Copyright (c) 2003 Association of Mizar Users
environ vocabulary KURATO_2, WELLFND1, FRECHET2, REARRAN1, YELLOW_6, BOOLE, EUCLID, PRE_TOPC, COMPTS_1, SETFAM_1, METRIC_1, SUBSET_1, PCOMPS_1, RELAT_1, ORDINAL2, LATTICES, SEQ_2, PROB_1, SEQ_1, FUNCT_1, SEQM_3, FRECHET, ARYTM_3, CONNSP_2, ARYTM, TOPS_1, ARYTM_1, NEWTON, NORMSP_1, COMPLEX1, INT_1, FINSEQ_1, TOPREAL1, TOPREAL2, JORDAN9, JORDAN2C, SQUARE_1, GOBOARD9, ZF_REFLE, FUNCOP_1, MCART_1, WAYBEL_7; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, ZFMISC_1, XREAL_0, REAL_1, NAT_1, SETFAM_1, MCART_1, STRUCT_0, PRE_TOPC, TOPS_1, COMPTS_1, RELAT_1, FUNCT_1, INT_1, FINSEQ_1, SEQ_1, TBSP_1, PARTFUN1, FUNCT_2, METRIC_1, PCOMPS_1, EUCLID, NORMSP_1, BORSUK_1, PROB_1, PSCOMP_1, LIMFUNC1, CONNSP_2, TOPREAL1, TOPREAL2, JORDAN2C, SEQM_3, GOBOARD9, FRECHET, FRECHET2, TOPRNS_1, JORDAN9, FUNCT_6, YELLOW_6; constructors REAL_1, JORDAN2C, GROUP_1, TOPREAL2, CONNSP_1, PSCOMP_1, TOPS_1, WEIERSTR, PROB_1, TOPRNS_1, FRECHET, NAT_1, JORDAN9, TBSP_1, CQC_SIM1, LIMFUNC1, GOBOARD9, YELLOW_6, FUNCT_6, FRECHET2; clusters XREAL_0, RELSET_1, NAT_1, INT_1, JORDAN1B, SUBSET_1, STRUCT_0, TOPS_1, BORSUK_1, EUCLID, FINSET_1, METRIC_1, TOPREAL6, GOBRD14, HAUSDORF, SEQ_1, SEQM_3, MEMBERED, FUNCT_2, PARTFUN1, ORDINAL2; requirements REAL, SUBSET, BOOLE, NUMERALS, ARITHM; definitions XBOOLE_0, TARSKI; theorems SETFAM_1, FRECHET, EUCLID, METRIC_1, SEQ_2, REAL_1, TOPREAL3, YELLOW_6, TOPMETR, CONNSP_2, GOBOARD6, XBOOLE_1, PRE_TOPC, SUBSET_1, NAT_1, REAL_2, FUNCT_2, RELSET_1, TOPRNS_1, NORMSP_1, XBOOLE_0, FUNCT_1, INT_1, AXIOMS, SQUARE_1, XREAL_0, SPPOL_1, TOPS_1, TOPREAL6, JORDAN2C, TBSP_1, METRIC_6, FIN_TOPO, ZFMISC_1, RELAT_1, TARSKI, FUNCOP_1, GOBRD14, XCMPLX_0, FUNCT_6, SEQM_3, XCMPLX_1, MCART_1, BORSUK_1, PROB_1, FRECHET2; schemes XBOOLE_0, COMPTS_1, FUNCT_2, COMPLSP1, NAT_1, LATTICE5; begin :: Preliminaries theorem Th1: for X, x being set, A being Subset of X st not x in A & x in X holds x in A` proof let X, x be set, A be Subset of X; assume not x in A & x in X; then x in X \ A by XBOOLE_0:def 4; hence thesis by SUBSET_1:def 5; end; theorem for F being Function, i being set st i in dom F holds meet F c= F.i proof let F be Function, i be set; assume A1: i in dom F; meet F c= F.i proof let x be set; assume x in meet F; hence x in F.i by A1,FUNCT_6:37,RELAT_1:60; end; hence meet F c= F.i; end; theorem Th3: for T being non empty 1-sorted, S1, S2 being SetSequence of the carrier of T holds S1 = S2 iff for n being Nat holds S1.n = S2.n proof let T be non empty 1-sorted, S1, S2 be SetSequence of the carrier of T; thus S1 = S2 implies for n being Nat holds S1.n = S2.n; assume A1: for n being Nat holds S1.n = S2.n; A2: dom S1 = NAT by FUNCT_2:def 1 .= dom S2 by FUNCT_2:def 1; for x being set st x in dom S1 holds S1.x = S2.x proof let x be set; assume x in dom S1; then reconsider n = x as Nat by FUNCT_2:def 1; S1.x = S2.n by A1 .= S2.x; hence thesis; end; hence thesis by A2,FUNCT_1:9; end; theorem Th4: for A, B, C, D being set st A meets B & C meets D holds [: A, C :] meets [: B, D :] proof let A, B, C, D be set; assume that A1: A meets B and A2: C meets D; consider x being set such that A3: x in A & x in B by A1,XBOOLE_0:3; consider y being set such that A4: y in C & y in D by A2,XBOOLE_0:3; A5: [x,y] in [: A, C :] by A3,A4,ZFMISC_1:106; [x,y] in [: B, D :] by A3,A4,ZFMISC_1:106; hence thesis by A5,XBOOLE_0:3; end; definition let X be 1-sorted; cluster -> non empty SetSequence of the carrier of X; coherence; end; definition let T be non empty 1-sorted; cluster non-empty SetSequence of the carrier of T; existence proof consider a being Element of T; set X = NAT --> {a}; reconsider X as SetSequence of the carrier of T; take X; A1: rng X = {{a}} by FUNCOP_1:14; not {} in rng X by A1,TARSKI:def 1; hence thesis by RELAT_1:def 9; end; end; definition let T be non empty 1-sorted; mode SetSequence of T is SetSequence of the carrier of T; canceled; end; scheme LambdaSSeq { X() -> non empty 1-sorted, F(set) -> Subset of X() } : ex f being SetSequence of X() st for n being Nat holds f.n = F(n) proof deffunc G(set)=F($1); set Y = bool the carrier of X(); A1: for x be set st x in NAT holds G(x) in Y; ex f be Function of NAT, Y st for x be set st x in NAT holds f.x = G(x) from Lambda1(A1); then consider f be Function of NAT, Y such that A2: for x be set st x in NAT holds f.x = F(x); reconsider f as SetSequence of X(); take f; thus thesis by A2; end; scheme ExTopStrSeq { R() -> non empty TopSpace, F(set) -> Subset of R() } : ex S be SetSequence of the carrier of R() st for n be Nat holds S.n = F(n) proof deffunc G(set)=F($1); ex f being Function of NAT, bool the carrier of R() st for x being Element of NAT holds f.x = G(x) from LambdaD; then consider f being Function of NAT, bool the carrier of R() such that A1: for x being Element of NAT holds f.x = F(x); take f; thus thesis by A1; end; theorem Th5: for X being non empty 1-sorted, F being SetSequence of the carrier of X holds rng F is Subset-Family of X by SETFAM_1:def 7; definition let X be non empty 1-sorted, F be SetSequence of the carrier of X; redefine func Union F -> Subset of X; coherence proof Union F c= the carrier of X; hence thesis; end; redefine func meet F -> Subset of X; coherence proof reconsider G = rng F as Subset-Family of X by Th5; meet G c= the carrier of X; hence thesis by FUNCT_6:def 4; end; end; begin :: Lower and Upper Limit of Sequences of Subsets definition let X be non empty set; let S be Function of NAT, X; let k be Nat; func S ^\ k -> Function of NAT, X means :Def2: for n being Nat holds it.n = S.(n + k); existence proof deffunc F(Nat)=S.($1 + k); thus ex IT being Function of NAT, X st for n being Nat holds IT.n = F(n) from LambdaD; end; uniqueness proof deffunc F(Nat)=S.($1 + k); thus for f,g being Function of NAT, X st (for n being Nat holds f.n = F(n)) & (for n being Nat holds g.n = F(n)) holds f=g from FuncDefUniq; end; end; definition let X be non empty 1-sorted, F be SetSequence of the carrier of X; func lim_inf F -> Subset of X means :Def3: ex f being SetSequence of X st it = Union f & for n being Nat holds f.n = meet (F ^\ n); existence proof deffunc F(Nat) = meet (F ^\ $1); consider f being SetSequence of X such that A1: for n being Nat holds f.n = F(n) from LambdaSSeq; take Union f; thus thesis by A1; end; uniqueness proof let A1, A2 be Subset of X; given f1 being SetSequence of X such that A2: A1 = Union f1 & for n being Nat holds f1.n = meet (F ^\ n); given f2 being SetSequence of X such that A3: A2 = Union f2 & for n being Nat holds f2.n = meet (F ^\ n); for n being Nat holds f1.n = f2.n proof let n be Nat; f1.n = meet (F ^\ n) by A2 .= f2.n by A3; hence thesis; end; hence thesis by A2,A3,Th3; end; func lim_sup F -> Subset of X means :Def4: ex f being SetSequence of X st it = meet f & for n being Nat holds f.n = Union (F ^\ n); existence proof deffunc F(Nat) = Union (F ^\ $1); consider f being SetSequence of X such that A4: for n being Nat holds f.n = F(n) from LambdaSSeq; take meet f; thus thesis by A4; end; uniqueness proof let A1, A2 be Subset of X; given f1 being SetSequence of X such that A5: A1 = meet f1 & for n being Nat holds f1.n = Union (F ^\ n); given f2 being SetSequence of X such that A6: A2 = meet f2 & for n being Nat holds f2.n = Union (F ^\ n); for n being Nat holds f1.n = f2.n proof let n be Nat; f1.n = Union (F ^\ n) by A5 .= f2.n by A6; hence thesis; end; hence thesis by A5,A6,Th3; end; end; theorem Th6: for X being non empty 1-sorted, F being SetSequence of the carrier of X, x being set holds x in meet F iff for z being Nat holds x in F.z proof let X be non empty 1-sorted, F be SetSequence of the carrier of X, x be set; hereby assume A1: x in meet F; let z be Nat; z in NAT; then z in dom F by FUNCT_2:def 1; hence x in F.z by A1,FUNCT_6:37; end; assume A2: for z being Nat holds x in F.z; for y being set st y in dom F holds x in F.y proof let y be set; assume y in dom F; then reconsider n = y as Nat by FUNCT_2:def 1; x in F.n by A2; hence thesis; end; hence thesis by FUNCT_6:37; end; theorem Th7: for X being non empty 1-sorted, F being SetSequence of the carrier of X, x being set holds x in lim_inf F iff ex n being Nat st for k being Nat holds x in F.(n+k) proof let X be non empty 1-sorted, F be SetSequence of the carrier of X, x be set; hereby assume A1: x in lim_inf F; consider f being SetSequence of X such that A2: lim_inf F = Union f & for n being Nat holds f.n = meet (F ^\ n) by Def3; consider n being Nat such that A3: x in f.n by A1,A2,PROB_1:25; take n; let k be Nat; A4: x in meet (F ^\ n) by A2,A3; set G = F ^\ n; G.k = F.(n + k) by Def2; hence x in F.(n+k) by A4,Th6; end; given n being Nat such that A5: for k being Nat holds x in F.(n+k); set G = F ^\ n; consider f being SetSequence of X such that A6: lim_inf F = Union f & for n being Nat holds f.n = meet (F ^\ n) by Def3; for z being Nat holds x in G.z proof let z be Nat; G.z = F.(n + z) by Def2; hence thesis by A5; end; then x in meet G by Th6; then x in f.n by A6; hence thesis by A6,PROB_1:25; end; theorem Th8: for X being non empty 1-sorted, F being SetSequence of the carrier of X, x being set holds x in lim_sup F iff for n being Nat ex k being Nat st x in F.(n+k) proof let X be non empty 1-sorted, F be SetSequence of the carrier of X, x be set; hereby assume A1: x in lim_sup F; consider f being SetSequence of X such that A2: lim_sup F = meet f & for n being Nat holds f.n = Union (F ^\ n) by Def4; let n be Nat; set G = F ^\ n; f.n = Union G by A2; then x in Union G by A1,A2,Th6; then consider k being Nat such that A3: x in G.k by PROB_1:25; take k; thus x in F.(n+k) by A3,Def2; end; assume A4: for n being Nat ex k being Nat st x in F.(n+k); consider f being SetSequence of X such that A5: lim_sup F = meet f & for n being Nat holds f.n = Union (F ^\ n) by Def4; for z being Nat holds x in f.z proof let z be Nat; set G = F ^\ z; A6: f.z = Union G by A5; consider k being Nat such that A7: x in F.(z+k) by A4; G.k = F.(z + k) by Def2; hence thesis by A6,A7,PROB_1:25; end; hence thesis by A5,Th6; end; theorem for X being non empty 1-sorted, F being SetSequence of the carrier of X holds lim_inf F c= lim_sup F proof let X be non empty 1-sorted, F be SetSequence of the carrier of X; let x be set; assume A1: x in lim_inf F; consider n be Nat such that A2: for k being Nat holds x in F.(n+k) by A1,Th7; now let k be Nat; x in F.(n+k) by A2; hence ex n being Nat st x in F.(k+n); end; hence thesis by Th8; end; theorem Th10: for X being non empty 1-sorted, F being SetSequence of the carrier of X holds meet F c= lim_inf F proof let X be non empty 1-sorted, F be SetSequence of the carrier of X; let x be set; assume x in meet F; then for k being Nat holds x in F.(0+k) by Th6; hence thesis by Th7; end; theorem Th11: for X being non empty 1-sorted, F being SetSequence of the carrier of X holds lim_sup F c= Union F proof let X be non empty 1-sorted, F be SetSequence of the carrier of X; let x be set; assume A1: x in lim_sup F; consider k being Nat such that A2: x in F.(0+k) by A1,Th8; thus thesis by A2,PROB_1:25; end; theorem for X being non empty 1-sorted, F being SetSequence of the carrier of X holds lim_inf F = (lim_sup Complement F)` proof let X be non empty 1-sorted, F be SetSequence of the carrier of X; set G = Complement F; thus lim_inf F c= (lim_sup Complement F)` proof let x be set; assume A1: x in lim_inf F; then consider n being Nat such that A2: for k being Nat holds x in F.(n+k) by Th7; for k being Nat holds not x in G.(n+k) proof let k be Nat; assume A3: x in G.(n+k); G.(n+k) = (F.(n+k))` by PROB_1:def 4; then not x in F.(n+k) by A3,SUBSET_1:54; hence thesis by A2; end; then not x in lim_sup G by Th8; then x in (lim_sup Complement F)` by A1,Th1; hence thesis; end; thus (lim_sup Complement F)` c= lim_inf F proof let x be set; assume A4: x in (lim_sup Complement F)`; then x in (lim_sup Complement F)`; then not x in lim_sup Complement F by SUBSET_1:54; then consider n being Nat such that A5: for k being Nat holds not x in G.(n+k) by Th8; for k being Nat holds x in F.(n+k) proof let k be Nat; assume not x in F.(n+k); then x in (F.(n+k))` by A4,Th1; then x in G.(n+k) by PROB_1:def 4; hence thesis by A5; end; hence thesis by Th7; end; end; theorem for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n /\ B.n) holds lim_inf C = lim_inf A /\ lim_inf B proof let X be non empty 1-sorted, A, B, C be SetSequence of the carrier of X; assume A1: for n being Nat holds C.n = A.n /\ B.n; thus lim_inf C c= lim_inf A /\ lim_inf B proof let x be set; assume x in lim_inf C; then consider n being Nat such that A2: for k being Nat holds x in C.(n+k) by Th7; for k being Nat holds x in A.(n+k) proof let k be Nat; A3: C.(n+k) = A.(n+k) /\ B.(n+k) by A1; x in C.(n+k) by A2; hence thesis by A3,XBOOLE_0:def 3; end; then A4: x in lim_inf A by Th7; for k being Nat holds x in B.(n+k) proof let k be Nat; A5: C.(n+k) = A.(n+k) /\ B.(n+k) by A1; x in C.(n+k) by A2; hence thesis by A5,XBOOLE_0:def 3; end; then x in lim_inf B by Th7; hence thesis by A4,XBOOLE_0:def 3; end; thus lim_inf A /\ lim_inf B c= lim_inf C proof let x be set; assume x in lim_inf A /\ lim_inf B; then A6: x in lim_inf A & x in lim_inf B by XBOOLE_0:def 3; then consider n1 being Nat such that A7: for k being Nat holds x in A.(n1+k) by Th7; consider n2 being Nat such that A8: for k being Nat holds x in B.(n2+k) by A6,Th7; set n = max (n1, n2); A9: for k being Nat holds x in A.(n+k) proof let k be Nat; n >= n1 by SQUARE_1:46; then consider g being Nat such that A10: n = n1 + g by NAT_1:28; x in A.(n1+(g+k)) by A7; hence thesis by A10,XCMPLX_1:1; end; A11: for k being Nat holds x in B.(n+k) proof let k be Nat; n >= n2 by SQUARE_1:46; then consider g being Nat such that A12: n = n2 + g by NAT_1:28; x in B.(n2+(g+k)) by A8; hence thesis by A12,XCMPLX_1:1; end; for k being Nat holds x in C.(n+k) proof let k be Nat; A13: x in A.(n+k) by A9; x in B.(n+k) by A11; then x in A.(n+k) /\ B.(n+k) by A13,XBOOLE_0:def 3; hence thesis by A1; end; hence thesis by Th7; end; end; theorem for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n \/ B.n) holds lim_sup C = lim_sup A \/ lim_sup B proof let X be non empty 1-sorted, A, B, C be SetSequence of the carrier of X; assume A1: for n being Nat holds C.n = A.n \/ B.n; thus lim_sup C c= lim_sup A \/ lim_sup B proof let x be set; assume A2: x in lim_sup C; (for n being Nat ex k being Nat st x in A.(n+k)) or (for n being Nat ex k being Nat st x in B.(n+k)) proof given n1 being Nat such that A3: for k being Nat holds not x in A.(n1+k); given n2 being Nat such that A4: for k being Nat holds not x in B.(n2+k); set n = max (n1, n2); n >= n1 by SQUARE_1:46; then consider g being Nat such that A5: n = n1 + g by NAT_1:28; n >= n2 by SQUARE_1:46; then consider h being Nat such that A6: n = n2 + h by NAT_1:28; consider k being Nat such that A7: x in C.(n+k) by A2,Th8; A8: x in A.(n+k) \/ B.(n+k) by A1,A7; per cases by A8,XBOOLE_0:def 2; suppose x in A.(n+k); then x in A.(n1+(g+k)) by A5,XCMPLX_1:1; hence thesis by A3; suppose x in B.(n+k); then x in B.(n2+(h+k)) by A6,XCMPLX_1:1; hence thesis by A4; end; then x in lim_sup A or x in lim_sup B by Th8; hence thesis by XBOOLE_0:def 2; end; thus lim_sup A \/ lim_sup B c= lim_sup C proof let x be set; assume A9: x in lim_sup A \/ lim_sup B; per cases by A9,XBOOLE_0:def 2; suppose A10: x in lim_sup A; for n being Nat ex k being Nat st x in C.(n+k) proof let n be Nat; consider k being Nat such that A11: x in A.(n+k) by A10,Th8; take k; x in A.(n+k) \/ B.(n+k) by A11,XBOOLE_0:def 2; hence thesis by A1; end; hence thesis by Th8; suppose A12: x in lim_sup B; for n being Nat ex k being Nat st x in C.(n+k) proof let n be Nat; consider k being Nat such that A13: x in B.(n+k) by A12,Th8; take k; x in A.(n+k) \/ B.(n+k) by A13,XBOOLE_0:def 2; hence thesis by A1; end; hence thesis by Th8; end; end; theorem for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n \/ B.n) holds lim_inf A \/ lim_inf B c= lim_inf C proof let X be non empty 1-sorted, A, B, C be SetSequence of the carrier of X; assume A1: for n being Nat holds C.n = A.n \/ B.n; let x be set; assume A2: x in lim_inf A \/ lim_inf B; per cases by A2,XBOOLE_0:def 2; suppose x in lim_inf A; then consider n being Nat such that A3: for k being Nat holds x in A.(n+k) by Th7; for k being Nat holds x in C.(n+k) proof let k be Nat; x in A.(n+k) by A3; then x in A.(n+k) \/ B.(n+k) by XBOOLE_0:def 2; hence thesis by A1; end; hence thesis by Th7; suppose x in lim_inf B; then consider n being Nat such that A4: for k being Nat holds x in B.(n+k) by Th7; for k being Nat holds x in C.(n+k) proof let k be Nat; x in B.(n+k) by A4; then x in A.(n+k) \/ B.(n+k) by XBOOLE_0:def 2; hence thesis by A1; end; hence thesis by Th7; end; theorem for X being non empty 1-sorted, A, B, C being SetSequence of the carrier of X st (for n being Nat holds C.n = A.n /\ B.n) holds lim_sup C c= lim_sup A /\ lim_sup B proof let X be non empty 1-sorted, A, B, C be SetSequence of the carrier of X; assume A1: for n being Nat holds C.n = A.n /\ B.n; let x be set; assume A2: x in lim_sup C; for n being Nat ex k being Nat st x in A.(n+k) proof let n be Nat; consider k being Nat such that A3: x in C.(n+k) by A2,Th8; take k; x in A.(n+k) /\ B.(n+k) by A1,A3; hence thesis by XBOOLE_0:def 3; end; then A4: x in lim_sup A by Th8; for n being Nat ex k being Nat st x in B.(n+k) proof let n be Nat; consider k being Nat such that A5: x in C.(n+k) by A2,Th8; take k; x in A.(n+k) /\ B.(n+k) by A1,A5; hence thesis by XBOOLE_0:def 3; end; then x in lim_sup B by Th8; hence thesis by A4,XBOOLE_0:def 3; end; theorem Th17: for X being non empty 1-sorted, A being SetSequence of the carrier of X, B being Subset of X st (for n being Nat holds A.n = B) holds lim_sup A = B proof let X be non empty 1-sorted, A be SetSequence of the carrier of X, B be Subset of X; assume A1: for n being Nat holds A.n = B; thus lim_sup A c= B proof let x be set; assume x in lim_sup A; then consider k being Nat such that A2: x in A.(0+k) by Th8; thus thesis by A1,A2; end; thus B c= lim_sup A proof let x be set; assume A3: x in B; for m being Nat ex k being Nat st x in A.(m+k) proof let m be Nat; take 0; thus thesis by A1,A3; end; hence thesis by Th8; end; end; theorem Th18: for X being non empty 1-sorted, A being SetSequence of the carrier of X, B being Subset of X st (for n being Nat holds A.n = B) holds lim_inf A = B proof let X be non empty 1-sorted, A be SetSequence of the carrier of X, B be Subset of X; assume A1: for n being Nat holds A.n = B; thus lim_inf A c= B proof let x be set; assume x in lim_inf A; then consider m being Nat such that A2: for k being Nat holds x in A.(m+k) by Th7; x in A.(m+0) by A2; hence thesis by A1; end; thus B c= lim_inf A proof let x be set; assume A3: x in B; ex m being Nat st for k being Nat holds x in A.(m+k) proof take 0; let k be Nat; thus thesis by A1,A3; end; hence thesis by Th7; end; end; theorem for X being non empty 1-sorted, A, B being SetSequence of the carrier of X, C being Subset of X st (for n being Nat holds B.n = C \+\ A.n) holds C \+\ lim_inf A c= lim_sup B proof let X be non empty 1-sorted, A, B be SetSequence of the carrier of X, C be Subset of X; assume A1: for n being Nat holds B.n = C \+\ A.n; let x be set; assume A2: x in C \+\ lim_inf A; per cases by A2,XBOOLE_0:1; suppose A3: x in C & not x in lim_inf A; for n being Nat ex k being Nat st x in B.(n+k) proof let n be Nat; consider k being Nat such that A4: not x in A.(n+k) by A3,Th7; take k; x in C \+\ A.(n+k) by A3,A4,XBOOLE_0:1; hence thesis by A1; end; hence thesis by Th8; suppose A5: not x in C & x in lim_inf A; then consider n being Nat such that A6: for k being Nat holds x in A.(n+k) by Th7; for m being Nat ex k being Nat st x in B.(m+k) proof let m be Nat; take k = n; x in A.(m+k) by A6; then x in C \+\ A.(m+k) by A5,XBOOLE_0:1; hence thesis by A1; end; hence thesis by Th8; end; theorem for X being non empty 1-sorted, A, B being SetSequence of the carrier of X, C being Subset of X st (for n being Nat holds B.n = C \+\ A.n) holds C \+\ lim_sup A c= lim_sup B proof let X be non empty 1-sorted, A, B be SetSequence of the carrier of X, C be Subset of X; assume A1: for n being Nat holds B.n = C \+\ A.n; let x be set; assume A2: x in C \+\ lim_sup A; per cases by A2,XBOOLE_0:1; suppose A3: x in C & not x in lim_sup A; then consider n being Nat such that A4: for k being Nat holds not x in A.(n+k) by Th8; for m being Nat ex k being Nat st x in B.(m+k) proof let m be Nat; take k = n; not x in A.(m+k) by A4; then x in C \+\ A.(m+k) by A3,XBOOLE_0:1; hence thesis by A1; end; hence thesis by Th8; suppose A5: not x in C & x in lim_sup A; for m being Nat ex k being Nat st x in B.(m+k) proof let m be Nat; consider k being Nat such that A6: x in A.(m+k) by A5,Th8; take k; x in C \+\ A.(m+k) by A5,A6,XBOOLE_0:1; hence thesis by A1; end; hence thesis by Th8; end; begin :: Ascending and Descending Families of Subsets definition let T be non empty 1-sorted, S be SetSequence of T; attr S is descending means :Def5: for i being Nat holds S.(i+1) c= S.i; attr S is ascending means :Def6: for i being Nat holds S.i c= S.(i+1); end; theorem Th21: for f being Function st (for i being Nat holds f.(i+1) c= f.i) for i, j being Nat st i <= j holds f.j c= f.i proof let f be Function; assume A1: for i being Nat holds f.(i+1) c= f.i; let i, j be Nat; assume A2: i <= j; defpred P[Nat] means i + $1 <= j implies f.(i + $1) c= f.i; A3: P[0]; A4: now let k be Nat; A5: i + k + 1 = i + (k + 1) by XCMPLX_1:1; assume A6: P[k]; f.(i + (k + 1)) c= f.(i + k) by A1,A5; hence P[k+1] by A5,A6,NAT_1:38,XBOOLE_1:1; end; A7: for k being Nat holds P[k] from Ind(A3,A4); consider k be Nat such that A8: i + k = j by A2,NAT_1:28; thus f.j c= f.i by A7,A8; end; theorem Th22: for T being non empty 1-sorted, C being SetSequence of T st C is descending holds for i, m being Nat st i >= m holds C.i c= C.m proof let T be non empty 1-sorted, C be SetSequence of T; assume C is descending; then A1: for i being Nat holds C.(i+1) c= C.i by Def5; let i, m be Nat; assume i >= m; hence thesis by A1,Th21; end; theorem Th23: for T being non empty 1-sorted, C being SetSequence of T st C is ascending holds for i, m being Nat st i >= m holds C.m c= C.i proof let T be non empty 1-sorted, C be SetSequence of T; assume C is ascending; then A1: for i being Nat holds C.i c= C.(i+1) by Def6; let i, m be Nat; assume m <= i; hence thesis by A1,FIN_TOPO:5; end; theorem Th24: for T being non empty 1-sorted, F being SetSequence of T, x being set st F is descending & ex k being Nat st for n being Nat st n > k holds x in F.n holds x in meet F proof let T be non empty 1-sorted, F be SetSequence of T, x be set; assume that A1: F is descending; A2: rng F <> {} by RELAT_1:64; given k being Nat such that A3: for n being Nat st n > k holds x in F.n; k + 1 > k by NAT_1:38; then A4: x in F.(k + 1) by A3; assume not x in meet F; then not x in meet rng F by FUNCT_6:def 4; then consider Y being set such that A5: Y in rng F & not x in Y by A2,SETFAM_1:def 1; consider y being set such that A6: y in dom F & Y = F.y by A5,FUNCT_1:def 5; reconsider y as Nat by A6,FUNCT_2:def 1; per cases; suppose y > k; hence thesis by A3,A5,A6; suppose y <= k; then F.k c= F.y by A1,Th22; then A7: not x in F.k by A5,A6; F.(k + 1) c= F.k by A1,Def5; hence thesis by A4,A7; end; theorem for T being non empty 1-sorted, F being SetSequence of T st F is descending holds lim_inf F = meet F proof let T be non empty 1-sorted, F be SetSequence of T; assume A1: F is descending; thus lim_inf F c= meet F proof let x be set; assume x in lim_inf F; then consider n being Nat such that A2: for k being Nat holds x in F.(n+k) by Th7; ex k being Nat st for n being Nat st n > k holds x in F.n proof take k = n; let n be Nat; assume n > k; then consider h being Nat such that A3: n = k + h by NAT_1:28; thus thesis by A2,A3; end; hence thesis by A1,Th24; end; thus meet F c= lim_inf F by Th10; end; theorem for T being non empty 1-sorted, F being SetSequence of T st F is ascending holds lim_sup F = Union F proof let T be non empty 1-sorted, F be SetSequence of T; assume A1: F is ascending; thus lim_sup F c= Union F by Th11; let x be set; assume x in Union F; then consider n being Nat such that A2: x in F.n by PROB_1:25; assume not x in lim_sup F; then consider m being Nat such that A3: for k being Nat holds not x in F.(m+k) by Th8; A4: not x in F.(m+0) by A3; per cases; suppose n <= m; then F.n c= F.m by A1,Th23; hence thesis by A2,A4; suppose n > m; then consider h being Nat such that A5: n = m + h by NAT_1:28; thus thesis by A2,A3,A5; end; begin :: Constant and Convergent Sequences definition let T be non empty 1-sorted, S be SetSequence of T; attr S is convergent means :Def7: lim_sup S = lim_inf S; end; theorem Th27: for T being non empty 1-sorted, S being SetSequence of T st S is constant holds the_value_of S is Subset of T proof let T be non empty 1-sorted, S be SetSequence of T; assume S is constant; then consider x being set such that A1: x in dom S & the_value_of S = S.x by YELLOW_6:def 1; reconsider n = x as Nat by A1,FUNCT_2:def 1; S.n in bool the carrier of T; hence thesis by A1; end; definition let T be non empty 1-sorted, S be SetSequence of T; redefine attr S is constant means :Def8: ex A being Subset of T st for n being Nat holds S.n = A; compatibility proof hereby assume A1: S is constant; then reconsider X = the_value_of S as Subset of T by Th27; take X; let n be Nat; n in NAT; then A2: n in dom S by FUNCT_2:def 1; consider x being set such that A3: x in dom S & X = S.x by A1,YELLOW_6:def 1; thus S.n = X by A1,A2,A3,SEQM_3:def 5; end; given A being Subset of T such that A4: for n being Nat holds S.n = A; for n1, n2 being set st n1 in dom S & n2 in dom S holds S.n1 = S.n2 proof let n1, n2 be set; assume n1 in dom S & n2 in dom S; then n1 in NAT & n2 in NAT by FUNCT_2:def 1; then S.n1 = A & S.n2 = A by A4; hence thesis; end; hence thesis by SEQM_3:def 5; end; end; definition let T be non empty 1-sorted; cluster constant -> convergent ascending descending SetSequence of T; coherence proof let S be SetSequence of T; assume S is constant; then consider A being Subset of T such that A1: for n being Nat holds S.n = A by Def8; A2: lim_sup S = A by A1,Th17; A3: lim_inf S = A by A1,Th18; A4: now let n be Nat; S.n = A & S.(n+1) = A by A1; hence S.n c= S.(n+1); end; now let n be Nat; S.n = A & S.(n+1) = A by A1; hence S.(n+1) c= S.n; end; hence thesis by A2,A3,A4,Def5,Def6,Def7; end; end; definition let T be non empty 1-sorted; cluster constant non empty SetSequence of T; existence proof consider A being Subset of T; set E = NAT --> A; reconsider E as SetSequence of T; E is constant; hence thesis; end; end; definition let T be non empty 1-sorted, S be convergent SetSequence of T; func Lim_K S -> Subset of T means :Def9: it = lim_sup S & it = lim_inf S; existence proof take L = lim_sup S; thus thesis by Def7; end; uniqueness; end; theorem for X being non empty 1-sorted, F being convergent SetSequence of X, x being set holds x in Lim_K F iff ex n being Nat st for k being Nat holds x in F.(n+k) proof let X be non empty 1-sorted, F be convergent SetSequence of X, x be set; Lim_K F = lim_inf F by Def9; hence thesis by Th7; end; begin :: Topological Lemmas reserve n for Nat; definition let f be FinSequence of the carrier of TOP-REAL 2; cluster L~f -> closed; coherence by SPPOL_1:49; end; theorem Th29: for r being real number, M being non empty Reflexive MetrStruct, x being Element of M st 0 < r holds x in Ball (x,r) proof let r be real number, M be non empty Reflexive MetrStruct, x be Element of M; assume 0 < r; then dist (x,x) < r by METRIC_1:1; hence thesis by METRIC_1:12; end; theorem Th30: for x being Point of Euclid n, r being real number holds Ball (x, r) is open Subset of TOP-REAL n proof let x be Point of Euclid n, r be real number; reconsider A = Ball (x, r) as Subset of TOP-REAL n by TOPREAL3:13; reconsider A as Subset of TOP-REAL n; r is Real by XREAL_0:def 1; then A is open by GOBOARD6:6; hence thesis; end; theorem Th31: for p, q being Point of TOP-REAL n, p', q' being Point of Euclid n st p = p' & q = q' holds dist (p', q') = |. p - q .| proof let p, q be Point of TOP-REAL n, p', q' be Point of Euclid n; assume A1: p = p' & q = q'; reconsider p'' = p, q'' = q as Element of REAL n by EUCLID:25; consider y be FinSequence of REAL such that A2: p - q = y & |. p - q .| = |. y .| by TOPRNS_1:def 6; p - q = p'' - q'' by EUCLID:def 13; hence thesis by A1,A2,SPPOL_1:20; end; theorem Th32: for p being Point of Euclid n, x, p' being Point of TOP-REAL n, r being real number st p = p' & x in Ball (p, r) holds |. x - p' .| < r proof let p be Point of Euclid n, x, p' be Point of TOP-REAL n, r be real number; reconsider x' = x as Point of Euclid n by TOPREAL3:13; assume A1: p = p' & x in Ball (p, r); then dist (x', p) < r by METRIC_1:12; hence thesis by A1,Th31; end; theorem Th33: for p being Point of Euclid n, x, p' being Point of TOP-REAL n, r being real number st p = p' & |. x - p' .| < r holds x in Ball (p, r) proof let p be Point of Euclid n, x, p' be Point of TOP-REAL n, r be real number; reconsider x' = x as Point of Euclid n by TOPREAL3:13; assume p = p' & |. x - p' .| < r; then dist (x', p) < r by Th31; hence thesis by METRIC_1:12; end; theorem for n being Nat, r being Point of TOP-REAL n, X being Subset of TOP-REAL n st r in Cl X holds ex seq being Real_Sequence of n st rng seq c= X & seq is convergent & lim seq = r proof let n be Nat, r be Point of TOP-REAL n, X be Subset of TOP-REAL n; assume A1: r in Cl X; reconsider r' = r as Point of Euclid n by TOPREAL3:13; defpred P[set,set] means ex z being Nat st $1 = z & $2 = choose (X /\ Ball (r', 1/(z+1))); A2: now let x be set; assume x in NAT; then reconsider k = x as Nat; set n1 = k+1; set oi = Ball (r', 1/n1); reconsider oi as open Subset of TOP-REAL n by Th30; reconsider u = choose (X /\ oi) as set; take u; n1 > 0 by NAT_1:19; then 1/n1 > 0 by REAL_2:127; then dist (r',r') < 1/n1 by METRIC_1:1; then r in oi by METRIC_1:12; then X meets oi by A1,PRE_TOPC:54; then X /\ oi is non empty by XBOOLE_0:def 7; then u in X /\ oi; hence u in the carrier of TOP-REAL n; thus P[x,u]; end; consider seq being Function such that A3: dom seq = NAT and A4: rng seq c= the carrier of TOP-REAL n and A5: for x being set st x in NAT holds P[x,seq.x] from NonUniqBoundFuncEx(A2); seq is Function of NAT, the carrier of TOP-REAL n by A3,A4,FUNCT_2:def 1,RELSET_1:11; then reconsider seq as Real_Sequence of n by NORMSP_1:def 3; take seq; thus rng seq c= X proof let y be set; assume y in rng seq; then consider x being set such that A6: x in dom seq & seq.x = y by FUNCT_1:def 5; consider k being Nat such that A7: x = k & seq.x = choose (X /\ Ball (r',1/(k+1))) by A3,A5,A6; set n1 = k+1; reconsider oi = Ball (r',1/n1) as open Subset of TOP-REAL n by Th30; n1 > 0 by NAT_1:19; then 1/n1 > 0 by REAL_2:127; then dist (r',r') < 1/n1 by METRIC_1:1; then r in oi by METRIC_1:12; then X meets oi by A1,PRE_TOPC:54; then X /\ oi is non empty by XBOOLE_0:def 7; hence y in X by A6,A7,XBOOLE_0:def 3; end; A8:now let p be Real; assume A9: 0 < p; set cp = [/ 1/p \]; A10: 0 < 1/p by A9,REAL_2:127; A11: 1/p <= cp by INT_1:def 5; A12: 0 < cp by A10,INT_1:def 5; then reconsider cp as Nat by INT_1:16; take k = cp; let m be Nat such that A13: k <= m; consider m' being Nat such that A14: m' = m & seq.m = choose (X /\ Ball (r',1/(m'+1))) by A5; set m1 = m+1; set oi = Ball (r',1/m1); reconsider oi as open Subset of TOP-REAL n by Th30; m1 > 0 by NAT_1:19; then 1/m1 > 0 by REAL_2:127; then dist (r',r') < 1/m1 by METRIC_1:1; then r in oi by METRIC_1:12; then X meets oi by A1,PRE_TOPC:54; then X /\ oi is non empty by XBOOLE_0:def 7; then A15: seq.m in oi by A14,XBOOLE_0:def 3; reconsider s = seq.m as Point of TOP-REAL n; A16: |. s - r .| < 1/m1 by A15,Th32; A17: k+1 > 0 by NAT_1:18; k+1 <= m+1 by A13,AXIOMS:24; then A18: 1/m1 <= 1/(k+1) by A17,REAL_2:152; k < k+1 by NAT_1:38; then A19: 1/(k+1) < 1/k by A12,REAL_2:151; 1/(1/p) >= 1/cp by A10,A11,REAL_2:152; then 1/k <= p by XCMPLX_1:56; then 1/(k+1) < p by A19,AXIOMS:22; then 1/m1 < p by A18,AXIOMS:22; hence |. seq.m - r .| < p by A16,AXIOMS:22; end; hence seq is convergent by TOPRNS_1:def 9; hence lim seq = r by A8,TOPRNS_1:def 10; end; definition let M be non empty MetrSpace; cluster TopSpaceMetr M -> first-countable; coherence by FRECHET:21; end; definition let n be Nat; cluster TOP-REAL n -> first-countable; coherence proof TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8; hence thesis; end; end; theorem Th35: for p being Point of Euclid n, q being Point of TOP-REAL n, r being real number st p = q & r > 0 holds Ball (p, r) is a_neighborhood of q proof let p be Point of Euclid n, q be Point of TOP-REAL n, r be real number; Ball (p, r) is Subset of TOP-REAL n by TOPREAL3:13; then reconsider A = Ball (p, r) as Subset of TOP-REAL n; assume p = q & r > 0; then A1: q in A by GOBOARD6:4; TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8; then A is open by TOPMETR:21; hence thesis by A1,CONNSP_2:5; end; theorem for A being Subset of TOP-REAL n, p being Point of TOP-REAL n, p' being Point of Euclid n st p = p' holds p in Cl A iff for r being real number st r > 0 holds Ball (p', r) meets A proof let A be Subset of TOP-REAL n, p be Point of TOP-REAL n, p' be Point of Euclid n; assume A1: p = p'; hereby assume A2: p in Cl A; let r be real number; assume A3: r > 0; the carrier of TOP-REAL n = the carrier of Euclid n by TOPREAL3:13; then reconsider B = Ball (p', r) as Subset of TOP-REAL n ; B is a_neighborhood of p by A1,A3,Th35; hence Ball (p', r) meets A by A2,YELLOW_6:6; end; assume A4: for r being real number st r > 0 holds Ball (p', r) meets A; for G being a_neighborhood of p holds G meets A proof let G be a_neighborhood of p; p in Int G by CONNSP_2:def 1; then consider r being real number such that A5: r > 0 & Ball (p', r) c= G by A1,GOBOARD6:8; Ball (p', r) meets A by A4,A5; hence thesis by A5,XBOOLE_1:63; end; hence thesis by YELLOW_6:6; end; theorem for x, y being Point of TOP-REAL n, x' being Point of Euclid n st x' = x & x <> y ex r being Real st not y in Ball (x', r) proof let x, y be Point of TOP-REAL n, x' be Point of Euclid n; assume that A1: x' = x and A2: x <> y; reconsider y' = y as Point of Euclid n by TOPREAL3:13; reconsider r = dist (x', y')/2 as Real; take r; A3: dist (x', y') <> 0 by A1,A2,METRIC_1:2; dist (x', y') >= 0 by METRIC_1:5; then dist (x', y') > r by A3,SEQ_2:4; hence thesis by METRIC_1:12; end; theorem Th38: for S being Subset of TOP-REAL n holds S is non Bounded iff for r being Real st r > 0 holds ex x, y being Point of Euclid n st x in S & y in S & dist (x, y) > r proof let S be Subset of TOP-REAL n; S is Subset of Euclid n by TOPREAL3:13; then reconsider S' = S as Subset of Euclid n; hereby assume S is non Bounded; then S' is non bounded by JORDAN2C:def 2; hence for r being Real st r > 0 ex x, y being Point of Euclid n st x in S & y in S & dist (x, y) > r by TBSP_1:def 9; end; assume A1: for r being Real st r > 0 ex x, y being Point of Euclid n st x in S & y in S & dist (x, y) > r; S is non Bounded proof assume S is Bounded; then consider C being Subset of Euclid n such that A2: C = S & C is bounded by JORDAN2C:def 2; thus thesis by A1,A2,TBSP_1:def 9; end; hence thesis; end; theorem Th39: for a, b being real number, x, y being Point of Euclid n st Ball (x, a) meets Ball (y, b) holds dist (x, y) < a + b proof let a, b be real number, x, y be Point of Euclid n; assume Ball (x, a) meets Ball (y, b); then consider z being set such that A1: z in Ball (x, a) & z in Ball (y, b) by XBOOLE_0:3; reconsider z as Point of Euclid n by A1; A2: dist (x, z) < a by A1,METRIC_1:12; dist (y, z) < b by A1,METRIC_1:12; then A3: dist (x, z) + dist (y, z) < dist (x, z) + b by REAL_1:67; dist (x, z) + b < a + b by A2,REAL_1:67; then A4: dist (x, z) + dist (y, z) < a + b by A3,AXIOMS:22; dist (x, z) + dist (y, z) >= dist (x, y) by METRIC_1:4; hence thesis by A4,AXIOMS:22; end; theorem Th40: for a, b, c being real number, x, y, z being Point of Euclid n st Ball (x, a) meets Ball (z, c) & Ball (z, c) meets Ball (y, b) holds dist (x, y) < a + b + 2*c proof let a, b, c be real number, x, y, z be Point of Euclid n; assume that A1: Ball (x, a) meets Ball (z, c) and A2: Ball (z, c) meets Ball (y, b); A5: dist (x, z) < a + c by A1,Th39; A6: dist (z, y) < c + b by A2,Th39; A7: dist (x, z) + dist (z, y) < (a + c) + dist (z, y) by A5,REAL_1:67; (a + c) + dist (z, y) < (a + c) + (c + b) by A6,REAL_1:67; then A8: dist (x, z) + dist (z, y) < (a + c) + (c + b) by A7,AXIOMS:22; dist (x, y) <= dist (x, z) + dist (z, y) by METRIC_1:4; then dist (x, y) < (a + c) + (c + b) by A8,AXIOMS:22; then dist (x, y) < (a + c) + c + b by XCMPLX_1:1; then dist (x, y) < a + (c + c) + b by XCMPLX_1:1; then dist (x, y) < a + 2*c + b by XCMPLX_1:11; hence thesis by XCMPLX_1:1; end; theorem Th41: for X, Y being non empty TopSpace, x being Point of X, y being Point of Y, V being Subset of [: X, Y :] holds V is a_neighborhood of [: {x}, {y} :] iff V is a_neighborhood of [ x, y ] proof let X, Y be non empty TopSpace, x be Point of X, y be Point of Y, V be Subset of [: X, Y :]; hereby assume V is a_neighborhood of [: {x}, {y} :]; then V is a_neighborhood of { [x, y] } by ZFMISC_1:35; hence V is a_neighborhood of [ x, y ] by CONNSP_2:10; end; assume V is a_neighborhood of [ x, y ]; then V is a_neighborhood of { [ x, y ] } by CONNSP_2:10; hence thesis by ZFMISC_1:35; end; scheme TSubsetEx { S() -> non empty TopStruct, P[set] } : ex X being Subset of S() st for x being Point of S() holds x in X iff P[x] proof defpred G[set] means P[$1]; consider A being set such that A1: for x being set holds x in A iff x in the carrier of S() & G[x] from Separation; A c= the carrier of S() proof let x be set; assume x in A; hence thesis by A1; end; then reconsider A as Subset of S(); take A; thus thesis by A1; end; scheme TSubsetUniq { S() -> TopStruct, P[set] } : for A1, A2 being Subset of S() st (for x being Point of S() holds x in A1 iff P[x]) & (for x being Point of S() holds x in A2 iff P[x]) holds A1 = A2 proof let A, B be Subset of S() such that A1: for p being Point of S() holds p in A iff P[p] and A2: for p being Point of S() holds p in B iff P[p]; hereby let x be set; assume A3: x in A; then P[x] by A1; hence x in B by A2,A3; end; let x be set; assume A4: x in B; then P[x] by A2; hence x in A by A1,A4; end; definition let T be non empty TopStruct; let S be SetSequence of the carrier of T; let i be Nat; redefine func S.i -> Subset of T; coherence proof S.i in bool the carrier of T; hence thesis; end; end; theorem Th42: for T be non empty 1-sorted, S being SetSequence of the carrier of T, R being Seq_of_Nat holds S * R is SetSequence of T proof let T be non empty 1-sorted, S be SetSequence of the carrier of T, R be Seq_of_Nat; A1: dom S = NAT by FUNCT_2:def 1; A2: dom R = NAT by FUNCT_2:def 1; rng R c= NAT by SEQM_3:def 8; then A3: dom (S * R) = NAT by A1,A2,RELAT_1:46; rng (S * R) c= rng S by RELAT_1:45; then rng (S * R) c= bool the carrier of T by XBOOLE_1:1; then S * R is Function of NAT, bool the carrier of T by A3,FUNCT_2:4; hence thesis; end; theorem Th43: id NAT is increasing Seq_of_Nat proof id NAT is Function of NAT, REAL by FUNCT_2:9; hence thesis by FRECHET2:2; end; definition cluster id NAT -> real-yielding; coherence by Th43; end; Lm1: for T being non empty 1-sorted, S being SetSequence of the carrier of T holds ex NS being increasing Seq_of_Nat st S = S * NS proof let T be non empty 1-sorted, S be SetSequence of the carrier of T; reconsider NS = id NAT as increasing Seq_of_Nat by Th43; take NS; thus S = S * NS by FUNCT_2:23; end; begin :: Subsequences definition let T be non empty 1-sorted, S be SetSequence of the carrier of T; mode subsequence of S -> SetSequence of T means :Def10: ex NS being increasing Seq_of_Nat st it = S * NS; existence proof take S; thus ex NS being increasing Seq_of_Nat st S = S * NS by Lm1; end; end; theorem Th44: for T being non empty 1-sorted, S being SetSequence of the carrier of T holds S is subsequence of S proof let T be non empty 1-sorted, S be SetSequence of the carrier of T; ex NS being increasing Seq_of_Nat st S = S * NS by Lm1; hence thesis by Def10; end; theorem for T being non empty 1-sorted, S being SetSequence of T, S1 being subsequence of S holds rng S1 c= rng S proof let T be non empty 1-sorted, S be SetSequence of T, S1 be subsequence of S; let y be set; assume y in rng S1; then consider x being set such that A1: x in dom S1 and A2: y = S1.x by FUNCT_1:def 5; consider NS being increasing Seq_of_Nat such that A3: S1 = S * NS by Def10; reconsider x as Nat by A1,FUNCT_2:def 1; NS.x in NAT; then A4: NS.x in dom S by FUNCT_2:def 1; y = S.(NS.x) by A1,A2,A3,FUNCT_1:22; hence y in rng S by A4,FUNCT_1:def 5; end; theorem for T being non empty 1-sorted, S1 being SetSequence of the carrier of T, S2 being subsequence of S1, S3 being subsequence of S2 holds S3 is subsequence of S1 proof let T be non empty 1-sorted, S1 be SetSequence of the carrier of T, S2 be subsequence of S1, S3 be subsequence of S2; consider NS2 being increasing Seq_of_Nat such that A1: S3 = S2 * NS2 by Def10; consider NS1 being increasing Seq_of_Nat such that A2: S2 = S1 * NS1 by Def10; S3 = S1 * (NS1 * NS2) by A1,A2,RELAT_1:55; hence S3 is subsequence of S1 by Def10; end; theorem Th47: for T being non empty 1-sorted, F, G being SetSequence of the carrier of T, A being Subset of T st G is subsequence of F & for i being Nat holds F.i = A holds G = F proof let T be non empty 1-sorted, F, G be SetSequence of the carrier of T, A be Subset of T; assume that A1: G is subsequence of F and A2: for i being Nat holds F.i = A; consider NS being increasing Seq_of_Nat such that A3: G = F * NS by A1,Def10; A4: for i being Nat holds G.i = F.i proof let i be Nat; dom NS = NAT by FUNCT_2:def 1; then G.i = F.(NS.i) by A3,FUNCT_1:23 .= A by A2 .= F.i by A2; hence thesis; end; A5: NAT = dom G & NAT = dom F by FUNCT_2:def 1; for x being set st x in dom F holds F.x = G.x proof let x be set; assume x in dom F; then reconsider n = x as Nat by FUNCT_2:def 1; F.x = G.n by A4 .= G.x; hence thesis; end; hence thesis by A5,FUNCT_1:9; end; theorem for T being non empty 1-sorted, A being constant SetSequence of T, B being subsequence of A holds A = B proof let T be non empty 1-sorted, A be constant SetSequence of T, B be subsequence of A; consider X being Subset of T such that A1: for n being Nat holds A.n = X by Def8; thus thesis by A1,Th47; end; theorem Th49: for T being non empty 1-sorted, S being SetSequence of the carrier of T, R being subsequence of S, n being Nat holds ex m being Nat st m >= n & R.n = S.m proof let T being non empty 1-sorted, S being SetSequence of the carrier of T, R being subsequence of S, n being Nat; consider NS being increasing Seq_of_Nat such that A1: R = S * NS by Def10; n in NAT; then A2: n in dom NS by FUNCT_2:def 1; take m = NS.n; thus m >= n by SEQM_3:33; thus thesis by A1,A2,FUNCT_1:23; end; definition let T be non empty 1-sorted, X be constant SetSequence of T; cluster -> constant subsequence of X; coherence proof let P be subsequence of X; consider A being Subset of T such that A1: for n being Nat holds X.n = A by Def8; ex A being Subset of T st for n being Nat holds P.n = A proof take A; let n be Nat; consider m being Nat such that A2: m >= n & P.n = X.m by Th49; thus thesis by A1,A2; end; hence thesis by Def8; end; end; scheme SubSeqChoice { T() -> non empty TopSpace, S() -> SetSequence of the carrier of T(), P[set]} : ex S1 being subsequence of S() st for n being Nat holds P[S1.n] provided A1: for n being Nat ex m being Nat st n <= m & P[S().m] proof consider n0 being Nat such that 0 <= n0 and A2: P[S().n0] by A1; defpred P1[set,set,set] means $3 in NAT & (for m,k being Nat st m = $2 & k = $3 holds m < k & P[S().k]); A3: for n being Nat for x being set ex y being set st P1[n,x,y] proof let n be Nat, x be set; per cases; suppose x in NAT; then reconsider mx = x as Nat; consider my being Nat such that A4: mx + 1 <= my & P[S().my] by A1; take my; thus my in NAT; thus for m,k being Nat st m=x & k=my holds m<k & P[S().k] by A4,NAT_1:38; suppose A5: not x in NAT; set y = 0; take 0; thus y in NAT; let m, k be Nat; assume m = x & k = y; hence m < k & P[S().k] by A5; end; consider g being Function such that A6: dom g = NAT and A7:g.0 = n0 and A8:for n being Nat holds P1[n,g.n,g.(n+1)] from RecChoice(A3); A9:rng g c= NAT proof let y be set; assume y in rng g; then consider x being set such that A10: x in dom g and A11: g.x = y by FUNCT_1:def 5; reconsider n = x as Nat by A6,A10; defpred P[Nat] means g.$1 in NAT; A12: P[0] by A7; A13: for k being Nat holds P[k] implies P[k+1] by A8; for k being Nat holds P[k] from Ind(A12,A13); then g.n in NAT; hence y in NAT by A11; end; then rng g c= REAL by XBOOLE_1:1; then reconsider g as Function of NAT,REAL by A6,FUNCT_2:4; reconsider g as Real_Sequence; for n being Nat holds g.n<g.(n+1) proof let n be Nat; g.(n+1) in rng g by A6,FUNCT_1:def 5; then reconsider k = g.(n+1) as Nat by A9; g.n in rng g by A6,FUNCT_1:def 5; then reconsider m = g.n as Nat by A9; m < k by A8; hence g.n<g.(n+1); end; then reconsider g as increasing Seq_of_Nat by A9,SEQM_3:def 8,def 11; reconsider S1 = S() * g as SetSequence of T() by Th42; A14:dom S1 = NAT by FUNCT_2:def 1; reconsider S1 as subsequence of S() by Def10; take S1; thus for n being Nat holds P[S1.n] proof let n be Nat; per cases; suppose n = 0; hence P[S1.n] by A2,A7,A14,FUNCT_1:22; suppose n <> 0; then n > 0 by NAT_1:19; then n >= 0 + 1 by NAT_1:38; then reconsider n' = n-1 as Nat by INT_1:18; A15: for m,k being Nat st m = g.n' & k = g.(n'+1) holds P[S().k] by A8; reconsider k = g.(n'+1) as Nat; A16: P[S().k] by A15; n + (-1) = n' by XCMPLX_0:def 8; then n = n'-(-1) by XCMPLX_1:26; then n = n'+1 by XCMPLX_1:151; hence P[S1.n] by A14,A16,FUNCT_1:22; end; end; begin :: Lower Topological Limit definition let T be non empty TopSpace; let S be SetSequence of the carrier of T; func Lim_inf S -> Subset of T means :Def11: for p being Point of T holds p in it iff for G being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds S.m meets G; existence proof defpred P[Point of T] means for G being a_neighborhood of $1 ex k being Nat st for m being Nat st m > k holds S.m meets G; thus ex IT being Subset of T st for p being Point of T holds p in IT iff P[p] from TSubsetEx; end; uniqueness proof defpred P[Point of T] means for G being a_neighborhood of $1 ex k being Nat st for m being Nat st m > k holds S.m meets G; thus for a,b being Subset of T st (for p being Point of T holds p in a iff P[p]) & (for p being Point of T holds p in b iff P[p]) holds a=b from TSubsetUniq; end; end; theorem Th50: for S being SetSequence of the carrier of TOP-REAL n, p being Point of TOP-REAL n, p' being Point of Euclid n st p = p' holds p in Lim_inf S iff for r being real number st r > 0 ex k being Nat st for m being Nat st m > k holds S.m meets Ball (p', r) proof let S be SetSequence of the carrier of TOP-REAL n, p be Point of TOP-REAL n, p' be Point of Euclid n; assume A1: p = p'; hereby assume A2: p in Lim_inf S; let r be real number; assume r > 0; then reconsider G = Ball (p', r) as a_neighborhood of p by A1,Th35; consider k being Nat such that A3: for m being Nat st m > k holds S.m meets G by A2,Def11; thus ex k being Nat st for m being Nat st m > k holds S.m meets Ball (p', r) by A3; end; assume A4:for r being real number st r > 0 ex k being Nat st for m being Nat st m > k holds S.m meets Ball (p', r); now let G be a_neighborhood of p; reconsider G' = Int G as Subset of TopSpaceMetr Euclid n by EUCLID:def 8; A5: TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8; p' in G' by A1,CONNSP_2:def 1; then consider r being real number such that A6: r > 0 & Ball (p', r) c= G' by A5,TOPMETR:22; G' c= G by TOPS_1:44; then A7: Ball (p', r) c= G by A6,XBOOLE_1:1; consider k being Nat such that A8: for m being Nat st m > k holds S.m meets Ball (p', r) by A4,A6; take k; let m be Nat; assume m > k; then S.m meets Ball (p', r) by A8; hence S.m meets G by A7,XBOOLE_1:63; end; hence thesis by Def11; end; theorem Th51: for T being non empty TopSpace, S being SetSequence of the carrier of T holds Cl Lim_inf S = Lim_inf S proof let T be non empty TopSpace; let S be SetSequence of the carrier of T; thus Cl Lim_inf S c= Lim_inf S proof let x be set; assume A1: x in Cl Lim_inf S; then reconsider x' = x as Point of T; now let G be a_neighborhood of x'; set H = Int G; H is open & x' in H by CONNSP_2:def 1; then Lim_inf S meets H by A1,PRE_TOPC:54; then consider z being set such that A2: z in Lim_inf S & z in H by XBOOLE_0:3; reconsider z as Point of T by A2; z in Int H by A2,TOPS_1:45; then H is a_neighborhood of z by CONNSP_2:def 1; then consider k being Nat such that A3: for m being Nat st m > k holds S.m meets H by A2,Def11; take k; let m be Nat; A4: H c= G by TOPS_1:44; assume m > k; then S.m meets H by A3; hence S.m meets G by A4,XBOOLE_1:63; end; hence x in Lim_inf S by Def11; end; thus thesis by PRE_TOPC:48; end; theorem Th52: for T being non empty TopSpace, S being SetSequence of the carrier of T holds Lim_inf S is closed proof let T be non empty TopSpace; let S be SetSequence of the carrier of T; Lim_inf S = Cl Lim_inf S by Th51; hence thesis; end; theorem for T being non empty TopSpace, R, S being SetSequence of the carrier of T st R is subsequence of S holds Lim_inf S c= Lim_inf R proof let T be non empty TopSpace, R, S be SetSequence of the carrier of T; assume A1: R is subsequence of S; let x be set; assume A2: x in Lim_inf S; then reconsider p = x as Point of T; consider Nseq being increasing Seq_of_Nat such that A3: R = S * Nseq by A1,Def10; for G being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds R.m meets G proof let G be a_neighborhood of p; consider k being Nat such that A4: for m being Nat st m > k holds S.m meets G by A2,Def11; take k; let m1 be Nat; assume A5: m1 > k; dom Nseq = NAT by FUNCT_2:def 1; then A6: R.m1 = S.(Nseq.m1) by A3,FUNCT_1:23; m1 <= Nseq.m1 by SEQM_3:33; then Nseq.m1 > k by A5,AXIOMS:22; hence thesis by A4,A6; end; hence thesis by Def11; end; theorem Th54: for T being non empty TopSpace, A, B being SetSequence of the carrier of T st for i being Nat holds A.i c= B.i holds Lim_inf A c= Lim_inf B proof let T be non empty TopSpace, A, B be SetSequence of the carrier of T; assume A1: for i being Nat holds A.i c= B.i; let x be set; assume A2: x in Lim_inf A; then reconsider p = x as Point of T; for G being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds B.m meets G proof let G be a_neighborhood of p; consider k being Nat such that A3: for m being Nat st m > k holds A.m meets G by A2,Def11; take k; let m1 be Nat; assume m1 > k; then A4: A.m1 meets G by A3; A.m1 c= B.m1 by A1; hence thesis by A4,XBOOLE_1:63; end; hence thesis by Def11; end; theorem for T being non empty TopSpace, A, B, C being SetSequence of the carrier of T st for i being Nat holds C.i = A.i \/ B.i holds Lim_inf A \/ Lim_inf B c= Lim_inf C proof let T be non empty TopSpace, A, B, C be SetSequence of the carrier of T; assume A1: for i being Nat holds C.i = A.i \/ B.i; let x be set; assume A2: x in Lim_inf A \/ Lim_inf B; reconsider p = x as Point of T by A2; per cases by A2,XBOOLE_0:def 2; suppose A3: x in Lim_inf A; for H being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds C.m meets H proof let H be a_neighborhood of p; consider k being Nat such that A4: for m being Nat st m > k holds A.m meets H by A3,Def11; take k; let m be Nat; assume m > k; then A5: A.m meets H by A4; C.m = A.m \/ B.m by A1; then A.m c= C.m by XBOOLE_1:7; hence thesis by A5,XBOOLE_1:63; end; hence thesis by Def11; suppose A6: x in Lim_inf B; for H being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds C.m meets H proof let H be a_neighborhood of p; consider k being Nat such that A7: for m being Nat st m > k holds B.m meets H by A6,Def11; take k; let m be Nat; assume m > k; then A8: B.m meets H by A7; C.m = A.m \/ B.m by A1; then B.m c= C.m by XBOOLE_1:7; hence thesis by A8,XBOOLE_1:63; end; hence thesis by Def11; end; theorem for T being non empty TopSpace, A, B, C being SetSequence of the carrier of T st for i being Nat holds C.i = A.i /\ B.i holds Lim_inf C c= Lim_inf A /\ Lim_inf B proof let T be non empty TopSpace, A, B, C be SetSequence of the carrier of T; assume A1: for i being Nat holds C.i = A.i /\ B.i; let x be set; assume A2: x in Lim_inf C; then reconsider p = x as Point of T; for H being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds A.m meets H proof let H be a_neighborhood of p; consider k being Nat such that A3: for m being Nat st m > k holds C.m meets H by A2,Def11; take k; let m be Nat; assume m > k; then A4: C.m meets H by A3; C.m = A.m /\ B.m by A1; then C.m c= A.m by XBOOLE_1:17; hence thesis by A4,XBOOLE_1:63; end; then A5: x in Lim_inf A by Def11; for H being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds B.m meets H proof let H be a_neighborhood of p; consider k being Nat such that A6: for m being Nat st m > k holds C.m meets H by A2,Def11; take k; let m be Nat; assume m > k; then A7: C.m meets H by A6; C.m = A.m /\ B.m by A1; then C.m c= B.m by XBOOLE_1:17; hence thesis by A7,XBOOLE_1:63; end; then x in Lim_inf B by Def11; hence thesis by A5,XBOOLE_0:def 3; end; theorem Th57: for T being non empty TopSpace, F, G being SetSequence of the carrier of T st for i being Nat holds G.i = Cl (F.i) holds Lim_inf G = Lim_inf F proof let T be non empty TopSpace, F, G be SetSequence of the carrier of T; assume A1: for i being Nat holds G.i = Cl (F.i); thus Lim_inf G c= Lim_inf F proof let x be set; assume A2: x in Lim_inf G; then reconsider p = x as Point of T; for H being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds F.m meets H proof let H be a_neighborhood of p; consider H1 being non empty Subset of T such that A3: H1 is a_neighborhood of p & H1 is open & H1 c= H by CONNSP_2:7; consider k being Nat such that A4: for m being Nat st m > k holds G.m meets H1 by A2,A3,Def11; take k; let m be Nat; assume m > k; then G.m meets H1 by A4; then consider y being set such that A5: y in G.m & y in H1 by XBOOLE_0:3; reconsider y as Point of T by A5; H1 is a_neighborhood of y by A3,A5,CONNSP_2:5; then consider H' being non empty Subset of T such that A6: H' is a_neighborhood of y & H' is open & H' c= H1 by CONNSP_2:7; y in Cl (F.m) by A1,A5; then H' meets F.m by A6,YELLOW_6:6; then H1 meets F.m by A6,XBOOLE_1:63; hence thesis by A3,XBOOLE_1:63; end; hence thesis by Def11; end; thus Lim_inf F c= Lim_inf G proof for i being Nat holds F.i c= G.i proof let i be Nat; G.i = Cl (F.i) by A1; hence thesis by PRE_TOPC:48; end; hence thesis by Th54; end; end; theorem for S being SetSequence of the carrier of TOP-REAL n, p being Point of TOP-REAL n holds (ex s being Real_Sequence of n st s is convergent & (for x being Nat holds s.x in S.x) & p = lim s) implies p in Lim_inf S proof let S be SetSequence of the carrier of TOP-REAL n, p be Point of TOP-REAL n; reconsider p' = p as Point of Euclid n by TOPREAL3:13; given s being Real_Sequence of n such that A1: s is convergent and A2: for x being Nat holds s.x in S.x and A3: p = lim s; for r being real number st r > 0 ex k being Nat st for m being Nat st m > k holds S.m meets Ball (p', r) proof let r be real number; reconsider r' = r as Real by XREAL_0:def 1; assume A4: r > 0; consider l being Nat such that A5: for m being Nat st l <= m holds |. s.m - p .| < r' by A1,A3,A4,TOPRNS_1:def 10; take v = max (0, l); let m be Nat; assume v < m; then l <= m by SQUARE_1:50; then |. s.m - p .| < r' by A5; then A6: s.m in Ball (p', r) by Th33; s.m in S.m by A2; hence thesis by A6,XBOOLE_0:3; end; hence thesis by Th50; end; theorem Th59: for T being non empty TopSpace, P being Subset of T, s being SetSequence of the carrier of T st (for i being Nat holds s.i c= P) holds Lim_inf s c= Cl P proof let T be non empty TopSpace, P be Subset of T, s be SetSequence of the carrier of T; assume A1: for i being Nat holds s.i c= P; let x be set; assume A2: x in Lim_inf s; then reconsider p = x as Point of T; for G being Subset of T st G is open holds p in G implies P meets G proof let G be Subset of T; assume A3: G is open; assume p in G; then reconsider G' = G as a_neighborhood of p by A3,CONNSP_2:5; consider k being Nat such that A4: for m being Nat st m > k holds s.m meets G' by A2,Def11; set m = k + 1; m > k by NAT_1:38; then A5: s.m meets G' by A4; s.m c= P by A1; hence thesis by A5,XBOOLE_1:63; end; hence thesis by PRE_TOPC:def 13; end; theorem Th60: for T being non empty TopSpace, F being SetSequence of the carrier of T, A being Subset of T st for i being Nat holds F.i = A holds Lim_inf F = Cl A proof let T be non empty TopSpace, F be SetSequence of the carrier of T, A be Subset of T; assume A1: for i being Nat holds F.i = A; then for i being Nat holds F.i c= A; hence Lim_inf F c= Cl A by Th59; thus Cl A c= Lim_inf F proof let x be set; assume A2: x in Cl A; then reconsider p = x as Point of T; for G being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds F.m meets G proof let G being a_neighborhood of p; take k = 1; let m be Nat; assume m > k; F.m = A by A1; hence thesis by A2,YELLOW_6:6; end; hence thesis by Def11; end; end; theorem for T being non empty TopSpace, F being SetSequence of the carrier of T, A being closed Subset of T st for i being Nat holds F.i = A holds Lim_inf F = A proof let T be non empty TopSpace, F be SetSequence of the carrier of T, A be closed Subset of T; assume for i being Nat holds F.i = A; then Lim_inf F = Cl A by Th60; hence thesis by PRE_TOPC:52; end; theorem Th62: for S being SetSequence of the carrier of TOP-REAL n, P being Subset of TOP-REAL n st P is Bounded & (for i being Nat holds S.i c= P) holds Lim_inf S is Bounded proof let S be SetSequence of the carrier of TOP-REAL n; let P be Subset of TOP-REAL n; assume that A1: P is Bounded and A2: for i being Nat holds S.i c= P; assume A3: Lim_inf S is non Bounded; consider P' being Subset of Euclid n such that A4: P' = P & P' is bounded by A1,JORDAN2C:def 2; consider t being Real, z being Point of Euclid n such that A5: 0 < t & P' c= Ball (z,t) by A4,METRIC_6:def 10; set r = 1, R = r + r + 3*t; 3*t > 3*0 by A5,REAL_1:70; then R > 3*0 by JORDAN2C:6; then consider x, y being Point of Euclid n such that A6: x in Lim_inf S & y in Lim_inf S & dist (x, y) > R by A3,Th38; consider k1 being Nat such that A7: for m being Nat st m > k1 holds S.m meets Ball (x, r) by A6,Th50; consider k2 being Nat such that A8: for m being Nat st m > k2 holds S.m meets Ball (y, r) by A6,Th50; set k = max (k1, k2) + 1; S.k c= P' by A2,A4; then A9: S.k c= Ball (z,t) by A5,XBOOLE_1:1; max (k1,k2) >= k1 by SQUARE_1:46; then k > k1 by NAT_1:38; then S.k meets Ball (x, r) by A7; then A10: Ball (z,t) meets Ball (x, r) by A9,XBOOLE_1:63; max (k1,k2) >= k2 by SQUARE_1:46; then k > k2 by NAT_1:38; then S.k meets Ball (y, r) by A8; then A11: Ball (z,t) meets Ball (y, r) by A9,XBOOLE_1:63; 2*t < 3*t by A5,REAL_1:70; then A12: r + r + 2*t < r + r + 3*t by REAL_1:67; dist (x,y) < r + r + 2*t by A10,A11,Th40; hence thesis by A6,A12,AXIOMS:22; end; theorem for S being SetSequence of the carrier of TOP-REAL 2, P being Subset of TOP-REAL 2 st P is Bounded & (for i being Nat holds S.i c= P) & for i being Nat holds S.i is compact holds Lim_inf S is compact proof let S be SetSequence of the carrier of TOP-REAL 2, P be Subset of TOP-REAL 2; assume that A1: P is Bounded and A2: (for i being Nat holds S.i c= P) and for i being Nat holds S.i is compact; A3: Lim_inf S is closed by Th52; Lim_inf S is Bounded by A1,A2,Th62; hence thesis by A3,TOPREAL6:88; end; theorem Th64: for A, B being SetSequence of the carrier of TOP-REAL n, C being SetSequence of the carrier of [: TOP-REAL n, TOP-REAL n :] st for i being Nat holds C.i = [:A.i, B.i:] holds [: Lim_inf A, Lim_inf B :] = Lim_inf C proof let A, B be SetSequence of the carrier of TOP-REAL n, C be SetSequence of the carrier of [: TOP-REAL n, TOP-REAL n :]; assume A1: for i being Nat holds C.i = [:A.i, B.i:]; thus [: Lim_inf A, Lim_inf B :] c= Lim_inf C proof let x be set; assume A2: x in [: Lim_inf A, Lim_inf B :]; then consider x1, x2 being set such that A3: x1 in Lim_inf A & x2 in Lim_inf B & x = [x1,x2] by ZFMISC_1:def 2; reconsider x1, x2 as Point of TOP-REAL n by A3; reconsider p = x as Point of [: TOP-REAL n, TOP-REAL n :] by A2; for G being a_neighborhood of p ex k being Nat st for m being Nat st m > k holds C.m meets G proof let G be a_neighborhood of p; G is a_neighborhood of [:{x1},{x2}:] by A3,Th41; then consider V being a_neighborhood of {x1}, W being a_neighborhood of x2 such that A4: [:V,W:] c= G by BORSUK_1:67; V is a_neighborhood of x1 by CONNSP_2:10; then consider k1 being Nat such that A5: for m being Nat st m > k1 holds A.m meets V by A3,Def11; consider k2 being Nat such that A6: for m being Nat st m > k2 holds B.m meets W by A3,Def11; take k = max (k1, k2); A7: k >= k1 & k >= k2 by SQUARE_1:46; let m be Nat; assume A8: m > k; then m > k1 by A7,AXIOMS:22; then A9: A.m meets V by A5; m > k2 by A7,A8,AXIOMS:22; then B.m meets W by A6; then [: A.m, B.m :] meets [: V, W :] by A9,Th4; then C.m meets [: V, W :] by A1; hence thesis by A4,XBOOLE_1:63; end; hence thesis by Def11; end; thus Lim_inf C c= [: Lim_inf A, Lim_inf B :] proof let x be set; assume A10: x in Lim_inf C; x in the carrier of [: TOP-REAL n, TOP-REAL n :] by A10; then A11: x in [: the carrier of TOP-REAL n, the carrier of TOP-REAL n :] by BORSUK_1:def 5; then reconsider p1 = x`1, p2 = x`2 as Point of TOP-REAL n by MCART_1:10; A12: x = [p1,p2] by A11,MCART_1:23; consider H being a_neighborhood of p2; consider F being a_neighborhood of p1; for G being a_neighborhood of p1 ex k being Nat st for m being Nat st m > k holds A.m meets G proof let G be a_neighborhood of p1; consider k being Nat such that A13: for m being Nat st m > k holds C.m meets [: G, H :] by A10,A12,Def11; take k; let m be Nat; assume m > k; then C.m meets [: G, H :] by A13; then consider y being set such that A14: y in C.m & y in [: G, H :] by XBOOLE_0:3; y in [:A.m, B.m:] by A1,A14; then A15: y`1 in A.m by MCART_1:10; y`1 in G by A14,MCART_1:10; hence thesis by A15,XBOOLE_0:3; end; then A16: p1 in Lim_inf A by Def11; for G being a_neighborhood of p2 ex k being Nat st for m being Nat st m > k holds B.m meets G proof let G be a_neighborhood of p2; consider k being Nat such that A17: for m being Nat st m > k holds C.m meets [: F, G :] by A10,A12,Def11; take k; let m be Nat; assume m > k; then C.m meets [: F, G :] by A17; then consider y being set such that A18: y in C.m & y in [: F, G :] by XBOOLE_0:3; y in [:A.m, B.m:] by A1,A18; then A19: y`2 in B.m by MCART_1:10; y`2 in G by A18,MCART_1:10; hence thesis by A19,XBOOLE_0:3; end; then p2 in Lim_inf B by Def11; hence thesis by A12,A16,ZFMISC_1:106; end; end; theorem for S being SetSequence of TOP-REAL 2 holds lim_inf S c= Lim_inf S proof let S be SetSequence of TOP-REAL 2; let x be set; assume A1: x in lim_inf S; then consider k being Nat such that A2: for n being Nat holds x in S.(k+n) by Th7; reconsider p = x as Point of Euclid 2 by A1,TOPREAL3:13; reconsider y = x as Point of TOP-REAL 2 by A1; for r being real number st r > 0 ex k being Nat st for m being Nat st m > k holds S.m meets Ball (p, r) proof let r be real number; assume A3: r > 0; take k; let m be Nat; assume m > k; then consider h being Nat such that A4: m = k + h by NAT_1:28; A5: x in S.m by A2,A4; x in Ball (p, r) by A3,Th29; hence thesis by A5,XBOOLE_0:3; end; then y in Lim_inf S by Th50; hence thesis; end; theorem for C being Simple_closed_curve, i being Nat holds Fr (UBD L~Cage (C,i))` = L~Cage (C,i) proof let C be Simple_closed_curve, i be Nat; set K = (UBD L~Cage (C,i))`; set R = L~Cage (C,i); UBD R c= R` by JORDAN2C:30; then A1: UBD R misses R by SUBSET_1:43; UBD R misses BDD R by JORDAN2C:28; then A2: UBD R misses (BDD R) \/ R by A1,XBOOLE_1:70; A3: [#] TOP-REAL 2 = R` \/ R by PRE_TOPC:18 .= (BDD R) \/ (UBD R) \/ R by JORDAN2C:31; K = [#] TOP-REAL 2 \ UBD L~Cage (C,i) by PRE_TOPC:17 .= ((UBD R) \/ ((BDD R) \/ R)) \ UBD R by A3,XBOOLE_1:4 .= R \/ BDD R by A2,XBOOLE_1:88; then A4: BDD R c= K by XBOOLE_1:7; (BDD R) \/ (BDD R)` = [#] TOP-REAL 2 by PRE_TOPC:18; then A5: (BDD R) \/ (BDD R)` = the carrier of TOP-REAL 2 by PRE_TOPC:def 3; ((BDD R) \/ (UBD R))` = R`` by JORDAN2C:31; then (BDD R)` /\ (UBD R)` = R by SUBSET_1:29; then (BDD R) \/ R = ((BDD R) \/ (BDD R)`) /\ ((BDD R) \/ K) by XBOOLE_1:24 .= (BDD R) \/ K by A5,XBOOLE_1:28 .= K by A4,XBOOLE_1:12; then A6: Cl K = (BDD L~Cage (C,i)) \/ L~Cage (C,i) by PRE_TOPC:52; K` = LeftComp Cage (C,i) by GOBRD14:46; then A7: Cl K` = LeftComp Cage (C,i) \/ R by GOBRD14:32; BDD L~Cage (C,i) misses UBD L~Cage (C,i) by JORDAN2C:28; then A8: (BDD L~Cage (C,i)) /\ (UBD L~Cage (C,i)) = {} by XBOOLE_0:def 7; Fr K = Cl K /\ Cl K` by TOPS_1:def 2 .= ((BDD L~Cage (C,i)) \/ L~Cage (C,i)) /\ ((UBD L~Cage (C,i)) \/ L~Cage (C,i)) by A6,A7,GOBRD14:46 .= ((BDD L~Cage (C,i)) /\ (UBD L~Cage (C,i))) \/ L~Cage (C,i) by XBOOLE_1:24 .= L~Cage (C,i) by A8; hence thesis; end; begin :: Upper Topological Limit definition let T be non empty TopSpace; let S be SetSequence of the carrier of T; func Lim_sup S -> Subset of T means :Def12: for x being set holds x in it iff ex A being subsequence of S st x in Lim_inf A; existence proof defpred P[set] means ex A being subsequence of S st $1 in Lim_inf A; consider X being set such that A1: for x being set holds x in X iff x in the carrier of T & P[x] from Separation; X c= the carrier of T proof let x be set; assume x in X; hence thesis by A1; end; then reconsider X as Subset of T; take X; thus thesis by A1; end; uniqueness proof let A1, A2 be Subset of T; defpred P[set] means ex A being subsequence of S st $1 in Lim_inf A; assume that A2: for x being set holds x in A1 iff P[x] and A3: for x being set holds x in A2 iff P[x]; A1 = A2 from Extensionality(A2, A3); hence thesis; end; end; theorem for N being Nat, F being sequence of TOP-REAL N, x being Point of TOP-REAL N, x' being Point of Euclid N st x = x' holds x is_a_cluster_point_of F iff for r being real number, n being Nat st r > 0 holds ex m being Nat st n <= m & F.m in Ball (x', r) proof let N be Nat, F be sequence of TOP-REAL N, x be Point of TOP-REAL N, x' be Point of Euclid N; assume A1: x = x'; hereby assume A2: x is_a_cluster_point_of F; let r be real number, n be Nat; assume A3: r > 0; reconsider O = Ball (x', r) as open Subset of TOP-REAL N by Th30; x in O by A1,A3,Th29; then consider m being Nat such that A4: n <= m & F.m in O by A2,FRECHET2:def 4; take m; thus n <= m & F.m in Ball (x', r) by A4; end; assume A5: for r being real number, n being Nat st r > 0 holds ex m being Nat st n <= m & F.m in Ball (x', r); for O being Subset of TOP-REAL N, n being Nat st O is open & x in O ex m being Nat st n <= m & F.m in O proof let O be Subset of TOP-REAL N, n be Nat; assume A6: O is open & x in O; reconsider G' = O as Subset of TopSpaceMetr Euclid N by EUCLID:def 8; A7: TOP-REAL N = TopSpaceMetr Euclid N by EUCLID:def 8; consider r being real number such that A8: r > 0 & Ball (x', r) c= G' by A1,A6,A7,TOPMETR:22; consider m being Nat such that A9: n <= m & F.m in Ball (x', r) by A5,A8; take m; thus thesis by A8,A9; end; hence thesis by FRECHET2:def 4; end; theorem Th68: for T being non empty TopSpace, A being SetSequence of the carrier of T holds Lim_inf A c= Lim_sup A proof let T be non empty TopSpace, A be SetSequence of the carrier of T; let x be set; assume A1: x in Lim_inf A; ex K being subsequence of A st x in Lim_inf K proof reconsider B = A as subsequence of A by Th44; take B; thus thesis by A1; end; hence thesis by Def12; end; theorem Th69: for A, B, C being SetSequence of the carrier of TOP-REAL 2 st (for i being Nat holds A.i c= B.i) & C is subsequence of A holds ex D being subsequence of B st for i being Nat holds C.i c= D.i proof let A, B, C be SetSequence of the carrier of TOP-REAL 2; assume that A1: for i being Nat holds A.i c= B.i and A2: C is subsequence of A; consider NS being increasing Seq_of_Nat such that A3: C = A * NS by A2,Def10; set D = B * NS; reconsider D as SetSequence of TOP-REAL 2 by Th42; reconsider D as subsequence of B by Def10; take D; for i being Nat holds C.i c= D.i proof let i be Nat; A4: dom NS = NAT by FUNCT_2:def 1; C.i c= D.i proof let x be set; assume x in C.i; then A5: x in A.(NS.i) by A3,A4,FUNCT_1:23; A.(NS.i) c= B.(NS.i) by A1; then x in B.(NS.i) by A5; hence thesis by A4,FUNCT_1:23; end; hence thesis; end; hence thesis; end; theorem for A, B, C being SetSequence of the carrier of TOP-REAL 2 st (for i being Nat holds A.i c= B.i) & C is subsequence of B holds ex D being subsequence of A st for i being Nat holds D.i c= C.i proof let A, B, C be SetSequence of the carrier of TOP-REAL 2; assume that A1: for i being Nat holds A.i c= B.i and A2: C is subsequence of B; consider NS being increasing Seq_of_Nat such that A3: C = B * NS by A2,Def10; set D = A * NS; reconsider D as SetSequence of TOP-REAL 2 by Th42; reconsider D as subsequence of A by Def10; take D; for i being Nat holds D.i c= C.i proof let i be Nat; A4: dom NS = NAT by FUNCT_2:def 1; D.i c= C.i proof let x be set; assume x in D.i; then A5: x in A.(NS.i) by A4,FUNCT_1:23; A.(NS.i) c= B.(NS.i) by A1; then x in B.(NS.i) by A5; hence thesis by A3,A4,FUNCT_1:23; end; hence thesis; end; hence thesis; end; theorem Th71: :: (2) for A, B being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds A.i c= B.i holds Lim_sup A c= Lim_sup B proof let A, B be SetSequence of the carrier of TOP-REAL 2; assume A1: for i being Nat holds A.i c= B.i; Lim_sup A c= Lim_sup B proof let x be set; assume x in Lim_sup A; then consider A1 being subsequence of A such that A2: x in Lim_inf A1 by Def12; consider D being subsequence of B such that A3: for i being Nat holds A1.i c= D.i by A1,Th69; Lim_inf A1 c= Lim_inf D by A3,Th54; hence thesis by A2,Def12; end; hence thesis; end; theorem :: (3) for A, B, C being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds C.i = A.i \/ B.i holds Lim_sup A \/ Lim_sup B c= Lim_sup C proof let A, B, C be SetSequence of the carrier of TOP-REAL 2; assume A1: for i being Nat holds C.i = A.i \/ B.i; A2: for i being Nat holds A.i c= C.i proof let i be Nat; C.i = A.i \/ B.i by A1; hence thesis by XBOOLE_1:7; end; A3: for i being Nat holds B.i c= C.i proof let i be Nat; C.i = A.i \/ B.i by A1; hence thesis by XBOOLE_1:7; end; thus Lim_sup A \/ Lim_sup B c= Lim_sup C proof let x be set; assume A4: x in Lim_sup A \/ Lim_sup B; per cases by A4,XBOOLE_0:def 2; suppose x in Lim_sup A; then consider A1 being subsequence of A such that A5: x in Lim_inf A1 by Def12; consider C1 being subsequence of C such that A6: for i being Nat holds A1.i c= C1.i by A2,Th69; Lim_inf A1 c= Lim_inf C1 by A6,Th54; hence thesis by A5,Def12; suppose x in Lim_sup B; then consider B1 being subsequence of B such that A7: x in Lim_inf B1 by Def12; consider C1 being subsequence of C such that A8: for i being Nat holds B1.i c= C1.i by A3,Th69; Lim_inf B1 c= Lim_inf C1 by A8,Th54; hence thesis by A7,Def12; end; end; theorem :: (4) for A, B, C being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds C.i = A.i /\ B.i holds Lim_sup C c= Lim_sup A /\ Lim_sup B proof let A, B, C be SetSequence of the carrier of TOP-REAL 2; assume A1: for i being Nat holds C.i = A.i /\ B.i; A2: for i being Nat holds C.i c= A.i proof let i be Nat; C.i = A.i /\ B.i by A1; hence thesis by XBOOLE_1:17; end; A3: for i being Nat holds C.i c= B.i proof let i be Nat; C.i = A.i /\ B.i by A1; hence thesis by XBOOLE_1:17; end; let x be set; assume x in Lim_sup C; then consider C1 being subsequence of C such that A4: x in Lim_inf C1 by Def12; consider D1 being subsequence of A such that A5: for i being Nat holds C1.i c= D1.i by A2,Th69; Lim_inf C1 c= Lim_inf D1 by A5,Th54; then A6: x in Lim_sup A by A4,Def12; consider E1 being subsequence of B such that A7: for i being Nat holds C1.i c= E1.i by A3,Th69; Lim_inf C1 c= Lim_inf E1 by A7,Th54; then x in Lim_sup B by A4,Def12; hence thesis by A6,XBOOLE_0:def 3; end; theorem Th74: for A, B being SetSequence of the carrier of TOP-REAL 2, C, C1 being SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :] st (for i being Nat holds C.i = [: A.i, B.i :]) & C1 is subsequence of C holds ex A1, B1 being SetSequence of the carrier of TOP-REAL 2 st A1 is subsequence of A & B1 is subsequence of B & for i being Nat holds C1.i = [: A1.i, B1.i :] proof let A, B be SetSequence of the carrier of TOP-REAL 2, C, C1 be SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :]; assume that A1: for i being Nat holds C.i = [: A.i, B.i :] and A2: C1 is subsequence of C; consider NS being increasing Seq_of_Nat such that A3: C1 = C * NS by A2,Def10; set A1 = A * NS; reconsider A1 as SetSequence of TOP-REAL 2 by Th42; set B1 = B * NS; reconsider B1 as SetSequence of TOP-REAL 2 by Th42; take A1, B1; for i being Nat holds C1.i = [: A1.i, B1.i :] proof let i be Nat; A4: dom NS = NAT by FUNCT_2:def 1; then A5: A1.i = A.(NS.i) by FUNCT_1:23; A6: B1.i = B.(NS.i) by A4,FUNCT_1:23; C1.i = C.(NS.i) by A3,A4,FUNCT_1:23 .= [: A1.i, B1.i :] by A1,A5,A6; hence thesis; end; hence thesis by Def10; end; theorem for A, B being SetSequence of the carrier of TOP-REAL 2, C being SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :] st for i being Nat holds C.i = [: A.i, B.i :] holds Lim_sup C c= [: Lim_sup A, Lim_sup B :] proof let A, B be SetSequence of the carrier of TOP-REAL 2, C be SetSequence of the carrier of [: TOP-REAL 2, TOP-REAL 2 :]; assume A1: for i being Nat holds C.i = [: A.i, B.i :]; let x be set; assume x in Lim_sup C; then consider C1 being subsequence of C such that A2: x in Lim_inf C1 by Def12; consider A1, B1 being SetSequence of the carrier of TOP-REAL 2 such that A3: A1 is subsequence of A & B1 is subsequence of B & for i being Nat holds C1.i = [: A1.i, B1.i :] by A1,Th74; x in the carrier of [: TOP-REAL 2, TOP-REAL 2 :] by A2; then x in [: the carrier of TOP-REAL 2, the carrier of TOP-REAL 2 :] by BORSUK_1:def 5; then consider x1, x2 being set such that A4: x = [x1, x2] by ZFMISC_1:102; x in [: Lim_inf A1, Lim_inf B1 :] by A2,A3,Th64; then x1 in Lim_inf A1 & x2 in Lim_inf B1 by A4,ZFMISC_1:106; then x1 in Lim_sup A & x2 in Lim_sup B by A3,Def12; hence thesis by A4,ZFMISC_1:106; end; theorem Th76: for T being non empty TopSpace, F being SetSequence of the carrier of T, A being Subset of T st for i being Nat holds F.i = A holds Lim_inf F = Lim_sup F proof let T be non empty TopSpace, F be SetSequence of the carrier of T, A be Subset of T; assume A1: for i being Nat holds F.i = A; thus Lim_inf F c= Lim_sup F by Th68; thus Lim_sup F c= Lim_inf F proof let x be set; assume x in Lim_sup F; then consider G being subsequence of F such that A2: x in Lim_inf G by Def12; thus thesis by A1,A2,Th47; end; end; theorem for F being SetSequence of the carrier of TOP-REAL 2, A being Subset of TOP-REAL 2 st for i being Nat holds F.i = A holds Lim_sup F = Cl A proof let F be SetSequence of the carrier of TOP-REAL 2, A be Subset of TOP-REAL 2; assume A1: for i being Nat holds F.i = A; then Lim_inf F = Lim_sup F by Th76; hence thesis by A1,Th60; end; theorem for F, G being SetSequence of the carrier of TOP-REAL 2 st for i being Nat holds G.i = Cl (F.i) holds Lim_sup G = Lim_sup F proof let F, G be SetSequence of the carrier of TOP-REAL 2; assume A1: for i being Nat holds G.i = Cl (F.i); A2: for i being Nat holds F.i c= G.i proof let i be Nat; G.i = Cl (F.i) by A1; hence thesis by PRE_TOPC:48; end; thus Lim_sup G c= Lim_sup F proof let x be set; assume x in Lim_sup G; then consider H being subsequence of G such that A3: x in Lim_inf H by Def12; consider NS being increasing Seq_of_Nat such that A4: H = G * NS by Def10; set P = F * NS; reconsider P as SetSequence of TOP-REAL 2 by Th42; reconsider P as subsequence of F by Def10; for i being Nat holds H.i = Cl (P.i) proof let i be Nat; A5: dom NS = NAT by FUNCT_2:def 1; then H.i = G.(NS.i) by A4,FUNCT_1:23 .= Cl (F.(NS.i)) by A1 .= Cl (P.i) by A5,FUNCT_1:23; hence thesis; end; then Lim_inf H = Lim_inf P by Th57; hence thesis by A3,Def12; end; thus Lim_sup F c= Lim_sup G by A2,Th71; end;