Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

### The abstract of the Mizar article:

### Some Properties of Cells and Gauges

**by****Adam Grabowski,****Artur Kornilowicz, and****Andrzej Trybulec**- Received October 13, 2000
- MML identifier: JORDAN1C

- [ Mizar article, MML identifier index ]

environ vocabulary TOPREAL2, PRE_TOPC, EUCLID, JORDAN2C, ARYTM, ARYTM_1, INT_1, ARYTM_3, PCOMPS_1, MATRIX_1, JORDAN8, METRIC_1, ABSVALUE, MCART_1, TREES_1, FINSEQ_1, PSCOMP_1, GROUP_1, GOBOARD5, FUNCT_5, RELAT_1, LATTICES, SQUARE_1, RCOMP_1, COMPTS_1, BOOLE, SEQ_2, ORDINAL2, REALSET1, FUNCT_1, JORDAN1A, CONNSP_1, SUBSET_1, RELAT_2, CONNSP_3, TOPS_1, SPPOL_1; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XCMPLX_0, XREAL_0, REAL_1, INT_1, SQUARE_1, STRUCT_0, NAT_1, FINSEQ_1, MATRIX_1, ABSVALUE, RELAT_1, RCOMP_1, TOPS_1, SEQ_4, FUNCT_2, PRE_TOPC, CARD_4, BINARITH, CONNSP_1, COMPTS_1, EUCLID, PCOMPS_1, METRIC_1, METRIC_6, TOPREAL2, GOBOARD5, JORDAN8, JORDAN2C, CONNSP_3, SPPOL_1, PSCOMP_1, GOBRD14, SEQ_2, JORDAN1A; constructors JORDAN8, REAL_1, CARD_4, PSCOMP_1, BINARITH, JORDAN2C, CONNSP_1, JORDAN9, GOBRD14, JORDAN1A, WSIERP_1, TOPREAL2, TBSP_1, CONNSP_3, TOPS_1, JORDAN1, ABSVALUE, SQUARE_1, GOBOARD2, RCOMP_1, PARTFUN1, INT_1; clusters XREAL_0, JORDAN8, INT_1, EUCLID, GOBRD14, BORSUK_2, JORDAN1A, JORDAN1B, STRUCT_0, PSCOMP_1, PRE_TOPC, WAYBEL_2, SPRECT_4, SEQ_2, RELSET_1, SEQ_1, TOPREAL6, MEMBERED, ORDINAL2, NUMBERS; requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM; begin reserve C for Simple_closed_curve, i, j, n for Nat, p for Point of TOP-REAL 2; canceled; theorem :: JORDAN1C:2 [i,j] in Indices Gauge(C,n) & [i+1,j] in Indices Gauge(C,n) implies dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) = Gauge(C,n)*(i+1,j)`1 - Gauge(C,n)*(i,j)`1; theorem :: JORDAN1C:3 [i,j] in Indices Gauge(C,n) & [i,j+1] in Indices Gauge(C,n) implies dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) = Gauge(C,n)*(i,j+1)`2 - Gauge(C,n)*(i,j)`2; theorem :: JORDAN1C:4 :: JORDAN1A:27 for S being Subset of TOP-REAL 2 st S is Bounded holds proj1.:S is bounded; theorem :: JORDAN1C:5 for C1 being non empty compact Subset of TOP-REAL 2, C2, S being non empty Subset of TOP-REAL 2 holds S = C1 \/ C2 & proj1.:C2 is non empty bounded_below implies W-bound S = min(W-bound C1, W-bound C2); theorem :: JORDAN1C:6 :: PSCOMP_1:68 for X being Subset of TOP-REAL 2 holds p in X & X is Bounded implies W-bound X <= p`1 & p`1 <= E-bound X & S-bound X <= p`2 & p`2 <= N-bound X; theorem :: JORDAN1C:7 p in west_halfline p & p in east_halfline p & p in north_halfline p & p in south_halfline p; theorem :: JORDAN1C:8 west_halfline p is non Bounded; theorem :: JORDAN1C:9 east_halfline p is non Bounded; theorem :: JORDAN1C:10 north_halfline p is non Bounded; theorem :: JORDAN1C:11 south_halfline p is non Bounded; definition let C be compact Subset of TOP-REAL 2; cluster UBD C -> non empty; end; theorem :: JORDAN1C:12 for C being compact Subset of TOP-REAL 2 holds UBD C is_a_component_of C`; theorem :: JORDAN1C:13 for C being compact Subset of TOP-REAL 2 for WH being connected Subset of TOP-REAL 2 st WH is non Bounded & WH misses C holds WH c= UBD C; theorem :: JORDAN1C:14 for C being compact Subset of TOP-REAL 2 for p being Point of TOP-REAL 2 st west_halfline p misses C holds west_halfline p c= UBD C; theorem :: JORDAN1C:15 for C being compact Subset of TOP-REAL 2 for p being Point of TOP-REAL 2 st east_halfline p misses C holds east_halfline p c= UBD C; theorem :: JORDAN1C:16 for C being compact Subset of TOP-REAL 2 for p being Point of TOP-REAL 2 st south_halfline p misses C holds south_halfline p c= UBD C; theorem :: JORDAN1C:17 for C being compact Subset of TOP-REAL 2 for p being Point of TOP-REAL 2 st north_halfline p misses C holds north_halfline p c= UBD C; theorem :: JORDAN1C:18 for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies W-bound C <= W-bound BDD C; theorem :: JORDAN1C:19 for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies E-bound C >= E-bound BDD C; theorem :: JORDAN1C:20 for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies S-bound C <= S-bound BDD C; theorem :: JORDAN1C:21 for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies N-bound C >= N-bound BDD C; theorem :: JORDAN1C:22 for C being compact non vertical Subset of TOP-REAL 2 for I being Integer st p in BDD C & I = [\ ((p`1 - W-bound C) / (E-bound C - W-bound C) * 2|^n) + 2 /] holds 1 < I; theorem :: JORDAN1C:23 for C being compact non vertical Subset of TOP-REAL 2 for I being Integer st p in BDD C & I = [\ ((p`1 - W-bound C) / (E-bound C - W-bound C) * 2|^n) + 2 /] holds I + 1 <= len Gauge (C, n); theorem :: JORDAN1C:24 for C being compact non horizontal Subset of TOP-REAL 2 for J being Integer st p in BDD C & J = [\ ((p`2 - S-bound C) / (N-bound C - S-bound C) * 2|^n) + 2 /] holds 1 < J & J+1 <= width Gauge (C, n); theorem :: JORDAN1C:25 for I being Integer st I = [\ ((p`1 - W-bound C) / (E-bound C - W-bound C) * 2|^n) + 2 /] holds (W-bound C) + (((E-bound C)-(W-bound C))/(2|^n))*(I-2) <= p`1; theorem :: JORDAN1C:26 for I being Integer st I = [\ ((p`1 - W-bound C) / (E-bound C - W-bound C) * 2|^n) + 2 /] holds p`1 < (W-bound C) + (((E-bound C)-(W-bound C))/(2|^n))*(I-1); theorem :: JORDAN1C:27 for J being Integer st J = [\ ((p`2 - S-bound C) / (N-bound C - S-bound C) * 2|^n) + 2 /] holds (S-bound C) + ((N-bound C - S-bound C)/(2|^n))*(J-2) <= p`2; theorem :: JORDAN1C:28 for J being Integer st J = [\ ((p`2 - S-bound C) / (N-bound C - S-bound C) * 2|^n) + 2 /] holds p`2 < (S-bound C) + ((N-bound C - S-bound C)/(2|^n))*(J-1); theorem :: JORDAN1C:29 for C being closed Subset of TOP-REAL 2, p being Point of Euclid 2 st p in BDD C ex r being Real st r > 0 & Ball(p,r) c= BDD C; theorem :: JORDAN1C:30 for p, q being Point of TOP-REAL 2, r being real number st dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < r & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < r & p in cell (Gauge (C, n), i, j) & q in cell (Gauge (C, n), i, j) & 1 <= i & i+1 <= len Gauge (C,n) & 1 <= j & j+1 <= width Gauge(C,n) holds dist (p,q) < 2 * r; theorem :: JORDAN1C:31 for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`2 <> N-bound BDD C; theorem :: JORDAN1C:32 for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`1 <> E-bound BDD C; theorem :: JORDAN1C:33 for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`2 <> S-bound BDD C; theorem :: JORDAN1C:34 for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`1 <> W-bound BDD C; theorem :: JORDAN1C:35 p in BDD C implies ex n,i,j being Nat st 1 < i & i < len Gauge(C,n) & 1 < j & j < width Gauge(C,n) & p`1 <> (Gauge(C,n)*(i,j))`1 & p in cell(Gauge(C,n),i,j) & cell(Gauge(C,n),i,j) c= BDD C; theorem :: JORDAN1C:36 for C being Subset of TOP-REAL 2 st C is Bounded holds UBD C is non empty;

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