intersection (superior_setsequence B)) /\ (Intersection (superior_setsequence A2))
proof
let B, A2, A3 be SetSequence of X; :: thesis: ( ( for n being Nat holds A3 . n = (B . n) /\ (A2 . n) ) implies Intersection (superior_setsequence A3) c= (Intersection (superior_setsequence B)) /\ (Intersection (superior_setsequence A2)) )
A2: ( lim_sup B = Intersection (superior_setsequence B) & lim_sup A2 = Intersection (superior_setsequence A2) ) ;
A3: lim_sup A3 = Intersection (superior_setsequence A3) ;
assume for n being Nat holds A3 . n = (B . n) /\ (A2 . n) ; :: thesis: Intersection (superior_setsequence A3) c= (Intersection (superior_setsequence B)) /\ (Intersection (superior_setsequence A2))
hence Intersection (superior_setsequence A3) c= (Intersection (superior_setsequence B)) /\ (Intersection (superior_setsequence A2)) by A2, A3, KURATO_0:13; :: thesis: verum
end;
assume for n being Nat holds S3 . n = (S1 . n) /\ (S2 . n) ; :: thesis: lim_sup S3 c= (lim_sup S1) /\ (lim_sup S2)
hence lim_sup S3 c= (lim_sup S1) /\ (lim_sup S2) by A1; :: thesis: verum