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_inf S1) \/ (lim_inf S2) c= lim_inf S3

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_inf S1) \/ (lim_inf S2) c= lim_inf S3

let S1, S2, S3 be SetSequence of Si; :: thesis: ( ( for n being Nat holds S3 . n = (S1 . n) \/ (S2 . n) ) implies (lim_inf S1) \/ (lim_inf S2) c= lim_inf S3 )

A1: for B, A2, A3 being SetSequence of X st ( for n being Nat holds A3 . n = (B . n) \/ (A2 . n) ) holds

(Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3)

hence (lim_inf S1) \/ (lim_inf S2) c= lim_inf S3 by A1; :: thesis: verum

for S1, S2, S3 being SetSequence of Si st ( for n being Nat holds S3 . n = (S1 . n) \/ (S2 . n) ) holds

(lim_inf S1) \/ (lim_inf S2) c= lim_inf S3

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_inf S1) \/ (lim_inf S2) c= lim_inf S3

let S1, S2, S3 be SetSequence of Si; :: thesis: ( ( for n being Nat holds S3 . n = (S1 . n) \/ (S2 . n) ) implies (lim_inf S1) \/ (lim_inf S2) c= lim_inf S3 )

A1: for B, A2, A3 being SetSequence of X st ( for n being Nat holds A3 . n = (B . n) \/ (A2 . n) ) holds

(Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3)

proof

assume
for n being Nat holds S3 . n = (S1 . n) \/ (S2 . n)
; :: thesis: (lim_inf S1) \/ (lim_inf S2) c= lim_inf S3
let B, A2, A3 be SetSequence of X; :: thesis: ( ( for n being Nat holds A3 . n = (B . n) \/ (A2 . n) ) implies (Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3) )

A2: ( lim_inf B = Union (inferior_setsequence B) & lim_inf A2 = Union (inferior_setsequence A2) ) ;

A3: lim_inf A3 = Union (inferior_setsequence A3) ;

assume for n being Nat holds A3 . n = (B . n) \/ (A2 . n) ; :: thesis: (Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3)

hence (Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3) by A2, A3, KURATO_0:12; :: thesis: verum

end;A2: ( lim_inf B = Union (inferior_setsequence B) & lim_inf A2 = Union (inferior_setsequence A2) ) ;

A3: lim_inf A3 = Union (inferior_setsequence A3) ;

assume for n being Nat holds A3 . n = (B . n) \/ (A2 . n) ; :: thesis: (Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3)

hence (Union (inferior_setsequence B)) \/ (Union (inferior_setsequence A2)) c= Union (inferior_setsequence A3) by A2, A3, KURATO_0:12; :: thesis: verum

hence (lim_inf S1) \/ (lim_inf S2) c= lim_inf S3 by A1; :: thesis: verum