Copyright (c) 2000 Association of Mizar Users
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; definitions TARSKI, XBOOLE_0; theorems AXIOMS, EUCLID, JORDAN8, PSCOMP_1, JORDAN1A, NAT_1, TOPREAL6, REAL_1, GOBOARD5, GOBRD14, JORDAN2C, CONNSP_1, GOBOARD6, TOPS_1, INT_1, REAL_2, TOPREAL5, HEINE, SQUARE_1, JORDAN1, STRUCT_0, JORDAN9, GOBOARD7, METRIC_1, PCOMPS_1, ABSVALUE, SEQ_2, SPRECT_1, RELAT_1, SEQ_4, METRIC_6, TOPREAL3, UNIFORM1, JCT_MISC, EXTENS_1, JORDAN3, AMI_5, FUNCT_1, SUBSET_1, TBSP_1, JORDAN1B, XREAL_0, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1; begin reserve C for Simple_closed_curve, i, j, n for Nat, p for Point of TOP-REAL 2; Lm1: for i being real number holds i + 1 - 2 = i - 1 proof let i be real number; i + 1 - 2 = i + (1 - (1 + 1)) by XCMPLX_1:29 .= i +- 1 .= i - 1 by XCMPLX_0:def 8; hence thesis; end; Lm2: for i being real number holds i - 1 + 2 = i + 1 proof let i be real number; i - 1 + 2 = i - (1 - 2) by XCMPLX_1:37 .= i -- 1 .= i + 1 by XCMPLX_1:151; hence thesis; end; Lm3: now let r be real number, j; thus [\ r + j /] - j = [\ r /] + j - j by INT_1:51 .= [\ r /] by XCMPLX_1:26; end; Lm4: for a, b, c being real number st a <> 0 & b <> 0 holds (a/b)*((c/a) * b) = c proof let a, b, c be real number; assume A1: a <> 0 & b <> 0; (a/b)*((c/a) * b) = (a/b)*b * (c/a) by XCMPLX_1:4 .= a * (c/a) by A1,XCMPLX_1:88 .= c by A1,XCMPLX_1:88; hence thesis; end; Lm5: for p being Point of TOP-REAL 2 holds p is Point of Euclid 2 proof let p be Point of TOP-REAL 2; TopSpaceMetr Euclid 2 = TOP-REAL 2 by EUCLID:def 8; then TOP-REAL 2 = TopStruct (#the carrier of Euclid 2, Family_open_set Euclid 2#) by PCOMPS_1:def 6; hence thesis; end; Lm6: for r being real number st r > 0 holds 2*(r/4) < r proof let r be real number; assume A1: r > 0; 2*(r/4) = 2*r/(2*2) by XCMPLX_1:75 .= 2*r/2/2 by XCMPLX_1:79 .= r/2 by XCMPLX_1:90; hence thesis by A1,SEQ_2:4; end; canceled; theorem Th2: [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 proof set G = Gauge(C,n); assume A1: [i,j] in Indices G & [i+1,j] in Indices G; then 1 <= j & j <= width G & 1 <= i & i < i+1 & i+1 <= len G by GOBOARD5:1,REAL_1:69; then A2: 1 <= width G & i <= len G & 1 <= i+1 by AXIOMS:22; len G >= 4 by JORDAN8:13; then 2 <= len G & 1 <= len G by AXIOMS:22; then A3: [1,1] in Indices G & [2,1] in Indices G by A2,GOBOARD7:10; dist(G*(i,j),G*(i+1,j)) = (E-bound C - W-bound C)/2|^n by A1,GOBRD14:20; then dist(G*(1,1),G*(1+1,1)) = dist(G*(i,j),G*(i+1,j)) by A3,GOBRD14:20 .= G*(i+1,j)`1 - G*(i,j)`1 by A1,GOBRD14:15; hence thesis; end; theorem Th3: [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 proof set G = Gauge(C,n); assume A1: [i,j] in Indices G & [i,j+1] in Indices G; then A2: 1 <= j & j < j+1 & j + 1 <= width G & 1 <= i & i <= len G by GOBOARD5:1,REAL_1:69; 1 <= j+1 by A1,GOBOARD5:1; then A3: 1 <= width G & 1 <= i+1 by A2,AXIOMS:22,NAT_1:29; A4: len G >= 4 by JORDAN8:13; 2|^n + 3 >= 3 by NAT_1:29; then width G >= 3 by JORDAN1A:49; then 2 <= width G & 1 <= len G by A4,AXIOMS:22; then A5: [1,1] in Indices G & [1,2] in Indices G by A3,GOBOARD7:10; dist(G*(i,j),G*(i,j+1)) = (N-bound C - S-bound C)/2|^n by A1,GOBRD14:19; then dist(G*(1,1),G*(1,1+1)) = dist(G*(i,j),G*(i,j+1)) by A5,GOBRD14:19 .= G*(i,j+1)`2 - G*(i,j)`2 by A1,GOBRD14:16; hence thesis; end; TopSpaceMetr Euclid 2 = TOP-REAL 2 by EUCLID:def 8; then Lm7:TOP-REAL 2 = TopStruct (#the carrier of Euclid 2, Family_open_set Euclid 2#) by PCOMPS_1:def 6; Lm8: for r being real number, q being Point of Euclid 2 st 1 <= i & i+1 <= len Gauge (C,n) & 1 <= j & j+1 <= width Gauge (C,n) & r > 0 & p = q & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < r/4 & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < r/4 & p in cell (Gauge(C,n),i,j) & Gauge(C,n)*(i,j) in Ball (q, r) & Gauge(C,n)*(i+1,j) in Ball (q, r) & Gauge(C,n)*(i,j+1) in Ball (q, r) & Gauge(C,n)*(i+1,j+1) in Ball (q, r) holds cell(Gauge(C,n),i,j) c= Ball (q,r) proof let r be real number, q be Point of Euclid 2; assume that A1: 1 <= i & i+1 <= len Gauge (C,n) & 1 <= j & j+1 <= width Gauge (C,n) and A2: r > 0 & p = q & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < r/4 & dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < r/4 & p in cell (Gauge(C,n),i,j) & Gauge(C,n)*(i,j) in Ball (q, r) & Gauge(C,n)*(i+1,j) in Ball (q, r) & Gauge(C,n)*(i,j+1) in Ball (q, r) & Gauge(C,n)*(i+1,j+1) in Ball (q, r); let x be set; assume A3: x in cell(Gauge(C,n),i,j); set I = i, J = j, l = r/4; reconsider Q = q, X = x as Point of TOP-REAL 2 by A3,Lm7; reconsider x' = x as Point of Euclid 2 by A3,Lm5; set G = Gauge(C,n); A4: 1 <= 1 + i by NAT_1:29; A5:i < i + 1 & j < j+1 by REAL_1:69; then A6:i <= len G & j <= width G by A1,AXIOMS:22; then A7: [i,j] in Indices G & [i+1,j] in Indices G by A1,A4,GOBOARD7:10; then A8: G*(I+1,J)`1 - G*(I,J)`1 < l by A2,Th2; 1 <= j+1 by A1,A5,AXIOMS:22; then [i,j+1] in Indices G by A1,A6,GOBOARD7:10; then A9: G*(I,J+1)`2 - G*(I,J)`2 < l by A2,A7,Th3; A10: G*(I,J)`1 <= Q`1 & Q`1 <= G*(I+1,J)`1 by A1,A2,JORDAN9:19; A11: G*(I,J)`2 <= Q`2 & Q`2 <= G*(I,J+1)`2 by A1,A2,JORDAN9:19; A12: G*(I,J)`1 <= X`1 & X`1 <= G*(I+1,J)`1 by A1,A3,JORDAN9:19; G*(I,J)`2 <= X`2 & X`2 <= G*(I,J+1)`2 by A1,A3,JORDAN9:19; then A13: dist (Q,X) <= (G*(I+1,J)`1 - G*(I,J)`1 ) + ( G*(I,J+1)`2 - G*(I,J)`2 ) by A10,A11,A12,GOBRD14:12; (G*(I+1,J)`1 - G*(I,J)`1 ) + ( G*(I,J+1)`2 - G*(I,J)`2 ) <= l + l by A8,A9,REAL_1:55; then dist (p,X) <= l + l by A2,A13,AXIOMS:22; then A14: dist (p,X) <= 2*(r/4) by XCMPLX_1:11; 2*(r/4) < r by A2,Lm6; then dist (X, p) < r by A14,AXIOMS:22; then dist (x', q) < r by A2,GOBRD14:def 1; hence thesis by METRIC_1:12; end; theorem Th4: :: JORDAN1A:27 for S being Subset of TOP-REAL 2 st S is Bounded holds proj1.:S is bounded proof let S be Subset of TOP-REAL 2; assume S is Bounded; then consider C being Subset of Euclid 2 such that A1: C = S and A2: C is bounded by JORDAN2C:def 2; consider r being Real, x being Point of Euclid 2 such that A3: 0 < r and A4: C c= Ball(x,r) by A2,METRIC_6:def 10; A5: the carrier of TOP-REAL 2 = the carrier of Euclid 2 by TOPREAL3:13; reconsider p = x as Point of TOP-REAL 2 by TOPREAL3:13; reconsider P = Ball(x,r) as Subset of TOP-REAL 2 by A5; set t = max(abs(p`1-r),abs(p`1+r)); now assume abs(p`1-r) <= 0 & abs(p`1+r) <= 0; then abs(p`1-r) = 0 & abs(p`1+r) = 0 by ABSVALUE:5; then abs(r-p`1) = 0 & abs(p`1+r) = 0 by UNIFORM1:13; then r-p`1 = 0 & p`1+r = 0 by ABSVALUE:7; then r-p`1+p`1+r = 0; then r-(p`1-p`1)+r = 0 by XCMPLX_1:37; then A6: r-0+r = 0 by XCMPLX_1:14; r+r > 0+0 by A3,REAL_1:67; hence contradiction by A6; end; then A7: t > 0 by SQUARE_1:50; A8: proj1.:P = ].p`1-r,p`1+r.[ by TOPREAL6:51; for s being real number st s in proj1.:S holds abs(s) < t proof let s being real number; proj1.:S c= proj1.:P by A1,A4,RELAT_1:156; hence thesis by A8,JCT_MISC:12; end; hence proj1.:S is bounded by A7,SEQ_4:14; end; theorem 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) proof let C1 be non empty compact Subset of TOP-REAL 2, C2, S be non empty Subset of TOP-REAL 2; assume A1: S = C1 \/ C2 & proj1.:C2 is non empty bounded_below; set P1 = proj1.:C1, P2 = proj1.:C2, PS = proj1.:S; A2: W-bound C1 = inf P1 & W-bound C2 = inf P2 by SPRECT_1:48; A3: P1 is non empty bounded_below & P2 is non empty bounded_below by A1,SPRECT_1:46; thus W-bound S = inf PS by SPRECT_1:48 .= inf(P1 \/ P2) by A1,RELAT_1:153 .= min(W-bound C1, W-bound C2) by A2,A3,SPRECT_1:52; end; Lm9: for X being Subset of TOP-REAL 2 holds p in X & X is Bounded implies inf (proj1||X) <= p`1 & p`1 <= sup (proj1||X) proof let X be Subset of TOP-REAL 2; set T = TOP-REAL 2; assume A1: p in X & X is Bounded; then reconsider X as non empty Subset of TOP-REAL 2; reconsider p' = p as Point of T|X by A1,JORDAN1:1; A2: the carrier of T|X = X by JORDAN1:1; A3: (proj1||X).:X = (proj1|X).:X by PSCOMP_1:def 26 .= proj1.:X by EXTENS_1:1; A4: proj1.:X is bounded by A1,Th4; then A5: (proj1||X).:X is bounded_above by A3,SEQ_4:def 3; (proj1||X).:X is bounded_below by A3,A4,SEQ_4:def 3; then A6: proj1||X is bounded_below by A2,PSCOMP_1:def 11; proj1||X is bounded_above by A2,A5,PSCOMP_1:def 12; then reconsider f = proj1||X as bounded RealMap of T|X by A6,SEQ_2:def 5; (proj1||X).p' = p`1 by A1,PSCOMP_1:69; then inf f <= p`1 & p`1 <= sup f by PSCOMP_1:47,50; hence thesis; end; Lm10: for X being Subset of TOP-REAL 2 holds p in X & X is Bounded implies inf (proj2||X) <= p`2 & p`2 <= sup (proj2||X) proof let X be Subset of TOP-REAL 2; set T = TOP-REAL 2; assume A1: p in X & X is Bounded; then reconsider X as non empty Subset of TOP-REAL 2; reconsider p' = p as Point of T|X by A1,JORDAN1:1; A2: the carrier of T|X = X by JORDAN1:1; A3: (proj2||X).:X = (proj2|X).:X by PSCOMP_1:def 26 .= proj2.:X by EXTENS_1:1; A4: proj2.:X is bounded by A1,JORDAN1A:27; then A5: (proj2||X).:X is bounded_above by A3,SEQ_4:def 3; (proj2||X).:X is bounded_below by A3,A4,SEQ_4:def 3; then A6: proj2||X is bounded_below by A2,PSCOMP_1:def 11; proj2||X is bounded_above by A2,A5,PSCOMP_1:def 12; then reconsider f = proj2||X as bounded RealMap of T|X by A6,SEQ_2:def 5; (proj2||X).p' = p`2 by A1,PSCOMP_1:70; then inf f <= p`2 & p`2 <= sup f by PSCOMP_1:47,50; hence thesis; end; theorem Th6: :: 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 proof let X be Subset of TOP-REAL 2; assume A1: p in X & X is Bounded; W-bound X = inf (proj1||X) by PSCOMP_1:def 30; hence W-bound X <= p`1 by A1,Lm9; E-bound X = sup (proj1||X) by PSCOMP_1:def 32; hence E-bound X >= p`1 by A1,Lm9; S-bound X = inf (proj2||X) by PSCOMP_1:def 33; hence S-bound X <= p`2 by A1,Lm10; N-bound X = sup (proj2||X) by PSCOMP_1:def 31; hence thesis by A1,Lm10; end; theorem p in west_halfline p & p in east_halfline p & p in north_halfline p & p in south_halfline p proof p`1 <= p`1 & p`2 = p`2; hence thesis by JORDAN1A:def 2,def 3,def 4,def 5; end; Lm11: for C being Subset of TOP-REAL 2 st C is Bounded for h being real number st BDD C <> {} & h > W-bound BDD C & (for p st p in BDD C holds p`1 >= h) holds contradiction proof let C be Subset of TOP-REAL 2 such that A1: C is Bounded; let h be real number; assume that A2: BDD C <> {} and A3: h > W-bound BDD C and A4: (for p st p in BDD C holds p`1 >= h); the carrier of ((TOP-REAL 2)|BDD C) <> {} by A2,JORDAN1:1; then reconsider T = (TOP-REAL 2)|BDD C as non empty TopSpace by STRUCT_0: def 1; set X = proj1||BDD C; A5: X = proj1|BDD C by PSCOMP_1:def 26; the carrier of T = BDD C by JORDAN1:1; then A6: X.:the carrier of T = proj1.:BDD C by A5,EXTENS_1:1; BDD C is Bounded by A1,JORDAN2C:114; then X.:the carrier of T is bounded by A6,Th4; then X.:the carrier of T is bounded_below by SEQ_4:def 3; then reconsider X as bounded_below RealMap of T by PSCOMP_1:def 11; A7: inf X = W-bound BDD C by PSCOMP_1:def 30; A8: h is Real by XREAL_0:def 1; for p being Point of T holds X.p >= h proof let p be Point of T; A9: the carrier of T = BDD C by JORDAN1:1; then p in BDD C; then reconsider p' = p as Point of TOP-REAL 2; X.p = proj1.p' by A5,A9,FUNCT_1:72; then X.p = p'`1 by PSCOMP_1:def 28; hence thesis by A4,A9; end; hence thesis by A3,A7,A8,PSCOMP_1:48; end; Lm12: for C being Subset of TOP-REAL 2 st C is Bounded for h being real number st BDD C <> {} & h < E-bound BDD C & (for p st p in BDD C holds p`1 <= h) holds contradiction proof let C be Subset of TOP-REAL 2 such that A1: C is Bounded; let h be real number; assume that A2: BDD C <> {} and A3: h < E-bound BDD C and A4: (for p st p in BDD C holds p`1 <= h); the carrier of ((TOP-REAL 2)|BDD C) <> {} by A2,JORDAN1:1; then reconsider T = (TOP-REAL 2)|BDD C as non empty TopSpace by STRUCT_0: def 1; set X = proj1||BDD C; A5: X = proj1|BDD C by PSCOMP_1:def 26; the carrier of T = BDD C by JORDAN1:1; then A6: X.:the carrier of T = proj1.:BDD C by A5,EXTENS_1:1; BDD C is Bounded by A1,JORDAN2C:114; then X.:the carrier of T is bounded by A6,Th4; then X.:the carrier of T is bounded_above by SEQ_4:def 3; then reconsider X as bounded_above RealMap of T by PSCOMP_1:def 12; A7: sup X = E-bound BDD C by PSCOMP_1:def 32; for p being Point of T holds X.p <= h proof let p be Point of T; A8: the carrier of T = BDD C by JORDAN1:1; then p in BDD C; then reconsider p' = p as Point of TOP-REAL 2; X.p = proj1.p' by A5,A8,FUNCT_1:72; then X.p = p'`1 by PSCOMP_1:def 28; hence thesis by A4,A8; end; hence thesis by A3,A7,PSCOMP_1:51; end; Lm13: for C being Subset of TOP-REAL 2 st C is Bounded for h being real number st BDD C <> {} & h < N-bound BDD C & (for p st p in BDD C holds p`2 <= h) holds contradiction proof let C be Subset of TOP-REAL 2 such that A1: C is Bounded; let h be real number; assume that A2: BDD C <> {} and A3: h < N-bound BDD C and A4: (for p st p in BDD C holds p`2 <= h); the carrier of ((TOP-REAL 2)|BDD C) <> {} by A2,JORDAN1:1; then reconsider T = (TOP-REAL 2)|BDD C as non empty TopSpace by STRUCT_0: def 1; set X = proj2||BDD C; A5: X = proj2|BDD C by PSCOMP_1:def 26; the carrier of T = BDD C by JORDAN1:1; then A6: X.:the carrier of T = proj2.:BDD C by A5,EXTENS_1:1; BDD C is Bounded by A1,JORDAN2C:114; then X.:the carrier of T is bounded by A6,JORDAN1A:27; then X.:the carrier of T is bounded_above by SEQ_4:def 3; then reconsider X as bounded_above RealMap of T by PSCOMP_1:def 12; A7: sup X = N-bound BDD C by PSCOMP_1:def 31; for p being Point of T holds X.p <= h proof let p be Point of T; A8: the carrier of T = BDD C by JORDAN1:1; then p in BDD C; then reconsider p' = p as Point of TOP-REAL 2; X.p = proj2.p' by A5,A8,FUNCT_1:72; then X.p = p'`2 by PSCOMP_1:def 29; hence thesis by A4,A8; end; hence thesis by A3,A7,PSCOMP_1:51; end; Lm14: for C being Subset of TOP-REAL 2 st C is Bounded for h being real number st BDD C <> {} & h > S-bound BDD C & (for p st p in BDD C holds p`2 >= h) holds contradiction proof let C be Subset of TOP-REAL 2 such that A1: C is Bounded; let h be real number; assume that A2: BDD C <> {} and A3: h > S-bound BDD C and A4: (for p st p in BDD C holds p`2 >= h); the carrier of ((TOP-REAL 2)|BDD C) <> {} by A2,JORDAN1:1; then reconsider T = (TOP-REAL 2)|BDD C as non empty TopSpace by STRUCT_0: def 1; set X = proj2||BDD C; A5: X = proj2|BDD C by PSCOMP_1:def 26; the carrier of T = BDD C by JORDAN1:1; then A6: X.:the carrier of T = proj2.:BDD C by A5,EXTENS_1:1; BDD C is Bounded by A1,JORDAN2C:114; then X.:the carrier of T is bounded by A6,JORDAN1A:27; then X.:the carrier of T is bounded_below by SEQ_4:def 3; then reconsider X as bounded_below RealMap of T by PSCOMP_1:def 11; A7: inf X = S-bound BDD C by PSCOMP_1:def 33; A8: h is Real by XREAL_0:def 1; for p being Point of T holds X.p >= h proof let p be Point of T; A9: the carrier of T = BDD C by JORDAN1:1; then p in BDD C; then reconsider p' = p as Point of TOP-REAL 2; X.p = proj2.p' by A5,A9,FUNCT_1:72; then X.p = p'`2 by PSCOMP_1:def 29; hence thesis by A4,A9; end; hence thesis by A3,A7,A8,PSCOMP_1:48; end; theorem Th8: west_halfline p is non Bounded proof set Wp = west_halfline p; assume Wp is Bounded; then consider C being Subset of Euclid 2 such that A1: C = Wp & C is bounded by JORDAN2C:def 2; consider r being Real such that A2: 0 < r & for x,y being Point of Euclid 2 st x in C & y in C holds dist(x,y) <= r by A1,TBSP_1:def 9; reconsider p1 = p, EX = |[p`1-2*r, p`2]| as Point of Euclid 2 by Lm5; A3: 0 <= 2*r by A2,SQUARE_1:19; then 0 + p`1 <= 2*r + p`1 by AXIOMS:24; then p`1 - 2*r <= p`1 by REAL_1:86; then |[p`1-2*r, p`2]|`1 <= p`1 & |[p`1-2*r, p`2]|`2 = p`2 by EUCLID:56; then A4: EX in Wp & p1 in Wp by JORDAN1A:def 5; set EX1 = p`1-2*r; set p11 = p`1, p12 = p`2; A5: p = |[p11,p12]| by EUCLID:57; A6: p11 - EX1 = 2*r by XCMPLX_1:18; A7: r < 2 * r by A2,REAL_2:144; dist (p1, EX) = sqrt ((p11 - EX1)^2 + (p12 - p`2)^2) by A5,GOBOARD6:9 .= sqrt ((p11 - EX1)^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= 2*r by A3,A6,SQUARE_1:89; hence thesis by A1,A2,A4,A7; end; theorem Th9: east_halfline p is non Bounded proof set Wp = east_halfline p; assume Wp is Bounded; then consider C being Subset of Euclid 2 such that A1: C = Wp & C is bounded by JORDAN2C:def 2; consider r being Real such that A2: 0 < r & for x,y being Point of Euclid 2 st x in C & y in C holds dist(x,y) <= r by A1,TBSP_1:def 9; reconsider p1 = p, EX = |[p`1+2*r, p`2]| as Point of Euclid 2 by Lm5; set EX1 = p`1+2*r, EX2 = p`2; set p11 = p`1, p12 = p`2; A3: 0 <= 2*r by A2,SQUARE_1:19; then 0 + p`1 <= 2*r + p`1 by AXIOMS:24; then |[EX1, p`2]|`1 >= p`1 & |[EX1, p`2]|`2 = p`2 by EUCLID:56; then A4: EX in Wp & p1 in Wp by JORDAN1A:def 3; A5: p = |[p11,p12]| by EUCLID:57; A6: EX1 - p11 = 2*r by XCMPLX_1:26; A7: r < 2 * r by A2,REAL_2:144; dist (p1, EX) = sqrt ((p11 - EX1)^2 + (p12 - EX2)^2) by A5,GOBOARD6:9 .= sqrt ((p11 - EX1)^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= sqrt ((-(EX1 - p11))^2 + 0) by XCMPLX_1:143 .= sqrt ((EX1 - p11)^2 + 0) by SQUARE_1:61 .= 2*r by A3,A6,SQUARE_1:89; hence thesis by A1,A2,A4,A7; end; theorem Th10: north_halfline p is non Bounded proof set Wp = north_halfline p; assume Wp is Bounded; then consider C being Subset of Euclid 2 such that A1: C = Wp & C is bounded by JORDAN2C:def 2; consider r being Real such that A2: 0 < r & for x,y being Point of Euclid 2 st x in C & y in C holds dist(x,y) <= r by A1,TBSP_1:def 9; reconsider p1 = p, EX = |[p`1, p`2+2*r]| as Point of Euclid 2 by Lm5; set EX2 = p`2+2*r, EX1 = p`1; set p11 = p`1, p12 = p`2; A3: 0 <= 2*r by A2,SQUARE_1:19; then 0 + p`2 <= 2*r + p`2 by AXIOMS:24; then |[p`1, EX2]|`1 = p`1 & |[p`1, EX2]|`2 >= p`2 by EUCLID:56; then A4: EX in Wp & p1 in Wp by JORDAN1A:def 2; A5: p = |[p11,p12]| by EUCLID:57; A6: EX2 - p12 = 2*r by XCMPLX_1:26; A7: r < 2 * r by A2,REAL_2:144; dist (p1, EX) = sqrt ((p11 - EX1)^2 + (p12 - EX2)^2) by A5,GOBOARD6:9 .= sqrt ((p12 - EX2)^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= sqrt ((-(EX2 - p12))^2 + 0) by XCMPLX_1:143 .= sqrt ((EX2 - p12)^2 + 0) by SQUARE_1:61 .= 2*r by A3,A6,SQUARE_1:89; hence thesis by A1,A2,A4,A7; end; theorem Th11: south_halfline p is non Bounded proof set Wp = south_halfline p; assume Wp is Bounded; then consider C being Subset of Euclid 2 such that A1: C = Wp & C is bounded by JORDAN2C:def 2; consider r being Real such that A2: 0 < r & for x,y being Point of Euclid 2 st x in C & y in C holds dist(x,y) <= r by A1,TBSP_1:def 9; reconsider p1 = p, EX = |[p`1, p`2-2*r]| as Point of Euclid 2 by Lm5; set EX2 = p`2-2*r, EX1 = p`1; set p11 = p`1, p12 = p`2; A3: 0 <= 2*r by A2,SQUARE_1:19; then 0 + p`2 <= 2*r + p`2 by AXIOMS:24; then p`2 - 2*r <= p`2 by REAL_1:86; then |[p`1, EX2]|`1 = p`1 & |[p`1, EX2]|`2 <= p`2 by EUCLID:56; then A4: EX in Wp & p1 in Wp by JORDAN1A:def 4; A5: p = |[p11,p12]| by EUCLID:57; A6: p12 - EX2 = 2*r by XCMPLX_1:18; A7: r < 2 * r by A2,REAL_2:144; dist (p1, EX) = sqrt ((p11 - EX1)^2 + (p12 - EX2)^2) by A5,GOBOARD6:9 .= sqrt ((p12 - EX2)^2 + 0) by SQUARE_1:60,XCMPLX_1:14 .= 2*r by A3,A6,SQUARE_1:89; hence thesis by A1,A2,A4,A7; end; Lm15: now let C be Subset of TOP-REAL 2; assume A1: C is Bounded; then consider B being Subset of TOP-REAL 2 such that A2: B is_outside_component_of C and A3: B = UBD C by JORDAN2C:76; A4: UBD C is_a_component_of C` by A2,A3,JORDAN2C:def 4; C` <> {} by A1,JORDAN2C:74; hence UBD C is non empty by A4,SPRECT_1:6; end; definition let C be compact Subset of TOP-REAL 2; cluster UBD C -> non empty; coherence proof A1: C is Bounded by JORDAN2C:73; then consider B being Subset of TOP-REAL 2 such that A2: B is_outside_component_of C and A3: B = UBD C by JORDAN2C:76; A4: UBD C is_a_component_of C` by A2,A3,JORDAN2C:def 4; C` <> {} by A1,JORDAN2C:74; hence thesis by A4,SPRECT_1:6; end; end; theorem Th12: for C being compact Subset of TOP-REAL 2 holds UBD C is_a_component_of C` proof let C be compact Subset of TOP-REAL 2; C is Bounded by JORDAN2C:73; then reconsider C1 = C` as non empty Subset of TOP-REAL 2 by JORDAN2C:74; set Om1 = (TOP-REAL 2)|C1; A1: the carrier of ((TOP-REAL 2)|C1) = C1 by JORDAN1:1; UBD C c= C` by JORDAN2C:30; then reconsider UB = UBD C as Subset of Om1 by A1; A2: UB is connected by CONNSP_1:24; UB is non empty a_union_of_components of Om1 by JORDAN2C:25; then UB is_a_component_of Om1 by A2,JORDAN1B:1; hence thesis by CONNSP_1:def 6; end; theorem Th13: 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 proof let C be compact Subset of TOP-REAL 2; let WH be connected Subset of TOP-REAL 2; assume A1: WH is non Bounded & WH misses C; A2: WH meets UBD C proof assume A3: WH misses UBD C; A4: (BDD C) \/ (UBD C) = C` by JORDAN2C:31; [#]the carrier of TOP-REAL 2 = C \/ C` by SUBSET_1:25; then the carrier of TOP-REAL 2 = C \/ C` by SUBSET_1:def 4; then WH c= (UBD C) \/ BDD C by A1,A4,XBOOLE_1:73; then A5: WH c= BDD C by A3,XBOOLE_1:73; C is Bounded by JORDAN2C:73; then BDD C is Bounded by JORDAN2C:114; hence thesis by A1,A5,JORDAN2C:16; end; A6: UBD C is_a_component_of C` by Th12; WH c= C` by A1,SUBSET_1:43; hence thesis by A2,A6,JORDAN1A:8; end; theorem Th14: 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 proof let C be compact Subset of TOP-REAL 2; let p be Point of TOP-REAL 2; set WH = west_halfline p; A1: WH is non Bounded non empty connected by Th8; assume WH misses C; hence thesis by A1,Th13; end; theorem Th15: 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 proof let C be compact Subset of TOP-REAL 2; let p be Point of TOP-REAL 2; set WH = east_halfline p; A1: WH is non Bounded non empty connected by Th9; assume WH misses C; hence thesis by A1,Th13; end; theorem Th16: 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 proof let C be compact Subset of TOP-REAL 2; let p be Point of TOP-REAL 2; set WH = south_halfline p; A1: WH is non Bounded non empty connected by Th11; assume WH misses C; hence thesis by A1,Th13; end; theorem Th17: 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 proof let C be compact Subset of TOP-REAL 2; let p be Point of TOP-REAL 2; set WH = north_halfline p; A1: WH is non Bounded non empty connected by Th10; assume WH misses C; hence thesis by A1,Th13; end; theorem Th18: for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies W-bound C <= W-bound BDD C proof let C be compact Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set WC = W-bound C, WB = W-bound BDD C; set hal = (WB + WC)/2; assume that A2: BDD C <> {} and A3: WC > WB; A4: hal > WB & hal < WC by A3,TOPREAL3:3; now per cases; suppose for q1 being Point of TOP-REAL 2 st q1 in BDD C holds q1`1 >= hal; hence contradiction by A1,A2,A4,Lm11; suppose ex q1 being Point of TOP-REAL 2 st q1 in BDD C & q1`1 < hal; then consider q1 being Point of TOP-REAL 2 such that A5: q1 in BDD C & q1`1 < hal; A6: q1`1 < WC by A4,A5,AXIOMS:22; set Q = |[(WC + q1`1)/2,q1`2]|; set WH = west_halfline Q; A7: Q`1 = (WC + q1`1)/2 by EUCLID:56; then q1`2 = Q`2 & q1`1 < Q`1 by A6,EUCLID:56,TOPREAL3:3; then A8: q1 in WH by JORDAN1A:def 5; WH misses C proof assume WH meets C; then consider y being set such that A9: y in WH /\ C by XBOOLE_0:4; A10: y in WH & y in C by A9,XBOOLE_0:def 3; reconsider y as Point of TOP-REAL 2 by A9; A11: y`1 <= Q`1 & y`2 = Q`2 by A10,JORDAN1A:def 5; Q`1 < WC by A6,A7,TOPREAL3:3; then y`1 < WC by A11,AXIOMS:22; hence thesis by A10,PSCOMP_1:71; end; then WH c= UBD C by Th14; then q1 in BDD C /\ UBD C by A5,A8,XBOOLE_0:def 3; then BDD C meets UBD C by XBOOLE_0:def 7; hence contradiction by JORDAN2C:28; end; hence thesis; end; theorem Th19: for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies E-bound C >= E-bound BDD C proof let C be compact Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set WC = E-bound BDD C, WB = E-bound C; set hal = (WB + WC)/2; assume that A2: BDD C <> {} and A3: WC > WB; A4: hal > WB & hal < WC by A3,TOPREAL3:3; now per cases; suppose for q1 being Point of TOP-REAL 2 st q1 in BDD C holds q1`1 <= hal; hence contradiction by A1,A2,A4,Lm12; suppose ex q1 being Point of TOP-REAL 2 st q1 in BDD C & q1`1 > hal; then consider q1 being Point of TOP-REAL 2 such that A5: q1 in BDD C & q1`1 > hal; A6: q1`1 > WB by A4,A5,AXIOMS:22; set Q = |[(WB + q1`1)/2,q1`2]|; set WH = east_halfline Q; A7: Q`1 = (WB + q1`1)/2 by EUCLID:56; then q1`2 = Q`2 & q1`1 > Q`1 by A6,EUCLID:56,TOPREAL3:3; then A8: q1 in WH by JORDAN1A:def 3; WH misses C proof assume WH /\ C <> {}; then consider y being set such that A9: y in WH /\ C by XBOOLE_0:def 1; A10: y in WH & y in C by A9,XBOOLE_0:def 3; reconsider y as Point of TOP-REAL 2 by A9; A11: y`1 >= Q`1 & y`2 = Q`2 by A10,JORDAN1A:def 3; Q`1 > WB by A6,A7,TOPREAL3:3; then y`1 > WB by A11,AXIOMS:22; hence thesis by A10,PSCOMP_1:71; end; then WH c= UBD C by Th15; then q1 in BDD C /\ UBD C by A5,A8,XBOOLE_0:def 3; then BDD C meets UBD C by XBOOLE_0:4; hence contradiction by JORDAN2C:28; end; hence thesis; end; theorem Th20: for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies S-bound C <= S-bound BDD C proof let C be compact Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set WC = S-bound C, WB = S-bound BDD C; set hal = (WB + WC)/2; assume that A2: BDD C <> {} and A3: WC > WB; A4: hal > WB & hal < WC by A3,TOPREAL3:3; now per cases; suppose for q1 being Point of TOP-REAL 2 st q1 in BDD C holds q1`2 >= hal; hence contradiction by A1,A2,A4,Lm14; suppose ex q1 being Point of TOP-REAL 2 st q1 in BDD C & q1`2 < hal; then consider q1 being Point of TOP-REAL 2 such that A5: q1 in BDD C & q1`2 < hal; A6: q1`2 < WC by A4,A5,AXIOMS:22; set Q = |[q1`1,(WC + q1`2)/2]|; set WH = south_halfline Q; A7: Q`2 = (WC + q1`2)/2 by EUCLID:56; then q1`1 = Q`1 & q1`2 < Q`2 by A6,EUCLID:56,TOPREAL3:3; then A8: q1 in WH by JORDAN1A:def 4; WH misses C proof assume WH /\ C <> {}; then consider y being set such that A9: y in WH /\ C by XBOOLE_0:def 1; A10: y in WH & y in C by A9,XBOOLE_0:def 3; reconsider y as Point of TOP-REAL 2 by A9; A11: y`2 <= Q`2 & y`1 = Q`1 by A10,JORDAN1A:def 4; Q`2 < WC by A6,A7,TOPREAL3:3; then y`2 < WC by A11,AXIOMS:22; hence thesis by A10,PSCOMP_1:71; end; then WH c= UBD C by Th16; then q1 in BDD C /\ UBD C by A5,A8,XBOOLE_0:def 3; then BDD C meets UBD C by XBOOLE_0:4; hence contradiction by JORDAN2C:28; end; hence thesis; end; theorem Th21: for C being compact Subset of TOP-REAL 2 holds BDD C <> {} implies N-bound C >= N-bound BDD C proof let C be compact Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set WC = N-bound BDD C, WB = N-bound C; set hal = (WB + WC)/2; assume that A2: BDD C <> {} and A3: WC > WB; A4: hal > WB & hal < WC by A3,TOPREAL3:3; now per cases; suppose for q1 being Point of TOP-REAL 2 st q1 in BDD C holds q1`2 <= hal; hence contradiction by A1,A2,A4,Lm13; suppose ex q1 being Point of TOP-REAL 2 st q1 in BDD C & q1`2 > hal; then consider q1 being Point of TOP-REAL 2 such that A5: q1 in BDD C & q1`2 > hal; A6: q1`2 > WB by A4,A5,AXIOMS:22; set Q = |[q1`1,(WB + q1`2)/2]|; set WH = north_halfline Q; A7: Q`2 = (WB + q1`2)/2 by EUCLID:56; then q1`1 = Q`1 & q1`2 > Q`2 by A6,EUCLID:56,TOPREAL3:3; then A8: q1 in WH by JORDAN1A:def 2; WH misses C proof assume WH /\ C <> {}; then consider y being set such that A9: y in WH /\ C by XBOOLE_0:def 1; A10: y in WH & y in C by A9,XBOOLE_0:def 3; reconsider y as Point of TOP-REAL 2 by A9; A11: y`2 >= Q`2 & y`1 = Q`1 by A10,JORDAN1A:def 2; Q`2 > WB by A6,A7,TOPREAL3:3; then y`2 > WB by A11,AXIOMS:22; hence thesis by A10,PSCOMP_1:71; end; then WH c= UBD C by Th17; then q1 in BDD C /\ UBD C by A5,A8,XBOOLE_0:def 3; then BDD C meets UBD C by XBOOLE_0:4; hence contradiction by JORDAN2C:28; end; hence thesis; end; theorem Th22: 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 proof let C be compact non vertical Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set W = W-bound C, E = E-bound C; set pW = p`1 - W, EW = E - W; let I be Integer; assume A2: p in BDD C & I = [\ (pW / EW * 2|^n) + 2 /]; then A3: W <= W-bound BDD C by Th18; E > W by SPRECT_1:33; then A4: EW > 0 by SQUARE_1:11; A5: 2|^n > 0 by HEINE:5; set K = [\ pW / EW * 2|^n /]; BDD C is Bounded by A1,JORDAN2C:114; then p`1 >= W-bound BDD C by A2,Th6; then p`1 >= W by A3,AXIOMS:22; then pW >= 0 by SQUARE_1:12; then pW / EW >= 0 by A4,REAL_2:125; then pW / EW * 2|^n >= 0 by A5,REAL_2:121; then A6: pW / EW * 2|^n + 1 >= 0 + 1 by AXIOMS:24; pW / EW * 2|^n - 1 < K by INT_1:def 4; then pW / EW * 2|^n - 1 + 2 < K + 2 by REAL_1:53; then pW / EW * 2|^n + 1 < K + 2 by Lm2; then 1 < K + 2 by A6,AXIOMS:22; hence thesis by A2,INT_1:51; end; theorem Th23: 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) proof let C be compact non vertical Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set W = W-bound C, E = E-bound C; set EW = E-W, pW = p`1 - W; let I be Integer; assume A2: p in BDD C & I = [\ (pW / EW * 2|^n) + 2 /]; A3: len Gauge (C, n) = 2|^n + 3 by JORDAN8:def 1; A4: 2|^n > 0 by HEINE:5; E > W by SPRECT_1:33; then A5: EW > 0 by SQUARE_1:11; A6: E >= E-bound BDD C by A2,Th19; BDD C is Bounded by A1,JORDAN2C:114; then p`1 <= E-bound BDD C by A2,Th6; then p`1 <= E by A6,AXIOMS:22; then p`1 - W <= EW by REAL_1:49; then pW / EW <= 1 by A5,REAL_2:117; then pW / EW * 2|^n <= 1 * 2|^n by A4,AXIOMS:25; then A7: pW / EW * 2|^n + 3 <= 2|^n + 3 by REAL_1:55; I <= (pW / EW * 2|^n) + 2 by A2,INT_1:def 4; then I + 1 <= (pW / EW * 2|^n) + 2 + 1 by AXIOMS:24; then I + 1 <= (pW / EW * 2|^n) + (2 + 1) by XCMPLX_1:1; hence thesis by A3,A7,AXIOMS:22; end; theorem Th24: 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) proof let C be compact non horizontal Subset of TOP-REAL 2; A1: C is Bounded by JORDAN2C:73; set W = S-bound C, E = N-bound C; set EW = E-W, pW = p`2 - W; let I be Integer; assume A2: p in BDD C & I = [\ (pW / EW * 2|^n) + 2 /]; A3: len Gauge (C, n) = 2|^n + 3 & len Gauge (C, n) = width Gauge (C, n) by JORDAN8:def 1; A4: 2|^n > 0 by HEINE:5; E > W by SPRECT_1:34; then A5: EW > 0 by SQUARE_1:11; A6: E >= N-bound BDD C & W <= S-bound BDD C by A2,Th20,Th21; set K = [\ pW / EW * 2|^n /]; BDD C is Bounded by A1,JORDAN2C:114; then p`2 >= S-bound BDD C by A2,Th6; then p`2 >= W by A6,AXIOMS:22; then pW >= 0 by SQUARE_1:12; then pW / EW >= 0 by A5,REAL_2:125; then pW / EW * 2|^n >= 0 by A4,REAL_2:121; then A7: pW / EW * 2|^n + 1 >= 0 + 1 by AXIOMS:24; pW / EW * 2|^n - 1 < K by INT_1:def 4; then pW / EW * 2|^n - 1 + 2 < K + 2 by REAL_1:53; then pW / EW * 2|^n + 1 < K + 2 by Lm2; then 1 < K + 2 by A7,AXIOMS:22; hence 1 < I by A2,INT_1:51; BDD C is Bounded by A1,JORDAN2C:114; then p`2 <= N-bound BDD C by A2,Th6; then p`2 <= E by A6,AXIOMS:22; then p`2 - W <= EW by REAL_1:49; then pW / EW <= 1 by A5,REAL_2:117; then pW / EW * 2|^n <= 1 * 2|^n by A4,AXIOMS:25; then A8: pW / EW * 2|^n + 3 <= 2|^n + 3 by REAL_1:55; I <= (pW / EW * 2|^n) + 2 by A2,INT_1:def 4; then I + 1 <= (pW / EW * 2|^n) + 2 + 1 by AXIOMS:24; then I + 1 <= (pW / EW * 2|^n) + (2 + 1) by XCMPLX_1:1; hence I+1 <= width Gauge (C, n) by A3,A8,AXIOMS:22; end; theorem Th25: 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 proof set E = E-bound C, W = W-bound C, EW = E-bound C - W-bound C; set PW = p`1 - W; set KI = [\ (PW / EW * 2|^n) /]; let I be Integer; assume I = [\ (PW / EW * 2|^n) + 2 /]; then A1: I - 2 = [\ (PW / EW * 2|^n) /] by Lm3; E > W by TOPREAL5:23; then A2: EW > 0 & 2|^n > 0 by HEINE:5,SQUARE_1:11; then A3: EW/(2|^n) > 0 by REAL_2:127; A4: (EW/(2|^n))*(PW / EW * 2|^n) = PW by A2,Lm4; A5: W + PW = p`1 by XCMPLX_1:27; KI <= PW / EW * 2|^n by INT_1:def 4; then (EW/(2|^n))*KI <= (EW/(2|^n))*(PW / EW * 2|^n) by A3,AXIOMS:25; hence thesis by A1,A4,A5,AXIOMS:24; end; theorem Th26: 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) proof set W = W-bound C, E = E-bound C; set EW = E - W, PW = p`1 - W; let I be Integer; assume A1: I = [\ (PW / EW * 2|^n) + 2 /]; set KI = I - 1; I > (PW / EW * 2|^n) + 2 - 1 by A1,INT_1:def 4; then I > (PW / EW * 2|^n) + (2 - 1) by XCMPLX_1:29; then I - 1 > (PW / EW * 2|^n) + 1 - 1 by REAL_1:54; then A2: I - 1 > (PW / EW * 2|^n) by XCMPLX_1:26; E > W by TOPREAL5:23; then A3: EW > 0 & 2|^n > 0 by HEINE:5,SQUARE_1:11; then A4: EW/(2|^n) > 0 by REAL_2:127; A5: (EW/(2|^n))*(PW / EW * 2|^n) = PW by A3,Lm4; A6: W + PW = p`1 by XCMPLX_1:27; (EW/(2|^n))*KI > (EW/(2|^n))*(PW / EW * 2|^n) by A2,A4,REAL_1:70; hence thesis by A5,A6,REAL_1:53; end; theorem Th27: 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 proof set E = N-bound C, W = S-bound C, EW = N-bound C - S-bound C; set PW = p`2 - W; set KI = [\ (PW / EW * 2|^n) /]; let I be Integer; assume I = [\ (PW / EW * 2|^n) + 2 /]; then A1: I - 2 = [\ (PW / EW * 2|^n) /] by Lm3; E > W by TOPREAL5:22; then A2: EW > 0 & 2|^n > 0 by HEINE:5,SQUARE_1:11; then A3: EW/(2|^n) > 0 by REAL_2:127; A4: (EW/(2|^n))*(PW / EW * 2|^n) = PW by A2,Lm4; A5: W + PW = p`2 by XCMPLX_1:27; KI <= PW / EW * 2|^n by INT_1:def 4; then (EW/(2|^n))*KI <= (EW/(2|^n))*(PW / EW * 2|^n) by A3,AXIOMS:25; hence thesis by A1,A4,A5,AXIOMS:24; end; theorem Th28: 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) proof set W = S-bound C, E = N-bound C; set EW = E - W, PW = p`2 - W; let I be Integer; assume A1: I = [\ (PW / EW * 2|^n) + 2 /]; set KI = I - 1; I > (PW / EW * 2|^n) + 2 - 1 by A1,INT_1:def 4; then I > (PW / EW * 2|^n) + (2 - 1) by XCMPLX_1:29; then I - 1 > (PW / EW * 2|^n) + 1 - 1 by REAL_1:54; then A2: I - 1 > (PW / EW * 2|^n) by XCMPLX_1:26; E > W by TOPREAL5:22; then A3: EW > 0 & 2|^n > 0 by HEINE:5,SQUARE_1:11; then A4: EW/(2|^n) > 0 by REAL_2:127; A5: (EW/(2|^n)) * (PW / EW * 2|^n) = PW by A3,Lm4; A6: W + PW = p`2 by XCMPLX_1:27; (EW/(2|^n))*KI > PW by A2,A4,A5,REAL_1:70; hence thesis by A6,REAL_1:53; end; theorem Th29: 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 proof let C be closed Subset of TOP-REAL 2, p be Point of Euclid 2; assume A1: p in BDD C; set W = Int BDD C; p in W by A1,TOPS_1:55; then consider r being real number such that A2: r > 0 & Ball(p,r) c= BDD C by GOBOARD6:8; reconsider r as Real by XREAL_0:def 1; take r; thus thesis by A2; end; theorem Th30: 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 proof set G = Gauge (C, n); let p, q be Point of TOP-REAL 2, r be real number; assume that A1: dist(G*(1,1),G*(1,2)) < r and A2: dist(G*(1,1),G*(2,1)) < r and A3: p in cell (G, i, j) and A4: q in cell (G, i, j) and A5: 1 <= i & i+1 <= len G and A6: 1 <= j & j+1 <= width G; A7: i <= i+1 & j <= j+1 & 1 <= i+1 & 1 <= j+1 by NAT_1:29; then i <= len G & j <= width G by A5,A6,AXIOMS:22; then A8: [i,j+1] in Indices G & [i,j] in Indices G & [i+1,j] in Indices G by A5,A6,A7,GOBOARD7:10; A9: G*(i,j)`1 <= p`1 & p`1 <= G*(i+1,j)`1 & G*(i,j)`2 <= p`2 & p`2 <= G*(i,j+1)`2 by A3,A5,A6,JORDAN9:19; G*(i,j)`1 <= q`1 & q`1 <= G*(i+1,j)`1 & G*(i,j)`2 <= q`2 & q`2 <= G*(i,j+1)`2 by A4,A5,A6,JORDAN9:19; then A10: dist (p,q) <= (G*(i+1,j)`1 - G*(i,j)`1 ) + ( G*(i,j+1)`2 - G*(i,j)`2 ) by A9,GOBRD14:12; A11: G*(i+1,j)`1 - G*(i,j)`1 < r by A2,A8,Th2; G*(i,j+1)`2 - G*(i,j)`2 < r by A1,A8,Th3; then (G*(i+1,j)`1 - G*(i,j)`1 ) + ( G*(i,j+1)`2 - G*(i,j)`2 ) < r + r by A11,REAL_1:67; then dist (q,p) < r + r by A10,AXIOMS:22; hence thesis by XCMPLX_1:11; end; theorem for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`2 <> N-bound BDD C proof let C be compact Subset of TOP-REAL 2; assume A1: p in BDD C; reconsider P = p as Point of Euclid 2 by Lm5; consider r being Real such that A2: r > 0 & Ball(P,r) c= BDD C by A1,Th29; assume A3: p`2 = N-bound BDD C; C is Bounded by JORDAN2C:73; then A4: BDD C is Bounded by JORDAN2C:114; A5: 0 < r/2 by A2,REAL_2:127; A6: r/2 < r by A2,SEQ_2:4; A7: p`2 + r/2 > p`2 + 0 by A5,REAL_1:53; set EX = |[p`1, p`2 + r/2]|; P = |[p`1,p`2]| by EUCLID:57; then A8: EX in Ball(P,r) by A5,A6,GOBOARD6:11; not EX in BDD C proof assume EX in BDD C; then EX`2 <= N-bound BDD C by A4,Th6; hence thesis by A3,A7,EUCLID:56; end; hence thesis by A2,A8; end; theorem for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`1 <> E-bound BDD C proof let C be compact Subset of TOP-REAL 2; assume A1: p in BDD C; reconsider P = p as Point of Euclid 2 by Lm5; consider r being Real such that A2: r > 0 & Ball(P,r) c= BDD C by A1,Th29; assume A3: p`1 = E-bound BDD C; C is Bounded by JORDAN2C:73; then A4: BDD C is Bounded by JORDAN2C:114; A5: 0 < r/2 by A2,REAL_2:127; A6: r/2 < r by A2,SEQ_2:4; A7: p`1 + r/2 > p`1 + 0 by A5,REAL_1:53; set EX = |[p`1 + r/2, p`2]|; P = |[p`1,p`2]| by EUCLID:57; then A8: EX in Ball(P,r) by A5,A6,GOBOARD6:10; not EX in BDD C proof assume EX in BDD C; then EX`1 <= E-bound BDD C by A4,Th6; hence thesis by A3,A7,EUCLID:56; end; hence thesis by A2,A8; end; theorem for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`2 <> S-bound BDD C proof let C be compact Subset of TOP-REAL 2; assume A1: p in BDD C; reconsider P = p as Point of Euclid 2 by Lm5; consider r being Real such that A2: r > 0 & Ball(P,r) c= BDD C by A1,Th29; assume A3: p`2 = S-bound BDD C; C is Bounded by JORDAN2C:73; then A4: BDD C is Bounded by JORDAN2C:114; A5: 0 < r/2 by A2,REAL_2:127; A6: r/2 < r by A2,SEQ_2:4; p`2 + r/2 > p`2 + 0 by A5,REAL_1:53; then A7: p`2 - r/2 < p`2 by REAL_1:84; set EX = |[p`1, p`2 - r/2]|; P = |[p`1,p`2]| by EUCLID:57; then A8: EX in Ball(P,r) by A5,A6,GOBOARD6:13; not EX in BDD C proof assume EX in BDD C; then EX`2 >= S-bound BDD C by A4,Th6; hence thesis by A3,A7,EUCLID:56; end; hence thesis by A2,A8; end; theorem Th34: for C being compact Subset of TOP-REAL 2 holds p in BDD C implies p`1 <> W-bound BDD C proof let C be compact Subset of TOP-REAL 2; assume A1: p in BDD C; reconsider P = p as Point of Euclid 2 by Lm5; consider r being Real such that A2: r > 0 & Ball(P,r) c= BDD C by A1,Th29; assume A3: p`1 = W-bound BDD C; C is Bounded by JORDAN2C:73; then A4: BDD C is Bounded by JORDAN2C:114; A5: 0 < r/2 by A2,REAL_2:127; A6: r/2 < r by A2,SEQ_2:4; p`1 + r/2 > p`1 + 0 by A5,REAL_1:53; then A7: p`1 - r/2 < p`1 by REAL_1:84; set EX = |[p`1 - r/2, p`2]|; P = |[p`1,p`2]| by EUCLID:57; then A8: EX in Ball(P,r) by A5,A6,GOBOARD6:12; not EX in BDD C proof assume EX in BDD C; then EX`1 >= W-bound BDD C by A4,Th6; hence thesis by A3,A7,EUCLID:56; end; hence thesis by A2,A8; end; theorem 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 proof assume A1: p in BDD C; reconsider P = p as Point of Euclid 2 by Lm5; consider r being Real such that A2: r > 0 & Ball(P,r) c= BDD C by A1,Th29; set l = r/4; l > 0 by A2,REAL_2:127; then consider n being Nat such that 1 < n and A3: dist(Gauge(C,n)*(1,1),Gauge(C,n)*(1,2)) < l and A4: dist(Gauge(C,n)*(1,1),Gauge(C,n)*(2,1)) < l by GOBRD14:21; take n; set G = Gauge(C,n); set W = W-bound C, E = E-bound C, S = S-bound C, N = N-bound C; set EW = E-W, NS = N-S; set I = [\ ((p`1 - W) / EW * 2|^n) + 2 /], J = [\ ((p`2 - S) / NS * 2|^n) + 2 /]; A5: 1 < I & I+1 <= len G by A1,Th22,Th23; A6: 1 < J & J+1 <= width G by A1,Th24; then 0 <= I & 0 <= J by A5,AXIOMS:22; then reconsider I, J as Nat by INT_1:16; A7: S + (NS/(2|^n))*(J-1) > p`2 by Th28; A8: 1 <= I + 1 & 1 <= J + 1 by NAT_1:29; A9: I < I + 1 & J < J + 1 by REAL_1:69; then A10: I <= len G & J <= width G by A5,A6,AXIOMS:22; then [I,J] in Indices G by A5,A6,GOBOARD7:10; then G*(I,J) = |[W+(EW/(2|^n))*(I-2), S+(NS/(2|^n))*(J-2)]| by JORDAN8:def 1; then G*(I,J)`1 = W+(EW/(2|^n))*(I-2) & G*(I,J)`2 = S+(NS/(2|^n))*(J-2) by EUCLID:56; then A11: G*(I,J)`1 <= p`1 & G*(I,J)`2 <= p`2 by Th25,Th27; [I+1,J] in Indices G by A5,A6,A8,A10,GOBOARD7:10; then G*(I+1,J) = |[W+(EW/(2|^n))*(I+1-2), S+(NS/(2|^n))*(J-2)]| by JORDAN8:def 1; then G*(I+1,J)`1 = W+(EW/(2|^n))*(I+1-2) by EUCLID:56 .= W+(EW/(2|^n))*(I-1) by Lm1; then A12: p`1 < G*(I+1,J)`1 by Th26; [I,J+1] in Indices G by A5,A6,A8,A10,GOBOARD7:10; then G*(I,J+1) = |[W+(EW/(2|^n))*(I-2), S+(NS/(2|^n))*(J+1-2)]| by JORDAN8:def 1; then A13: G*(I,J+1)`2 = S+(NS/(2|^n))*(J+1-2) by EUCLID:56 .= S+(NS/(2|^n))*(J-1) by Lm1; then A14: p`2 < G*(I,J+1)`2 by Th28; A15: p in cell(G,I,J) by A5,A6,A7,A11,A12,A13,JORDAN9:19; reconsider GIJ = G*(I,J) as Point of Euclid 2 by Lm5; GIJ in cell (G,I,J) by A5,A6,JORDAN9:22; then A16: dist (G*(I,J),p) <= 2*(r/4) by A3,A4,A5,A6,A15,Th30; A17: 2*(r/4) < r by A2,Lm6; then dist (G*(I,J), p) < r by A16,AXIOMS:22; then dist (GIJ, P) < r by GOBRD14:def 1; then A18: GIJ in Ball (P, r) by METRIC_1:12; reconsider GIJ = G*(I+1,J) as Point of Euclid 2 by Lm5; GIJ in cell (G,I,J) by A5,A6,JORDAN9:22; then dist (G*(I+1,J),p) <= 2*(r/4) by A3,A4,A5,A6,A15,Th30; then dist (G*(I+1,J), p) < r by A17,AXIOMS:22; then dist (GIJ, P) < r by GOBRD14:def 1; then A19: GIJ in Ball (P, r) by METRIC_1:12; reconsider GIJ = G*(I+1,J+1) as Point of Euclid 2 by Lm7; GIJ in cell (G,I,J) by A5,A6,JORDAN9:22; then dist (G*(I+1,J+1),p) <= 2*(r/4) by A3,A4,A5,A6,A15,Th30; then dist (G*(I+1,J+1), p) < r by A17,AXIOMS:22; then A20: dist (GIJ, P) < r by GOBRD14:def 1; reconsider GIJ = G*(I,J+1) as Point of Euclid 2 by Lm7; GIJ in cell (G,I,J) by A5,A6,JORDAN9:22; then dist (G*(I,J+1),p) <= 2*(r/4) by A3,A4,A5,A6,A15,Th30; then dist (G*(I,J+1), p) < r by A17,AXIOMS:22; then dist (GIJ, P) < r by GOBRD14:def 1; then A21: G*(I,J+1) in Ball(P,r) & G*(I+1,J+1) in Ball(P,r) by A20,METRIC_1:12; per cases; suppose A22: p`1 <> G*(I,J)`1; take I, J; thus 1 < I & I < len G & 1 < J & J < width G by A5,A6,A9,AXIOMS:22; cell(G,I,J) c= Ball(P,r) by A2,A3,A4,A5,A6,A15,A18,A19,A21,Lm8; hence thesis by A2,A5,A6,A7,A11,A12,A13,A22,JORDAN9:19,XBOOLE_1:1; suppose A23: p`1 = G*(I,J)`1; set q = G*(I-'1,J); reconsider GIJ = G*(I-'1,J) as Point of Euclid 2 by Lm5; A24: I-'1+1 = I by A5,AMI_5:4; A25: 1 <= I-'1 & I-'1+1 <= len G & 1 <= J & J + 1 <= width G by A1,A5,A10,Th24,AMI_5:4,JORDAN3:12; then A26: GIJ in cell (G,I-'1,J) by JORDAN9:22; A27: I-'1 < I by A25,JORDAN3:14; then A28: p`1 <= G*(I-'1+1,J)`1 & p`1 > G*(I-'1,J)`1 by A5,A10,A23,A25,AMI_5:4,GOBOARD5:4; A29: G*(I-'1,J)`2 = G*(I,J)`2 by A24,A25,JORDAN9:18; A30: G*(I-'1,J+1)`2 = G*(I,J+1)`2 by A24,A25,JORDAN9:18; then A31: p in cell (G,I-'1,J) by A11,A14,A25,A28,A29,JORDAN9:19; then dist (p,q) < 2*l by A3,A4,A25,A26,Th30; then dist (q, p) < r by A17,AXIOMS:22; then dist (GIJ, P) < r by GOBRD14:def 1; then A32: GIJ in Ball (P, r) by METRIC_1:12; take I-'1, J; len G = width G by JORDAN8:def 1; then A33: 1 <= J & J <= len G by A6,A9,AXIOMS:22; I-'1 <> 1 proof assume I-'1 = 1; then 1 = I - 1 by JORDAN3:1; then I = 1 + 1 by XCMPLX_1:27; then G*(I,J)`1 = W-bound C by A33,JORDAN8:14; then A34: p`1 <= W-bound BDD C by A1,A23,Th18; p`1 <> W-bound BDD C by A1,Th34; then A35: p`1 < W-bound BDD C by A34,REAL_1:def 5; C is Bounded by JORDAN2C:73; then BDD C is Bounded by JORDAN2C:114; hence thesis by A1,A35,Th6; end; hence 1 < I-'1 & I-'1 < len G & 1 < J & J < width G by A1,A9,A10,A25,A27,Th24,AXIOMS:22,REAL_1:def 5; set q = G*(I-'1,J+1); reconsider GIJ = G*(I-'1,J+1) as Point of Euclid 2 by Lm5; A36: 1 <= I-'1 & I-'1+1 <= len G & 1 <= J & J + 1 <= width G by A1,A5,A10,Th24,AMI_5:4,JORDAN3:12; then A37: GIJ in cell (G,I-'1,J) by JORDAN9:22; I-'1 < I by A36,JORDAN3:14; then A38: p`1 <= G*(I-'1+1,J)`1 & p`1 > G*(I-'1,J)`1 by A5,A10,A23,A36,AMI_5:4,GOBOARD5:4; then p in cell (G,I-'1,J) by A11,A14,A29,A30,A36,JORDAN9:19; then dist (q, p) < 2*(r/4) by A3,A4,A36,A37,Th30; then dist (q, p) < r by A17,AXIOMS:22; then dist (GIJ, P) < r by GOBRD14:def 1; then G*(I-'1+1,J) in Ball(P,r) & G*(I-'1,J+1) in Ball(P,r) & G*(I-'1+1,J+1) in Ball(P,r) by A5,A18,A21,AMI_5:4,METRIC_1:12; then cell(G,I-'1,J) c= Ball(P,r) by A2,A3,A4,A31,A32,A36,Lm8; hence thesis by A2,A11,A14,A29,A30,A36,A38,JORDAN9:19,XBOOLE_1:1; end; theorem for C being Subset of TOP-REAL 2 st C is Bounded holds UBD C is non empty by Lm15;