### The Mizar article:

### Gauges and Cages. Part II

**by****Artur Kornilowicz, and****Robert Milewski**

- Received November 6, 2000
Copyright (c) 2000 Association of Mizar Users

- MML identifier: JORDAN1D
- [ MML identifier index ]

environ vocabulary ARYTM, EUCLID, COMPTS_1, RELAT_2, SPPOL_1, NAT_1, BOOLE, TARSKI, MATRIX_2, INT_1, GROUP_1, ARYTM_3, ARYTM_1, FINSEQ_1, JORDAN8, MCART_1, PSCOMP_1, TREES_1, MATRIX_1, GOBOARD5, SETFAM_1, JORDAN9, PRE_TOPC, JORDAN1A, TOPREAL1, GOBOARD1, FINSEQ_4, RELAT_1, FUNCT_1; notation TARSKI, XBOOLE_0, ENUMSET1, SETFAM_1, ORDINAL1, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1, FUNCT_1, STRUCT_0, FINSEQ_1, FINSEQ_4, BINARITH, NEWTON, PRE_TOPC, COMPTS_1, CONNSP_1, MATRIX_1, EUCLID, WSIERP_1, GOBOARD1, TOPREAL1, GOBOARD5, PSCOMP_1, SPPOL_1, ABIAN, GOBRD13, JORDAN8, JORDAN9, JORDAN1A; constructors JORDAN8, REAL_1, CARD_4, PSCOMP_1, BINARITH, CONNSP_1, JORDAN9, JORDAN1A, WSIERP_1, ABSVALUE, FINSEQ_4, GOBRD13, TOPREAL2, ENUMSET1, ABIAN, REALSET1, INT_1; clusters XREAL_0, TOPREAL6, JORDAN8, INT_1, NEWTON, RELSET_1, EUCLID, JORDAN1A, ABIAN, BINARITH, GRAPH_3, NAT_1, SPRECT_1, STRUCT_0, MEMBERED; requirements NUMERALS, SUBSET, REAL, BOOLE, ARITHM; definitions TARSKI, SETFAM_1, XBOOLE_0; theorems AXIOMS, BINARITH, EUCLID, GOBRD11, JORDAN8, PSCOMP_1, JORDAN1A, NAT_1, REAL_1, GOBOARD5, FINSEQ_2, SPRECT_2, FINSEQ_4, SPPOL_2, TOPREAL1, SPRECT_3, AMI_5, JORDAN3, GOBRD13, JORDAN9, GOBOARD1, TARSKI, JORDAN10, ENUMSET1, NEWTON, NAT_2, INT_1, REAL_2, SCMFSA9A, INT_3, WSIERP_1, HEINE, GOBOARD7, GOBOARD9, POLYNOM4, SPRECT_1, SQUARE_1, SPPOL_1, ZFMISC_1, ABIAN, GROUP_4, SETFAM_1, XBOOLE_0, XBOOLE_1, CQC_THE1, XCMPLX_0, XCMPLX_1; schemes NAT_1; begin :: Preliminaries reserve a, b, i, k, m, n for Nat, r, s for real number, D for non empty Subset of TOP-REAL 2, C for compact connected non vertical non horizontal Subset of TOP-REAL 2; 1 = 2 * 0 + 1; then Lm1: 1 div 2 = 0 by NAT_1:def 1; 2 = 2 * 1 + 0; then Lm2: 2 div 2 = 1 by NAT_1:def 1; Lm3: for x, A, B, C, D being set holds x in A \/ B \/ C \/ D iff x in A or x in B or x in C or x in D proof let x, A, B, C, D be set; hereby assume x in A \/ B \/ C \/ D; then x in A \/ B \/ C or x in D by XBOOLE_0:def 2; then x in A \/ B or x in C or x in D by XBOOLE_0:def 2; hence x in A or x in B or x in C or x in D by XBOOLE_0:def 2; end; assume x in A or x in B or x in C or x in D; then x in A \/ B or x in C or x in D by XBOOLE_0:def 2; then x in A \/ B \/ C or x in D by XBOOLE_0:def 2; hence thesis by XBOOLE_0:def 2; end; Lm4: for A, B, C, D being set holds union {A,B,C,D} = A \/ B \/ C \/ D proof let A, B, C, D be set; hereby let x be set; assume x in union {A,B,C,D}; then consider Z being set such that A1: x in Z and A2: Z in {A,B,C,D} by TARSKI:def 4; Z = A or Z = B or Z = C or Z = D by A2,ENUMSET1:18; hence x in A \/ B \/ C \/ D by A1,Lm3; end; let x be set; assume x in A \/ B \/ C \/ D; then A3: x in A or x in B or x in C or x in D by Lm3; A in {A,B,C,D} & B in {A,B,C,D} & C in {A,B,C,D} & D in {A,B,C,D} by ENUMSET1:19; hence thesis by A3,TARSKI:def 4; end; theorem Th1: for A, B being set st for x being set st x in A ex K being set st K c= B & x c= union K holds union A c= union B proof let A, B be set such that A1: for x being set st x in A ex K being set st K c= B & x c= union K; let a be set; assume a in union A; then consider Z being set such that A2: a in Z and A3: Z in A by TARSKI:def 4; consider K being set such that A4: K c= B and A5: Z c= union K by A1,A3; ex S being set st a in S & S in K by A2,A5,TARSKI:def 4; hence a in union B by A4,TARSKI:def 4; end; definition let m be even Integer; cluster m + 2 -> even; coherence proof 2 = 2*1; then reconsider t = 2 as even Integer; m + t is even; hence thesis; end; end; definition let m be odd Integer; cluster m + 2 -> odd; coherence proof 2 = 2*1; then reconsider t = 2 as even Integer; m + t is odd; hence thesis; end; end; definition let m be non empty Nat; cluster 2|^m -> even; coherence proof defpred P[Nat] means $1 is non empty implies 2|^$1 is even; A1: P[0]; A2: P[k] implies P[k+1] proof assume P[k] & k+1 is non empty; 2|^(k+1) = 2*2|^k by NEWTON:11; hence thesis; end; P[k] from Ind(A1,A2); hence thesis; end; end; definition let n be even Nat, m be non empty Nat; cluster n|^m -> even; coherence proof defpred P[Nat] means $1 is non empty implies n|^$1 is even; A1: P[0]; A2: P[k] implies P[k+1] proof assume P[k] & k+1 is non empty; n|^(k+1) = n*n|^k by NEWTON:11; hence thesis; end; P[k] from Ind(A1,A2); hence thesis; end; end; theorem Th2: r <> 0 implies 1/r * r|^(i+1) = r|^i proof assume A1: r <> 0; thus 1/r * r|^(i+1) = 1/r * (r|^i * r) by NEWTON:11 .= 1/r * r * r|^i by XCMPLX_1:4 .= 1*r|^i by A1,XCMPLX_1:107 .= r|^i; end; theorem Th3: r/s is not Integer implies - [\ r/s /] = [\ (-r) / s /] + 1 proof assume A1: r/s is not Integer; r/s - 1 < [\ r/s /] by INT_1:def 4; then - (r/s - 1) > - [\ r/s /] by REAL_1:50; then -r/s + 1 > - [\ r/s /] by XCMPLX_1:162; then - [\ r/s /] <= (-r) / s + 1 by XCMPLX_1:188; then - [\ r/s /] - 1 <= (-r) / s + 1 - 1 by REAL_1:49; then A2: - [\ r/s /] - 1 <= (-r) / s + (1-1) by XCMPLX_1:29; [\ r/s /] < r/s by A1,INT_1:48; then -r/s < - [\ r/s /] by REAL_1:50; then -r/s - 1 < - [\ r/s /] - 1 by REAL_1:54; then (-r)/s - 1 < - [\ r/s /] - 1 by XCMPLX_1:188; then - [\ r/s /] - 1 + 1 = [\ (-r) / s /] + 1 by A2,INT_1:def 4; then - [\ r/s /] - (1 - 1) = [\ (-r) / s /] + 1 by XCMPLX_1:37; hence - [\ r/s /] = [\ (-r) / s /] + 1; end; theorem Th4: r/s is Integer implies - [\ r/s /] = [\ (-r) / s /] proof assume r/s is Integer; then A1: [\ r/s /] = r/s by INT_1:47; A2: -r/s = (-r)/s by XCMPLX_1:188; then (-r) / s - 1 < - [\ r/s /] - 0 by A1,REAL_1:92; hence - [\ r/s /] = [\ (-r) / s /] by A1,A2,INT_1:def 4; end; theorem n > 0 & k mod n <> 0 implies - (k div n) = (-k) div n + 1 proof assume A1: n > 0; assume k mod n <> 0; then k qua Integer mod n <> 0 by SCMFSA9A:5; then not n qua Integer divides k by A1,INT_3:11; then A2: k/n is not Integer by A1,WSIERP_1:22; thus - (k div n) = - (k qua Integer div n) by SCMFSA9A:5 .= - [\ k / n /] by INT_1:def 7 .= [\ (-k) / n /] + 1 by A2,Th3 .= (-k) div n + 1 by INT_1:def 7; end; theorem n > 0 & k mod n = 0 implies - (k div n) = (-k) div n proof assume A1: n > 0; assume k mod n = 0; then k qua Integer mod n = 0 by SCMFSA9A:5; then n qua Integer divides k by A1,INT_3:11; then A2: k/n is Integer by A1,WSIERP_1:22; thus - (k div n) = - (k qua Integer div n) by SCMFSA9A:5 .= - [\ k / n /] by INT_1:def 7 .= [\ (-k) / n /] by A2,Th4 .= (-k) div n by INT_1:def 7; end; Lm5: now let m be real number; assume 2 <= m; then 2*m >= 2*2 by REAL_2:197; then 2*m - 2 >= 4 - 2 by REAL_1:49; hence 2*m - 2 >= 0 by AXIOMS:22; end; Lm6: now let m be real number; assume 1 <= m; then 2*m >= 2*1 by REAL_2:197; then 2*m - 1 >= 2 - 1 by REAL_1:49; hence 2*m - 1 >= 0 by AXIOMS:22; end; Lm7: now let m; assume 2 <= m; then 2*m - 2 >= 0 by Lm5; hence 2*m-2 = 2*m-'2 by BINARITH:def 3; end; Lm8: now let m; assume 1 <= m; then 2*m - 1 >= 0 by Lm6; hence 2*m-1 = 2*m-'1 by BINARITH:def 3; end; Lm9: now let m; assume A1: m >= 1; then 2*m >= 2*1 by AXIOMS:25; then 2*m-1 >= 2-1 by REAL_1:49; then A2: 2*m-'1 >= 1 by A1,Lm8; thus 2*m-'2+1 = 2*m-'1-'1+1 by JORDAN3:8 .= 2*m-'1 by A2,AMI_5:4; end; Lm10: for x being real number st 2 <= m holds x/(2|^i)*(m-2) = x/(2|^(i+1))*(2*m-'2-2) proof let x be real number; assume 2 <= m; then A1: 2*m - 2 >= 0 by Lm5; thus x/(2|^i)*(m-2) = x/((2|^i)/(m-2)) by XCMPLX_1:82 .= x/((2|^i)*2/((m-2)*2)) by XCMPLX_1:92 .= x/((2|^i)*2)*((m-2)*2) by XCMPLX_1:82 .= x/(2|^(i+1))*((m-2)*2) by NEWTON:11 .= x/(2|^(i+1))*(2*m-2*2) by XCMPLX_1:40 .= x/(2|^(i+1))*(2*m-(2+2)) .= x/(2|^(i+1))*(2*m-2-2) by XCMPLX_1:36 .= x/(2|^(i+1))*(2*m-'2-2) by A1,BINARITH:def 3; end; Lm11: 2 <= m implies 1 <= 2*m-'2 proof assume A1: 2 <= m; then 2*2 <= 2*m by REAL_2:197; then A2: 4-2 <= 2*m-2 by REAL_1:49; 2*m-'2 = 2*m-2 by A1,Lm7; hence thesis by A2,AXIOMS:22; end; Lm12: 1 <= m implies 1 <= 2*m-'1 proof assume A1: 1 <= m; then 2*1 <= 2*m by REAL_2:197; then 2-1 <= 2*m-1 by REAL_1:49; hence thesis by A1,Lm8; end; Lm13: m < 2|^i+3 implies 2*m-'2 < 2|^(i+1) + 3 proof per cases by CQC_THE1:3; suppose A1: m = 0 or m = 1; A2: 2*0-'2 = 0 by POLYNOM4:1; A3: 2*1-'2 = 0 by GOBOARD9:1; 0+0 < 2|^(i+1) by HEINE:5; hence thesis by A1,A2,A3,REAL_1:67; suppose A4: 2 <= m; assume m < 2|^i+3; then m+1 <= 2|^i + 3 by NAT_1:38; then 2*(m+1) <= 2*(2|^i + 3) by AXIOMS:25; then 2*m+2*1 <= 2*(2|^i + 3) by XCMPLX_1:8; then 2*m+2*1 <= 2*(2|^i) + 2*3 by XCMPLX_1:8; then 2*m+2*1 <= 2|^(i+1) + 6 by NEWTON:11; then 2*m+2-4 <= 2|^(i+1) + 6 - 4 by REAL_1:49; then 2*m+(2-4) <= 2|^(i+1) + 6 - 4 by XCMPLX_1:29; then 2*m+-2 <= 2|^(i+1) + (6 - 4) by XCMPLX_1:29; then 2*m-2 <= 2|^(i+1) + 2 by XCMPLX_0:def 8; then A5: 2*m-'2 <= 2|^(i+1) + 2 by A4,Lm7; 2|^(i+1) + 2 < 2|^(i+1) + 3 by REAL_1:53; hence 2*m-'2 < 2|^(i+1) + 3 by A5,AXIOMS:22; end; Lm14: now let m; assume 2 <= m; hence 2*m-'2+1-2 = 2*m-2+1-2 by Lm7 .= 2*m+-2+1-2 by XCMPLX_0:def 8 .= 2*m+(-2+1)-2 by XCMPLX_1:1 .= 2*m+(-1-2) by XCMPLX_1:29 .= 2*m+-3 .= 2*m-3 by XCMPLX_0:def 8; end; begin :: Gauges and Cages theorem Th7: 2 <= m & m < len Gauge(D,i) & 1 <= a & a <= len Gauge(D,i) & 1 <= b & b <= len Gauge(D,i+1) implies Gauge(D,i)*(m,a)`1 = Gauge(D,i+1)*(2*m-'2,b)`1 proof set I = Gauge(D,i), J = Gauge(D,i+1), z = N-bound D, e = E-bound D, s = S-bound D, w = W-bound D; assume that A1: 2 <= m and A2: m < len I and A3: 1 <= a & a <= len I and A4: 1 <= b & b <= len J; A5: len I = width I by JORDAN8:def 1; A6: len J = width J by JORDAN8:def 1; A7: 1 <= 2*m-'2 by A1,Lm11; m < 2|^i + 3 by A2,JORDAN8:def 1; then 2*m-'2 <= 2|^(i+1) + 3 by Lm13; then 2*m-'2 <= len J by JORDAN8:def 1; then A8: [2*m-'2,b] in Indices J by A4,A6,A7,GOBOARD7:10; 1 <= m by A1,AXIOMS:22; then [m,a] in Indices I by A2,A3,A5,GOBOARD7:10; hence I*(m,a)`1 = |[w+((e-w)/(2|^i))*(m-2),s+((z-s)/(2|^i))*(a-2)]|`1 by JORDAN8:def 1 .= w+((e-w)/(2|^i))*(m-2) by EUCLID:56 .= w+((e-w)/(2|^(i+1)))*(2*m-'2-2) by A1,Lm10 .= |[w+((e-w)/(2|^(i+1)))*(2*m-'2-2),s+((z-s)/(2|^(i+1)))*(b-2)]|`1 by EUCLID:56 .= J*(2*m-'2,b)`1 by A8,JORDAN8:def 1; end; theorem Th8: 2 <= n & n < len Gauge(D,i) & 1 <= a & a <= len Gauge(D,i) & 1 <= b & b <= len Gauge(D,i+1) implies Gauge(D,i)*(a,n)`2 = Gauge(D,i+1)*(b,2*n-'2)`2 proof set I = Gauge(D,i), J = Gauge(D,i+1), z = N-bound D, e = E-bound D, s = S-bound D, w = W-bound D; assume that A1: 2 <= n and A2: n < len I and A3: 1 <= a & a <= len I and A4: 1 <= b & b <= len J; A5: len I = width I by JORDAN8:def 1; A6: len J = width J by JORDAN8:def 1; A7: 1 <= 2*n-'2 by A1,Lm11; n < 2|^i + 3 by A2,JORDAN8:def 1; then 2*n-'2 <= 2|^(i+1) + 3 by Lm13; then 2*n-'2 <= len J by JORDAN8:def 1; then A8: [b,2*n-'2] in Indices J by A4,A6,A7,GOBOARD7:10; 1 <= n by A1,AXIOMS:22; then [a,n] in Indices I by A2,A3,A5,GOBOARD7:10; hence I*(a,n)`2 = |[w+((e-w)/(2|^i))*(a-2),s+((z-s)/(2|^i))*(n-2)]|`2 by JORDAN8:def 1 .= s+((z-s)/(2|^i))*(n-2) by EUCLID:56 .= s+((z-s)/(2|^(i+1)))*(2*n-'2-2) by A1,Lm10 .= |[w+((e-w)/(2|^(i+1)))*(b-2),s+((z-s)/(2|^(i+1)))*(2*n-'2-2)]|`2 by EUCLID:56 .= J*(b,2*n-'2)`2 by A8,JORDAN8:def 1; end; Lm15: m+1 < len Gauge(D,i) implies 2*m-'1 < len Gauge(D,i+1) proof assume m+1 < len Gauge(D,i); then m+1 < 2|^i + 3 by JORDAN8:def 1; then 2*(m+1)-'2 < 2|^(i+1) + 3 by Lm13; then 2*m+2*1-'2 < 2|^(i+1) + 3 by XCMPLX_1:8; then A1: 2*m < 2|^(i+1) + 3 by BINARITH:39; 2*m-'1 <= 2*m by JORDAN3:7; then 2*m-'1 < 2|^(i+1) + 3 by A1,AXIOMS:22; hence thesis by JORDAN8:def 1; end; theorem Th9: for D being compact non vertical non horizontal Subset of TOP-REAL 2 holds 2 <= m & m+1 < len Gauge(D,i) & 2 <= n & n+1 < len Gauge(D,i) implies cell(Gauge(D,i),m,n) = cell(Gauge(D,i+1),2*m-'2,2*n-'2) \/ cell(Gauge(D,i+1),2*m-'1,2*n-'2) \/ cell(Gauge(D,i+1),2*m-'2,2*n-'1) \/ cell(Gauge(D,i+1),2*m-'1,2*n-'1) proof let D be compact non vertical non horizontal Subset of TOP-REAL 2; set I = Gauge(D,i), J = Gauge(D,i+1), z = N-bound D, e = E-bound D, s = S-bound D, w = W-bound D; assume that A1: 2 <= m and A2: m+1 < len I and A3: 2 <= n and A4: n+1 < len I; A5: len I = width I by JORDAN8:def 1; A6: len J = width J by JORDAN8:def 1; A7: 1 <= m & 1 <= n by A1,A3,AXIOMS:22; then A8: 2*m-'2+1 = 2*m-'1 by Lm9; A9: 2*n-'2+1 = 2*n-'1 by A7,Lm9; A10: 2*m-'2 = 2*m-2 by A1,Lm7; A11: 2*m-'1 = 2*m-1 by A7,Lm8; A12: 2*n-'2 = 2*n-2 by A3,Lm7; A13: 2*n-'1 = 2*n-1 by A7,Lm8; A14: 2*m-'2+1-2 = 2*m-3 by A1,Lm14; A15: 2*n-'2+1-2 = 2*n-3 by A3,Lm14; A16: 2*(m+1)-'2-2 = 2*m+2*1-'2-2 by XCMPLX_1:8 .= 2*m-2 by BINARITH:39; A17: 2*(n+1)-'2-2 = 2*n+2*1-'2-2 by XCMPLX_1:8 .= 2*n-2 by BINARITH:39; A18: 1 <= 2*n-'1 by A7,Lm12; A19: 1 <= 2*m-'1 by A7,Lm12; then A20: 2*m-'1+1 = 2*m by JORDAN3:6; A21: 2*n-'1+1 = 2*n by A18,JORDAN3:6; A22: m < len I & n < width I by A2,A4,A5,NAT_1:38; then A23: cell(I,m,n) = { |[r,q]| where r, q is Real: I*(m,1)`1 <= r & r <= I*(m+1,1)`1 & I*(1,n)`2 <= q & q <= I*(1,n+1)`2 } by A7,GOBRD11:32; A24: 1 < len I by A7,A22,AXIOMS:22; A25: 1 <= len J by GOBRD11:34; A26: 1 <= 2*m-'2 & 1 <= 2*n-'2 by A1,A3,Lm11; A27: 2*m-'2 < 2*m-'1 & 2*n-'2 < 2*n-'1 by A10,A11,A12,A13,REAL_1:92; m < 2|^i + 3 by A22,JORDAN8:def 1; then 2*m-'2 < 2|^(i+1) + 3 by Lm13; then A28: 2*m-'2 < len J by JORDAN8:def 1; n < 2|^i + 3 by A5,A22,JORDAN8:def 1; then 2*n-'2 < 2|^(i+1) + 3 by Lm13; then A29: 2*n-'2 < width J by A6,JORDAN8:def 1; then A30: cell(J,2*m-'2,2*n-'2) = { |[r,q]| where r, q is Real: J*(2*m-'2,1)`1 <= r & r <= J* (2*m-'2+1,1)`1 & J*(1,2*n-'2)`2 <= q & q <= J*(1,2*n-'2+1)`2 } by A26,A28,GOBRD11:32; A31: 2*n-'1 < len J by A4,Lm15; A32: 2*m-'1 < len J by A2,Lm15; then A33: cell(J,2*m-'1,2*n-'2) = { |[r,q]| where r, q is Real: J*(2*m-'1,1)`1 <= r & r <= J* (2*m-'1+1,1)`1 & J*(1,2*n-'2)`2 <= q & q <= J* (1,2*n-'2+1)`2 } by A19,A26,A29,GOBRD11:32; A34: cell(J,2*m-'2,2*n-'1) = { |[r,q]| where r, q is Real: J*(2*m-'2,1)`1 <= r & r <= J* (2*m-'2+1,1)`1 & J*(1,2*n-'1)`2 <= q & q <= J*(1,2*n-'1+1)`2 } by A6,A18,A26,A28,A31,GOBRD11:32; A35: cell(J,2*m-'1,2*n-'1) = { |[r,q]| where r, q is Real: J*(2*m-'1,1)`1 <= r & r <= J* (2*m-'1+1,1)`1 & J*(1,2*n-'1)`2 <= q & q <= J*(1,2*n-'1+1)`2 } by A6,A18,A19,A31,A32,GOBRD11:32; A36: 2|^(i+1) > 0 by HEINE:5; e >= w by SPRECT_1:23; then e-w >= 0 by SQUARE_1:12; then A37: (e-w)/(2|^(i+1)) >= 0 by A36,REAL_2:125; 2*m-3 < 2*m-2 by REAL_1:92; then (e-w)/(2|^(i+1))*(2*m-3) <= (e-w)/(2|^(i+1))*(2*m-2) by A37,AXIOMS:25 ; then A38: w+(e-w)/(2|^(i+1))*(2*m-3) <= w+(e-w)/(2|^(i+1))*(2*m-2) by AXIOMS:24 ; z >= s by SPRECT_1:24; then z-s >= 0 by SQUARE_1:12; then A39: (z-s)/(2|^(i+1)) >= 0 by A36,REAL_2:125; 2*n-3 < 2*n-2 by REAL_1:92; then (z-s)/(2|^(i+1))*(2*n-3) <= (z-s)/(2|^(i+1))*(2*n-2) by A39,AXIOMS:25 ; then A40: s+(z-s)/(2|^(i+1))*(2*n-3) <= s+(z-s)/(2|^(i+1))*(2*n-2) by AXIOMS:24 ; A41: 2*m-'2+1 <= len J by A28,NAT_1:38; 1 <= 2*m-'2+1 by NAT_1:29; then [2*m-'2+1,1] in Indices J by A6,A25,A41,GOBOARD7:10; then A42: J*(2*m-'2+1,1)`1 = |[w+(e-w)/(2|^(i+1))*(2*m-'2+1-2),s+(z-s)/(2|^(i+1))*(1-2)]|`1 by JORDAN8:def 1 .= w+(e-w)/(2|^(i+1))*(2*m-'2+1-2) by EUCLID:56; 2*m-'1 <= 2*m by GOBOARD9:2; then A43: 1 <= 2*m by A19,AXIOMS:22; 2*n-'1 <= 2*n by GOBOARD9:2; then A44: 1 <= 2*n by A18,AXIOMS:22; 2*m-'1+1 <= len J by A32,NAT_1:38; then [2*m,1] in Indices J by A6,A20,A25,A43,GOBOARD7:10; then A45: J*(2*m,1)`1 = |[w+(e-w)/(2|^(i+1))*(2*m-2),s+(z-s)/(2|^(i+1))*(1-2)]|`1 by JORDAN8:def 1 .= w+(e-w)/(2|^(i+1))*(2*m-2) by EUCLID:56; 2*n-'1+1 <= len J by A31,NAT_1:38; then [1,2*n] in Indices J by A6,A21,A25,A44,GOBOARD7:10; then A46: J*(1,2*n)`2 = |[w+(e-w)/(2|^(i+1))*(1-2),s+(z-s)/(2|^(i+1))*(2*n-2)]|`2 by JORDAN8:def 1 .= s+(z-s)/(2|^(i+1))*(2*n-2) by EUCLID:56; A47: 2*n-'2+1 <= len J by A6,A29,NAT_1:38; 1 <= 2*n-'2+1 by NAT_1:29; then [1,2*n-'2+1] in Indices J by A6,A25,A47,GOBOARD7:10; then A48: J*(1,2*n-'2+1)`2 = |[w+(e-w)/(2|^(i+1))*(1-2),s+(z-s)/(2|^(i+1))*(2*n-'2+1-2)]|`2 by JORDAN8:def 1 .= s+(z-s)/(2|^(i+1))*(2*n-'2+1-2) by EUCLID:56; m <= m+1 by NAT_1:29; then A49: 2 <= m+1 by A1,AXIOMS:22; n <= n+1 by NAT_1:29; then A50: 2 <= n+1 by A3,AXIOMS:22; 1 <= m+1 by NAT_1:29; then [m+1,1] in Indices I by A2,A5,A24,GOBOARD7:10; then A51: I*(m+1,1)`1 = |[w+(e-w)/(2|^i)*(m+1-2),s+(z-s)/(2|^i)*(1-2)]|`1 by JORDAN8:def 1 .= w+(e-w)/(2|^i)*(m+1-2) by EUCLID:56 .= w+(e-w)/(2|^(i+1))*(2*(m+1)-'2-2) by A49,Lm10; 1 <= n+1 by NAT_1:29; then [1,n+1] in Indices I by A4,A5,A24,GOBOARD7:10; then A52: I*(1,n+1)`2 = |[w+(e-w)/(2|^i)*(1-2),s+(z-s)/(2|^i)*(n+1-2)]|`2 by JORDAN8:def 1 .= s+(z-s)/(2|^i)*(n+1-2) by EUCLID:56 .= s+(z-s)/(2|^(i+1))*(2*(n+1)-'2-2) by A50,Lm10; A53: I*(m,1)`1 = J*(2*m-'2,1)`1 by A1,A22,A24,A25,Th7; A54: I*(1,n)`2 = J*(1,2*n-'2)`2 by A3,A5,A22,A24,A25,Th8; A55: J*(2*m-'2,1)`1 < J*(2*m-'1,1)`1 by A6,A25,A26,A27,A32,GOBOARD5:4; A56: J*(1,2*n-'2)`2 < J*(1,2*n-'1)`2 by A6,A25,A26,A27,A31,GOBOARD5:5; thus cell(Gauge(D,i),m,n) c= cell(Gauge(D,i+1),2*m-'2,2*n-'2) \/ cell(Gauge(D,i+1),2*m-'1,2*n-'2) \/ cell(Gauge(D,i+1),2*m-'2,2*n-'1) \/ cell(Gauge(D,i+1),2*m-'1,2*n-'1) proof let x be set; assume x in cell(I,m,n); then consider r, q being Real such that A57: x = |[r,q]| and A58: I*(m,1)`1 <= r & r <= I*(m+1,1)`1 & I*(1,n)`2 <= q & q <= I* (1,n+1)`2 by A23; r <= J*(2*m-'1,1)`1 & q <= J*(1,2*n-'1)`2 or r >= J*(2*m-'1,1)`1 & q <= J*(1,2*n-'1)`2 or r <= J*(2*m-'1,1)`1 & q >= J*(1,2*n-'1)`2 or r >= J*(2*m-'1,1)`1 & q >= J*(1,2*n-'1)`2; then |[r,q]| in cell(J,2*m-'2,2*n-'2) or |[r,q]| in cell(J,2*m-'1,2*n-'2) or |[r,q]| in cell(J,2*m-'2,2*n-'1) or |[r,q]| in cell(J,2*m-'1,2*n-'1) by A8,A9,A16,A17,A20,A21,A30,A33,A34,A35,A45,A46,A51,A52,A53,A54,A58; hence thesis by A57,Lm3; end; let x be set; assume A59: x in cell(Gauge(D,i+1),2*m-'2,2*n-'2) \/ cell(Gauge(D,i+1),2*m-'1,2*n-'2) \/ cell(Gauge(D,i+1),2*m-'2,2*n-'1) \/ cell(Gauge(D,i+1),2*m-'1,2*n-'1); per cases by A59,Lm3; suppose x in cell(Gauge(D,i+1),2*m-'2,2*n-'2); then consider r, q being Real such that A60: x = |[r,q]| and A61: J*(2*m-'2,1)`1 <= r and A62: r <= J*(2*m-'2+1,1)`1 and A63: J*(1,2*n-'2)`2 <= q and A64: q <= J*(1,2*n-'2+1)`2 by A30; A65: r <= I*(m+1,1)`1 by A14,A16,A38,A42,A51,A62,AXIOMS:22; q <= I*(1,n+1)`2 by A15,A17,A40,A48,A52,A64,AXIOMS:22; hence thesis by A23,A53,A54,A60,A61,A63,A65; suppose x in cell(Gauge(D,i+1),2*m-'1,2*n-'2); then consider r, q being Real such that A66: x = |[r,q]| and A67: J*(2*m-'1,1)`1 <= r and A68: r <= J*(2*m-'1+1,1)`1 and A69: J*(1,2*n-'2)`2 <= q and A70: q <= J*(1,2*n-'2+1)`2 by A33; A71: I*(m,1)`1 <= r by A53,A55,A67,AXIOMS:22; q <= I*(1,n+1)`2 by A15,A17,A40,A48,A52,A70,AXIOMS:22; hence thesis by A16,A20,A23,A45,A51,A54,A66,A68,A69,A71; suppose x in cell(Gauge(D,i+1),2*m-'2,2*n-'1); then consider r, q being Real such that A72: x = |[r,q]| and A73: J*(2*m-'2,1)`1 <= r and A74: r <= J*(2*m-'2+1,1)`1 and A75: J*(1,2*n-'1)`2 <= q and A76: q <= J*(1,2*n-'1+1)`2 by A34; A77: r <= I*(m+1,1)`1 by A14,A16,A38,A42,A51,A74,AXIOMS:22; I*(1,n)`2 <= q by A54,A56,A75,AXIOMS:22; hence thesis by A17,A21,A23,A46,A52,A53,A72,A73,A76,A77; suppose x in cell(Gauge(D,i+1),2*m-'1,2*n-'1); then consider r, q being Real such that A78: x = |[r,q]| and A79: J*(2*m-'1,1)`1 <= r and A80: r <= J*(2*m-'1+1,1)`1 and A81: J*(1,2*n-'1)`2 <= q and A82: q <= J*(1,2*n-'1+1)`2 by A35; A83: I*(m,1)`1 <= r by A53,A55,A79,AXIOMS:22; I*(1,n)`2 <= q by A54,A56,A81,AXIOMS:22; hence thesis by A16,A17,A20,A21,A23,A45,A46,A51,A52,A78,A80,A82,A83; end; theorem for D being compact non vertical non horizontal Subset of TOP-REAL 2, k being Nat holds 2 <= m & m+1 < len Gauge(D,i) & 2 <= n & n+1 < len Gauge(D,i) implies cell(Gauge(D,i),m,n) = union { cell(Gauge(D,i+k),a,b) where a, b is Nat: 2|^k*m - 2|^(k+1) + 2 <= a & a <= 2|^k*m - 2|^k + 1 & 2|^k*n - 2|^(k+1) + 2 <= b & b <= 2|^k*n - 2|^k + 1 } proof let D be compact non vertical non horizontal Subset of TOP-REAL 2; let k be Nat; assume that A1: 2 <= m and A2: m+1 < len Gauge(D,i) and A3: 2 <= n and A4: n+1 < len Gauge(D,i); deffunc F(Nat) = { cell(Gauge(D,i+$1),a,b) where a, b is Nat: 2|^$1*m - 2|^($1+1) + 2 <= a & a <= 2|^$1*m - 2|^$1 + 1 & 2|^$1*n - 2|^($1+1) + 2 <= b & b <= 2|^$1*n - 2|^$1 + 1 }; defpred P[Nat] means cell(Gauge(D,i),m,n) = union F($1); A5: P[0] proof A6: now let m; A7: 2|^0 * m = 1*m by NEWTON:9; hence 2|^0 * m - 2|^(0+1) + 2 = m - 2 + 2 by NEWTON:10 .= m by XCMPLX_1:27; thus 2|^0 * m - 2|^0 + 1 = m - 1 + 1 by A7,NEWTON:9 .= m by XCMPLX_1:27; end; F(0) = { cell(Gauge(D,i),m,n) } proof hereby let x be set; assume x in F(0); then consider a, b such that A8: x = cell(Gauge(D,i+0),a,b) and A9: 2|^0 * m - 2|^(0+1) + 2 <= a & a <= 2|^0 * m - 2|^0 + 1 & 2|^0 * n - 2|^(0+1) + 2 <= b & b <= 2|^0 * n - 2|^0 + 1; now let a, m; assume A10: 2|^0 * m - 2|^(0+1) + 2 <= a & a <= 2|^0 * m - 2|^0 + 1; 2|^0 * m - 2|^(0+1) + 2= m & 2|^0 * m - 2|^0 + 1 = m by A6; hence a = m by A10,AXIOMS:21; end; then a = m & b = n by A9; hence x in { cell(Gauge(D,i),m,n) } by A8,TARSKI:def 1; end; let x be set; assume x in { cell(Gauge(D,i),m,n) }; then A11: x = cell(Gauge(D,i+0),m,n) by TARSKI:def 1; 2|^0 * m - 2|^(0+1) + 2 <= m & m <= 2|^0 * m - 2|^0 + 1 & 2|^0 * n - 2|^(0+1) + 2 <= n & n <= 2|^0 * n - 2|^0 + 1 by A6; hence x in F(0) by A11; end; hence thesis by ZFMISC_1:31; end; A12:now let m; thus m+1-2 = m+(1-2) by XCMPLX_1:29 .= m+-1 .= m-1 by XCMPLX_0:def 8; end; A13:for w being Nat st P[w] holds P[w+1] proof let w be Nat such that A14: P[w]; A15: i+w+1 = i+(w+1) by XCMPLX_1:1; A16: len Gauge(D,i+w) = 2|^(i+w) + 3 by JORDAN8:def 1; A17: len Gauge(D,i) = 2|^i + 3 by JORDAN8:def 1; A18: 2|^w > 0 by HEINE:5; A19: 2|^(w+1) > 0 by HEINE:5; for x being set st x in F(w) ex K being set st K c= F(w+1) & x c= union K proof let x be set; assume x in F(w); then consider a, b such that A20: x = cell(Gauge(D,i+w),a,b) and A21: 2|^w*m - 2|^(w+1) + 2 <= a and A22: a <= 2|^w*m - 2|^w + 1 and A23: 2|^w*n - 2|^(w+1) + 2 <= b and A24: b <= 2|^w*n - 2|^w + 1; now let m; assume 2 <= m; then 2|^w*m >= 2|^w*2 by A18,AXIOMS:25; then 2|^w*m >= 2|^(w+1) by NEWTON:11; then 0 <= 2|^w*m - 2|^(w+1) by SQUARE_1:12; hence 0 + 2 <= 2|^w*m - 2|^(w+1) + 2 by AXIOMS:24; end; then A25: 2 <= 2|^w*m - 2|^(w+1) + 2 & 2 <= 2|^w*n - 2|^(w+1) + 2 by A1,A3; then A26: 2 <= a by A21,AXIOMS:22; A27: 2 <= b by A23,A25,AXIOMS:22; A28: 1 <= a by A26,AXIOMS:22; then A29: 2*a-'1 = 2*a-1 by Lm8; A30: 1 <= b by A27,AXIOMS:22; then A31: 2*b-'1 = 2*b-1 by Lm8; A32: 2*a-'2 = 2*a-2 by A26,Lm7; A33: 2*b-'2 = 2*b-2 by A27,Lm7; A34: 2*a-'2 < 2*a-'1 by A29,A32,REAL_1:92; A35: 2*b-'2 < 2*b-'1 by A31,A33,REAL_1:92; take K = { cell(Gauge(D,i+w+1),2*a-'2,2*b-'2), cell(Gauge(D,i+w+1),2*a-'1,2*b-'2), cell(Gauge(D,i+w+1),2*a-'2,2*b-'1), cell(Gauge(D,i+w+1),2*a-'1,2*b-'1) }; hereby let q be set; assume A36: q in K; A37: now let a,m; assume A38: 2 <= a; assume a <= 2|^w*m - 2|^w + 1; then 2*a <= 2*(2|^w*m - 2|^w + 1) by AXIOMS:25; then 2*a <= 2*(2|^w*m) - 2*(2|^w) + 2*1 by XCMPLX_1:44; then 2*a <= 2*2|^w*m - 2*(2|^w) + 2 by XCMPLX_1:4; then 2*a <= 2|^(w+1)*m - 2*(2|^w) + 2 by NEWTON:11; then 2*a <= 2|^(w+1)*m - 2|^(w+1) + 2 by NEWTON:11; then 2*a-2 <= 2|^(w+1)*m - 2|^(w+1) + 2 - 2 by REAL_1:49; then 2*a-'2 <= 2|^(w+1)*m - 2|^(w+1) + 2 - 2 by A38,Lm7; then A39: 2*a-'2 <= 2|^(w+1)*m - 2|^(w+1) + (2 - 2) by XCMPLX_1:29; 2|^(w+1)*m - 2|^(w+1) + 0 < 2|^(w+1)*m - 2|^(w+1) + 1 by REAL_1:53; hence 2*a-'2 < 2|^(w+1)*m - 2|^(w+1) + 1 by A39,AXIOMS:22; end; A40: now let a,m; assume A41: 2 <= a; assume 2|^w*m - 2|^(w+1) + 2 <= a; then 2 * (2|^w*m - 2|^(w+1) + 2) <= 2 * a by AXIOMS:25; then 2*(2|^w*m) - 2 * (2|^(w+1)) + 2*2 <= 2 * a by XCMPLX_1:44; then 2*2|^w*m - 2 * (2|^(w+1)) + 4 <= 2 * a by XCMPLX_1:4; then 2|^(w+1)*m - 2 * (2|^(w+1)) + 4 <= 2 * a by NEWTON:11; then 2|^(w+1)*m - 2|^(w+1+1) + 4 <= 2 * a by NEWTON:11; then 2|^(w+1)*m - 2|^(w+1+1) + 4 - 2 <= 2 * a - 2 by REAL_1:49 ; then 2|^(w+1)*m - 2|^(w+1+1) + 4 - 2 <= 2 * a -' 2 by A41,Lm7; hence 2|^(w+1)*m - 2|^(w+1+1) + (4 - 2) <= 2 * a -' 2 by XCMPLX_1:29; end; then A42: 2|^(w+1)*m - 2|^(w+1+1) + 2 <= 2*a-'2 by A21,A26; A43: 2*a-'2 < 2|^(w+1)*m - 2|^(w+1) + 1 by A22,A26,A37; then 2*a-'2+1 <= 2|^(w+1)*m - 2|^(w+1) + 1 by INT_1:20; then A44: 2*a-'1 <= 2|^(w+1)*m - 2|^(w+1) + 1 by A28,Lm9; A45: 2|^(w+1)*n - 2|^(w+1+1) + 2 <= 2*b-'2 by A23,A27,A40; A46: 2*b-'2 < 2|^(w+1)*n - 2|^(w+1) + 1 by A24,A27,A37; then 2*b-'2+1 <= 2|^(w+1)*n - 2|^(w+1) + 1 by INT_1:20; then A47: 2*b-'1 <= 2|^(w+1)*n - 2|^(w+1) + 1 by A30,Lm9; A48: 2|^(w+1)*m - 2|^(w+1+1) + 2 <= 2*a-'1 by A34,A42,AXIOMS:22; A49: 2|^(w+1)*n - 2|^(w+1+1) + 2 <= 2*b-'1 by A35,A45,AXIOMS:22; q = cell(Gauge(D,i+(w+1)),2*a-'2,2*b-'2) or q = cell(Gauge(D,i+(w+1)),2*a-'1,2*b-'2) or q = cell(Gauge(D,i+(w+1)),2*a-'2,2*b-'1) or q = cell(Gauge(D,i+(w+1)),2*a-'1,2*b-'1) by A15,A36,ENUMSET1:18; hence q in F(w+1) by A42,A43,A44,A45,A46,A47,A48,A49; end; now let a, m; assume m+1 < len Gauge(D,i); then m+1-2 < 2|^i + 3 - 2 by A17,REAL_1:54 ; then m-1 < 2|^i + 3 - 2 by A12; then m-1 < 2|^i + (3 - 2) by XCMPLX_1:29; then m-1 <= 2|^i + 0 by INT_1:20; then 2|^w*(m-1) <= 2|^w*2|^i by A18,AXIOMS:25; then 2|^w*(m-1) <= 2|^(w+i) by NEWTON:13; then A50: 2|^w*(m-1)+3 <= 2|^(w+i)+3 by AXIOMS:24; assume a <= 2|^w*m - 2|^w + 1; then a+1 <= 2|^w*m - 2|^w + 1 + 1 by AXIOMS:24; then a+1 < 2|^w*m - 2|^w + 1 + 1 + 1 by SPPOL_1:5; then a+1 < 2|^w*m - 2|^w + 1 + (1 + 1) by XCMPLX_1:1; then a+1 < 2|^w*m - 1*2|^w + (1 + 2) by XCMPLX_1:1; then a+1 < 2|^w*(m - 1) + 3 by XCMPLX_1:40; hence a+1 < len Gauge(D,i+w) by A16,A50,AXIOMS:22; end; then a+1 < len Gauge(D,i+w) & b+1 < len Gauge(D,i+w) by A2,A4,A22,A24; then cell(Gauge(D,i+w),a,b) = cell(Gauge(D,i+w+1),2*a-'2,2*b-'2) \/ cell(Gauge(D,i+w+1),2*a-'1,2*b-'2) \/ cell(Gauge(D,i+w+1),2*a-'2,2*b-'1) \/ cell(Gauge(D,i+w+1),2*a-'1,2*b-'1) by A26,A27,Th9; hence x c= union K by A20,Lm4; end; hence cell(Gauge(D,i),m,n) c= union F(w+1) by A14,Th1; F(w+1) is_finer_than F(w) proof let X be set; assume X in F(w+1); then consider a, b being Nat such that A51: X = cell(Gauge(D,i+(w+1)),a,b) and A52: 2|^(w+1)*m - 2|^(w+1+1) + 2 <= a and A53: a <= 2|^(w+1)*m - 2|^(w+1) + 1 and A54: 2|^(w+1)*n - 2|^(w+1+1) + 2 <= b and A55: b <= 2|^(w+1)*n - 2|^(w+1) + 1; A56: now let a be even Nat; A57: ex e being Nat st a = 2*e by ABIAN:def 2; thus 2*(a div 2 + 1) = 2*(a div 2) + 2*1 by XCMPLX_1:8 .= a + 2 by A57,AMI_5:3; end; A58: now let a be odd Nat; consider e being Nat such that A59: a = 2*e+1 by SCMFSA9A:1; A60: 2*e mod 2 = 0 by GROUP_4:101; thus 2*(a div 2 + 1) = 2*(a div 2) + 2*1 by XCMPLX_1:8 .= 2*(2*e div 2 + (1 div 2)) + 2 by A59,A60,GROUP_4:106 .= 2*(e + 0) + (1+1) by Lm1,AMI_5:3 .= a + 1 by A59,XCMPLX_1:1; end; deffunc G(Nat,Nat)= cell(Gauge(D,i+w+1),2*(a div 2 + 1)-'$1,2*(b div 2 + 1)-'$2); A61: now let a, m; assume A62: 2 <= m; 2|^(w+1+1) = 2|^(w+1) * 2|^1 by NEWTON:13 .= 2|^(w+1) * 2 by NEWTON:10; then 2|^(w+1)*m >= 2|^(w+1+1) by A19,A62,AXIOMS:25; then 0 <= 2|^(w+1)*m - 2|^(w+1+1) by SQUARE_1:12; hence 0 + 2 <= 2|^(w+1)*m - 2|^(w+1+1) + 2 by AXIOMS:24; end; then 2 <= 2|^(w+1)*m - 2|^(w+1+1) + 2 by A1; then A63: 2 <= a by A52,AXIOMS:22; 2 <= 2|^(w+1)*n - 2|^(w+1+1) + 2 by A3,A61; then A64: 2 <= b by A54,AXIOMS:22; take Y = cell(Gauge(D,i+w),a div 2 + 1,b div 2 + 1); 2 div 2 <= a div 2 by A63,NAT_2:26; then A65: 1 + 1 <= a div 2 + 1 by Lm2,AXIOMS:24; A66: now let m; thus 2 * (2|^w*m - 2|^(w+1) + 2) = 2*(2|^w*m - 2|^(w+1)) + 2*2 by XCMPLX_1:8 .= 2*(2|^w*m) - 2*2|^(w+1) + (2+2) by XCMPLX_1:40 .= 2*2|^w*m - 2*2|^(w+1) + (2+2) by XCMPLX_1:4 .= 2*2|^w*m - 2|^(w+1+1) + (2+2) by NEWTON:11 .= 2|^(w+1)*m - 2|^(w+1+1) + (2+2) by NEWTON:11 .= 2|^(w+1)*m - 2|^(w+1+1) + 2+2 by XCMPLX_1:1; end; A67: now let m; thus 2 * (2|^w*m - 2|^w + 1) = 2*(2|^w*m - 2|^w) + 2*1 by XCMPLX_1:8 .= 2*(2|^w*m) - 2*2|^w + (1+1) by XCMPLX_1:40 .= 2*2|^w*m - 2*2|^w + (1+1) by XCMPLX_1:4 .= 2*2|^w*m - 2|^(w+1) + (1+1) by NEWTON:11 .= 2|^(w+1)*m - 2|^(w+1) + (1+1) by NEWTON:11 .= 2|^(w+1)*m - 2|^(w+1) + 1+1 by XCMPLX_1:1; end; A68: now let a, m; assume A69: m+1 < len Gauge(D,i); assume a <= 2|^(w+1)*m - 2|^(w+1) + 1; then a+3 <= 2|^(w+1)*m - 2|^(w+1) + 1 + 3 by AXIOMS:24; then A70: a+3+0 < 2|^(w+1)*m - 2|^(w+1) + 1 + 3 + 1 by REAL_1:67; then a+3+1 <= 2|^(w+1)*m - 2|^(w+1) + 1 + 3 + 1 by INT_1:20; then A71: a+(3+1) <= 2|^(w+1)*m - 2|^(w+1) + 1 + 3 + 1 by XCMPLX_1:1; m+1 < 2|^i + 3 by A69,JORDAN8:def 1; then 2*(m+1)-'2 < 2|^(i+1) + 3 by Lm13; then 2*m+2*1-'2 < 2|^(i+1) + 3 by XCMPLX_1:8; then 2*m < 2|^(i+1) + 3 by BINARITH:39; then 1/2*(2*m) < 1/2*(2|^(i+1) + 3) by REAL_1:70; then 1/2*2*m < 1/2*(2|^(i+1) + 3) by XCMPLX_1:4; then m < 1/2*2|^(i+1) + 1/2*3 by XCMPLX_1:8; then A72: m < 2|^i + 1/2*3 by Th2; 2|^i + 3/2 < 2|^i + 2 by REAL_1:53; then m < 2|^i + 2 by A72,AXIOMS:22; then m+1 <= 2|^i + 2 by NAT_1:38; then m+1-2 <= 2|^i + 2 - 2 by REAL_1:49; then m-1 <= 2|^i + 2 - 2 by A12; then m-1 <= 2|^i + (2 - 2) by XCMPLX_1:29; then 2|^(w+1)*(m-1) <= 2|^(w+1)*2|^i by A19,AXIOMS:25; then 2|^(w+1)*(m - 1) + 5 < 2|^(w+1) * 2|^i + 6 by REAL_1:67; then 2|^(w+1)*(m - 1) + 5 < 2|^(w+1+i) + 6 by NEWTON:13; then 2|^(w+1)*(m - 1) + (1 + 4) < 2*2|^(i+w) + 6 by A15,NEWTON:11; then 2|^(w+1)*(m - 1) + 1 + (3 + 1) < 2*2|^(i+w) + 6 by XCMPLX_1:1; then 2|^(w+1)*(m - 1) + 1 + 3 + 1 < 2*2|^(i+w) + 2*3 by XCMPLX_1:1; then 2|^(w+1)*(m - 1) + 1 + 3 + 1 < 2 * (2|^(i+w) + 3) by XCMPLX_1:8 ; then A73: 2|^(w+1)*m - 2|^(w+1)*1 + 1 + 3 + 1 < 2 * (2|^(i+w) + 3) by XCMPLX_1:40; now per cases; suppose A74: a is odd; 2 * (a div 2 + 1 + 1) = 2*(a div 2 + 1) + 2*1 by XCMPLX_1:8 .= a+1+2 by A58,A74 .= a+(1+2) by XCMPLX_1:1; hence 2 * (a div 2 + 1 + 1) < 2 * (2|^(i+w) + 3) by A70,A73,AXIOMS:22; suppose A75: a is even; 2 * (a div 2 + 1 + 1) = 2*(a div 2 + 1) + 2*1 by XCMPLX_1:8 .= a+2+2 by A56,A75 .= a+(2+2) by XCMPLX_1:1; hence 2 * (a div 2 + 1 + 1) < 2 * (2|^(i+w) + 3) by A71,A73,AXIOMS:22; end; hence a div 2 + 1 + 1 < len Gauge(D,i+w) by A16,AXIOMS:25; end; then A76: a div 2 + 1+1 < len Gauge(D,i+w) by A2,A53; 2 div 2 <= b div 2 by A64,NAT_2:26; then A77: 1 + 1 <= b div 2 + 1 by Lm2,AXIOMS:24; b div 2 + 1+1 < len Gauge(D,i+w) by A4,A55,A68; then A78: Y = G(2,2) \/ G(1,2) \/ G(2,1) \/ G(1,1) by A65,A76,A77,Th9; A79: now let m; let a be even Nat; assume A80: 2|^(w+1)*m - 2|^(w+1+1) + 2 <= a; A81: 2 * (2|^w*m - 2|^(w+1) + 2) = 2|^(w+1)*m - 2|^(w+1+1) + 2+2 by A66; 2|^(w+1)*m - 2|^(w+1+1) + 2 + 2 <= a + 2 by A80,AXIOMS:24; then 2 * (2|^w*m - 2|^(w+1) + 2) <= 2 * (a div 2 + 1) by A56,A81; hence 2|^w*m - 2|^(w+1) + 2 <= a div 2 + 1 by REAL_1:70; end; A82: now let m; let a be even Nat; assume a <= 2|^(w+1)*m - 2|^(w+1) + 1; then A83: a < 2|^(w+1)*m - 2|^(w+1) + 1 by REAL_1:def 5; A84: 2 * (2|^w*m - 2|^w + 1) = 2|^(w+1)*m - 2|^(w+1) + 1+1 by A67; a + 1 <= 2|^(w+1)*m - 2|^(w+1) + 1 by A83,INT_1:20; then a + 1 + 1 <= 2|^(w+1)*m - 2|^(w+1) + 1 + 1 by AXIOMS:24; then a + (1+1) <= 2|^(w+1)*m - 2|^(w+1) + 1 + 1 by XCMPLX_1:1; then 2*(a div 2 + 1) <= 2*(2|^w*m - 2|^w + 1) by A56,A84; hence a div 2 + 1 <= 2|^w*m - 2|^w + 1 by REAL_1:70; end; A85: now let m; let a be odd Nat; assume 2|^(w+1)*m - 2|^(w+1+1) + 2 <= a; then A86: 2|^(w+1)*m - 2|^(w+1+1) + 2 < a by REAL_1:def 5; A87: 2 * (2|^w*m - 2|^(w+1) + 2) = 2|^(w+1)*m - 2|^(w+1+1) + 2+2 by A66; 2|^(w+1)*m - 2|^(w+1+1) + 2 + 1 < a + 1 by A86,REAL_1:53; then 2|^(w+1)*m - 2|^(w+1+1) + 2 + 1 + 1 <= a + 1 by INT_1:20; then 2|^(w+1)*m - 2|^(w+1+1) + 2 + (1 + 1) <= a + 1 by XCMPLX_1:1; then 2 * (2|^w*m - 2|^(w+1) + 2) <= 2 * (a div 2 + 1) by A58,A87; hence 2|^w*m - 2|^(w+1) + 2 <= a div 2 + 1 by REAL_1:70; end; A88: now let m; let a be odd Nat; assume A89: a <= 2|^(w+1)*m - 2|^(w+1) + 1; A90: 2 * (2|^w*m - 2|^w + 1) = 2|^(w+1)*m - 2|^(w+1) + 1+1 by A67; a + 1 <= 2|^(w+1)*m - 2|^(w+1) + 1 + 1 by A89,AXIOMS:24; then 2*(a div 2 + 1) <= 2*(2|^w*m - 2|^w + 1) by A58,A90; hence a div 2 + 1 <= 2|^w*m - 2|^w + 1 by REAL_1:70; end; per cases; suppose A91: a is odd & b is odd; then A92: 2*(a div 2 + 1)-'1 = a+1-'1 by A58 .= a by BINARITH:39; A93: 2|^w*m - 2|^(w+1) + 2 <= a div 2 + 1 by A52,A85,A91; A94: a div 2 + 1 <= 2|^w*m - 2|^w + 1 by A53,A88,A91; A95: 2|^w*n - 2|^(w+1) + 2 <= b div 2 + 1 by A54,A85,A91; b div 2 + 1 <= 2|^w*n - 2|^w + 1 by A55,A88,A91; hence Y in F(w) by A93,A94,A95; 2*(b div 2 + 1)-'1 = b+1-'1 by A58,A91 .= b by BINARITH:39; hence X c= Y by A15,A51,A78,A92,XBOOLE_1:7; suppose A96: a is odd & b is even; then A97: 2*(a div 2 + 1)-'1 = a+1-'1 by A58 .= a by BINARITH:39; A98: 2*(b div 2 + 1)-'2 = b+2-'2 by A56,A96 .= b by BINARITH:39; A99: 2|^w*m - 2|^(w+1) + 2 <= a div 2 + 1 by A52,A85,A96; A100: a div 2 + 1 <= 2|^w*m - 2|^w + 1 by A53,A88,A96; A101: 2|^w*n - 2|^(w+1) + 2 <= b div 2 + 1 by A54,A79,A96; b div 2 + 1 <= 2|^w*n - 2|^w + 1 by A55,A82,A96; hence Y in F(w) by A99,A100,A101; A102: G(1,2) c= G(2,2) \/ G(1,2) by XBOOLE_1:7; G(2,2) \/ G(1,2) c= G(2,2) \/ G(1,2) \/ G(2,1) by XBOOLE_1:7; then A103: G(1,2) c= G(2,2) \/ G(1,2) \/ G(2,1) by A102,XBOOLE_1:1; G(2,2) \/ G(1,2) \/ G(2,1) c= Y by A78,XBOOLE_1:7; hence X c= Y by A15,A51,A97,A98,A103,XBOOLE_1:1; suppose A104: a is even & b is odd; then A105: 2*(a div 2 + 1)-'2 = a+2-'2 by A56 .= a by BINARITH:39; A106: 2*(b div 2 + 1)-'1 = b+1-'1 by A58,A104 .= b by BINARITH:39; A107: 2|^w*m - 2|^(w+1) + 2 <= a div 2 + 1 by A52,A79,A104; A108: a div 2 + 1 <= 2|^w*m - 2|^w + 1 by A53,A82,A104; A109: 2|^w*n - 2|^(w+1) + 2 <= b div 2 + 1 by A54,A85,A104; b div 2 + 1 <= 2|^w*n - 2|^w + 1 by A55,A88,A104; hence Y in F(w) by A107,A108,A109; A110: G(2,1) c= G(2,2) \/ G(1,2) \/ G(2,1) by XBOOLE_1:7; G(2,2) \/ G(1,2) \/ G(2,1) c= Y by A78,XBOOLE_1:7; hence X c= Y by A15,A51,A105,A106,A110,XBOOLE_1:1; suppose A111: a is even & b is even; then A112: 2*(a div 2 + 1)-'2 = a+2-'2 by A56 .= a by BINARITH:39; A113: 2|^w*m - 2|^(w+1) + 2 <= a div 2 + 1 by A52,A79,A111; A114: a div 2 + 1 <= 2|^w*m - 2|^w + 1 by A53,A82,A111; A115: 2|^w*n - 2|^(w+1) + 2 <= b div 2 + 1 by A54,A79,A111; b div 2 + 1 <= 2|^w*n - 2|^w + 1 by A55,A82,A111; hence Y in F(w) by A113,A114,A115; 2*(b div 2 + 1)-'2 = b+2-'2 by A56,A111 .= b by BINARITH:39; then X c= G(2,2) \/ (G(1,2) \/ G(2,1) \/ G(1,1)) by A15,A51,A112,XBOOLE_1:7; then X c= G(2,2) \/ (G(1,2) \/ G(2,1)) \/ G(1,1) by XBOOLE_1:4; hence X c= Y by A78,XBOOLE_1:4; end; then A116: union F(w+1) c= union F(w) by SETFAM_1:18; let d be set; assume d in union F(w+1); hence thesis by A14,A116; end; for w being Nat holds P[w] from Ind(A5,A13); hence thesis; end; theorem Th11: ex i being Nat st 1 <= i & i < len Cage(C,n) & N-max C in right_cell(Cage(C,n),i,Gauge(C,n)) proof N-max C in N-most C by PSCOMP_1:101; then consider p be Point of TOP-REAL 2 such that A1: north_halfline N-max C /\ L~Cage(C,n) = {p} by JORDAN1A:107; A2: p in north_halfline N-max C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in north_halfline N-max C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: N-max C in N-most C by PSCOMP_1:101; then A14: p`2 = N-bound L~Cage(C,n) by A2,JORDAN1A:103; A15: LSeg(Cage(C,n),i) is horizontal by A3,A5,A6,A13,JORDAN1A:99; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`2 = p`2 by A5,A15,SPRECT_3:19; A18: (Cage(C,n)/.(i+1))`2 = p`2 by A5,A15,A16,SPRECT_3:19; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: j1 = j2 proof assume j1 <> j2; then j1 < j2 or j2 < j1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A22,A23,GOBOARD5:5; end; (Gauge(C,n)*(i1,j1))`2 = Gauge(C,n)*(i1,len Gauge(C,n))`2 by A9,A14,A17,A20,JORDAN1A:91; then A25: len Gauge(C,n) <= j1 by A19,A20,A22,GOBOARD5:5; then A26: j1 = len Gauge(C,n) by A19,A23,AXIOMS:21; A27: 1 <= i1 & i1 < len Gauge(C,n) by A4,A8,A9,A10,A11,A12,A19,A21,A24,A25,GOBOARD5:1,JORDAN10:4,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: (N-max C)`2 = N-bound C by PSCOMP_1:94 .= Gauge(C,n)*(1,len Gauge(C,n)-'1)`2 by A29,JORDAN8:17; A31: len Gauge(C,n)-'1 <= len Gauge(C,n) by JORDAN3:7; len Gauge(C,n) >= 1+1 by A28,AXIOMS:22; then A32: len Gauge(C,n)-1 >= 1 by REAL_1:84; then len Gauge(C,n)-1 >= 0 by AXIOMS:22; then A33: len Gauge(C,n)-'1 >= 1 by A32,BINARITH:def 3; then A34: Gauge(C,n)*(1,j1)`2 >= (N-max C)`2 by A19,A26,A29,A30,A31,SPRECT_3:24; i1 <= i1+1 by NAT_1:29; then (Cage(C,n)/.i)`1 <= (Cage(C,n)/.(i+1))`1 by A4,A8,A9,A10,A11,A12,A19, A20,A21,A22,A24,A25,JORDAN10:4,JORDAN1A:39; then (Cage(C,n)/.i)`1 <= p`1 & p`1 <= (Cage(C,n)/.(i+1))`1 by A5,A16,TOPREAL1:9; then Gauge(C,n)*(i1,len Gauge(C,n))`1 <= (N-max C)`1 & (N-max C)`1 <= Gauge(C,n)*(i1+1,len Gauge(C,n))`1 by A3,A4,A8,A9,A10,A11, A12,A19,A24,A26,JORDAN10:4,JORDAN1A:def 2,NAT_1:38; then A35: Gauge(C,n)*(i1,1)`1 <= (N-max C)`1 & (N-max C)`1 <= Gauge(C,n)*(i1+1,1)`1 by A4,A8,A9,A10,A11,A12,A19,A20,A21, A24,A25,A29,GOBOARD5:3,JORDAN10:4,NAT_1:38; A36: len Gauge(C,n)-'1+1 = len Gauge(C,n) by A29,AMI_5:4; then A37: len Gauge(C,n)-'1 < len Gauge(C,n) by NAT_1:38; N-max C = |[(N-max C)`1,(N-max C)`2]| by EUCLID:57; then N-max C in { |[r,s]| where r,s is Real : Gauge(C,n)*(i1,1)`1 <= r & r <= Gauge(C,n)*(i1+1,1)`1 & Gauge(C,n)*(1,j1-'1)`2 <= s & s <= Gauge(C,n)*(1,j1)`2 } by A26,A30,A34,A35; then N-max C in cell(Gauge(C,n),i1,j1-'1) by A19,A26,A27,A33,A36,A37,GOBRD11:32; hence N-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11,A12,A19,A24,A25,GOBRD13:25,JORDAN10:4,NAT_1:38; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & N-max C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: N-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th11; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence N-max C in right_cell(Cage(C,n),i) by A2; end; theorem Th13: ex i being Nat st 1 <= i & i < len Cage(C,n) & E-min C in right_cell(Cage(C,n),i,Gauge(C,n)) proof E-min C in E-most C by PSCOMP_1:111; then consider p be Point of TOP-REAL 2 such that A1: east_halfline E-min C /\ L~Cage(C,n) = {p} by JORDAN1A:108; A2: p in east_halfline E-min C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in east_halfline E-min C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: E-min C in E-most C by PSCOMP_1:111; then A14: p`1 = E-bound L~Cage(C,n) by A2,JORDAN1A:104; A15: LSeg(Cage(C,n),i) is vertical by A3,A5,A6,A13,JORDAN1A:100; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`1 = p`1 by A5,A15,SPRECT_3:20; A18: (Cage(C,n)/.(i+1))`1 = p`1 by A5,A15,A16,SPRECT_3:20; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: i1 = i2 proof assume i1 <> i2; then i1 < i2 or i2 < i1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A21,A22,GOBOARD5:4; end; (Gauge(C,n)*(i1,j1))`1 = Gauge(C,n)*(len Gauge(C,n),j1)`1 by A9,A14,A17,A19,A22,A23,JORDAN1A:92; then A25: len Gauge(C,n) <= i1 by A20,A22,A23,GOBOARD5:4; then A26: i1 = len Gauge(C,n) by A20,AXIOMS:21; A27: 1 <= j2 & j2 < width Gauge(C,n) by A4,A8,A9,A10,A11,A12,A23,A24,A25, JORDAN10:1,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: (E-min C)`1 = E-bound C by PSCOMP_1:104 .= Gauge(C,n)*(len Gauge(C,n)-'1,1)`1 by A29,JORDAN8:15; A31: len Gauge(C,n)-'1 <= len Gauge(C,n) by JORDAN3:7; len Gauge(C,n) >= 1+1 by A28,AXIOMS:22; then A32: len Gauge(C,n)-1 >= 1 by REAL_1:84; then A33: len Gauge(C,n)-1 >= 0 by AXIOMS:22; then len Gauge(C,n)-'1 >= 1 by A32,BINARITH:def 3; then A34: Gauge(C,n)*(i1,1)`1 >= (E-min C)`1 by A19,A26,A29,A30,A31,SPRECT_3:25; j2 <= j2+1 by NAT_1:29; then (Cage(C,n)/.i)`2 >= (Cage(C,n)/.(i+1))`2 by A4,A8,A9,A10,A11,A12,A20,A23,A24,A25,JORDAN10:1,JORDAN1A:40; then (Cage(C,n)/.(i+1))`2 <= p`2 & p`2 <= (Cage(C,n)/.i)`2 by A5,A16,TOPREAL1:10; then Gauge(C,n)*(len Gauge(C,n),j2)`2 <= (E-min C)`2 & (E-min C)`2 <= Gauge(C,n)*(len Gauge(C,n),j2+1)`2 by A3,A4,A8,A9,A10,A11, A12,A24,A26,JORDAN10:1,JORDAN1A:def 3,NAT_1:38; then A35: Gauge(C,n)*(1,j2)`2 <= (E-min C)`2 & (E-min C)`2 <= Gauge(C,n)* (1,j2+1)`2 by A4,A8,A9,A10,A11,A12,A21,A22,A23,A24,A26,GOBOARD5:2,JORDAN10:1, NAT_1:38; len Gauge(C,n) < len Gauge(C,n)+1 by NAT_1:38; then len Gauge(C,n)-1 < len Gauge(C,n) by REAL_1:84; then A36: 1 <= len Gauge(C,n)-'1 & len Gauge(C,n)-'1 < len Gauge(C,n) by A32,A33,BINARITH:def 3; A37: len Gauge(C,n)-'1+1 = len Gauge(C,n) by A29,AMI_5:4; E-min C = |[(E-min C)`1,(E-min C)`2]| by EUCLID:57; then E-min C in { |[r,s]| where r,s is Real : Gauge(C,n)*(len Gauge(C,n)-'1,1)`1 <= r & r <= Gauge(C,n)*(len Gauge(C,n),1)`1 & Gauge(C,n)*(1,j2)`2 <= s & s <= Gauge(C,n)*(1,j2+1)`2 } by A26,A30,A34,A35; then E-min C in cell(Gauge(C,n),i2-'1,j2) by A24,A26,A27,A36,A37,GOBRD11:32; hence E-min C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11, A12,A24,A25,GOBRD13:29,JORDAN10:1,NAT_1:38; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & E-min C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: E-min C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th13; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence E-min C in right_cell(Cage(C,n),i) by A2; end; theorem Th15: ex i being Nat st 1 <= i & i < len Cage(C,n) & E-max C in right_cell(Cage(C,n),i,Gauge(C,n)) proof E-max C in E-most C by PSCOMP_1:111; then consider p be Point of TOP-REAL 2 such that A1: east_halfline E-max C /\ L~Cage(C,n) = {p} by JORDAN1A:108; A2: p in east_halfline E-max C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in east_halfline E-max C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: E-max C in E-most C by PSCOMP_1:111; then A14: p`1 = E-bound L~Cage(C,n) by A2,JORDAN1A:104; A15: LSeg(Cage(C,n),i) is vertical by A3,A5,A6,A13,JORDAN1A:100; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`1 = p`1 by A5,A15,SPRECT_3:20; A18: (Cage(C,n)/.(i+1))`1 = p`1 by A5,A15,A16,SPRECT_3:20; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: i1 = i2 proof assume i1 <> i2; then i1 < i2 or i2 < i1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A21,A22,GOBOARD5:4; end; (Gauge(C,n)*(i1,j1))`1 = Gauge(C,n)*(len Gauge(C,n),j1)`1 by A9,A14,A17,A19,A22,A23,JORDAN1A:92; then A25: len Gauge(C,n) <= i1 by A20,A22,A23,GOBOARD5:4; then A26: i1 = len Gauge(C,n) by A20,AXIOMS:21; A27: 1 <= j2 & j2 < width Gauge(C,n) by A4,A8,A9,A10,A11,A12,A23,A24,A25,JORDAN10:1,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: (E-max C)`1 = E-bound C by PSCOMP_1:104 .= Gauge(C,n)*(len Gauge(C,n)-'1,1)`1 by A29,JORDAN8:15; A31: len Gauge(C,n)-'1 <= len Gauge(C,n) by JORDAN3:7; len Gauge(C,n) >= 1+1 by A28,AXIOMS:22; then A32: len Gauge(C,n)-1 >= 1 by REAL_1:84; then A33: len Gauge(C,n)-1 >= 0 by AXIOMS:22; then len Gauge(C,n)-'1 >= 1 by A32,BINARITH:def 3; then A34: Gauge(C,n)*(i1,1)`1 >= (E-max C)`1 by A19,A26,A29,A30,A31,SPRECT_3:25; j2 <= j2+1 by NAT_1:29; then (Cage(C,n)/.i)`2 >= (Cage(C,n)/.(i+1))`2 by A4,A8,A9,A10,A11,A12,A20, A23,A24,A25,JORDAN10:1,JORDAN1A:40; then (Cage(C,n)/.(i+1))`2 <= p`2 & p`2 <= (Cage(C,n)/.i)`2 by A5,A16,TOPREAL1:10; then Gauge(C,n)*(len Gauge(C,n),j2)`2 <= (E-max C)`2 & (E-max C)`2 <= Gauge(C,n)*(len Gauge(C,n),j2+1)`2 by A3,A4,A8,A9,A10,A11, A12,A24,A26,JORDAN10:1,JORDAN1A:def 3,NAT_1:38; then A35: Gauge(C,n)*(1,j2)`2 <= (E-max C)`2 & (E-max C)`2 <= Gauge(C,n)* (1,j2+1)`2 by A4,A8,A9,A10,A11,A12,A22,A23,A24,A25,A29,GOBOARD5:2,JORDAN10:1, NAT_1:38; len Gauge(C,n) < len Gauge(C,n)+1 by NAT_1:38; then len Gauge(C,n)-1 < len Gauge(C,n) by REAL_1:84; then A36: 1 <= len Gauge(C,n)-'1 & len Gauge(C,n)-'1 < len Gauge(C,n) by A32,A33,BINARITH:def 3; A37: len Gauge(C,n)-'1+1 = len Gauge(C,n) by A29,AMI_5:4; E-max C = |[(E-max C)`1,(E-max C)`2]| by EUCLID:57; then E-max C in { |[r,s]| where r,s is Real : Gauge(C,n)*(len Gauge(C,n)-'1,1)`1 <= r & r <= Gauge(C,n)*(len Gauge(C,n),1)`1 & Gauge(C,n)*(1,j2)`2 <= s & s <= Gauge(C,n)*(1,j2+1)`2 } by A26,A30,A34,A35; then E-max C in cell(Gauge(C,n),i2-'1,j2) by A24,A26,A27,A36,A37,GOBRD11:32; hence E-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11, A12,A24,A25,GOBRD13:29,JORDAN10:1,NAT_1:38; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & E-max C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: E-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th15; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence E-max C in right_cell(Cage(C,n),i) by A2; end; theorem Th17: ex i being Nat st 1 <= i & i < len Cage(C,n) & S-min C in right_cell(Cage(C,n),i,Gauge(C,n)) proof S-min C in S-most C by PSCOMP_1:121; then consider p be Point of TOP-REAL 2 such that A1: south_halfline S-min C /\ L~Cage(C,n) = {p} by JORDAN1A:109; A2: p in south_halfline S-min C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in south_halfline S-min C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: S-min C in S-most C by PSCOMP_1:121; then A14: p`2 = S-bound L~Cage(C,n) by A2,JORDAN1A:105; A15: LSeg(Cage(C,n),i) is horizontal by A3,A5,A6,A13,JORDAN1A:101; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`2 = p`2 by A5,A15,SPRECT_3:19; A18: (Cage(C,n)/.(i+1))`2 = p`2 by A5,A15,A16,SPRECT_3:19; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: j1 = j2 proof assume j1 <> j2; then j1 < j2 or j2 < j1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A22,A23,GOBOARD5:5; end; (Gauge(C,n)*(i1,j1))`2 = Gauge(C,n)*(i1,1)`2 by A9,A14,A17,A20,JORDAN1A:93; then A25: 1 >= j1 by A20,A23,GOBOARD5:5; then A26: j1 = 1 by A22,AXIOMS:21; A27: 1 <= i2 & i2 < len Gauge(C,n) by A4,A8,A9,A10,A11,A12,A20,A24,A25, GOBOARD5:1,JORDAN10:3,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: 1+1 <= len Gauge(C,n) by A28,AXIOMS:22; A31: (S-min C)`2 = S-bound C by PSCOMP_1:114 .= Gauge(C,n)*(1,2)`2 by A29,JORDAN8:16; then A32: Gauge(C,n)* (1,j1)`2 <= (S-min C)`2 by A19,A26,A29,A30,SPRECT_3:24; i2 <= i2+1 by NAT_1:29; then (Cage(C,n)/.i)`1 >= (Cage(C,n)/.(i+1))`1 by A4,A8,A9,A10,A11,A12,A20, A21,A22,A23,A25,JORDAN10:3,JORDAN1A:39; then (Cage(C,n)/.(i+1))`1 <= p`1 & p`1 <= (Cage(C,n)/.i)`1 by A5,A16,TOPREAL1:9; then A33: Gauge(C,n)*(i2,1)`1 <= (S-min C)`1 & (S-min C)`1 <= Gauge(C,n)*(i2+1,1)`1 by A3,A4,A8,A9,A10,A11,A12,A24,A26,JORDAN10: 3,JORDAN1A:def 4; S-min C = |[(S-min C)`1,(S-min C)`2]| by EUCLID:57; then S-min C in { |[r,s]| where r,s is Real : Gauge(C,n)*(i2,1)`1 <= r & r <= Gauge(C,n)*(i2+1,1)`1 & Gauge(C,n)*(1,j1)`2 <= s & s <= Gauge(C,n)*(1,j1+1)`2 } by A26,A31,A32,A33; then S-min C in cell(Gauge(C,n),i2,j1) by A19,A26,A27,A29,GOBRD11:32; hence S-min C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11,A12,A24,A26,GOBRD13:27,JORDAN10:3 ; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & S-min C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: S-min C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th17; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence S-min C in right_cell(Cage(C,n),i) by A2; end; theorem Th19: ex i being Nat st 1 <= i & i < len Cage(C,n) & S-max C in right_cell(Cage(C,n),i,Gauge(C,n)) proof S-max C in S-most C by PSCOMP_1:121; then consider p be Point of TOP-REAL 2 such that A1: south_halfline S-max C /\ L~Cage(C,n) = {p} by JORDAN1A:109; A2: p in south_halfline S-max C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in south_halfline S-max C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: S-max C in S-most C by PSCOMP_1:121; then A14: p`2 = S-bound L~Cage(C,n) by A2,JORDAN1A:105; A15: LSeg(Cage(C,n),i) is horizontal by A3,A5,A6,A13,JORDAN1A:101; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`2 = p`2 by A5,A15,SPRECT_3:19; A18: (Cage(C,n)/.(i+1))`2 = p`2 by A5,A15,A16,SPRECT_3:19; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: j1 = j2 proof assume j1 <> j2; then j1 < j2 or j2 < j1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A22,A23,GOBOARD5:5; end; (Gauge(C,n)*(i1,j1))`2 = Gauge(C,n)*(i1,1)`2 by A9,A14,A17,A20,JORDAN1A:93; then A25: 1 >= j1 by A20,A23,GOBOARD5:5; then A26: j1 = 1 by A22,AXIOMS:21; A27: 1 <= i2 & i2 < len Gauge(C,n) by A4,A8,A9,A10,A11,A12,A20,A24,A25, GOBOARD5:1,JORDAN10:3,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: 1+1 <= len Gauge(C,n) by A28,AXIOMS:22; A31: (S-max C)`2 = S-bound C by PSCOMP_1:114 .= Gauge(C,n)*(1,2)`2 by A29,JORDAN8:16; then A32: Gauge(C,n)* (1,j1)`2 <= (S-max C)`2 by A19,A26,A29,A30,SPRECT_3:24; i2 <= i2+1 by NAT_1:29; then (Cage(C,n)/.i)`1 >= (Cage(C,n)/.(i+1))`1 by A4,A8,A9,A10,A11,A12,A20, A21,A22,A23,A25,JORDAN10:3,JORDAN1A:39; then (Cage(C,n)/.(i+1))`1 <= p`1 & p`1 <= (Cage(C,n)/.i)`1 by A5,A16,TOPREAL1:9; then A33: Gauge(C,n)*(i2,1)`1 <= (S-max C)`1 & (S-max C)`1 <= Gauge(C,n)*(i2+1,1)`1 by A3,A4,A8,A9,A10,A11,A12,A24,A26,JORDAN10: 3,JORDAN1A:def 4; S-max C = |[(S-max C)`1,(S-max C)`2]| by EUCLID:57; then S-max C in { |[r,s]| where r,s is Real : Gauge(C,n)*(i2,1)`1 <= r & r <= Gauge(C,n)*(i2+1,1)`1 & Gauge(C,n)*(1,j1)`2 <= s & s <= Gauge(C,n)*(1,j1+1)`2 } by A26,A31,A32,A33; then S-max C in cell(Gauge(C,n),i2,j1) by A19,A26,A27,A29,GOBRD11:32; hence S-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11,A12,A24,A26,GOBRD13: 27,JORDAN10:3; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & S-max C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: S-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th19; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence S-max C in right_cell(Cage(C,n),i) by A2; end; theorem Th21: ex i being Nat st 1 <= i & i < len Cage(C,n) & W-min C in right_cell(Cage(C,n),i,Gauge(C,n)) proof W-min C in W-most C by PSCOMP_1:91; then consider p be Point of TOP-REAL 2 such that A1: west_halfline W-min C /\ L~Cage(C,n) = {p} by JORDAN1A:110; A2: p in west_halfline W-min C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in west_halfline W-min C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: W-min C in W-most C by PSCOMP_1:91; then A14: p`1 = W-bound L~Cage(C,n) by A2,JORDAN1A:106; A15: LSeg(Cage(C,n),i) is vertical by A3,A5,A6,A13,JORDAN1A:102; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`1 = p`1 by A5,A15,SPRECT_3:20; A18: (Cage(C,n)/.(i+1))`1 = p`1 by A5,A15,A16,SPRECT_3:20; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: i1 = i2 proof assume i1 <> i2; then i1 < i2 or i2 < i1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A21,A22,GOBOARD5:4; end; (Gauge(C,n)*(i1,j1))`1 = Gauge(C,n)*(1,j1)`1 by A9,A14,A17,A19,A22,A23,JORDAN1A:94; then A25: 1 >= i1 by A20,A22,A23,GOBOARD5:4; then A26: i1 = 1 by A20,AXIOMS:21; A27: 1 <= j1 & j1 < width Gauge(C,n) by A4,A8,A9,A10,A11,A12,A22,A24,A25, JORDAN10:2,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: 1+1 <= len Gauge(C,n) by A28,AXIOMS:22; A31: (W-min C)`1 = W-bound C by PSCOMP_1:84 .= Gauge(C,n)*(2,1)`1 by A29,JORDAN8:14; then A32: Gauge(C,n)* (i1,1)`1 <= (W-min C)`1 by A19,A26,A29,A30,SPRECT_3:25; j1 <= j1+1 by NAT_1:29; then (Cage(C,n)/.i)`2 <= (Cage(C,n)/.(i+1))`2 by A4,A8,A9,A10,A11,A12,A20, A22,A24,A25,JORDAN10:2,JORDAN1A:40; then (Cage(C,n)/.i)`2 <= p`2 & p`2 <= (Cage(C,n)/.(i+1))`2 by A5,A16,TOPREAL1:10; then A33: Gauge(C,n)*(1,j1)`2 <= (W-min C)`2 & (W-min C)`2 <= Gauge(C,n)*(1,j1+1)`2 by A3,A4,A8,A9,A10,A11,A12,A24,A26,JORDAN10: 2,JORDAN1A:def 5; W-min C = |[(W-min C)`1,(W-min C)`2]| by EUCLID:57; then W-min C in { |[r,s]| where r,s is Real : Gauge(C,n)*(i1,1)`1 <= r & r <= Gauge(C,n)*(i1+1,1)`1 & Gauge(C,n)*(1,j1)`2 <= s & s <= Gauge(C,n)*(1,j1+1)`2 } by A26,A31,A32,A33; then W-min C in cell(Gauge(C,n),i1,j1) by A26,A27,A29,GOBRD11:32; hence W-min C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11,A12,A24,A26,GOBRD13: 23,JORDAN10:2; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & W-min C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: W-min C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th21; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence W-min C in right_cell(Cage(C,n),i) by A2; end; theorem Th23: ex i being Nat st 1 <= i & i < len Cage(C,n) & W-max C in right_cell(Cage(C,n),i,Gauge(C,n)) proof W-max C in W-most C by PSCOMP_1:91; then consider p be Point of TOP-REAL 2 such that A1: west_halfline W-max C /\ L~Cage(C,n) = {p} by JORDAN1A:110; A2: p in west_halfline W-max C /\ L~Cage(C,n) by A1,TARSKI:def 1; then A3: p in west_halfline W-max C & p in L~Cage(C,n) by XBOOLE_0:def 3; then consider i be Nat such that A4: 1 <= i & i+1 <= len Cage(C,n) and A5: p in LSeg(Cage(C,n),i) by SPPOL_2:13; take i; thus A6: 1 <= i & i < len Cage(C,n) by A4,NAT_1:38; A7: Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i1,j1,i2,j2 be Nat such that A8: [i1,j1] in Indices Gauge(C,n) and A9: Cage(C,n)/.i = Gauge(C,n)*(i1,j1) and A10: [i2,j2] in Indices Gauge(C,n) and A11: Cage(C,n)/.(i+1) = Gauge(C,n)*(i2,j2) and A12: i1 = i2 & j1+1 = j2 or i1+1 = i2 & j1 = j2 or i1 = i2+1 & j1 = j2 or i1 = i2 & j1 = j2+1 by A4,JORDAN8:6; A13: W-max C in W-most C by PSCOMP_1:91; then A14: p`1 = W-bound L~Cage(C,n) by A2,JORDAN1A:106; A15: LSeg(Cage(C,n),i) is vertical by A3,A5,A6,A13,JORDAN1A:102; A16: LSeg(Cage(C,n),i) = LSeg(Cage(C,n)/.i,Cage(C,n)/.(i+1)) by A4,TOPREAL1:def 5; then A17: (Cage(C,n)/.i)`1 = p`1 by A5,A15,SPRECT_3:20; A18: (Cage(C,n)/.(i+1))`1 = p`1 by A5,A15,A16,SPRECT_3:20; A19: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; A20: 1 <= i1 & i1 <= len Gauge(C,n) by A8,GOBOARD5:1; A21: 1 <= i2 & i2 <= len Gauge(C,n) by A10,GOBOARD5:1; A22: 1 <= j1 & j2 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A23: 1 <= j2 & j1 <= width Gauge(C,n) by A8,A10,GOBOARD5:1; A24: i1 = i2 proof assume i1 <> i2; then i1 < i2 or i2 < i1 by REAL_1:def 5; hence contradiction by A9,A11,A12,A17,A18,A20,A21,A22,GOBOARD5:4; end; (Gauge(C,n)*(i1,j1))`1 = Gauge(C,n)*(1,j1)`1 by A9,A14,A17,A19,A22,A23,JORDAN1A:94; then A25: 1 >= i1 by A20,A22,A23,GOBOARD5:4; then A26: i1 = 1 by A20,AXIOMS:21; A27: 1 <= j1 & j1 < width Gauge(C,n) by A4,A8,A9,A10,A11,A12,A22,A24,A25, JORDAN10:2,NAT_1:38; A28: len Gauge(C,n) >= 4 by JORDAN8:13; then A29: 1 < len Gauge(C,n) by AXIOMS:22; A30: 1+1 <= len Gauge(C,n) by A28,AXIOMS:22; A31: (W-max C)`1 = W-bound C by PSCOMP_1:84 .= Gauge(C,n)*(2,1)`1 by A29,JORDAN8:14; then A32: Gauge(C,n)* (i1,1)`1 <= (W-max C)`1 by A19,A26,A29,A30,SPRECT_3:25; j1 <= j1+1 by NAT_1:29; then (Cage(C,n)/.i)`2 <= (Cage(C,n)/.(i+1))`2 by A4,A8,A9,A10,A11,A12,A20, A22,A24,A25,JORDAN10:2,JORDAN1A:40; then (Cage(C,n)/.i)`2 <= p`2 & p`2 <= (Cage(C,n)/.(i+1))`2 by A5,A16,TOPREAL1:10; then A33: Gauge(C,n)*(1,j1)`2 <= (W-max C)`2 & (W-max C)`2 <= Gauge(C,n)*(1,j1+1)`2 by A3,A4,A8,A9,A10,A11,A12,A24,A26,JORDAN10: 2,JORDAN1A:def 5; W-max C = |[(W-max C)`1,(W-max C)`2]| by EUCLID:57; then W-max C in { |[r,s]| where r,s is Real : Gauge(C,n)*(i1,1)`1 <= r & r <= Gauge(C,n)*(i1+1,1)`1 & Gauge(C,n)*(1,j1)`2 <= s & s <= Gauge(C,n)*(1,j1+1)`2 } by A26,A31,A32,A33; then W-max C in cell(Gauge(C,n),i1,j1) by A26,A27,A29,GOBRD11:32; hence W-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by A4,A7,A8,A9,A10,A11,A12,A24,A26,GOBRD13: 23,JORDAN10:2; end; theorem ex i being Nat st 1 <= i & i < len Cage(C,n) & W-max C in right_cell(Cage(C,n),i) proof consider i be Nat such that A1: 1 <= i & i < len Cage(C,n) and A2: W-max C in right_cell(Cage(C,n),i,Gauge(C,n)) by Th23; take i; thus 1 <= i & i < len Cage(C,n) by A1; A3: i+1 <= len Cage(C,n) by A1,NAT_1:38; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then right_cell(Cage(C,n),i,Gauge(C,n)) c= right_cell(Cage(C,n),i) by A1,A3,GOBRD13:34; hence W-max C in right_cell(Cage(C,n),i) by A2; end; theorem Th25: ex i being Nat st 1 <= i & i <= len Gauge(C,n) & N-min L~Cage(C,n) = Gauge(C,n)*(i,width Gauge(C,n)) proof N-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:43; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = N-min L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = N-min L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take i; thus A6: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; now assume j < width Gauge(C,n); then (N-min L~Cage(C,n))`2 < Gauge(C,n)*(i,width Gauge(C,n))`2 by A3,A5,A6,A7,GOBOARD5:5; then N-bound L~Cage(C,n) < Gauge(C,n)*(i,width Gauge(C,n))`2 by PSCOMP_1:94; hence contradiction by A6,A8,JORDAN1A:91; end; hence N-min L~Cage(C,n) = Gauge(C,n)*(i,width Gauge(C,n)) by A3,A5,A7,AXIOMS:21; end; theorem ex i being Nat st 1 <= i & i <= len Gauge(C,n) & N-max L~Cage(C,n) = Gauge(C,n)*(i,width Gauge(C,n)) proof N-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:44; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = N-max L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = N-max L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take i; thus A6: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; now assume j < width Gauge(C,n); then (N-max L~Cage(C,n))`2 < Gauge(C,n)*(i,width Gauge(C,n))`2 by A3,A5,A6,A7,GOBOARD5:5; then N-bound L~Cage(C,n) < Gauge(C,n)*(i,width Gauge(C,n))`2 by PSCOMP_1:94; hence contradiction by A6,A8,JORDAN1A:91; end; hence N-max L~Cage(C,n) = Gauge(C,n)*(i,width Gauge(C,n)) by A3,A5,A7,AXIOMS:21; end; theorem ex i being Nat st 1 <= i & i <= len Gauge(C,n) & Gauge(C,n)*(i,width Gauge(C,n)) in rng Cage(C,n) proof consider i be Nat such that A1: 1 <= i & i <= len Gauge(C,n) & N-min L~Cage(C,n) = Gauge(C,n)*(i,width Gauge(C,n)) by Th25; take i; thus thesis by A1,SPRECT_2:43; end; theorem Th28: ex j being Nat st 1 <= j & j <= width Gauge(C,n) & E-min L~Cage(C,n) = Gauge(C,n)*(len Gauge(C,n),j) proof E-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:49; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = E-min L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = E-min L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take j; thus A6: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; now assume i < len Gauge(C,n); then (E-min L~Cage(C,n))`1 < Gauge(C,n)*(len Gauge(C,n),j)`1 by A3,A5,A6,A7,GOBOARD5:4; then E-bound L~Cage(C,n) < Gauge(C,n)* (len Gauge(C,n),j)`1 by PSCOMP_1:104; hence contradiction by A6,A8,JORDAN1A:92; end; hence E-min L~Cage(C,n) = Gauge(C,n)*(len Gauge(C,n),j) by A3,A5,A7,AXIOMS:21; end; theorem ex j being Nat st 1 <= j & j <= width Gauge(C,n) & E-max L~Cage(C,n) = Gauge(C,n)*(len Gauge(C,n),j) proof E-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:50; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = E-max L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = E-max L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take j; thus A6: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; now assume i < len Gauge(C,n); then (E-max L~Cage(C,n))`1 < Gauge(C,n)*(len Gauge(C,n),j)`1 by A3,A5,A6,A7,GOBOARD5:4; then E-bound L~Cage(C,n) < Gauge(C,n)* (len Gauge(C,n),j)`1 by PSCOMP_1:104; hence contradiction by A6,A8,JORDAN1A:92; end; hence E-max L~Cage(C,n) = Gauge(C,n)*(len Gauge(C,n),j) by A3,A5,A7,AXIOMS:21; end; theorem ex j being Nat st 1 <= j & j <= width Gauge(C,n) & Gauge(C,n)*(len Gauge(C,n),j) in rng Cage(C,n) proof consider j be Nat such that A1: 1 <= j & j <= width Gauge(C,n) & E-min L~Cage(C,n) = Gauge(C,n)*(len Gauge(C,n),j) by Th28; take j; thus thesis by A1,SPRECT_2:49; end; theorem Th31: ex i being Nat st 1 <= i & i <= len Gauge(C,n) & S-min L~Cage(C,n) = Gauge(C,n)*(i,1) proof S-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:45; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = S-min L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = S-min L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take i; thus A6: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; now assume j > 1; then (S-min L~Cage(C,n))`2 > Gauge(C,n)*(i,1)`2 by A3,A5,A6,A7,GOBOARD5:5; then S-bound L~Cage(C,n) > Gauge(C,n)*(i,1)`2 by PSCOMP_1:114; hence contradiction by A6,JORDAN1A:93; end; hence S-min L~Cage(C,n) = Gauge(C,n)*(i,1) by A3,A5,A7,AXIOMS:21; end; theorem ex i being Nat st 1 <= i & i <= len Gauge(C,n) & S-max L~Cage(C,n) = Gauge(C,n)*(i,1) proof S-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:46; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = S-max L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = S-max L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take i; thus A6: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; now assume j > 1; then (S-max L~Cage(C,n))`2 > Gauge(C,n)*(i,1)`2 by A3,A5,A6,A7,GOBOARD5:5; then S-bound L~Cage(C,n) > Gauge(C,n)*(i,1)`2 by PSCOMP_1:114; hence contradiction by A6,JORDAN1A:93; end; hence S-max L~Cage(C,n) = Gauge(C,n)*(i,1) by A3,A5,A7,AXIOMS:21; end; theorem ex i being Nat st 1 <= i & i <= len Gauge(C,n) & Gauge(C,n)*(i,1) in rng Cage(C,n) proof consider i be Nat such that A1: 1 <= i & i <= len Gauge(C,n) & S-min L~Cage(C,n) = Gauge(C,n)*(i,1) by Th31; take i; thus thesis by A1,SPRECT_2:45; end; theorem Th34: ex j being Nat st 1 <= j & j <= width Gauge(C,n) & W-min L~Cage(C,n) = Gauge(C,n)*(1,j) proof W-min L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:47; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = W-min L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = W-min L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take j; thus A6: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; now assume i > 1; then (W-min L~Cage(C,n))`1 > Gauge(C,n)*(1,j)`1 by A3,A5,A6,A7,GOBOARD5:4; then W-bound L~Cage(C,n) > Gauge(C,n)*(1,j)`1 by PSCOMP_1:84; hence contradiction by A6,A8,JORDAN1A:94; end; hence W-min L~Cage(C,n) = Gauge(C,n)*(1,j) by A3,A5,A7,AXIOMS:21; end; theorem ex j being Nat st 1 <= j & j <= width Gauge(C,n) & W-max L~Cage(C,n) = Gauge(C,n)*(1,j) proof W-max L~Cage(C,n) in rng Cage(C,n) by SPRECT_2:48; then consider m be Nat such that A1: m in dom Cage(C,n) and A2: Cage(C,n).m = W-max L~Cage(C,n) by FINSEQ_2:11; A3: Cage(C,n)/.m = W-max L~Cage(C,n) by A1,A2,FINSEQ_4:def 4; Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1; then consider i,j be Nat such that A4: [i,j] in Indices Gauge(C,n) and A5: Cage(C,n)/.m = Gauge(C,n)*(i,j) by A1,GOBOARD1:def 11; take j; thus A6: 1 <= j & j <= width Gauge(C,n) by A4,GOBOARD5:1; A7: 1 <= i & i <= len Gauge(C,n) by A4,GOBOARD5:1; A8: len Gauge(C,n) = width Gauge(C,n) by JORDAN8:def 1; now assume i > 1; then (W-max L~Cage(C,n))`1 > Gauge(C,n)*(1,j)`1 by A3,A5,A6,A7,GOBOARD5:4; then W-bound L~Cage(C,n) > Gauge(C,n)*(1,j)`1 by PSCOMP_1:84; hence contradiction by A6,A8,JORDAN1A:94; end; hence W-max L~Cage(C,n) = Gauge(C,n)*(1,j) by A3,A5,A7,AXIOMS:21; end; theorem ex j being Nat st 1 <= j & j <= width Gauge(C,n) & Gauge(C,n)*(1,j) in rng Cage(C,n) proof consider j be Nat such that A1: 1 <= j & j <= width Gauge(C,n) & W-min L~Cage(C,n) = Gauge(C,n)*(1,j) by Th34; take j; thus thesis by A1,SPRECT_2:47; end;

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