### The Mizar article:

### Bounding Boxes for Special Sequences in $\calE^2$

**by****Yatsuka Nakamura, and****Adam Grabowski**

- Received June 8, 1998
Copyright (c) 1998 Association of Mizar Users

- MML identifier: JORDAN5D
- [ MML identifier index ]

environ vocabulary PRE_TOPC, EUCLID, SEQM_3, GOBOARD5, FINSEQ_1, SEQ_2, SEQ_4, ARYTM_1, TOPREAL1, NAT_1, RELAT_1, FUNCT_1, JORDAN4, GOBOARD2, TREES_1, MCART_1, GOBOARD1, MATRIX_1, PSCOMP_1, FUNCT_5, REALSET1, SUBSET_1, BOOLE, ORDINAL2, FINSET_1, SQUARE_1, SPPOL_1, JORDAN5D, FINSEQ_4, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XREAL_0, MCART_1, REAL_1, NAT_1, BINARITH, RELAT_1, FINSEQ_1, FUNCT_1, FINSEQ_4, FINSET_1, SEQ_4, MATRIX_1, CQC_SIM1, STRUCT_0, TOPREAL1, SPPOL_1, GOBOARD1, GOBOARD2, GOBOARD5, PRE_TOPC, EUCLID, PRE_CIRC, JORDAN4, PSCOMP_1; constructors REAL_1, SPPOL_1, CQC_SIM1, GOBOARD2, BINARITH, PRE_CIRC, JORDAN4, PSCOMP_1, FINSEQ_4, COMPTS_1, REALSET1; clusters EUCLID, GOBOARD2, GOBOARD5, RELSET_1, STRUCT_0, GROUP_2, PRELAMB, SPRECT_1, FINSEQ_1, XREAL_0, ARYTM_3, MEMBERED, PRE_CIRC, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions TARSKI, XBOOLE_0; theorems AXIOMS, BINARITH, EUCLID, FINSEQ_3, FINSEQ_4, FINSEQ_6, FUNCT_1, FUNCT_2, JORDAN3, GOBOARD1, GOBOARD2, GOBOARD5, GOBOARD7, FINSET_1, GOBOARD9, NAT_1, PRE_TOPC, REAL_1, SEQ_4, SCMFSA_7, SPPOL_1, CQC_SIM1, SPPOL_2, TOPREAL1, MATRIX_1, FINSEQ_1, RLVECT_1, ZFMISC_1, JORDAN4, PSCOMP_1, AMI_5, SPRECT_2, FINSEQ_2, PRE_CIRC, XBOOLE_0, XBOOLE_1, XCMPLX_1, ORDINAL2; schemes DOMAIN_1; begin :: Preliminaries reserve p, q for Point of TOP-REAL 2, r for Real, h for non constant standard special_circular_sequence, g for FinSequence of TOP-REAL 2, f for non empty FinSequence of TOP-REAL 2, I, i1, i, j, k for Nat; canceled 2; theorem Th3: for n be Nat, h be FinSequence of TOP-REAL n st len h >= 2 holds h/.len h in LSeg(h,len h-'1) proof let n be Nat; let h be FinSequence of TOP-REAL n; set i = len h; assume A1: len h >= 2; then i >= 1 by AXIOMS:22; then A2: i-'1+1 = len h by AMI_5:4; 2-1 <= i-1 by A1,REAL_1:49; then 1 <= i-'1 by JORDAN3:1; hence h/.i in LSeg(h,i-'1) by A2,TOPREAL1:27; end; theorem Th4: 3 <= i implies i mod (i-'1) = 1 proof A1: 3-'1 = 2+1-'1 .= 2 by BINARITH:39; assume A2: 3 <= i; then 1 <= i by AXIOMS:22; then A3: i-'1 = i-1 by SCMFSA_7:3; A4: 2 <= i-'1 by A1,A2,JORDAN3:5; then A5: 0 < i-'1 by AXIOMS:22; A6: i-'1 <= i by GOBOARD9:2; 1 < i-1 by A3,A4,AXIOMS:22; then 1+i < (i-1)+i by REAL_1:67; then 1+i-1 < (i-1)+i-1 by REAL_1:92; then 1-1+i < (i-1)+i-1 by XCMPLX_1:29; then i < (i-1) + (i-1) by XCMPLX_1:29; hence i mod (i-'1) = i - (i-1) by A3,A5,A6,JORDAN4:7 .= 1 by XCMPLX_1:18; end; theorem Th5: p in rng h implies ex i be Nat st 1 <= i & i+1 <= len h & h.i = p proof assume p in rng h; then consider x be set such that A1: x in dom h & p = h.x by FUNCT_1:def 5; reconsider i = x as Nat by A1; set j = S_Drop (i,h); A2: 1 <= i & i <= len h by A1,FINSEQ_3:27; A3: 4 < len h by GOBOARD7:36; then A4: 3 <= len h by AXIOMS:22; A5: 1 < len h & 1+1 <= len h by A3,AXIOMS:22; 1 <= j & j+1 <= len h & h.j = p proof A6: i <= len h -' 1 + 1 by A2,A5,AMI_5:4; per cases by A6,NAT_1:26; suppose A7: i <= len h-'1; then j = i by A2,JORDAN4:34; then j + 1 <= len h -' 1 + 1 by A7,REAL_1:55; hence thesis by A1,A2,A5,A7,AMI_5:4,JORDAN4:34; suppose A8: i = len h-'1+1; then i = len h by A5,AMI_5:4; then i mod (len h -'1) = 1 by A4,Th4; then A9: j = 1 by JORDAN4:def 1; 1 <= len h & len h <= len h by A3,AXIOMS:22; then A10: 1 in dom h & len h in dom h by FINSEQ_3:27; then h.1 = h/.1 by FINSEQ_4:def 4 .= h/.len h by FINSEQ_6:def 1 .= h.len h by A10,FINSEQ_4:def 4; hence thesis by A1,A5,A8,A9,AMI_5:4; end; hence thesis; end; theorem Th6: for g being FinSequence of REAL st r in rng g holds Incr g.1 <= r & r <= Incr g.len Incr g proof let g be FinSequence of REAL; assume r in rng g; then r in rng Incr g by GOBOARD2:def 2; then consider x being set such that A1: x in dom Incr g & Incr g.x=r by FUNCT_1:def 5; A2: x in Seg len Incr g by A1,FINSEQ_1:def 3; reconsider j=x as Nat by A1; A3:1 <= j & j <= len Incr g by A2,FINSEQ_1:3; then A4: 1 <= len Incr g by AXIOMS:22; then 1 in dom Incr g by FINSEQ_3:27; hence Incr g.1 <= r by A1,A3,GOBOARD2:18; len Incr g in dom Incr g by A4,FINSEQ_3:27; hence thesis by A1,A3,GOBOARD2:18; end; theorem Th7: 1 <= i & i <= len h & 1 <= I & I <= width GoB h implies (GoB h)*(1,I)`1 <= (h/.i)`1 & (h/.i)`1 <= (GoB h)*(len GoB h,I)`1 proof assume A1: 1<=i & i<=len h & 1 <= I & I <= width GoB h; A2:GoB h=GoB(Incr(X_axis h),Incr(Y_axis(h))) by GOBOARD2:def 3; then A3:1<=1 & 1<=len GoB(Incr(X_axis h),Incr(Y_axis h)) by GOBOARD7:34; A4: len GoB h <= len GoB(Incr(X_axis h),Incr(Y_axis h)) & I<=width GoB(Incr(X_axis(h)),Incr(Y_axis(h))) by A1,GOBOARD2:def 3; then A5: [1,I] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A1,A3,GOBOARD7: 10; 1<=len GoB h by GOBOARD7:34; then A6: [len GoB h,I] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A1,A4,GOBOARD7:10; (GoB h)*(len GoB h,I)=(GoB(Incr(X_axis h),Incr(Y_axis h)))* (len GoB h,I) by GOBOARD2:def 3 .=|[Incr(X_axis h).len GoB h,Incr(Y_axis(h)).I]| by A6,GOBOARD2:def 1; then A7:(GoB h)*(len GoB h,I)`1 = Incr(X_axis h).len GoB h by EUCLID:56; (GoB h)*(1,I)=(GoB(Incr(X_axis(h)),Incr(Y_axis(h))))*(1,I) by GOBOARD2:def 3 .=|[Incr(X_axis(h)).1,Incr(Y_axis(h)).I]| by A5,GOBOARD2:def 1; then A8:(GoB h)*(1,I)`1=Incr(X_axis(h)).1 by EUCLID:56; A9: len GoB h = len Incr (X_axis h) by A2,GOBOARD2:def 1; i<=len (X_axis h) by A1,GOBOARD1:def 3; then A10:i in dom (X_axis h) by A1,FINSEQ_3:27; then (X_axis(h)).i=(h/.i)`1 by GOBOARD1:def 3; then (h/.i)`1 in rng X_axis h by A10,FUNCT_1:def 5; hence thesis by A7,A8,A9,Th6; end; theorem Th8: 1 <= i & i <= len h & 1 <= I & I <= len GoB h implies (GoB h)*(I,1)`2 <= (h/.i)`2 & (h/.i)`2 <= (GoB h)*(I,width GoB h)`2 proof assume A1: 1<=i & i<=len h & 1 <= I & I <= len GoB h; A2:GoB h=GoB(Incr(X_axis h),Incr(Y_axis h)) by GOBOARD2:def 3; A3:1<=I & I<=len GoB(Incr(X_axis h),Incr(Y_axis h)) by A1,GOBOARD2:def 3; A4:1<=width GoB h by GOBOARD7:35; width GoB h <= width GoB(Incr(X_axis h),Incr(Y_axis h)) & 1<=width GoB(Incr(X_axis h),Incr(Y_axis h)) by A2,GOBOARD7:35; then A5:[I,1] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A3,GOBOARD7: 10; A6: [I,width GoB h] in Indices GoB(Incr(X_axis h),Incr(Y_axis h)) by A1,A2,A4,GOBOARD7:10; (GoB h)*(I,width GoB h)=(GoB(Incr(X_axis h),Incr(Y_axis h)))* (I,width GoB h) by GOBOARD2:def 3 .=|[Incr(X_axis(h)).I,Incr(Y_axis(h)).width GoB h]| by A6,GOBOARD2:def 1; then A7:(GoB h)*(I,width GoB h)`2 = Incr(Y_axis h).width GoB h by EUCLID: 56; (GoB h)*(I,1)=(GoB(Incr(X_axis h),Incr(Y_axis h)))*(I,1) by GOBOARD2:def 3 .= |[Incr(X_axis(h)).I,Incr(Y_axis(h)).1]| by A5,GOBOARD2:def 1; then A8:(GoB h)*(I,1)`2 = Incr(Y_axis h).1 by EUCLID:56; A9: width GoB h = len Incr Y_axis h by A2,GOBOARD2:def 1; i <= len Y_axis h by A1,GOBOARD1:def 4; then A10:i in dom Y_axis h by A1,FINSEQ_3:27; then (Y_axis h).i = (h/.i)`2 by GOBOARD1:def 4; then (h/.i)`2 in rng Y_axis h by A10,FUNCT_1:def 5; hence thesis by A7,A8,A9,Th6; end; theorem Th9: 1 <= i & i <= len GoB f implies ex k,j st k in dom f & [i,j] in Indices GoB f & f/.k = (GoB f)*(i,j) proof assume A1: 1 <= i & i <= len GoB f; A2: GoB f = GoB(Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 3; then len Incr X_axis f = len GoB f by GOBOARD2:def 1; then i in dom Incr X_axis f by A1,FINSEQ_3:27; then (Incr X_axis f).i in rng Incr X_axis f by FUNCT_1:def 5; then (Incr X_axis f).i in rng X_axis f by GOBOARD2:def 2; then consider k such that A3: k in dom X_axis f and A4: (X_axis f).k = (Incr X_axis f).i by FINSEQ_2:11; A5: len X_axis f = len f by GOBOARD1:def 3 .= len Y_axis f by GOBOARD1:def 4; then dom X_axis f = dom Y_axis f by FINSEQ_3:31; then (Y_axis f).k in rng Y_axis f by A3,FUNCT_1:def 5; then (Y_axis f).k in rng Incr Y_axis f by GOBOARD2:def 2; then consider j such that A6: j in dom Incr Y_axis f and A7: (Y_axis f).k = (Incr Y_axis f).j by FINSEQ_2:11; take k,j; len X_axis f = len f by GOBOARD1:def 3; hence k in dom f by A3,FINSEQ_3:31; A8: i in dom GoB f by A1,FINSEQ_3:27; j in Seg len Incr Y_axis f by A6,FINSEQ_1:def 3; then j in Seg width GoB(Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 1; then [i,j] in [:dom GoB f, Seg width GoB f:] by A2,A8,ZFMISC_1:106; hence A9: [i,j] in Indices GoB f by MATRIX_1:def 5; dom X_axis f = dom Y_axis f by A5,FINSEQ_3:31; then (X_axis f).k = (f/.k)`1 & (Y_axis f).k = (f/.k)`2 by A3,GOBOARD1:def 3,def 4; hence f/.k = |[Incr(X_axis f).i,Incr(Y_axis f).j]| by A4,A7,EUCLID:57 .= (GoB f)*(i,j) by A2,A9,GOBOARD2:def 1; end; theorem Th10: 1 <= j & j <= width GoB f implies ex k,i st k in dom f & [i,j] in Indices GoB f & f/.k = (GoB f)*(i,j) proof assume A1: 1 <= j & j <= width GoB f; A2: GoB f = GoB(Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 3; then len Incr Y_axis f = width GoB f by GOBOARD2:def 1; then j in dom Incr Y_axis f by A1,FINSEQ_3:27; then (Incr Y_axis f).j in rng Incr Y_axis f by FUNCT_1:def 5; then (Incr Y_axis f).j in rng Y_axis f by GOBOARD2:def 2; then consider k such that A3: k in dom Y_axis f and A4: (Y_axis f).k = (Incr Y_axis f).j by FINSEQ_2:11; A5: len X_axis f = len f by GOBOARD1:def 3 .= len Y_axis f by GOBOARD1:def 4; then k in dom X_axis f by A3,FINSEQ_3:31; then (X_axis f).k in rng X_axis f by FUNCT_1:def 5; then (X_axis f).k in rng Incr X_axis f by GOBOARD2:def 2; then consider i such that A6: i in dom Incr X_axis f and A7: (X_axis f).k = (Incr X_axis f).i by FINSEQ_2:11; take k,i; len Y_axis f = len f by GOBOARD1:def 4; hence k in dom f by A3,FINSEQ_3:31; A8: j in Seg width GoB f by A1,FINSEQ_1:3; len GoB(Incr X_axis f,Incr Y_axis f) = len Incr X_axis f by GOBOARD2:def 1; then i in dom GoB(Incr X_axis f,Incr Y_axis f) by A6,FINSEQ_3:31; then [i,j] in [:dom GoB f, Seg width GoB f:] by A2,A8,ZFMISC_1:106; hence A9: [i,j] in Indices GoB f by MATRIX_1:def 5; k in dom X_axis f & k in dom Y_axis f by A3,A5,FINSEQ_3:31; then (X_axis f).k = (f/.k)`1 & (Y_axis f).k = (f/.k)`2 by GOBOARD1:def 3,def 4; hence f/.k = |[Incr(X_axis f).i,Incr(Y_axis f).j]| by A4,A7,EUCLID:57 .= (GoB f)*(i,j) by A2,A9,GOBOARD2:def 1; end; theorem Th11: 1 <= i & i <= len GoB f & 1 <= j & j <= width GoB f implies ex k st k in dom f & [i,j] in Indices GoB f & (f/.k)`1 = (GoB f)*(i,j)`1 proof assume A1: 1 <= i & i <= len GoB f & 1 <= j & j <= width GoB f; A2: GoB f = GoB (Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 3; then len Incr X_axis f = len GoB f by GOBOARD2:def 1; then i in dom Incr X_axis f by A1,FINSEQ_3:27; then (Incr X_axis f).i in rng Incr X_axis f by FUNCT_1:def 5; then (Incr X_axis f).i in rng X_axis f by GOBOARD2:def 2; then consider k such that A3: k in dom X_axis f and A4: (X_axis f).k = (Incr X_axis f).i by FINSEQ_2:11; len Incr Y_axis f = width GoB f by A2,GOBOARD2:def 1; then A5: j in dom Incr Y_axis f by A1,FINSEQ_3:27; take k; len X_axis f = len f by GOBOARD1:def 3; hence k in dom f by A3,FINSEQ_3:31; A6: i in dom GoB f by A1,FINSEQ_3:27; j in Seg len Incr Y_axis f by A5,FINSEQ_1:def 3; then j in Seg width GoB(Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 1; then [i,j] in [:dom GoB f, Seg width GoB f:] by A2,A6,ZFMISC_1:106; hence [i,j] in Indices GoB f by MATRIX_1:def 5; then A7:(GoB f)*(i,j) = |[Incr(X_axis f).i,Incr(Y_axis f).j]| by A2,GOBOARD2:def 1; thus (f/.k)`1 = Incr(X_axis f).i by A3,A4,GOBOARD1:def 3 .= (GoB f)*(i,j)`1 by A7,EUCLID:56; end; theorem Th12: 1 <= i & i <= len GoB f & 1 <= j & j <= width GoB f implies ex k st k in dom f & [i,j] in Indices GoB f & (f/.k)`2 = (GoB f)*(i,j)`2 proof assume A1: 1 <= i & i <= len GoB f & 1 <= j & j <= width GoB f; A2: GoB f = GoB (Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 3; then len Incr Y_axis f = width GoB f by GOBOARD2:def 1; then j in dom Incr Y_axis f by A1,FINSEQ_3:27; then (Incr Y_axis f).j in rng Incr Y_axis f by FUNCT_1:def 5; then (Incr Y_axis f).j in rng Y_axis f by GOBOARD2:def 2; then consider k such that A3: k in dom Y_axis f and A4: (Y_axis f).k = (Incr Y_axis f).j by FINSEQ_2:11; len Incr Y_axis f = width GoB f by A2,GOBOARD2:def 1; then A5: j in dom Incr Y_axis f by A1,FINSEQ_3:27; take k; len Y_axis f = len f by GOBOARD1:def 4; hence k in dom f by A3,FINSEQ_3:31; A6: i in dom GoB f by A1,FINSEQ_3:27; j in Seg len Incr Y_axis f by A5,FINSEQ_1:def 3; then j in Seg width GoB(Incr X_axis f,Incr Y_axis f) by GOBOARD2:def 1; then [i,j] in [:dom GoB f, Seg width GoB f:] by A2,A6,ZFMISC_1:106; hence [i,j] in Indices GoB f by MATRIX_1:def 5; then A7: (GoB f)*(i,j) = |[Incr(X_axis f).i,Incr(Y_axis f).j]| by A2,GOBOARD2:def 1; thus (f/.k)`2 = Incr(Y_axis f).j by A3,A4,GOBOARD1:def 4 .= (GoB f)*(i,j)`2 by A7,EUCLID:56; end; begin :: Extrema of projections theorem Th13: 1 <= i & i <= len h implies S-bound L~h <= (h/.i)`2 & (h/.i)`2 <= N-bound L~h proof len h > 4 by GOBOARD7:36; then A1: len h >= 2 by AXIOMS:22; assume 1<=i & i<=len h; then i in dom h by FINSEQ_3:27; then h/.i in L~h by A1,GOBOARD1:16; hence thesis by PSCOMP_1:71; end; theorem Th14: 1 <= i & i <= len h implies W-bound L~h <= (h/.i)`1 & (h/.i)`1 <= E-bound L~h proof len h > 4 by GOBOARD7:36; then A1: len h >= 2 by AXIOMS:22; assume 1<=i & i<=len h; then i in dom h by FINSEQ_3:27; then h/.i in L~h by A1,GOBOARD1:16; hence thesis by PSCOMP_1:71; end; theorem Th15: for X being Subset of REAL st X = { q`2 : q`1 = W-bound L~h & q in L~h } holds X = (proj2 || W-most L~h).:the carrier of (TOP-REAL 2)|(W-most L~h) proof set T = (TOP-REAL 2)|(W-most L~h); set F = proj2 || W-most L~h; let X be Subset of REAL such that A1: X = { q`2 : q`1 = W-bound L~h & q in L~h }; thus X c= (proj2 || W-most L~h).:the carrier of T proof let x be set; assume x in X; then consider q1 being Point of TOP-REAL 2 such that A2: q1`2 = x & q1`1 = W-bound L~h & q1 in L~h by A1; A3: dom F = the carrier of T by FUNCT_2:def 1 .= [#] ((TOP-REAL 2)|(W-most L~h)) by PRE_TOPC:12 .= W-most L~h by PRE_TOPC:def 10; q1 in W-most L~h by A2,SPRECT_2:16; then q1 in dom F & q1 in the carrier of T & x = F.q1 by A2,A3,FUNCT_2:def 1,PSCOMP_1:70; hence thesis by FUNCT_1:def 12; end; thus (proj2 || W-most L~h).:the carrier of T c= X proof let x be set; assume x in (proj2 || W-most L~h).:the carrier of T; then consider x1 be set such that A4: x1 in dom F & x1 in the carrier of T & x = F.x1 by FUNCT_1:def 12; x1 in [#] ((TOP-REAL 2)|(W-most L~h)) by A4,PRE_TOPC:12; then A5: x1 in W-most L~h by PRE_TOPC:def 10; then reconsider x2 = x1 as Element of TOP-REAL 2; A6: x = x2`2 by A4,A5,PSCOMP_1:70; A7: x2`1 = (W-min L~h)`1 by A5,PSCOMP_1:88 .= W-bound L~h by PSCOMP_1:84; W-most L~h = LSeg(SW-corner L~h, NW-corner L~h) /\ L~h by PSCOMP_1:def 38; then W-most L~h c= L~h by XBOOLE_1:17; hence thesis by A1,A5,A6,A7; end; end; theorem Th16: for X being Subset of REAL st X = { q`2 : q`1 = E-bound L~h & q in L~h } holds X = (proj2 || E-most L~h).:the carrier of (TOP-REAL 2)|(E-most L~h) proof set T = (TOP-REAL 2)|(E-most L~h); set F = proj2 || E-most L~h; let X be Subset of REAL such that A1: X = { q`2 : q`1 = E-bound L~h & q in L~h }; thus X c= (proj2 || E-most L~h).:the carrier of T proof let x be set; assume x in X; then consider q1 being Point of TOP-REAL 2 such that A2: q1`2 = x & q1`1 = E-bound L~h & q1 in L~h by A1; A3: dom F = the carrier of T by FUNCT_2:def 1 .= [#] ((TOP-REAL 2)|(E-most L~h)) by PRE_TOPC:12 .= E-most L~h by PRE_TOPC:def 10; q1 in E-most L~h by A2,SPRECT_2:17; then q1 in dom F & q1 in the carrier of T & x = F.q1 by A2,A3,FUNCT_2:def 1,PSCOMP_1:70; hence thesis by FUNCT_1:def 12; end; thus (proj2 || E-most L~h).:the carrier of T c= X proof let x be set; assume x in (proj2 || E-most L~h).:the carrier of T; then consider x1 be set such that A4: x1 in dom F & x1 in the carrier of T & x = F.x1 by FUNCT_1:def 12; x1 in [#] ((TOP-REAL 2)|(E-most L~h)) by A4,PRE_TOPC:12; then A5: x1 in E-most L~h by PRE_TOPC:def 10; then reconsider x2 = x1 as Element of TOP-REAL 2; A6: x = x2`2 by A4,A5,PSCOMP_1:70; A7: x2`1 = (E-min L~h)`1 by A5,PSCOMP_1:108 .= E-bound L~h by PSCOMP_1:104; E-most L~h = LSeg(SE-corner L~h, NE-corner L~h) /\ L~h by PSCOMP_1:def 40; then E-most L~h c= L~h by XBOOLE_1:17; hence thesis by A1,A5,A6,A7; end; end; theorem Th17: for X being Subset of REAL st X = { q`1 : q`2 = N-bound L~h & q in L~h } holds X = (proj1 || N-most L~h).:the carrier of (TOP-REAL 2)|(N-most L~h) proof set T = (TOP-REAL 2)|(N-most L~h); set F = proj1 || N-most L~h; let X be Subset of REAL such that A1: X = { q`1 : q`2 = N-bound L~h & q in L~h }; thus X c= (proj1 || N-most L~h).:the carrier of T proof let x be set; assume x in X; then consider q1 being Point of TOP-REAL 2 such that A2: q1`1 = x & q1`2 = N-bound L~h & q1 in L~h by A1; A3: dom F = the carrier of T by FUNCT_2:def 1 .= [#] ((TOP-REAL 2)|(N-most L~h)) by PRE_TOPC:12 .= N-most L~h by PRE_TOPC:def 10; q1 in N-most L~h by A2,SPRECT_2:14; then q1 in dom F & q1 in the carrier of T & x = F.q1 by A2,A3,FUNCT_2:def 1,PSCOMP_1:69; hence thesis by FUNCT_1:def 12; end; thus (proj1 || N-most L~h).:the carrier of T c= X proof let x be set; assume x in (proj1 || N-most L~h).:the carrier of T; then consider x1 be set such that A4: x1 in dom F & x1 in the carrier of T & x = F.x1 by FUNCT_1:def 12; x1 in [#] ((TOP-REAL 2)|(N-most L~h)) by A4,PRE_TOPC:12; then A5: x1 in N-most L~h by PRE_TOPC:def 10; then reconsider x2 = x1 as Element of TOP-REAL 2; A6: x = x2`1 by A4,A5,PSCOMP_1:69; A7: x2`2 = (N-min L~h)`2 by A5,PSCOMP_1:98 .= N-bound L~h by PSCOMP_1:94; N-most L~h = LSeg(NW-corner L~h, NE-corner L~h) /\ L~h by PSCOMP_1:def 39; then N-most L~h c= L~h by XBOOLE_1:17; hence thesis by A1,A5,A6,A7; end; end; theorem Th18: for X being Subset of REAL st X = { q`1 : q`2 = S-bound L~h & q in L~h } holds X = (proj1 || S-most L~h).:the carrier of (TOP-REAL 2)|(S-most L~h) proof set T = (TOP-REAL 2)|(S-most L~h); set F = proj1 || S-most L~h; let X be Subset of REAL such that A1: X = { q`1 : q`2 = S-bound L~h & q in L~h }; thus X c= (proj1 || S-most L~h).:the carrier of T proof let x be set; assume x in X; then consider q1 being Point of TOP-REAL 2 such that A2: q1`1 = x & q1`2 = S-bound L~h & q1 in L~h by A1; A3: dom F = the carrier of T by FUNCT_2:def 1 .= [#] ((TOP-REAL 2)|(S-most L~h)) by PRE_TOPC:12 .= S-most L~h by PRE_TOPC:def 10; q1 in S-most L~h by A2,SPRECT_2:15; then q1 in dom F & q1 in the carrier of T & x = F.q1 by A2,A3,FUNCT_2:def 1,PSCOMP_1:69; hence thesis by FUNCT_1:def 12; end; thus (proj1 || S-most L~h).:the carrier of T c= X proof let x be set; assume x in (proj1 || S-most L~h).:the carrier of T; then consider x1 be set such that A4: x1 in dom F & x1 in the carrier of T & x = F.x1 by FUNCT_1:def 12; x1 in [#] ((TOP-REAL 2)|(S-most L~h)) by A4,PRE_TOPC:12; then A5: x1 in S-most L~h by PRE_TOPC:def 10; then reconsider x2 = x1 as Element of TOP-REAL 2; A6: x = x2`1 by A4,A5,PSCOMP_1:69; A7: x2`2 = (S-min L~h)`2 by A5,PSCOMP_1:118 .= S-bound L~h by PSCOMP_1:114; S-most L~h = LSeg(SW-corner L~h, SE-corner L~h) /\ L~h by PSCOMP_1:def 41; then S-most L~h c= L~h by XBOOLE_1:17; hence thesis by A1,A5,A6,A7; end; end; theorem Th19: for X being Subset of REAL st X = { q`1 : q in L~g } holds X = (proj1 || L~g).:the carrier of (TOP-REAL 2)|L~g proof set T = (TOP-REAL 2)|L~g; set F = proj1 || L~g; let X be Subset of REAL such that A1: X = { q`1 : q in L~g }; thus X c= (proj1 || L~g).:the carrier of T proof let x be set; assume x in X; then consider q1 being Point of TOP-REAL 2 such that A2: q1`1 = x & q1 in L~g by A1; dom F = the carrier of T by FUNCT_2:def 1 .= [#] ((TOP-REAL 2)|L~g) by PRE_TOPC:12 .= L~g by PRE_TOPC:def 10; then q1 in dom F & q1 in the carrier of T & x = F.q1 by A2,FUNCT_2:def 1,PSCOMP_1:69; hence thesis by FUNCT_1:def 12; end; thus (proj1 || L~g).:the carrier of T c= X proof let x be set; assume x in (proj1 || L~g).:the carrier of T; then consider x1 be set such that A3: x1 in dom F & x1 in the carrier of T & x = F.x1 by FUNCT_1:def 12; x1 in [#] ((TOP-REAL 2)|L~g) by A3,PRE_TOPC:12; then A4: x1 in L~g by PRE_TOPC:def 10; then reconsider x2 = x1 as Element of TOP-REAL 2; x = x2`1 by A3,A4,PSCOMP_1:69; hence thesis by A1,A4; end; end; theorem Th20: for X being Subset of REAL st X = { q`2 : q in L~g } holds X = (proj2 || L~g).:the carrier of (TOP-REAL 2)|L~g proof set T = (TOP-REAL 2)|L~g; set F = proj2 || L~g; let X be Subset of REAL such that A1: X = { q`2 : q in L~g }; thus X c= (proj2 || L~g).:the carrier of T proof let x be set; assume x in X; then consider q1 being Point of TOP-REAL 2 such that A2: q1`2 = x & q1 in L~g by A1; dom F = the carrier of T by FUNCT_2:def 1 .= [#] ((TOP-REAL 2)|L~g) by PRE_TOPC:12 .= L~g by PRE_TOPC:def 10; then q1 in dom F & q1 in the carrier of T & x = F.q1 by A2,FUNCT_2:def 1,PSCOMP_1:70; hence thesis by FUNCT_1:def 12; end; thus (proj2 || L~g).:the carrier of T c= X proof let x be set; assume x in (proj2 || L~g).:the carrier of T; then consider x1 be set such that A3: x1 in dom F & x1 in the carrier of T & x = F.x1 by FUNCT_1:def 12; x1 in [#] ((TOP-REAL 2)|L~g) by A3,PRE_TOPC:12; then A4: x1 in L~g by PRE_TOPC:def 10; then reconsider x2 = x1 as Element of TOP-REAL 2; x = x2`2 by A3,A4,PSCOMP_1:70; hence thesis by A1,A4; end; end; theorem Th21: for X being Subset of REAL st X = { q`2 : q`1 = W-bound L~h & q in L~h } holds lower_bound X = inf (proj2 || W-most L~h) proof set T = (TOP-REAL 2)|(W-most L~h); let X be Subset of REAL; assume X = { q`2 : q`1 = W-bound L~h & q in L~h }; then X = (proj2 || W-most L~h).:the carrier of T by Th15; hence thesis by PSCOMP_1:def 20; end; theorem Th22: for X being Subset of REAL st X = { q`2 : q`1 = W-bound L~h & q in L~h } holds upper_bound X = sup (proj2 || W-most L~h) proof set T = (TOP-REAL 2)|(W-most L~h); set F = proj2 || W-most L~h; let X be Subset of REAL; assume X = { q`2 : q`1 = W-bound L~h & q in L~h }; then X = F.:the carrier of T by Th15; hence thesis by PSCOMP_1:def 21; end; theorem Th23: for X being Subset of REAL st X = { q`2 : q`1 = E-bound L~h & q in L~h } holds lower_bound X = inf (proj2 || E-most L~h) proof set T = (TOP-REAL 2)|(E-most L~h); set F = proj2 || E-most L~h; let X be Subset of REAL; assume X = { q`2 : q`1 = E-bound L~h & q in L~h }; then X = F.:the carrier of T by Th16; hence thesis by PSCOMP_1:def 20; end; theorem Th24: for X being Subset of REAL st X = { q`2 : q`1 = E-bound L~h & q in L~h } holds upper_bound X = sup (proj2 || E-most L~h) proof set T = (TOP-REAL 2)|(E-most L~h); set F = proj2 || E-most L~h; let X be Subset of REAL; assume X = { q`2 : q`1 = E-bound L~h & q in L~h }; then X = F.:the carrier of T by Th16; hence thesis by PSCOMP_1:def 21; end; theorem Th25: for X being Subset of REAL st X = { q`1 : q in L~g } holds lower_bound X = inf (proj1 || L~g) proof set T = (TOP-REAL 2)|L~g; set F = proj1 || L~g; let X be Subset of REAL; assume X = { q`1 : q in L~g }; then X = F.:the carrier of T by Th19; hence thesis by PSCOMP_1:def 20; end; theorem Th26: for X being Subset of REAL st X = { q`1 : q`2 = S-bound L~h & q in L~h } holds lower_bound X = inf (proj1 || S-most L~h) proof set T = (TOP-REAL 2)|(S-most L~h); set F = proj1 || S-most L~h; let X be Subset of REAL; assume X = { q`1 : q`2 = S-bound L~h & q in L~h }; then X = F.:the carrier of T by Th18; hence thesis by PSCOMP_1:def 20; end; theorem Th27: for X being Subset of REAL st X = { q`1 : q`2 = S-bound L~h & q in L~h } holds upper_bound X = sup (proj1 || S-most L~h) proof set T = (TOP-REAL 2)|(S-most L~h); set F = proj1 || S-most L~h; let X be Subset of REAL; assume X = { q`1 : q`2 = S-bound L~h & q in L~h }; then X = F.:the carrier of T by Th18; hence thesis by PSCOMP_1:def 21; end; theorem Th28: for X being Subset of REAL st X = { q`1 : q`2 = N-bound L~h & q in L~h } holds lower_bound X = inf (proj1 || N-most L~h) proof set T = (TOP-REAL 2)|(N-most L~h); set F = proj1 || N-most L~h; let X be Subset of REAL; assume X = { q`1 : q`2 = N-bound L~h & q in L~h }; then X = F.:the carrier of T by Th17; hence thesis by PSCOMP_1:def 20; end; theorem Th29: for X being Subset of REAL st X = { q`1 : q`2 = N-bound L~h & q in L~h } holds upper_bound X = sup (proj1 || N-most L~h) proof set T = (TOP-REAL 2)|(N-most L~h); set F = proj1 || N-most L~h; let X be Subset of REAL; assume X = { q`1 : q`2 = N-bound L~h & q in L~h }; then X = F.:the carrier of T by Th17; hence thesis by PSCOMP_1:def 21; end; theorem Th30: for X being Subset of REAL st X = { q`2 : q in L~g } holds lower_bound X = inf (proj2 || L~g) proof set T = (TOP-REAL 2)|L~g; set F = proj2 || L~g; let X be Subset of REAL; assume X = { q`2 : q in L~g }; then X = F.:the carrier of T by Th20; hence thesis by PSCOMP_1:def 20; end; theorem Th31: for X being Subset of REAL st X = { q`1 : q in L~g } holds upper_bound X = sup (proj1 || L~g) proof set T = (TOP-REAL 2)|L~g; set F = proj1 || L~g; let X be Subset of REAL; assume X = { q`1 : q in L~g }; then X = F.:the carrier of T by Th19; hence thesis by PSCOMP_1:def 21; end; theorem Th32: for X being Subset of REAL st X = { q`2 : q in L~g } holds upper_bound X = sup (proj2 || L~g) proof set T = (TOP-REAL 2)|L~g; set F = proj2 || L~g; let X be Subset of REAL; assume X = { q`2 : q in L~g }; then X = F.:the carrier of T by Th20; hence thesis by PSCOMP_1:def 21; end; theorem Th33: p in L~h & 1 <= I & I <= width GoB h implies (GoB h)*(1,I)`1 <= p`1 proof assume A1: p in L~h & 1 <= I & I <= width GoB h; then consider i such that A2:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by SPPOL_2:14; i<=i+1 by NAT_1:29; then i<=len h by A2,AXIOMS:22; then A3: (GoB h)*(1,I)`1<=(h/.i)`1 by A1,A2,Th7; 1<=i+1 by NAT_1:29; then A4: (GoB h)*(1,I)`1<=(h/.(i+1))`1 by A1,A2,Th7; now per cases; case (h/.i)`1<=(h/.(i+1))`1; then (h/.i)`1<=p`1 by A2,TOPREAL1:9; hence (GoB h)*(1,I)`1<=p`1 by A3,AXIOMS:22; case (h/.i)`1>(h/.(i+1))`1; then (h/.(i+1))`1<=p`1 by A2,TOPREAL1:9; hence (GoB h)*(1,I)`1<=p`1 by A4,AXIOMS:22; end; hence (GoB h)*(1,I)`1<=p`1; end; theorem Th34: p in L~h & 1 <= I & I <= width GoB h implies p`1 <= (GoB h)*(len GoB h,I)`1 proof assume A1: p in L~h & 1 <= I & I <= width GoB h; then consider i such that A2:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by SPPOL_2:14; i<=i+1 by NAT_1:29; then A3: i<=len h by A2,AXIOMS:22; A4: 1<=i+1 by NAT_1:29; A5: (GoB h)*(len GoB h,I)`1>=(h/.i)`1 by A1,A2,A3,Th7; A6: (GoB h)*(len GoB h,I)`1>=(h/.(i+1))`1 by A1,A2,A4,Th7; now per cases; case (h/.i)`1<=(h/.(i+1))`1; then (h/.(i+1))`1>=p`1 by A2,TOPREAL1:9; hence (GoB h)*(len GoB h,I)`1>=p`1 by A6,AXIOMS:22; case (h/.i)`1>(h/.(i+1))`1; then (h/.i)`1>=p`1 by A2,TOPREAL1:9; hence (GoB h)*(len GoB h, I)`1>=p`1 by A5,AXIOMS:22; end; hence (GoB h)*(len GoB h, I)`1>=p`1; end; theorem Th35: p in L~h & 1 <= I & I <= len GoB h implies (GoB h)*(I,1)`2 <= p`2 proof assume A1: p in L~h & 1 <= I & I <= len GoB h; then consider i such that A2:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by SPPOL_2:14; i<=i+1 by NAT_1:29; then i<=len h by A2,AXIOMS:22; then A3: (GoB h)*(I,1)`2<=(h/.i)`2 by A1,A2,Th8; 1<=i+1 by NAT_1:29; then A4: (GoB h)*(I,1)`2<=(h/.(i+1))`2 by A1,A2,Th8; now per cases; case (h/.i)`2<=(h/.(i+1))`2; then (h/.i)`2<=p`2 by A2,TOPREAL1:10; hence (GoB h)*(I,1)`2<=p`2 by A3,AXIOMS:22; case (h/.i)`2>(h/.(i+1))`2; then (h/.(i+1))`2<=p`2 by A2,TOPREAL1:10; hence (GoB h)*(I,1)`2<=p`2 by A4,AXIOMS:22; end; hence (GoB h)*(I,1)`2<=p`2; end; theorem Th36: p in L~h & 1 <= I & I <= len GoB h implies p`2 <= (GoB h)*(I,width GoB h)`2 proof assume A1: p in L~h & 1 <= I & I <= len GoB h; then consider i such that A2:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by SPPOL_2:14; i<=i+1 by NAT_1:29; then A3: i<=len h by A2,AXIOMS:22; A4: 1<=i+1 by NAT_1:29; A5: (GoB h)*(I,width GoB h)`2>=(h/.i)`2 by A1,A2,A3,Th8; A6: (GoB h)*(I,width GoB h)`2>=(h/.(i+1))`2 by A1,A2,A4,Th8; now per cases; case (h/.i)`2<=(h/.(i+1))`2; then (h/.(i+1))`2>=p`2 by A2,TOPREAL1:10; hence (GoB h)*(I,width GoB h)`2>=p`2 by A6,AXIOMS:22; case (h/.i)`2>(h/.(i+1))`2; then (h/.i)`2>=p`2 by A2,TOPREAL1:10; hence (GoB h)*(I,width GoB h)`2>=p`2 by A5,AXIOMS:22; end; hence (GoB h)*(I,width GoB h)`2>=p`2; end; theorem Th37: 1 <= i & i <= len GoB h & 1 <= j & j <= width GoB h implies ex q st q`1 = (GoB h)*(i,j)`1 & q in L~h proof assume 1 <= i & i <= len GoB h & 1 <= j & j <= width GoB h; then consider k such that A1: k in dom h & [i,j] in Indices GoB h & (h/.k)`1 = (GoB h)* (i,j)`1 by Th11; take q = h/.k; thus q`1 = (GoB h)*(i,j)`1 by A1; 4 < len h by GOBOARD7:36; then 2 <= len h by AXIOMS:22; hence thesis by A1,GOBOARD1:16; end; theorem Th38: 1 <= i & i <= len GoB h & 1 <= j & j <= width GoB h implies ex q st q`2 = (GoB h)*(i,j)`2 & q in L~h proof assume 1 <= i & i <= len GoB h & 1 <= j & j <= width GoB h; then consider k such that A1: k in dom h & [i,j] in Indices GoB h & (h/.k)`2 = (GoB h)*(i,j)`2 by Th12; take q = h/.k; thus q`2 = (GoB h)*(i,j)`2 by A1; 4 < len h by GOBOARD7:36; then 2 <= len h by AXIOMS:22; hence thesis by A1,GOBOARD1:16; end; theorem Th39: W-bound L~h = (GoB h)*(1,1)`1 proof set X = { q`1 : q in L~h }, A = (GoB h)*(1,1)`1; consider a be set such that A1: a in L~h by XBOOLE_0:def 1; reconsider a as Point of TOP-REAL 2 by A1; A2: a`1 in X by A1; X c= REAL proof let b be set; assume b in X; then consider qq be Point of TOP-REAL 2 such that A3: b = qq`1 & qq in L~h; thus thesis by A3; end; then reconsider X as non empty Subset of REAL by A2; lower_bound X = A proof A4:1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; A5: for p be real number st p in X holds p >= A proof let p be real number; assume p in X; then consider s be Point of TOP-REAL 2 such that A6: p = s`1 & s in L~h; thus thesis by A4,A6,Th33; end; consider q1 be Point of TOP-REAL 2 such that A7: q1`1 = (GoB h)*(1,1)`1 & q1 in L~h by A4,Th37; reconsider q11 = q1`1 as Real; for q be real number st for p be real number st p in X holds p >= q holds A >= q proof let q be real number such that A8: for p be real number st p in X holds p >= q; q11 in X by A7; hence thesis by A7,A8; end; hence thesis by A5,PSCOMP_1:9; end; then A = inf (proj1 || L~h) by Th25; hence thesis by PSCOMP_1:def 30; end; theorem Th40: S-bound L~h = (GoB h)*(1,1)`2 proof set X = { q`2 : q in L~h }, A = (GoB h)*(1,1)`2; consider a be set such that A1: a in L~h by XBOOLE_0:def 1; reconsider a as Point of TOP-REAL 2 by A1; A2: a`2 in X by A1; X c= REAL proof let b be set; assume b in X; then consider qq be Point of TOP-REAL 2 such that A3: b = qq`2 & qq in L~h; thus thesis by A3; end; then reconsider X as non empty Subset of REAL by A2; lower_bound X = A proof A4:1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; A5: for p be real number st p in X holds p >= A proof let p be real number; assume p in X; then consider s be Point of TOP-REAL 2 such that A6: p = s`2 & s in L~h; thus thesis by A4,A6,Th35; end; consider q1 be Point of TOP-REAL 2 such that A7: q1`2 = (GoB h)*(1,1)`2 & q1 in L~h by A4,Th38; reconsider q11 = q1`2 as Real; for q be real number st for p be real number st p in X holds p >= q holds A >= q proof let q be real number such that A8: for p be real number st p in X holds p >= q; q11 in X by A7; hence thesis by A7,A8; end; hence thesis by A5,PSCOMP_1:9; end; then A = inf (proj2 || L~h) by Th30; hence thesis by PSCOMP_1:def 33; end; theorem Th41: E-bound L~h = (GoB h)*(len GoB h,1)`1 proof set X = { q`1 : q in L~h }, A = (GoB h)*(len GoB h,1)`1; consider a be set such that A1: a in L~h by XBOOLE_0:def 1; reconsider a as Point of TOP-REAL 2 by A1; A2: a`1 in X by A1; X c= REAL proof let b be set; assume b in X; then consider qq be Point of TOP-REAL 2 such that A3: b = qq`1 & qq in L~h; thus thesis by A3; end; then reconsider X as non empty Subset of REAL by A2; upper_bound X = A proof A4: for p be real number st p in X holds p <= A proof let p be real number; assume p in X; then consider s be Point of TOP-REAL 2 such that A5: p = s`1 & s in L~h; 1 <= width GoB h by GOBOARD7:35; hence thesis by A5,Th34; end; 1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; then consider q1 be Point of TOP-REAL 2 such that A6: q1`1 = (GoB h)*(len GoB h,1)`1 & q1 in L~h by Th37; reconsider q11 = q1`1 as Real; for q be real number st for p be real number st p in X holds p <= q holds A <= q proof let q be real number such that A7: for p be real number st p in X holds p <= q; q11 in X by A6; hence thesis by A6,A7; end; hence thesis by A4,PSCOMP_1:11; end; then A = sup (proj1 || L~h) by Th31; hence thesis by PSCOMP_1:def 32; end; theorem Th42: N-bound L~h = (GoB h)*(1,width GoB h)`2 proof set X = { q`2 : q in L~h }, A = (GoB h)*(1,width GoB h)`2; consider a be set such that A1: a in L~h by XBOOLE_0:def 1; reconsider a as Point of TOP-REAL 2 by A1; A2: a`2 in X by A1; X c= REAL proof let b be set; assume b in X; then consider qq be Point of TOP-REAL 2 such that A3: b = qq`2 & qq in L~h; thus thesis by A3; end; then reconsider X as non empty Subset of REAL by A2; upper_bound X = A proof A4:1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; A5: for p be real number st p in X holds p <= A proof let p be real number; assume p in X; then consider s be Point of TOP-REAL 2 such that A6: p = s`2 & s in L~h; thus thesis by A4,A6,Th36; end; consider q1 be Point of TOP-REAL 2 such that A7: q1`2 = (GoB h)*(1,width GoB h)`2 & q1 in L~h by A4,Th38; reconsider q11 = q1`2 as Real; for q be real number st for p be real number st p in X holds p <= q holds A <= q proof let q be real number such that A8: for p be real number st p in X holds p <= q; q11 in X by A7; hence thesis by A7,A8; end; hence thesis by A5,PSCOMP_1:11; end; then A = sup (proj2 || L~h) by Th32; hence thesis by PSCOMP_1:def 31; end; theorem Th43: for Y being non empty finite Subset of NAT st 1 <= i & i <= len f & 1 <= I & I <= len GoB f & Y = { j : [I,j] in Indices GoB f & ex k st k in dom f & f/.k = (GoB f)*(I,j) } & (f/.i)`1 = ((GoB f)*(I,1))`1 & i1 = min Y holds (GoB f)*(I,i1)`2 <= (f/.i)`2 proof let Y be non empty finite Subset of NAT; assume A1: 1<=i & i<=len f & 1<=I & I<=len GoB f & Y={j:[I,j] in Indices GoB f & ex k st k in dom f & f/.k=(GoB f)*(I,j)} & (f/.i)`1=((GoB f)*(I,1))`1 & i1=min Y; then i1 in Y by CQC_SIM1:def 8; then consider j such that A2:i1=j & [I,j] in Indices GoB f & ex k st k in dom f & f/.k=(GoB f)*(I,j) by A1; A3:i in dom f by A1,FINSEQ_3:27; then consider i2,j2 be Nat such that A4: [i2,j2] in Indices GoB f & f/.i=(GoB f)*(i2,j2) by GOBOARD5:12; A5:1<=i2 & i2<=len GoB f & 1<=j2 & j2<=width GoB f by A4,GOBOARD5:1; then A6:(f/.i)`1=((GoB f)*(I,j2))`1 by A1,GOBOARD5:3; A7:(f/.i)`2=((GoB f)*(1,j2))`2 by A4,A5,GOBOARD5:2 .=((GoB f)*(I,j2))`2 by A1,A5,GOBOARD5:2; f/.i=|[(f/.i)`1,(f/.i)`2]| by EUCLID:57; then A8: f/.i=(GoB f)*(I,j2) by A6,A7,EUCLID:57; [I,j2] in Indices GoB f by A1,A5,GOBOARD7:10; then j2 in Y by A1,A3,A8; then A9:1<=i1 & i1<=j2 & j2<=width GoB f & 1<=1 & 1<=len GoB f by A1,A2,A4,AXIOMS:22,CQC_SIM1:def 8,GOBOARD5:1; now per cases; case i1<j2; hence (GoB f)*(I,i1)`2<=(GoB f)*(I,j2)`2 by A1,A9,GOBOARD5:5; case i1>=j2; hence (GoB f)*(I,i1)`2<=(GoB f)*(I,j2)`2 by A9,AXIOMS:21; end; hence thesis by A7; end; theorem Th44: for Y being non empty finite Subset of NAT st 1 <= i & i <= len h & 1 <= I & I <= width GoB h & Y = { j : [j,I] in Indices GoB h & ex k st k in dom h & h/.k = (GoB h)*(j,I) } & (h/.i)`2 = ((GoB h)* (1,I))`2 & i1 = min Y holds (GoB h)*(i1,I)`1 <= (h/.i)`1 proof let Y be non empty finite Subset of NAT; assume A1: 1<=i & i<=len h & 1 <= I & I <= width GoB h & Y={j:[j,I] in Indices GoB h & ex k st k in dom h & h/.k=(GoB h)*(j,I)} & (h/.i)`2=((GoB h)*(1,I))`2 & i1=min Y; then i1 in Y by CQC_SIM1:def 8; then consider j such that A2:i1=j & [j,I] in Indices GoB h & ex k st k in dom h & h/.k=(GoB h)*(j,I) by A1; A3:i in dom h by A1,FINSEQ_3:27; then consider i2,j2 be Nat such that A4: [i2,j2] in Indices GoB h & h/.i=(GoB h)*(i2,j2) by GOBOARD5:12; A5:1<=i2 & i2<=len GoB h & 1<=j2 & j2<=width GoB h by A4,GOBOARD5:1; then A6:(h/.i)`2=((GoB h)*(i2,I))`2 by A1,GOBOARD5:2; A7:(h/.i)`1=((GoB h)*(i2,1))`1 by A4,A5,GOBOARD5:3 .=((GoB h)*(i2,I))`1 by A1,A5,GOBOARD5:3; h/.i=|[(h/.i)`1,(h/.i)`2]| by EUCLID:57; then A8: h/.i=(GoB h)*(i2,I) by A6,A7,EUCLID:57; [i2,I] in Indices GoB h by A1,A5,GOBOARD7:10; then i2 in Y by A1,A3,A8; then A9:1<=i1 & i1<=i2 & i2<=len GoB h & 1<=width GoB h by A1,A2,A4,CQC_SIM1:def 8,GOBOARD5:1,GOBOARD7:35; now per cases; case i1<i2; hence (GoB h)*(i1,I)`1<=(GoB h)*(i2,I)`1 by A1,A9,GOBOARD5:4; case i1>=i2; hence (GoB h)*(i1,I)`1<=(GoB h)*(i2,I)`1 by A9,AXIOMS:21; end; hence thesis by A7; end; theorem Th45: for Y being non empty finite Subset of NAT st 1 <= i & i <= len h & 1 <= I & I <= width GoB h & Y = { j : [j,I] in Indices GoB h & ex k st k in dom h & h/.k = (GoB h)*(j,I) } & (h/.i)`2 = ((GoB h)* (1,I))`2 & i1 = max Y holds (GoB h)*(i1,I)`1 >= (h/.i)`1 proof let Y be non empty finite Subset of NAT; assume A1: 1<=i & i<=len h & 1<=I & I<=width GoB h & Y={j:[j,I] in Indices GoB h & ex k st k in dom h & h/.k=(GoB h)*(j,I)} & (h/.i)`2=((GoB h)* (1,I))`2 & i1=max Y; i1 in Y by A1,PRE_CIRC:def 1; then consider j such that A2: i1=j & [j,I] in Indices GoB h & ex k st k in dom h & h/.k=(GoB h)*(j,I) by A1; A3:i in dom h by A1,FINSEQ_3:27; then consider i2,j2 be Nat such that A4: [i2,j2] in Indices GoB h & h/.i=(GoB h)*(i2,j2) by GOBOARD5:12; A5:1<=i2 & i2<=len GoB h & 1<=j2 & j2<=width GoB h by A4,GOBOARD5:1; then A6:(h/.i)`2=((GoB h)*(i2,I))`2 by A1,GOBOARD5:2; A7:(h/.i)`1=((GoB h)*(i2,1))`1 by A4,A5,GOBOARD5:3 .=((GoB h)*(i2,I))`1 by A1,A5,GOBOARD5:3; h/.i=|[(h/.i)`1,(h/.i)`2]| by EUCLID:57; then A8: h/.i=(GoB h)*(i2,I) by A6,A7,EUCLID:57; [i2,I] in Indices GoB h by A1,A5,GOBOARD7:10; then i2 in Y by A1,A3,A8; then A9:1<=i1 & i1>=i2 & i2<=len GoB h & 1<=width GoB h by A1,A2,A4,GOBOARD5:1,GOBOARD7:35,PRE_CIRC:def 1; A10: i1 <= len GoB h by A2,GOBOARD5:1; now per cases; case i1>i2; hence (GoB h)*(i1,I)`1>=(GoB h)*(i2,I)`1 by A1,A5,A10,GOBOARD5:4; case i1<=i2; hence (GoB h)*(i1,I)`1>=(GoB h)*(i2,I)`1 by A9,AXIOMS:21; end; hence thesis by A7; end; theorem Th46: for Y being non empty finite Subset of NAT st 1 <= i & i <= len f & 1 <= I & I <= len GoB f & Y = { j : [I,j] in Indices GoB f & ex k st k in dom f & f/.k = (GoB f)*(I,j) } & (f/.i)`1 = ((GoB f)* (I,1))`1 & i1 = max Y holds (GoB f)*(I,i1)`2 >= (f/.i)`2 proof let Y be non empty finite Subset of NAT; assume A1: 1<=i & i<=len f & 1 <= I & I <= len GoB f & Y={j:[I,j] in Indices GoB f & ex k st k in dom f & f/.k=(GoB f)*(I,j)} & (f/.i)`1=((GoB f)*(I,1))`1 & i1=max Y; i1 in Y by A1,PRE_CIRC:def 1; then consider j such that A2:i1=j & [I,j] in Indices GoB f & ex k st k in dom f & f/.k=(GoB f)*(I,j) by A1; A3:1<=I & I<=len GoB f & 1<=i1 & i1<=width GoB f by A2,GOBOARD5:1; A4:i in dom f by A1,FINSEQ_3:27; then consider i2,j2 be Nat such that A5: [i2,j2] in Indices GoB f & f/.i=(GoB f)*(i2,j2) by GOBOARD5:12; A6:1<=i2 & i2<=len GoB f & 1<=j2 & j2<=width GoB f by A5,GOBOARD5:1; then A7:(f/.i)`1=((GoB f)*(I,j2))`1 by A1,GOBOARD5:3; A8:(f/.i)`2=((GoB f)*(1,j2))`2 by A5,A6,GOBOARD5:2 .=((GoB f)*(I,j2))`2 by A1,A6,GOBOARD5:2; f/.i=|[(f/.i)`1,(f/.i)`2]| by EUCLID:57; then A9: f/.i=(GoB f)*(I,j2) by A7,A8,EUCLID:57; [I,j2] in Indices GoB f by A1,A6,GOBOARD7:10; then j2 in Y by A1,A4,A9; then A10: i1>=j2 by A1,PRE_CIRC:def 1; now per cases; case i1 > j2; hence (GoB f)*(I,i1)`2 >= (GoB f)*(I,j2)`2 by A3,A6,GOBOARD5:5; case i1<=j2; hence (GoB f)*(I,i1)`2 >= (GoB f)*(I,j2)`2 by A10,AXIOMS:21; end; hence thesis by A8; end; Lm1: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT, h st p`1=W-bound L~h & p in L~h & Y={j:[1,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(1,j)} & i1=min Y holds (GoB h)*(1,i1)`2<=p`2 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT,h;assume A1: p`1=W-bound L~h & p in L~h & Y={j:[1,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(1,j)} & i1=min Y; then A2: p`1=((GoB h)*(1,1))`1 by Th39; A3: 1 <= width GoB h by GOBOARD7:35; A4: 1 <= len GoB h by GOBOARD7:34; consider i such that A5:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A6:i<=len h by A5,AXIOMS:22; A7:1<=i+1 by A5,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A5,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A8:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A5,GOBOARD7:5; now per cases; case (h/.i)`2<=(h/.(i+1))`2; then A9: (h/.i)`2<=p`2 & p`2<=(h/.(i+1))`2 by A5,TOPREAL1:10; (GoB h)*(1,i1)`2<=(h/.i)`2 by A1,A2,A4,A5,A6,A8,Th43; hence (GoB h)*(1,i1)`2<=p`2 by A9,AXIOMS:22; case (h/.i)`2>(h/.(i+1))`2; then A10: (h/.(i+1))`2<=p`2 & p`2<=(h/.i)`2 by A5,TOPREAL1:10; (GoB h)*(1,i1)`2<=(h/.(i+1))`2 by A1,A2,A4,A5,A7,A8,Th43; hence (GoB h)*(1,i1)`2<=p`2 by A10,AXIOMS:22; end; hence (GoB h)*(1,i1)`2<=p`2; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A5,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A11:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A5,GOBOARD7:6; now per cases; case (h/.i)`1<=(h/.(i+1))`1; then A12:(h/.i)`1<=((GoB h)*(1,1))`1 by A2,A5,TOPREAL1:9; (h/.i)`1>=((GoB h)*(1,1))`1 by A3,A5,A6,Th7; then (h/.i)`1=((GoB h)*(1,1))`1 by A12,AXIOMS:21; hence (GoB h)*(1,i1)`2<=p`2 by A1,A4,A5,A6,A11,Th43; case (h/.i)`1>(h/.(i+1))`1; then A13:(h/.(i+1))`1<=((GoB h)*(1,1))`1 by A2,A5,TOPREAL1:9; (h/.(i+1))`1>=((GoB h)*(1,1))`1 by A3,A5,A7,Th7; then (h/.(i+1))`1=((GoB h)*(1,1))`1 by A13,AXIOMS:21; hence (GoB h)*(1,i1)`2<=p`2 by A1,A4,A5,A7,A11,Th43; end; hence (GoB h)*(1,i1)`2<=p`2; end; hence (GoB h)*(1,i1)`2<=p`2; end; Lm2: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT, h st p`1=W-bound L~h & p in L~h & Y={j:[1,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(1,j)} & i1=max Y holds (GoB h)*(1,i1)`2 >= p`2 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT,h;assume A1: p`1=W-bound L~h & p in L~h & Y={j:[1,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(1,j)} & i1=max Y; then A2: p`1=((GoB h)*(1,1))`1 by Th39; consider i such that A3:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; A4: 1 <= width GoB h by GOBOARD7:35; A5: 1 <= len GoB h by GOBOARD7:34; i<=i+1 by NAT_1:29; then A6:i<=len h by A3,AXIOMS:22; A7:1<=i+1 by A3,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A3,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A8:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A3,GOBOARD7:5; now per cases; case (h/.i)`2>=(h/.(i+1))`2; then A9: (h/.i)`2 >= p`2 & p`2 >= (h/.(i+1))`2 by A3,TOPREAL1:10; (GoB h)*(1,i1)`2>=(h/.i)`2 by A1,A2,A3,A5,A6,A8,Th46; hence (GoB h)*(1,i1)`2>=p`2 by A9,AXIOMS:22; case (h/.i)`2<(h/.(i+1))`2; then A10: (h/.(i+1))`2>=p`2 & p`2>=(h/.i)`2 by A3,TOPREAL1:10; (GoB h)*(1,i1)`2>=(h/.(i+1))`2 by A1,A2,A3,A5,A7,A8,Th46; hence (GoB h)*(1,i1)`2>=p`2 by A10,AXIOMS:22; end; hence (GoB h)*(1,i1)`2>=p`2; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A3,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A11:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A3,GOBOARD7:6; now per cases; case (h/.i)`1<=(h/.(i+1))`1; then A12:(h/.i)`1<=((GoB h)*(1,1))`1 by A2,A3,TOPREAL1:9; (h/.i)`1>=((GoB h)*(1,1))`1 by A3,A4,A6,Th7; then (h/.i)`1=((GoB h)*(1,1))`1 by A12,AXIOMS:21; hence (GoB h)*(1,i1)`2>=p`2 by A1,A3,A5,A6,A11,Th46; case (h/.i)`1>(h/.(i+1))`1; then A13:(h/.(i+1))`1<=((GoB h)*(1,1))`1 by A2,A3,TOPREAL1:9; (h/.(i+1))`1>=((GoB h)*(1,1))`1 by A3,A4,A7,Th7; then (h/.(i+1))`1=((GoB h)*(1,1))`1 by A13,AXIOMS:21; hence (GoB h)*(1,i1)`2>=p`2 by A1,A3,A5,A7,A11,Th46; end; hence (GoB h)*(1,i1)`2>=p`2; end; hence (GoB h)*(1,i1)`2>=p`2; end; Lm3: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT, h st p`1=E-bound L~h & p in L~h & Y={j:[len GoB h,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(len GoB h,j)} & i1=min Y holds (GoB h)*(len GoB h,i1)`2<=p`2 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT,h;assume A1: p`1=E-bound L~h & p in L~h & Y={j:[len GoB h,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(len GoB h,j)} & i1=min Y; then A2: p`1=((GoB h)*(len GoB h,1))`1 by Th41; A3: 1 <= width GoB h by GOBOARD7:35; consider i such that A4:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A5:i<=len h by A4,AXIOMS:22; A6:1<=i+1 by A4,SPPOL_1:5; A7: 1 <= len GoB h by GOBOARD7:34; now per cases by SPPOL_1:41; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A4,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A8:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A4,GOBOARD7:5; now per cases; case (h/.i)`2<=(h/.(i+1))`2; then A9: (h/.i)`2<=p`2 & p`2<=(h/.(i+1))`2 by A4,TOPREAL1:10; (GoB h)*(len GoB h,i1)`2<=(h/.i)`2 by A1,A2,A4,A5,A7,A8,Th43; hence (GoB h)*(len GoB h,i1)`2<=p`2 by A9,AXIOMS:22; case (h/.i)`2>(h/.(i+1))`2; then A10: (h/.(i+1))`2<=p`2 & p`2<=(h/.i)`2 by A4,TOPREAL1:10; (GoB h)*(len GoB h,i1)`2<=(h/.(i+1))`2 by A1,A2,A4,A6,A7,A8,Th43; hence (GoB h)*(len GoB h,i1)`2<=p`2 by A10,AXIOMS:22; end; hence (GoB h)*(len GoB h,i1)`2<=p`2; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A4,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A11:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A4,GOBOARD7:6; now per cases; case (h/.i)`1>=(h/.(i+1))`1; then A12:(h/.i)`1>=((GoB h)*(len GoB h,1))`1 by A2,A4,TOPREAL1:9; (h/.i)`1<=((GoB h)*(len GoB h,1))`1 by A3,A4,A5,Th7; then (h/.i)`1=((GoB h)*(len GoB h,1))`1 by A12,AXIOMS:21; hence (GoB h)*(len GoB h,i1)`2<=p`2 by A1,A4,A5,A7,A11,Th43; case (h/.i)`1<(h/.(i+1))`1; then A13:(h/.(i+1))`1>=((GoB h)*(len GoB h,1))`1 by A2,A4,TOPREAL1:9; (h/.(i+1))`1<=((GoB h)*(len GoB h,1))`1 by A3,A4,A6,Th7; then (h/.(i+1))`1=((GoB h)*(len GoB h,1))`1 by A13,AXIOMS:21; hence (GoB h)*(len GoB h,i1)`2<=p`2 by A1,A4,A6,A7,A11,Th43; end; hence (GoB h)*(len GoB h,i1)`2<=p`2; end; hence (GoB h)*(len GoB h,i1)`2<=p`2; end; Lm4: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT, h st p`1=E-bound L~h & p in L~h & Y={j:[len GoB h,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(len GoB h,j)} & i1=max Y holds (GoB h)*(len GoB h,i1)`2 >= p`2 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT,h;assume A1: p`1=E-bound L~h & p in L~h & Y={j:[len GoB h,j] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(len GoB h,j)} & i1=max Y; then A2: p`1=((GoB h)*(len GoB h,1))`1 by Th41; A3: 1 <= len GoB h by GOBOARD7:34; A4: 1 <= width GoB h by GOBOARD7:35; consider i such that A5:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A6:i<=len h by A5,AXIOMS:22; A7:1<=i+1 by A5,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A5,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A8:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A5,GOBOARD7:5; now per cases; case (h/.i)`2>=(h/.(i+1))`2; then A9: (h/.i)`2 >= p`2 & p`2 >= (h/.(i+1))`2 by A5,TOPREAL1:10; (GoB h)*(len GoB h,i1)`2>=(h/.i)`2 by A1,A2,A3,A5,A6,A8,Th46; hence (GoB h)*(len GoB h,i1)`2>=p`2 by A9,AXIOMS:22; case (h/.i)`2<(h/.(i+1))`2; then A10: (h/.(i+1))`2>=p`2 & p`2>=(h/.i)`2 by A5,TOPREAL1:10; (GoB h)*(len GoB h,i1)`2>=(h/.(i+1))`2 by A1,A2,A3,A5,A7,A8,Th46; hence (GoB h)*(len GoB h,i1)`2>=p`2 by A10,AXIOMS:22; end; hence (GoB h)*(len GoB h,i1)`2>=p`2; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A5,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A11:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A5,GOBOARD7:6; now per cases; case (h/.i)`1>=(h/.(i+1))`1; then A12:(h/.i)`1>=((GoB h)*(len GoB h,1))`1 by A2,A5,TOPREAL1:9; (h/.i)`1<=((GoB h)*(len GoB h,1))`1 by A4,A5,A6,Th7; then (h/.i)`1=((GoB h)*(len GoB h,1))`1 by A12,AXIOMS:21; hence (GoB h)*(len GoB h,i1)`2>=p`2 by A1,A3,A5,A6,A11,Th46; case (h/.i)`1<(h/.(i+1))`1; then A13:(h/.(i+1))`1>=((GoB h)*(len GoB h,1))`1 by A2,A5,TOPREAL1:9; (h/.(i+1))`1<=((GoB h)*(len GoB h,1))`1 by A4,A5,A7,Th7; then (h/.(i+1))`1=((GoB h)*(len GoB h,1))`1 by A13,AXIOMS:21; hence (GoB h)*(len GoB h,i1)`2>=p`2 by A1,A3,A5,A7,A11,Th46; end; hence (GoB h)*(len GoB h,i1)`2>=p`2; end; hence (GoB h)*(len GoB h,i1)`2>=p`2; end; Lm5: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT, h st p`2=S-bound L~h & p in L~h & Y={j:[j,1] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,1)} & i1=min Y holds (GoB h)*(i1,1)`1<=p`1 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT,h;assume A1: p`2=S-bound L~h & p in L~h & Y={j:[j,1] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,1)} & i1=min Y; then A2: p`2=((GoB h)*(1,1))`2 by Th40; consider i such that A3:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A4:i<=len h by A3,AXIOMS:22; A5: 1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; A6:1<=i+1 by A3,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A3,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A7:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A3,GOBOARD7:6; now per cases; case (h/.i)`1<=(h/.(i+1))`1; then A8: (h/.i)`1<=p`1 & p`1<=(h/.(i+1))`1 by A3,TOPREAL1:9; (GoB h)*(i1,1)`1<=(h/.i)`1 by A1,A2,A3,A4,A5,A7,Th44; hence (GoB h)*(i1,1)`1<=p`1 by A8,AXIOMS:22; case (h/.i)`1>(h/.(i+1))`1; then A9: (h/.(i+1))`1<=p`1 & p`1<=(h/.i)`1 by A3,TOPREAL1:9; (GoB h)*(i1,1)`1<=(h/.(i+1))`1 by A1,A2,A3,A5,A6,A7,Th44; hence (GoB h)*(i1,1)`1<=p`1 by A9,AXIOMS:22; end; hence (GoB h)*(i1,1)`1<=p`1; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A3,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A10:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A3,GOBOARD7:5; now per cases; case (h/.i)`2<=(h/.(i+1))`2; then A11:(h/.i)`2<=((GoB h)*(1,1))`2 by A2,A3,TOPREAL1:10; (h/.i)`2>=((GoB h)*(1,1))`2 by A3,A4,A5,Th8; then (h/.i)`2=((GoB h)*(1,1))`2 by A11,AXIOMS:21; hence (GoB h)*(i1,1)`1<=p`1 by A1,A3,A4,A5,A10,Th44; case (h/.i)`2>(h/.(i+1))`2; then A12:(h/.(i+1))`2<=((GoB h)*(1,1))`2 by A2,A3,TOPREAL1:10; (h/.(i+1))`2>=((GoB h)*(1,1))`2 by A3,A5,A6,Th8; then (h/.(i+1))`2=((GoB h)*(1,1))`2 by A12,AXIOMS:21; hence (GoB h)*(i1,1)`1<=p`1 by A1,A3,A5,A6,A10,Th44; end; hence (GoB h)*(i1,1)`1<=p`1; end; hence (GoB h)*(i1,1)`1<=p`1; end; Lm6: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT, h st p`2=N-bound L~h & p in L~h & Y={j:[j,width GoB h] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,width GoB h)} & i1=min Y holds (GoB h)*(i1,width GoB h)`1<=p`1 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT,h;assume A1: p`2=N-bound L~h & p in L~h & Y={j:[j,width GoB h] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,width GoB h)} & i1=min Y; then A2: p`2=((GoB h)*(1,width GoB h))`2 by Th42; set I = width GoB h; A3: 1 <= I by GOBOARD7:35; consider i such that A4:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A5:i<=len h by A4,AXIOMS:22; A6:1<=i+1 by A4,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A4,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A7:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A4,GOBOARD7:6; now per cases; case (h/.i)`1<=(h/.(i+1))`1; then A8: (h/.i)`1<=p`1 & p`1<=(h/.(i+1))`1 by A4,TOPREAL1:9; (GoB h)*(i1,I)`1<=(h/.i)`1 by A1,A2,A3,A4,A5,A7,Th44; hence (GoB h)*(i1,I)`1<=p`1 by A8,AXIOMS:22; case (h/.i)`1>(h/.(i+1))`1; then A9: (h/.(i+1))`1<=p`1 & p`1<=(h/.i)`1 by A4,TOPREAL1:9; (GoB h)*(i1,I)`1<=(h/.(i+1))`1 by A1,A2,A3,A4,A6,A7,Th44; hence (GoB h)*(i1,I)`1<=p`1 by A9,AXIOMS:22; end; hence (GoB h)*(i1,I)`1<=p`1; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A4,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A10:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A4,GOBOARD7:5; A11: 1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; now per cases; case (h/.i)`2>=(h/.(i+1))`2; then A12:(h/.i)`2>=((GoB h)*(1,I))`2 by A2,A4,TOPREAL1:10; (h/.i)`2<=((GoB h)*(1,I))`2 by A4,A5,A11,Th8; then (h/.i)`2=((GoB h)*(1,I))`2 by A12,AXIOMS:21; hence (GoB h)*(i1,I)`1<=p`1 by A1,A3,A4,A5,A10,Th44; case (h/.i)`2<(h/.(i+1))`2; then A13:(h/.(i+1))`2>=((GoB h)*(1,I))`2 by A2,A4,TOPREAL1:10; (h/.(i+1))`2 <= ((GoB h)*(1,I))`2 by A4,A6,A11,Th8; then (h/.(i+1))`2=((GoB h)*(1,I))`2 by A13,AXIOMS:21; hence (GoB h)*(i1,width GoB h)`1<=p`1 by A1,A3,A4,A6,A10,Th44; end; hence (GoB h)*(i1,width GoB h)`1<=p`1; end; hence (GoB h)*(i1,width GoB h)`1<=p`1; end; Lm7: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT st p`2=S-bound L~h & p in L~h & Y={j:[j,1] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,1)} & i1=max Y holds (GoB h)*(i1,1)`1>=p`1 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT;assume A1: p`2=S-bound L~h & p in L~h & Y={j:[j,1] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,1)} & i1=max Y; then A2: p`2=((GoB h)*(1,1))`2 by Th40; consider i such that A3:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A4:i<=len h by A3,AXIOMS:22; A5: 1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; A6:1<=i+1 by A3,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A3,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A7:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A3,GOBOARD7:6; now per cases; case (h/.i)`1>=(h/.(i+1))`1; then A8: (h/.i)`1>=p`1 & p`1>=(h/.(i+1))`1 by A3,TOPREAL1:9; (GoB h)*(i1,1)`1>=(h/.i)`1 by A1,A2,A3,A4,A5,A7,Th45; hence (GoB h)*(i1,1)`1>=p`1 by A8,AXIOMS:22; case (h/.i)`1<(h/.(i+1))`1; then A9: (h/.(i+1))`1>=p`1 & p`1>=(h/.i)`1 by A3,TOPREAL1:9; (GoB h)*(i1,1)`1>=(h/.(i+1))`1 by A1,A2,A3,A5,A6,A7,Th45; hence (GoB h)*(i1,1)`1>=p`1 by A9,AXIOMS:22; end; hence (GoB h)*(i1,1)`1>=p`1; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A3,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A10:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A3,GOBOARD7:5; now per cases; case (h/.i)`2<=(h/.(i+1))`2; then A11:(h/.i)`2<=((GoB h)*(1,1))`2 by A2,A3,TOPREAL1:10; (h/.i)`2>=((GoB h)*(1,1))`2 by A3,A4,A5,Th8; then (h/.i)`2=((GoB h)*(1,1))`2 by A11,AXIOMS:21; hence (GoB h)*(i1,1)`1>=p`1 by A1,A3,A4,A5,A10,Th45; case (h/.i)`2>(h/.(i+1))`2; then A12:(h/.(i+1))`2<=((GoB h)*(1,1))`2 by A2,A3,TOPREAL1:10; (h/.(i+1))`2>=((GoB h)*(1,1))`2 by A3,A5,A6,Th8; then (h/.(i+1))`2=((GoB h)*(1,1))`2 by A12,AXIOMS:21; hence (GoB h)*(i1,1)`1>=p`1 by A1,A3,A5,A6,A10,Th45; end; hence (GoB h)*(i1,1)`1>=p`1; end; hence (GoB h)*(i1,1)`1>=p`1; end; Lm8: for p being Point of TOP-REAL 2, Y being non empty finite Subset of NAT st p`2=N-bound L~h & p in L~h & Y={j:[j,width GoB h] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,width GoB h)} & i1=max Y holds (GoB h)*(i1,width GoB h)`1>=p`1 proof let p be Point of TOP-REAL 2,Y be non empty finite Subset of NAT;assume A1: p`2=N-bound L~h & p in L~h & Y={j:[j,width GoB h] in Indices GoB h & ex i st i in dom h & h/.i=(GoB h)*(j,width GoB h)} & i1=max Y; then A2: p`2=((GoB h)*(1,width GoB h))`2 by Th42; consider i such that A3:1<=i & i+1<=len h & p in LSeg(h/.i,h/.(i+1)) by A1,SPPOL_2:14; i<=i+1 by NAT_1:29; then A4:i<=len h by A3,AXIOMS:22; A5:1<=i+1 by A3,SPPOL_1:5; now per cases by SPPOL_1:41; case LSeg(h,i) is horizontal; then LSeg(h/.i,h/.(i+1)) is horizontal by A3,TOPREAL1:def 5; then (h/.i)`2=(h/.(i+1))`2 by SPPOL_1:36; then A6:p`2=(h/.(i+1))`2 & p`2=(h/.i)`2 by A3,GOBOARD7:6; now per cases; case (h/.i)`1>=(h/.(i+1))`1; then A7: (h/.i)`1>=p`1 & p`1>=(h/.(i+1))`1 by A3,TOPREAL1:9; 1 <= width GoB h by GOBOARD7:35; then (GoB h)*(i1,width GoB h)`1>=(h/.i)`1 by A1,A2,A3,A4,A6,Th45; hence (GoB h)*(i1,width GoB h)`1>=p`1 by A7,AXIOMS:22; case (h/.i)`1<(h/.(i+1))`1; then A8: (h/.(i+1))`1>=p`1 & p`1>=(h/.i)`1 by A3,TOPREAL1:9; 1 <= width GoB h by GOBOARD7:35; then (GoB h)*(i1,width GoB h)`1>=(h/.(i+1))`1 by A1,A2,A3,A5,A6,Th45; hence (GoB h)*(i1,width GoB h)`1>=p`1 by A8,AXIOMS:22; end; hence (GoB h)*(i1,width GoB h)`1>=p`1; case LSeg(h,i) is vertical; then LSeg(h/.i,h/.(i+1)) is vertical by A3,TOPREAL1:def 5; then (h/.i)`1=(h/.(i+1))`1 by SPPOL_1:37; then A9:p`1=(h/.(i+1))`1 & p`1=(h/.i)`1 by A3,GOBOARD7:5; A10: 1 <= len GoB h & 1 <= width GoB h by GOBOARD7:34,35; now per cases; case (h/.i)`2>=(h/.(i+1))`2; then A11:(h/.i)`2>=((GoB h)*(1,width GoB h))`2 by A2,A3,TOPREAL1:10; A12: 1 <= width GoB h by GOBOARD7:35; (h/.i)`2<=((GoB h)*(1,width GoB h))`2 by A3,A4,A10,Th8; then (h/.i)`2=((GoB h)*(1,width GoB h))`2 by A11,AXIOMS:21; hence (GoB h)*(i1,width GoB h)`1>=p`1 by A1,A3,A4,A9,A12,Th45; case (h/.i)`2 < (h/.(i+1))`2; then A13:(h/.(i+1))`2>=((GoB h)*(1,width GoB h))`2 by A2,A3,TOPREAL1:10; A14: 1 <= width GoB h by GOBOARD7:35; (h/.(i+1))`2<=((GoB h)*(1,width GoB h))`2 by A3,A5,A10,Th8; then (h/.(i+1))`2=((GoB h)*(1,width GoB h))`2 by A13,AXIOMS:21; hence (GoB h)*(i1,width GoB h)`1>=p`1 by A1,A3,A5,A9,A14,Th45; end; hence (GoB h)*(i1,width GoB h)`1>=p`1; end; hence (GoB h)*(i1,width GoB h)`1>=p`1; end; Lm9:len h >= 2 proof len h > 4 by GOBOARD7:36; hence thesis by AXIOMS:22; end; begin :: Coordinates of the special circular sequences bounding boxes definition let g be non constant standard special_circular_sequence; func i_s_w g -> Nat means :Def1: [1,it] in Indices GoB g & (GoB g)*(1,it) = W-min L~g; existence proof defpred P[Nat] means [1,$1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(1,$1); set Y={j:P[j]}; A1: 1 <= len GoB g by GOBOARD7:34; then consider i,j such that A2: i in dom g & [1,j] in Indices GoB g & g/.i = (GoB g)*(1,j) by Th9; A3: j in Y by A2; A4: Y c= Seg width GoB g proof let y be set; assume y in Y; then ex j st y=j & [1,j] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(1,j); then [1,y] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in Seg width GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A3,A4,FINSET_1:13; set i1 = min Y; i1 in Y by CQC_SIM1:def 8; then consider j such that A5:j=i1 & ([1,j] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(1,j)); consider i such that A6: i in dom g & g/.i=(GoB g)*(1,j) by A5; 1 <= 1 & 1 <= len GoB g & 1 <= i1 & i1 <= width (GoB g) by A5,GOBOARD5:1; then A7:(GoB g)*(1,i1)`1=(GoB g)*(1,1)`1 by GOBOARD5:3; then A8:(GoB g)*(1,i1)`1=W-bound L~g by Th39; {q`2:q`1=W-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`2:q`1=W-bound(L~g) & q in L~g}; then ex q st X=q`2 & q`1=W-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`2:q`1=W-bound L~g & q in L~g} as Subset of REAL; set s1=(GoB g)*(1,1)`2; for r be real number st r in B holds s1<=r proof let r be real number;assume r in B; then consider q such that A9: r=q`2 & q`1 = W-bound L~g & q in L~g; thus thesis by A1,A9,Th35; end; then A10:B is bounded_below by SEQ_4:def 2; A11:1<=i & i<=len g by A6,FINSEQ_3:27; now per cases by A11,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A11,TOPREAL1:27; hence (GoB g)*(1,i1) in L~g by A5,A6,SPPOL_2:17; case A12:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A12,Th3; hence (GoB g)*(1,i1) in L~g by A5,A6,SPPOL_2:17; end; then (GoB g)*(1,i1)`1=W-bound(L~g) & (GoB g)*(1,i1) in L~g by A7,Th39; then A13:(GoB g)*(1,i1)`2 in {q`2:q`1=W-bound(L~g) & q in L~g}; then A14:lower_bound B <=(GoB g)*(1,i1)`2 by A10,SEQ_4:def 5; for r being real number st r in B holds r>= (GoB g)*(1,i1)`2 proof let r be real number;assume r in B; then consider q such that A15:r=q`2 & (q`1=W-bound(L~g) & q in L~g); thus r>= (GoB g)*(1,i1)`2 by A15,Lm1; end; then lower_bound B >=(GoB g)*(1,i1)`2 by A13,PSCOMP_1:8; then A16: (GoB g)*(1,i1)`2=lower_bound B by A14,AXIOMS:21 .= inf (proj2 || W-most L~g) by Th21; (GoB g)*(1,i1)=|[(GoB g)*(1,i1)`1,(GoB g)*(1,i1)`2]| by EUCLID:57; then (GoB g)*(1,i1)=W-min(L~g) by A8,A16,PSCOMP_1:def 42; hence thesis by A5; end; uniqueness by GOBOARD1:21; func i_n_w g -> Nat means :Def2: [1,it] in Indices GoB g & (GoB g)*(1,it) = W-max L~g; existence proof defpred P[Nat] means [1,$1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(1,$1); set Y={j:P[j]}; A17: 1 <= len GoB g by GOBOARD7:34; then consider i,j such that A18: i in dom g & [1,j] in Indices GoB g & g/.i = (GoB g)*(1,j) by Th9; A19: j in Y by A18; A20: Y c= Seg width GoB g proof let y be set; assume y in Y; then ex j st y=j & [1,j] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(1,j); then [1,y] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in Seg width GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A19,A20,FINSET_1: 13; reconsider i1 = max Y as Nat by ORDINAL2:def 21; i1 in Y by PRE_CIRC:def 1; then consider j such that A21:j=i1 & [1,j] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(1,j); consider i such that A22: i in dom g & g/.i=(GoB g)*(1,j) by A21; 1 <= 1 & 1 <= len GoB g & 1 <= i1 & i1 <= width GoB g by A21,GOBOARD5:1; then A23:(GoB g)*(1,i1)`1=(GoB g)*(1,1)`1 by GOBOARD5:3; then A24:(GoB g)*(1,i1)`1=W-bound L~g by Th39; {q`2:q`1=W-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`2:q`1=W-bound(L~g) & q in L~g}; then ex q st X=q`2 & q`1=W-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`2:q`1=W-bound L~g & q in L~g} as Subset of REAL; set s1 = (GoB g)*(1,width GoB g)`2; for r be real number st r in B holds s1 >= r proof let r be real number; assume r in B; then consider q such that A25: r=q`2 & q`1 = W-bound L~g & q in L~g; thus thesis by A17,A25,Th36; end; then A26:B is bounded_above by SEQ_4:def 1; A27:1<=i & i<=len g by A22,FINSEQ_3:27; now per cases by A27,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A27,TOPREAL1:27; hence (GoB g)*(1,i1) in L~g by A21,A22,SPPOL_2:17; case A28:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A28,Th3; hence (GoB g)*(1,i1) in L~g by A21,A22,SPPOL_2:17; end; then (GoB g)*(1,i1)`1=W-bound L~g & (GoB g)*(1,i1) in L~g by A23,Th39; then A29:(GoB g)*(1,i1)`2 in {q`2:q`1=W-bound L~g & q in L~g}; then A30:upper_bound B >= (GoB g)*(1,i1)`2 by A26,SEQ_4:def 4; for r being real number st r in B holds r <= (GoB g)*(1,i1)`2 proof let r be real number;assume r in B; then consider q such that A31:r = q`2 & q`1 = W-bound L~g & q in L~g; thus r <= (GoB g)*(1,i1)`2 by A31,Lm2; end; then upper_bound B <= (GoB g)*(1,i1)`2 by A29,PSCOMP_1:10; then A32: (GoB g)*(1,i1)`2 = upper_bound B by A30,AXIOMS:21 .= sup (proj2 || W-most L~g) by Th22; (GoB g)*(1,i1)=|[(GoB g)*(1,i1)`1,(GoB g)*(1,i1)`2]| by EUCLID:57; then (GoB g)*(1,i1)=W-max L~g by A24,A32,PSCOMP_1:def 43; hence thesis by A21; end; uniqueness by GOBOARD1:21; func i_s_e g -> Nat means :Def3: [len GoB g,it] in Indices GoB g & (GoB g)*(len GoB g,it) = E-min L~g; existence proof defpred P[Nat] means [len GoB g,$1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(len GoB g,$1); set Y={j:P[j]}; A33: 1 <= len GoB g by GOBOARD7:34; then consider i,j such that A34: i in dom g & [len GoB g,j] in Indices GoB g & g/.i = (GoB g)*(len GoB g,j) by Th9; A35: j in Y by A34; A36: Y c= Seg width GoB g proof let y be set; assume y in Y; then ex j st y=j & [len GoB g,j] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(len GoB g,j); then [len GoB g,y] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in Seg width GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A35,A36,FINSET_1: 13; set i1 = min Y; i1 in Y by CQC_SIM1:def 8; then consider j such that A37: j=i1 & [len GoB g,j] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(len GoB g,j); consider i such that A38: i in dom g & g/.i=(GoB g)*(len GoB g,j) by A37; 1 <= 1 & 1 <= len GoB g & 1 <= i1 & i1 <= width GoB g by A37,GOBOARD5:1; then A39:(GoB g)*(len GoB g,i1)`1=(GoB g)*(len GoB g,1)`1 by GOBOARD5:3; then A40:(GoB g)*(len GoB g,i1)`1=E-bound L~g by Th41; {q`2:q`1=E-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`2:q`1=E-bound L~g & q in L~g}; then ex q st X=q`2 & q`1=E-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`2:q`1=E-bound L~g & q in L~g} as Subset of REAL; set s1=(GoB g)*(len GoB g,1)`2; for r be real number st r in B holds s1<=r proof let r be real number;assume r in B; then consider q such that A41: r=q`2 & q`1 = E-bound L~g & q in L~g; thus thesis by A33,A41,Th35; end; then A42:B is bounded_below by SEQ_4:def 2; A43:1<=i & i<=len g by A38,FINSEQ_3:27; now per cases by A43,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A43,TOPREAL1:27; hence (GoB g)*(len GoB g,i1) in L~g by A37,A38,SPPOL_2:17; case A44:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A44,Th3; hence (GoB g)*(len GoB g,i1) in L~g by A37,A38,SPPOL_2:17; end; then (GoB g)*(len GoB g,i1)`1=E-bound L~g & (GoB g)*(len GoB g,i1) in L~g by A39,Th41; then A45:(GoB g)*(len GoB g,i1)`2 in {q`2:q`1=E-bound L~g & q in L~g}; then A46:lower_bound B <=(GoB g)*(len GoB g,i1)`2 by A42,SEQ_4:def 5; for r be real number st r in B holds r>= (GoB g)*(len GoB g,i1)`2 proof let r be real number;assume r in B; then consider q such that A47:r=q`2 & q`1=E-bound L~g & q in L~g; thus r >= (GoB g)*(len GoB g,i1)`2 by A47,Lm3; end; then lower_bound B >= (GoB g)*(len GoB g,i1)`2 by A45,PSCOMP_1:8; then A48: (GoB g)*(len GoB g,i1)`2=lower_bound B by A46,AXIOMS:21 .= inf (proj2 || E-most L~g) by Th23; (GoB g)*(len GoB g,i1)=|[(GoB g)*(len GoB g,i1)`1, (GoB g)*(len GoB g,i1)`2]| by EUCLID:57; then (GoB g)*(len GoB g,i1)=E-min(L~g) by A40,A48,PSCOMP_1:def 47; hence thesis by A37; end; uniqueness by GOBOARD1:21; func i_n_e g -> Nat means :Def4: [len GoB g,it] in Indices GoB g & (GoB g)*(len GoB g,it) = E-max L~g; existence proof defpred P[Nat] means [len GoB g,$1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(len GoB g,$1); set Y={j:P[j]}; 0 <> len GoB g by GOBOARD1:def 5; then 1 <= len GoB g by RLVECT_1:99; then consider i,j such that A49: i in dom g & [len GoB g,j] in Indices GoB g & g/.i = (GoB g)*(len GoB g,j) by Th9; A50: j in Y by A49; A51: Y c= Seg width GoB g proof let y be set; assume y in Y; then ex j st y=j & [len GoB g,j] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(len GoB g,j); then [len GoB g,y] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in Seg width GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A50,A51,FINSET_1: 13; reconsider i1 = max Y as Nat by ORDINAL2:def 21; i1 in Y by PRE_CIRC:def 1; then consider j such that A52:j=i1 & [len GoB g,j] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(len GoB g,j); consider i such that A53: i in dom g & g/.i=(GoB g)*(len GoB g,j) by A52; 1 <= 1 & 1 <= len GoB g & 1 <= i1 & i1 <= width GoB g by A52,GOBOARD5:1; then A54:(GoB g)*(len GoB g,i1)`1=(GoB g)*(len GoB g,1)`1 by GOBOARD5:3; then A55:(GoB g)*(len GoB g,i1)`1 = E-bound L~g by Th41; {q`2:q`1=E-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`2:q`1=E-bound(L~g) & q in L~g}; then ex q st X=q`2 & q`1=E-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`2:q`1=E-bound L~g & q in L~g} as Subset of REAL; set s1 = (GoB g)*(len GoB g,width GoB g)`2; for r be real number st r in B holds s1 >= r proof let r be real number; assume r in B; then consider q such that A56: r=q`2 & q`1 = E-bound L~g & q in L~g; 1 <= len GoB g by GOBOARD7:34; hence thesis by A56,Th36; end; then A57:B is bounded_above by SEQ_4:def 1; A58:1<=i & i<=len g by A53,FINSEQ_3:27; now per cases by A58,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A58,TOPREAL1:27; hence (GoB g)*(len GoB g,i1) in L~g by A52,A53,SPPOL_2:17; case A59:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A59,Th3; hence (GoB g)*(len GoB g,i1) in L~g by A52,A53,SPPOL_2:17; end; then (GoB g)*(len GoB g,i1)`1=E-bound L~g & (GoB g)*(len GoB g,i1) in L~g by A54,Th41; then A60:(GoB g)*(len GoB g,i1)`2 in {q`2:q`1=E-bound L~g & q in L~g}; then A61:upper_bound B >= (GoB g)*(len GoB g,i1)`2 by A57,SEQ_4:def 4; for r being real number st r in B holds r <= (GoB g)*(len GoB g,i1)`2 proof let r be real number;assume r in B; then consider q such that A62:r = q`2 & q`1 = E-bound L~g & q in L~g; thus r <= (GoB g)*(len GoB g,i1)`2 by A62,Lm4; end; then upper_bound B <= (GoB g)*(len GoB g,i1)`2 by A60,PSCOMP_1:10; then A63: (GoB g)*(len GoB g,i1)`2 = upper_bound B by A61,AXIOMS:21 .= sup (proj2 || E-most L~g) by Th24; (GoB g)*(len GoB g,i1)=|[(GoB g)*(len GoB g,i1)`1,(GoB g)* (len GoB g,i1)`2]| by EUCLID:57; then (GoB g)*(len GoB g,i1)=E-max L~g by A55,A63,PSCOMP_1:def 46; hence thesis by A52; end; uniqueness by GOBOARD1:21; func i_w_s g -> Nat means :Def5: [it, 1] in Indices GoB g & (GoB g)*(it,1) = S-min L~g; existence proof defpred P[Nat] means [$1,1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*($1,1); set Y={j:P[j]}; A64: 1 <= width GoB g by GOBOARD7:35; then consider i,j such that A65: i in dom g & [j,1] in Indices GoB g & g/.i = (GoB g)*(j,1) by Th10; A66: j in Y by A65; A67: Y c= dom GoB g proof let y be set; assume y in Y; then ex j st y=j & [j,1] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(j,1); then [y,1] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in dom GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A66,A67,FINSET_1: 13; set i1 = min Y; i1 in Y by CQC_SIM1:def 8; then consider j such that A68:j=i1 & [j,1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(j,1); consider i such that A69: i in dom g & g/.i=(GoB g)*(j,1) by A68; 1 <= i1 & i1 <= len GoB g & 1 <= 1 & 1 <= width GoB g by A68,GOBOARD5:1; then A70:(GoB g)*(i1,1)`2=(GoB g)*(1,1)`2 by GOBOARD5:2; then A71:(GoB g)*(i1,1)`2=S-bound L~g by Th40; {q`1:q`2=S-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`1:q`2=S-bound L~g & q in L~g}; then ex q st X=q`1 & q`2=S-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`1:q`2=S-bound L~g & q in L~g} as Subset of REAL; set s1=(GoB g)*(1,1)`1; for r be real number st r in B holds s1<=r proof let r be real number;assume r in B; then consider q such that A72: r=q`1 & q`2 = S-bound L~g & q in L~g; thus thesis by A64,A72,Th33; end; then A73:B is bounded_below by SEQ_4:def 2; A74:1<=i & i<=len g by A69,FINSEQ_3:27; now per cases by A74,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A74,TOPREAL1:27; hence (GoB g)*(i1,1) in L~g by A68,A69,SPPOL_2:17; case A75:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A75,Th3; hence (GoB g)*(i1,1) in L~g by A68,A69,SPPOL_2:17; end; then (GoB g)*(i1,1)`2=S-bound L~g & (GoB g)*(i1,1) in L~g by A70,Th40; then A76:(GoB g)*(i1,1)`1 in {q`1:q`2=S-bound L~g & q in L~g}; then A77:lower_bound B <=(GoB g)*(i1,1)`1 by A73,SEQ_4:def 5; for r being real number st r in B holds r>= (GoB g)*(i1,1)`1 proof let r be real number;assume r in B; then consider q such that A78:r=q`1 & q`2=S-bound L~g & q in L~g; thus r >= (GoB g)*(i1,1)`1 by A78,Lm5; end; then lower_bound B >=(GoB g)*(i1,1)`1 by A76,PSCOMP_1:8; then A79: (GoB g)*(i1,1)`1=lower_bound B by A77,AXIOMS:21 .= inf (proj1 || S-most L~g) by Th26; (GoB g)*(i1,1)=|[(GoB g)*(i1,1)`1,(GoB g)*(i1,1)`2]| by EUCLID:57; then (GoB g)*(i1,1)=S-min L~g by A71,A79,PSCOMP_1:def 49; hence thesis by A68; end; uniqueness by GOBOARD1:21; func i_e_s g -> Nat means :Def6: [it, 1] in Indices GoB g & (GoB g)*(it,1) = S-max L~g; existence proof defpred P[Nat] means [$1,1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*($1,1); set Y={j:P[j]}; 1 <= width GoB g by GOBOARD7:35; then consider i,j such that A80: i in dom g & [j,1] in Indices GoB g & g/.i = (GoB g)*(j,1) by Th10; A81: j in Y by A80; A82: Y c= dom GoB g proof let y be set; assume y in Y; then ex j st y=j & [j,1] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(j,1); then [y,1] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in dom GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A81,A82,FINSET_1: 13; reconsider i1 = max Y as Nat by ORDINAL2:def 21; i1 in Y by PRE_CIRC:def 1; then consider j such that A83:j=i1 & [j,1] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(j,1); consider i such that A84: i in dom g & g/.i=(GoB g)*(j,1) by A83; 1 <= i1 & i1 <= len GoB g & 1 <= 1 & 1 <= width GoB g by A83,GOBOARD5:1; then A85:(GoB g)*(i1,1)`2=(GoB g)*(1,1)`2 by GOBOARD5:2; then A86:(GoB g)*(i1,1)`2 = S-bound L~g by Th40; {q`1:q`2=S-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`1:q`2=S-bound L~g & q in L~g}; then ex q st X=q`1 & q`2=S-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`1:q`2 = S-bound L~g & q in L~g} as Subset of REAL; set s1 = (GoB g)*(len GoB g,1)`1; for r be real number st r in B holds s1 >= r proof let r be real number; assume r in B; then consider q such that A87: r=q`1 & q`2 = S-bound L~g & q in L~g; 1 <= width GoB g by GOBOARD7:35; hence thesis by A87,Th34; end; then A88:B is bounded_above by SEQ_4:def 1; A89:1<=i & i<=len g by A84,FINSEQ_3:27; now per cases by A89,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A89,TOPREAL1:27; hence (GoB g)*(i1,1) in L~g by A83,A84,SPPOL_2:17; case A90:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A90,Th3; hence (GoB g)*(i1,1) in L~g by A83,A84,SPPOL_2:17; end; then (GoB g)*(i1,1)`2=S-bound L~g & (GoB g)*(i1,1) in L~g by A85,Th40; then A91:(GoB g)*(i1,1)`1 in {q`1:q`2=S-bound L~g & q in L~g}; then A92:upper_bound B >= (GoB g)*(i1,1)`1 by A88,SEQ_4:def 4; for r being real number st r in B holds r <= (GoB g)*(i1,1)`1 proof let r be real number;assume r in B; then consider q such that A93:r = q`1 & q`2 = S-bound L~g & q in L~g; thus r <= (GoB g)*(i1,1)`1 by A93,Lm7; end; then upper_bound B <= (GoB g)*(i1,1)`1 by A91,PSCOMP_1:10; then A94: (GoB g)*(i1,1)`1 = upper_bound B by A92,AXIOMS:21 .= sup (proj1 || S-most L~g) by Th27; (GoB g)*(i1,1)=|[(GoB g)*(i1,1)`1,(GoB g)*(i1,1)`2]| by EUCLID:57; then (GoB g)*(i1,1)=S-max L~g by A86,A94,PSCOMP_1:def 48; hence thesis by A83; end; uniqueness by GOBOARD1:21; func i_w_n g -> Nat means :Def7: [it, width GoB g] in Indices GoB g & (GoB g)*(it,width GoB g) = N-min L~g; existence proof defpred P[Nat] means [$1,width GoB g] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*($1,width GoB g); set Y={j:P[j]}; 1 <= width GoB g by GOBOARD7:35; then consider i,j such that A95: i in dom g & [j,width GoB g] in Indices GoB g & g/.i = (GoB g)*(j,width GoB g) by Th10; A96: j in Y by A95; A97: Y c= dom GoB g proof let y be set; assume y in Y; then ex j st y=j & [j,width GoB g] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(j,width GoB g); then [y,width GoB g] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in dom GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A96,A97,FINSET_1: 13; set i1 = min Y; i1 in Y by CQC_SIM1:def 8; then consider j such that A98:j=i1 & [j,width GoB g] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(j,width GoB g); consider i such that A99: i in dom g & g/.i=(GoB g)*(j,width GoB g) by A98; 1 <= i1 & i1 <= len GoB g & 1 <= 1 & 1 <= width GoB g by A98,GOBOARD5:1; then A100:(GoB g)*(i1,width GoB g)`2=(GoB g)*(1,width GoB g)`2 by GOBOARD5:2; then A101:(GoB g)*(i1,width GoB g)`2=N-bound L~g by Th42; {q`1:q`2=N-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`1:q`2=N-bound L~g & q in L~g}; then ex q st X=q`1 & q`2=N-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`1:q`2=N-bound L~g & q in L~g} as Subset of REAL; set s1=(GoB g)*(1,width GoB g)`1; for r be real number st r in B holds s1<=r proof let r be real number;assume r in B; then consider q1 be Point of TOP-REAL 2 such that A102: r=q1`1 & q1`2 = N-bound L~g & q1 in L~g; 1 <= width GoB g & width GoB g <= width GoB g by GOBOARD7:35; hence thesis by A102,Th33; end; then A103:B is bounded_below by SEQ_4:def 2; A104:1<=i & i<=len g by A99,FINSEQ_3:27; now per cases by A104,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A104,TOPREAL1:27; hence (GoB g)*(i1,width GoB g) in L~g by A98,A99,SPPOL_2:17; case A105:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A105,Th3; hence (GoB g)*(i1,width GoB g) in L~g by A98,A99,SPPOL_2:17; end; then (GoB g)*(i1,width GoB g)`2=N-bound L~g & (GoB g)*(i1,width GoB g) in L~g by A100,Th42; then A106:(GoB g)*(i1,width GoB g)`1 in {q`1:q`2=N-bound L~g & q in L~g}; then A107:lower_bound B <= (GoB g)*(i1,width GoB g)`1 by A103,SEQ_4:def 5; for r being real number st r in B holds r>= (GoB g)*(i1,width GoB g)`1 proof let r be real number;assume r in B; then consider q such that A108:r=q`1 & q`2=N-bound L~g & q in L~g; thus r >= (GoB g)*(i1,width GoB g)`1 by A108,Lm6; end; then lower_bound B >=(GoB g)*(i1,width GoB g)`1 by A106,PSCOMP_1:8; then A109: (GoB g)*(i1,width GoB g)`1=lower_bound B by A107,AXIOMS:21 .= inf (proj1 || N-most L~g) by Th28; (GoB g)*(i1,width GoB g)=|[(GoB g)*(i1,width GoB g)`1, (GoB g)*(i1,width GoB g)`2]| by EUCLID:57; then (GoB g)*(i1,width GoB g)=N-min L~g by A101,A109,PSCOMP_1:def 44; hence thesis by A98; end; uniqueness by GOBOARD1:21; func i_e_n g -> Nat means :Def8: [it, width GoB g] in Indices GoB g & (GoB g)*(it,width GoB g) = N-max L~g; existence proof defpred P[Nat] means [$1,width GoB g] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*($1,width GoB g); set Y={j:P[j]}; 1 <= width GoB g by GOBOARD7:35; then consider i,j such that A110: i in dom g & [j,width GoB g] in Indices GoB g & g/.i = (GoB g)*(j,width GoB g) by Th10; A111: j in Y by A110; A112: Y c= dom GoB g proof let y be set; assume y in Y; then ex j st y=j & [j,width GoB g] in Indices GoB g & ex i st i in dom g & g/.i = (GoB g)*(j,width GoB g); then [y,width GoB g] in [: dom GoB g, Seg width GoB g :] by MATRIX_1:def 5; hence y in dom GoB g by ZFMISC_1:106; end; Y is Subset of NAT from SubsetD; then reconsider Y as non empty finite Subset of NAT by A111,A112,FINSET_1 :13; reconsider i1 = max Y as Nat by ORDINAL2:def 21; i1 in Y by PRE_CIRC:def 1; then consider j such that A113:j=i1 & [j,width GoB g] in Indices GoB g & ex i st i in dom g & g/.i=(GoB g)*(j,width GoB g); consider i such that A114: i in dom g & g/.i=(GoB g)*(j,width GoB g) by A113; 1 <= i1 & i1 <= len GoB g & 1 <= 1 & 1 <= width GoB g by A113,GOBOARD5:1; then A115:(GoB g)*(i1,width GoB g)`2=(GoB g)*(1,width GoB g)`2 by GOBOARD5:2; then A116:(GoB g)*(i1,width GoB g)`2 = N-bound L~g by Th42; {q`1:q`2=N-bound L~g & q in L~g} c= REAL proof let X be set; assume X in {q`1:q`2=N-bound L~g & q in L~g}; then ex q st X=q`1 & q`2=N-bound L~g & q in L~g; hence thesis; end; then reconsider B={q`1:q`2 = N-bound L~g & q in L~g} as Subset of REAL; set s1 = (GoB g)*(len GoB g,width GoB g)`1; for r be real number st r in B holds s1 >= r proof let r be real number; assume r in B; then consider q such that A117: r=q`1 & q`2 = N-bound L~g & q in L~g ; 1 <= len GoB g & 1 <= width GoB g by GOBOARD7:34,35; hence thesis by A117,Th34; end; then A118:B is bounded_above by SEQ_4:def 1; A119:1<=i & i<=len g by A114,FINSEQ_3:27; now per cases by A119,REAL_1:def 5; case i<len g; then i+1<=len g by NAT_1:38; then g/.i in LSeg(g,i) by A119,TOPREAL1:27; hence (GoB g)*(i1,width GoB g) in L~g by A113,A114,SPPOL_2:17; case A120:i=len g; len g >= 2 by Lm9; then g/.i in LSeg(g,i-'1) by A120,Th3; hence (GoB g)*(i1,width GoB g) in L~g by A113,A114,SPPOL_2:17; end; then (GoB g)*(i1,width GoB g)`2=N-bound L~g & (GoB g)*(i1,width GoB g) in L~g by A115,Th42; then A121:(GoB g)*(i1,width GoB g)`1 in {q`1:q`2=N-bound L~g & q in L~g }; then A122:upper_bound B >= (GoB g)*(i1,width GoB g)`1 by A118,SEQ_4:def 4; for r being real number st r in B holds r <= (GoB g)*(i1,width GoB g)`1 proof let r be real number;assume r in B; then consider q such that A123:r = q`1 & q`2 = N-bound L~g & q in L~g; thus r <= (GoB g)*(i1,width GoB g)`1 by A123,Lm8; end; then upper_bound B <= (GoB g)*(i1,width GoB g)`1 by A121,PSCOMP_1:10; then A124: (GoB g)*(i1,width GoB g)`1 = upper_bound B by A122,AXIOMS: 21 .= sup (proj1 || N-most L~g) by Th29; (GoB g)*(i1,width GoB g)=|[(GoB g)*(i1,width GoB g)`1, (GoB g)*(i1,width GoB g)`2]| by EUCLID:57; then (GoB g)*(i1,width GoB g)=N-max L~g by A116,A124,PSCOMP_1:def 45; hence thesis by A113; end; uniqueness by GOBOARD1:21; end; theorem 1 <= i_w_n h & i_w_n h <= len GoB h & 1 <= i_e_n h & i_e_n h <= len GoB h & 1 <= i_w_s h & i_w_s h <= len GoB h & 1 <= i_e_s h & i_e_s h <= len GoB h proof A1: [i_w_n h, width GoB h] in Indices GoB h by Def7; A2: [i_e_n h, width GoB h] in Indices GoB h by Def8; A3: [i_w_s h, 1] in Indices GoB h by Def5; [i_e_s h, 1] in Indices GoB h by Def6; hence thesis by A1,A2,A3,GOBOARD5:1; end; theorem 1 <= i_n_e h & i_n_e h <= width GoB h & 1 <= i_s_e h & i_s_e h <= width GoB h & 1 <= i_n_w h & i_n_w h <= width GoB h & 1 <= i_s_w h & i_s_w h <= width GoB h proof A1: [len GoB h, i_n_e h] in Indices GoB h by Def4; A2: [len GoB h, i_s_e h] in Indices GoB h by Def3; A3: [1, i_n_w h] in Indices GoB h by Def2; [1, i_s_w h] in Indices GoB h by Def1; hence thesis by A1,A2,A3,GOBOARD5:1; end; Lm10: for i1, i2 be Nat st 1 <= i1 & i1+1<=len h & 1<=i2 & i2+1<=len h & h.i1=h.i2 holds i1 = i2 proof let i1, i2 be Nat; assume A1: 1 <= i1 & i1+1 <= len h; assume A2: 1 <= i2 & i2+1 <= len h & h.i1 = h.i2; assume A3: i1 <> i2; A4: i1 < len h & i2 < len h by A1,A2,NAT_1:38; then A5: i1 in dom h & i2 in dom h by A1,A2,FINSEQ_3:27; then A6: h/.i1 = h.i2 by A2,FINSEQ_4:def 4 .= h/.i2 by A5,FINSEQ_4:def 4; now per cases by A3,AXIOMS:21; suppose i1 < i2; hence h/.i1 <> h/.i2 by A1,A4,GOBOARD7:38; suppose i2 < i1; hence h/.i1 <> h/.i2 by A2,A4,GOBOARD7:38; end; hence thesis by A6; end; definition let g be non constant standard special_circular_sequence; func n_s_w g -> Nat means :Def9: 1 <= it & it+1 <= len g & g.it = W-min L~g; existence proof W-min L~g in rng g by SPRECT_2:47; then consider i be Nat such that A1: 1 <= i & i+1 <= len g & g.i = W-min L~g by Th5; thus thesis by A1; end; uniqueness by Lm10; func n_n_w g -> Nat means :Def10: 1 <= it & it+1 <= len g & g.it = W-max L~g; existence proof W-max L~g in rng g by SPRECT_2:48; then consider i be Nat such that A2: 1 <= i & i+1 <= len g & g.i = W-max L~g by Th5; thus thesis by A2; end; uniqueness by Lm10; func n_s_e g -> Nat means :Def11: 1 <= it & it+1 <= len g & g.it = E-min L~g; existence proof E-min L~g in rng g by SPRECT_2:49; then consider i be Nat such that A3: 1 <= i & i+1 <= len g & g.i = E-min L~g by Th5; thus thesis by A3; end; uniqueness by Lm10; func n_n_e g -> Nat means :Def12: 1 <= it & it+1 <= len g & g.it = E-max L~g; existence proof E-max L~g in rng g by SPRECT_2:50; then consider i be Nat such that A4: 1 <= i & i+1 <= len g & g.i = E-max L~g by Th5; thus thesis by A4; end; uniqueness by Lm10; func n_w_s g -> Nat means :Def13: 1 <= it & it+1 <= len g & g.it = S-min L~g; existence proof S-min L~g in rng g by SPRECT_2:45; then consider i be Nat such that A5: 1 <= i & i+1 <= len g & g.i = S-min L~g by Th5; thus thesis by A5; end; uniqueness by Lm10; func n_e_s g -> Nat means :Def14: 1 <= it & it+1 <= len g & g.it = S-max L~g; existence proof S-max L~g in rng g by SPRECT_2:46; then consider i be Nat such that A6: 1 <= i & i+1 <= len g & g.i = S-max L~g by Th5; thus thesis by A6; end; uniqueness by Lm10; func n_w_n g -> Nat means :Def15: 1 <= it & it+1 <= len g & g.it = N-min L~g; existence proof N-min L~g in rng g by SPRECT_2:43; then consider i be Nat such that A7: 1 <= i & i+1 <= len g & g.i = N-min L~g by Th5; thus thesis by A7; end; uniqueness by Lm10; func n_e_n g -> Nat means :Def16: 1 <= it & it+1 <= len g & g.it = N-max L~g; existence proof N-max L~g in rng g by SPRECT_2:44; then consider i be Nat such that A8: 1 <= i & i+1 <= len g & g.i = N-max L~g by Th5; thus thesis by A8; end; uniqueness by Lm10; end; theorem n_w_n h <> n_w_s h proof set i1 = n_w_n h, i2 = n_w_s h; A1: i1 <= i1 + 1 & i2 <= i2 + 1 by NAT_1:29; A2: 1 <= i1 & i1+1 <= len h & 1 <= i2 & i2+1 <= len h by Def13,Def15; then i1 <= len h & i2 <= len h by A1,AXIOMS:22; then A3: i1 in dom h & i2 in dom h by A2,FINSEQ_3:27; A4: h.i1 = N-min L~h by Def15; A5: h.i2 = S-min L~h by Def13; A6: h.i1 = h/.i1 & h.i2 = h/.i2 by A3,FINSEQ_4:def 4; thus i1 <> i2 proof assume i1 = i2; then A7: N-bound(L~h) = (h/.i2)`2 by A4,A6,PSCOMP_1:94 .= S-bound(L~h) by A5,A6,PSCOMP_1:114; len h > 4 by GOBOARD7:36; then 1 <= len h by AXIOMS:22; then (h/.1)`2 <= N-bound (L~h) & (h/.1)`2 >= S-bound (L~h) by Th13; then A8: (h/.1)`2 = N-bound (L~h) by A7,AXIOMS:21; consider ii be Nat such that A9: ii in dom h & (h/.ii)`2 <> (h/.1)`2 by GOBOARD7:33; A10: 1 <= ii & ii <= len h by A9,FINSEQ_3:27; then A11: (h/.ii)`2 <= N-bound (L~h) by Th13; (h/.ii)`2 >= S-bound L~h by A10,Th13; hence thesis by A7,A8,A9,A11,AXIOMS:21; end; end; theorem n_s_w h <> n_s_e h proof set i1 = n_s_w h, i2 = n_s_e h; A1: i1 <= i1 + 1 & i2 <= i2 + 1 by NAT_1:29; A2: 1 <= i1 & i1+1 <= len h & 1 <= i2 & i2+1 <= len h by Def9,Def11; then i1 <= len h & i2 <= len h by A1,AXIOMS:22; then A3: i1 in dom h & i2 in dom h by A2,FINSEQ_3:27; A4: h.i1 = W-min L~h by Def9; A5: h.i2 = E-min L~h by Def11; A6: h.i1 = h/.i1 & h.i2 = h/.i2 by A3,FINSEQ_4:def 4; thus i1 <> i2 proof assume i1 = i2; then A7: W-bound L~h = (h/.i2)`1 by A4,A6,PSCOMP_1:84 .= E-bound L~h by A5,A6,PSCOMP_1:104; len h > 4 by GOBOARD7:36; then 1 <= len h by AXIOMS:22; then (h/.1)`1 <= E-bound L~h & (h/.1)`1 >= W-bound L~h by Th14; then A8: (h/.1)`1 = W-bound L~h by A7,AXIOMS:21; consider ii be Nat such that A9: ii in dom h & (h/.ii)`1 <> (h/.1)`1 by GOBOARD7:32; A10: 1 <= ii & ii <= len h by A9,FINSEQ_3:27; then A11: (h/.ii)`1 <= E-bound L~h by Th14; (h/.ii)`1 >= W-bound L~h by A10,Th14; hence thesis by A7,A8,A9,A11,AXIOMS:21; end; end; theorem n_e_n h <> n_e_s h proof set i1 = n_e_n h, i2 = n_e_s h; A1: i1 <= i1 + 1 & i2 <= i2 + 1 by NAT_1:29; A2: 1 <= i1 & i1+1 <= len h & 1 <= i2 & i2+1 <= len h by Def14,Def16; then i1 <= len h & i2 <= len h by A1,AXIOMS:22; then A3: i1 in dom h & i2 in dom h by A2,FINSEQ_3:27; A4: h.i1 = N-max L~h by Def16; A5: h.i2 = S-max L~h by Def14; A6: h.i1 = h/.i1 & h.i2 = h/.i2 by A3,FINSEQ_4:def 4; thus i1 <> i2 proof assume i1 = i2; then A7: N-bound(L~h) = (h/.i2)`2 by A4,A6,PSCOMP_1:94 .= S-bound(L~h) by A5,A6,PSCOMP_1:114; len h > 4 by GOBOARD7:36; then 1 <= len h by AXIOMS:22; then (h/.1)`2 <= N-bound (L~h) & (h/.1)`2 >= S-bound (L~h) by Th13; then A8: (h/.1)`2 = N-bound (L~h) by A7,AXIOMS:21; consider ii be Nat such that A9: ii in dom h & (h/.ii)`2 <> (h/.1)`2 by GOBOARD7:33; A10: 1 <= ii & ii <= len h by A9,FINSEQ_3:27; then A11: (h/.ii)`2 <= N-bound (L~h) by Th13; (h/.ii)`2 >= S-bound L~h by A10,Th13; hence thesis by A7,A8,A9,A11,AXIOMS:21; end; end; theorem n_n_w h <> n_n_e h proof set i1 = n_n_w h, i2 = n_n_e h; A1: i1 <= i1 + 1 & i2 <= i2 + 1 by NAT_1:29; A2: 1 <= i1 & i1+1 <= len h & 1 <= i2 & i2+1 <= len h by Def10,Def12; then i1 <= len h & i2 <= len h by A1,AXIOMS:22; then A3: i1 in dom h & i2 in dom h by A2,FINSEQ_3:27; A4: h.i1 = W-max L~h by Def10; A5: h.i2 = E-max L~h by Def12; A6: h.i1 = h/.i1 & h.i2 = h/.i2 by A3,FINSEQ_4:def 4; thus i1 <> i2 proof assume i1 = i2; then A7: W-bound L~h = (h/.i2)`1 by A4,A6,PSCOMP_1:84 .= E-bound L~h by A5,A6,PSCOMP_1:104; len h > 4 by GOBOARD7:36; then 1 <= len h by AXIOMS:22; then (h/.1)`1 <= E-bound L~h & (h/.1)`1 >= W-bound L~h by Th14; then A8: (h/.1)`1 = W-bound L~h by A7,AXIOMS:21; consider ii be Nat such that A9: ii in dom h & (h/.ii)`1 <> (h/.1)`1 by GOBOARD7:32; A10: 1 <= ii & ii <= len h by A9,FINSEQ_3:27; then A11: (h/.ii)`1 <= E-bound L~h by Th14; (h/.ii)`1 >= W-bound L~h by A10,Th14; hence thesis by A7,A8,A9,A11,AXIOMS:21; end; end;

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