:: GOBOARD6 semantic presentation begin Lm1: for p, q being Point of (TOP-REAL 2) holds ( (p + q) `1 = (p `1) + (q `1) & (p + q) `2 = (p `2) + (q `2) ) proof let p, q be Point of (TOP-REAL 2); ::_thesis: ( (p + q) `1 = (p `1) + (q `1) & (p + q) `2 = (p `2) + (q `2) ) p + q = |[((p `1) + (q `1)),((p `2) + (q `2))]| by EUCLID:55; hence ( (p + q) `1 = (p `1) + (q `1) & (p + q) `2 = (p `2) + (q `2) ) by EUCLID:52; ::_thesis: verum end; Lm2: for p, q being Point of (TOP-REAL 2) holds ( (p - q) `1 = (p `1) - (q `1) & (p - q) `2 = (p `2) - (q `2) ) proof let p, q be Point of (TOP-REAL 2); ::_thesis: ( (p - q) `1 = (p `1) - (q `1) & (p - q) `2 = (p `2) - (q `2) ) p - q = |[((p `1) - (q `1)),((p `2) - (q `2))]| by EUCLID:61; hence ( (p - q) `1 = (p `1) - (q `1) & (p - q) `2 = (p `2) - (q `2) ) by EUCLID:52; ::_thesis: verum end; Lm3: for r being Real for p being Point of (TOP-REAL 2) holds ( (r * p) `1 = r * (p `1) & (r * p) `2 = r * (p `2) ) proof let r be Real; ::_thesis: for p being Point of (TOP-REAL 2) holds ( (r * p) `1 = r * (p `1) & (r * p) `2 = r * (p `2) ) let p be Point of (TOP-REAL 2); ::_thesis: ( (r * p) `1 = r * (p `1) & (r * p) `2 = r * (p `2) ) r * p = |[(r * (p `1)),(r * (p `2))]| by EUCLID:57; hence ( (r * p) `1 = r * (p `1) & (r * p) `2 = r * (p `2) ) by EUCLID:52; ::_thesis: verum end; theorem Th1: :: GOBOARD6:1 for M being non empty Reflexive MetrStruct for u being Point of M for r being real number st r > 0 holds u in Ball (u,r) proof let M be non empty Reflexive MetrStruct ; ::_thesis: for u being Point of M for r being real number st r > 0 holds u in Ball (u,r) let u be Point of M; ::_thesis: for r being real number st r > 0 holds u in Ball (u,r) let r be real number ; ::_thesis: ( r > 0 implies u in Ball (u,r) ) A1: ( Ball (u,r) = { q where q is Point of M : dist (u,q) < r } & dist (u,u) = 0 ) by METRIC_1:1, METRIC_1:17; assume r > 0 ; ::_thesis: u in Ball (u,r) hence u in Ball (u,r) by A1; ::_thesis: verum end; Lm4: for M being MetrSpace for B being Subset of (TopSpaceMetr M) for r being real number for u being Point of M st B = Ball (u,r) holds B is open proof let M be MetrSpace; ::_thesis: for B being Subset of (TopSpaceMetr M) for r being real number for u being Point of M st B = Ball (u,r) holds B is open let B be Subset of (TopSpaceMetr M); ::_thesis: for r being real number for u being Point of M st B = Ball (u,r) holds B is open let r be real number ; ::_thesis: for u being Point of M st B = Ball (u,r) holds B is open let u be Point of M; ::_thesis: ( B = Ball (u,r) implies B is open ) A1: ( TopSpaceMetr M = TopStruct(# the carrier of M,(Family_open_set M) #) & Ball (u,r) in Family_open_set M ) by PCOMPS_1:29, PCOMPS_1:def_5; assume B = Ball (u,r) ; ::_thesis: B is open hence B is open by A1, PRE_TOPC:def_2; ::_thesis: verum end; theorem Th2: :: GOBOARD6:2 for n being Nat for p being Point of (Euclid n) for q being Point of (TOP-REAL n) for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q proof let n be Nat; ::_thesis: for p being Point of (Euclid n) for q being Point of (TOP-REAL n) for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q let p be Point of (Euclid n); ::_thesis: for q being Point of (TOP-REAL n) for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q let q be Point of (TOP-REAL n); ::_thesis: for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q let r be real number ; ::_thesis: ( p = q & r > 0 implies Ball (p,r) is a_neighborhood of q ) reconsider A = Ball (p,r) as Subset of (TOP-REAL n) by TOPREAL3:8; A1: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider AA = A as Subset of (TopSpaceMetr (Euclid n)) ; AA is open by TOPMETR:14; then A2: A is open by A1, PRE_TOPC:30; assume ( p = q & r > 0 ) ; ::_thesis: Ball (p,r) is a_neighborhood of q hence Ball (p,r) is a_neighborhood of q by A2, Th1, CONNSP_2:3; ::_thesis: verum end; theorem Th3: :: GOBOARD6:3 for n being Nat for r being Real for B being Subset of (TOP-REAL n) for u being Point of (Euclid n) st B = Ball (u,r) holds B is open proof let n be Nat; ::_thesis: for r being Real for B being Subset of (TOP-REAL n) for u being Point of (Euclid n) st B = Ball (u,r) holds B is open let r be Real; ::_thesis: for B being Subset of (TOP-REAL n) for u being Point of (Euclid n) st B = Ball (u,r) holds B is open let B be Subset of (TOP-REAL n); ::_thesis: for u being Point of (Euclid n) st B = Ball (u,r) holds B is open let u be Point of (Euclid n); ::_thesis: ( B = Ball (u,r) implies B is open ) A1: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider BB = B as Subset of (TopSpaceMetr (Euclid n)) ; assume B = Ball (u,r) ; ::_thesis: B is open then BB is open by Lm4; hence B is open by A1, PRE_TOPC:30; ::_thesis: verum end; theorem Th4: :: GOBOARD6:4 for M being non empty MetrSpace for u being Point of M for P being Subset of (TopSpaceMetr M) holds ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) proof let M be non empty MetrSpace; ::_thesis: for u being Point of M for P being Subset of (TopSpaceMetr M) holds ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) let u be Point of M; ::_thesis: for P being Subset of (TopSpaceMetr M) holds ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) let P be Subset of (TopSpaceMetr M); ::_thesis: ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) hereby ::_thesis: ( ex r being real number st ( r > 0 & Ball (u,r) c= P ) implies u in Int P ) assume u in Int P ; ::_thesis: ex r being real number st ( r > 0 & Ball (u,r) c= P ) then consider r being real number such that A1: r > 0 and A2: Ball (u,r) c= Int P by TOPMETR:15; take r = r; ::_thesis: ( r > 0 & Ball (u,r) c= P ) thus r > 0 by A1; ::_thesis: Ball (u,r) c= P Int P c= P by TOPS_1:16; hence Ball (u,r) c= P by A2, XBOOLE_1:1; ::_thesis: verum end; given r being real number such that A3: r > 0 and A4: Ball (u,r) c= P ; ::_thesis: u in Int P TopSpaceMetr M = TopStruct(# the carrier of M,(Family_open_set M) #) by PCOMPS_1:def_5; then reconsider B = Ball (u,r) as Subset of (TopSpaceMetr M) ; A5: B is open by Lm4; u in Ball (u,r) by A3, Th1; hence u in Int P by A4, A5, TOPS_1:22; ::_thesis: verum end; Lm5: for T being TopSpace for A being Subset of T for B being Subset of TopStruct(# the carrier of T, the topology of T #) st A = B holds Int A = Int B proof let T be TopSpace; ::_thesis: for A being Subset of T for B being Subset of TopStruct(# the carrier of T, the topology of T #) st A = B holds Int A = Int B let A be Subset of T; ::_thesis: for B being Subset of TopStruct(# the carrier of T, the topology of T #) st A = B holds Int A = Int B let B be Subset of TopStruct(# the carrier of T, the topology of T #); ::_thesis: ( A = B implies Int A = Int B ) assume A1: A = B ; ::_thesis: Int A = Int B reconsider AA = Int A as Subset of TopStruct(# the carrier of T, the topology of T #) ; AA is open by PRE_TOPC:30; hence Int A c= Int B by A1, TOPS_1:16, TOPS_1:24; :: according to XBOOLE_0:def_10 ::_thesis: Int B c= Int A reconsider BB = Int B as Subset of T ; BB is open by PRE_TOPC:30; hence Int B c= Int A by A1, TOPS_1:16, TOPS_1:24; ::_thesis: verum end; theorem Th5: :: GOBOARD6:5 for n being Nat for u being Point of (Euclid n) for P being Subset of (TOP-REAL n) holds ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) proof let n be Nat; ::_thesis: for u being Point of (Euclid n) for P being Subset of (TOP-REAL n) holds ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) let u be Point of (Euclid n); ::_thesis: for P being Subset of (TOP-REAL n) holds ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) let P be Subset of (TOP-REAL n); ::_thesis: ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) A1: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8; then reconsider PP = P as Subset of (TopSpaceMetr (Euclid n)) ; ( u in Int PP iff ex r being real number st ( r > 0 & Ball (u,r) c= PP ) ) by Th4; hence ( u in Int P iff ex r being real number st ( r > 0 & Ball (u,r) c= P ) ) by A1, Lm5; ::_thesis: verum end; theorem Th6: :: GOBOARD6:6 for r1, s1, r2, s2 being Real for u, v being Point of (Euclid 2) st u = |[r1,s1]| & v = |[r2,s2]| holds dist (u,v) = sqrt (((r1 - r2) ^2) + ((s1 - s2) ^2)) proof let r1, s1, r2, s2 be Real; ::_thesis: for u, v being Point of (Euclid 2) st u = |[r1,s1]| & v = |[r2,s2]| holds dist (u,v) = sqrt (((r1 - r2) ^2) + ((s1 - s2) ^2)) let u, v be Point of (Euclid 2); ::_thesis: ( u = |[r1,s1]| & v = |[r2,s2]| implies dist (u,v) = sqrt (((r1 - r2) ^2) + ((s1 - s2) ^2)) ) assume A1: ( u = |[r1,s1]| & v = |[r2,s2]| ) ; ::_thesis: dist (u,v) = sqrt (((r1 - r2) ^2) + ((s1 - s2) ^2)) A2: ( |[r1,s1]| `1 = r1 & |[r1,s1]| `2 = s1 ) by EUCLID:52; A3: ( |[r2,s2]| `1 = r2 & |[r2,s2]| `2 = s2 ) by EUCLID:52; thus dist (u,v) = (Pitag_dist 2) . (u,v) by METRIC_1:def_1 .= sqrt (((r1 - r2) ^2) + ((s1 - s2) ^2)) by A1, A2, A3, TOPREAL3:7 ; ::_thesis: verum end; theorem Th7: :: GOBOARD6:7 for r, s, r2, r1 being Real for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= r2 & r2 < r1 holds |[(r + r2),s]| in Ball (u,r1) proof let r, s, r2, r1 be Real; ::_thesis: for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= r2 & r2 < r1 holds |[(r + r2),s]| in Ball (u,r1) let u be Point of (Euclid 2); ::_thesis: ( u = |[r,s]| & 0 <= r2 & r2 < r1 implies |[(r + r2),s]| in Ball (u,r1) ) assume that A1: u = |[r,s]| and A2: 0 <= r2 and A3: r2 < r1 ; ::_thesis: |[(r + r2),s]| in Ball (u,r1) reconsider v = |[(r + r2),s]| as Point of (Euclid 2) by TOPREAL3:8; dist (u,v) = sqrt (((r - (r + r2)) ^2) + ((s - s) ^2)) by A1, Th6 .= sqrt ((- (r - (r + r2))) ^2) .= r2 by A2, SQUARE_1:22 ; hence |[(r + r2),s]| in Ball (u,r1) by A3, METRIC_1:11; ::_thesis: verum end; theorem Th8: :: GOBOARD6:8 for r, s, s2, s1 being Real for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= s2 & s2 < s1 holds |[r,(s + s2)]| in Ball (u,s1) proof let r, s, s2, s1 be Real; ::_thesis: for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= s2 & s2 < s1 holds |[r,(s + s2)]| in Ball (u,s1) let u be Point of (Euclid 2); ::_thesis: ( u = |[r,s]| & 0 <= s2 & s2 < s1 implies |[r,(s + s2)]| in Ball (u,s1) ) assume that A1: u = |[r,s]| and A2: 0 <= s2 and A3: s2 < s1 ; ::_thesis: |[r,(s + s2)]| in Ball (u,s1) reconsider v = |[r,(s + s2)]| as Point of (Euclid 2) by TOPREAL3:8; dist (u,v) = sqrt (((r - r) ^2) + ((s - (s + s2)) ^2)) by A1, Th6 .= sqrt ((- (s - (s + s2))) ^2) .= s2 by A2, SQUARE_1:22 ; hence |[r,(s + s2)]| in Ball (u,s1) by A3, METRIC_1:11; ::_thesis: verum end; theorem Th9: :: GOBOARD6:9 for r, s, r2, r1 being Real for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= r2 & r2 < r1 holds |[(r - r2),s]| in Ball (u,r1) proof let r, s, r2, r1 be Real; ::_thesis: for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= r2 & r2 < r1 holds |[(r - r2),s]| in Ball (u,r1) let u be Point of (Euclid 2); ::_thesis: ( u = |[r,s]| & 0 <= r2 & r2 < r1 implies |[(r - r2),s]| in Ball (u,r1) ) assume that A1: u = |[r,s]| and A2: 0 <= r2 and A3: r2 < r1 ; ::_thesis: |[(r - r2),s]| in Ball (u,r1) reconsider v = |[(r - r2),s]| as Point of (Euclid 2) by TOPREAL3:8; dist (u,v) = sqrt (((r - (r - r2)) ^2) + ((s - s) ^2)) by A1, Th6 .= r2 by A2, SQUARE_1:22 ; hence |[(r - r2),s]| in Ball (u,r1) by A3, METRIC_1:11; ::_thesis: verum end; theorem Th10: :: GOBOARD6:10 for r, s, s2, s1 being Real for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= s2 & s2 < s1 holds |[r,(s - s2)]| in Ball (u,s1) proof let r, s, s2, s1 be Real; ::_thesis: for u being Point of (Euclid 2) st u = |[r,s]| & 0 <= s2 & s2 < s1 holds |[r,(s - s2)]| in Ball (u,s1) let u be Point of (Euclid 2); ::_thesis: ( u = |[r,s]| & 0 <= s2 & s2 < s1 implies |[r,(s - s2)]| in Ball (u,s1) ) assume that A1: u = |[r,s]| and A2: 0 <= s2 and A3: s2 < s1 ; ::_thesis: |[r,(s - s2)]| in Ball (u,s1) reconsider v = |[r,(s - s2)]| as Point of (Euclid 2) by TOPREAL3:8; dist (u,v) = sqrt (((s - (s - s2)) ^2) + ((r - r) ^2)) by A1, Th6 .= s2 by A2, SQUARE_1:22 ; hence |[r,(s - s2)]| in Ball (u,s1) by A3, METRIC_1:11; ::_thesis: verum end; theorem Th11: :: GOBOARD6:11 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds (G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j)) proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds (G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j)) let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies (G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j)) ) assume that A1: ( 1 <= i & i < len G ) and A2: ( 1 <= j & j < width G ) ; ::_thesis: (G * (i,j)) + (G * ((i + 1),(j + 1))) = (G * (i,(j + 1))) + (G * ((i + 1),j)) A3: ( 1 <= j + 1 & j + 1 <= width G ) by A2, NAT_1:13; A4: ( 1 <= i + 1 & i + 1 <= len G ) by A1, NAT_1:13; then A5: (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),1)) `1 by A3, GOBOARD5:2 .= (G * ((i + 1),j)) `1 by A2, A4, GOBOARD5:2 ; A6: (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A4, A3, GOBOARD5:1 .= (G * (i,(j + 1))) `2 by A1, A3, GOBOARD5:1 ; A7: (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A2, GOBOARD5:1 .= (G * ((i + 1),j)) `2 by A2, A4, GOBOARD5:1 ; A8: ((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2 = ((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2) by Lm1 .= ((G * (i,(j + 1))) + (G * ((i + 1),j))) `2 by A7, A6, Lm1 ; A9: (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A2, GOBOARD5:2 .= (G * (i,(j + 1))) `1 by A1, A3, GOBOARD5:2 ; ((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1 = ((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1) by Lm1 .= ((G * (i,(j + 1))) + (G * ((i + 1),j))) `1 by A9, A5, Lm1 ; hence (G * (i,j)) + (G * ((i + 1),(j + 1))) = |[(((G * (i,(j + 1))) + (G * ((i + 1),j))) `1),(((G * (i,(j + 1))) + (G * ((i + 1),j))) `2)]| by A8, EUCLID:53 .= (G * (i,(j + 1))) + (G * ((i + 1),j)) by EUCLID:53 ; ::_thesis: verum end; Lm6: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8 .= TopStruct(# the carrier of (Euclid 2),(Family_open_set (Euclid 2)) #) by PCOMPS_1:def_5 ; theorem Th12: :: GOBOARD6:12 for G being Go-board holds Int (v_strip (G,0)) = { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } proof let G be Go-board; ::_thesis: Int (v_strip (G,0)) = { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } 0 <> width G by GOBOARD1:def_3; then 1 <= width G by NAT_1:14; then A1: v_strip (G,0) = { |[r,s]| where r, s is Real : r <= (G * (1,1)) `1 } by GOBOARD5:10; thus Int (v_strip (G,0)) c= { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } c= Int (v_strip (G,0)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (v_strip (G,0)) or x in { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } ) assume A2: x in Int (v_strip (G,0)) ; ::_thesis: x in { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } then reconsider u = x as Point of (Euclid 2) by Lm6; consider r1 being real number such that A3: r1 > 0 and A4: Ball (u,r1) c= v_strip (G,0) by A2, Th5; reconsider p = u as Point of (TOP-REAL 2) by Lm6; A5: p = |[(p `1),(p `2)]| by EUCLID:53; reconsider r1 = r1 as Real by XREAL_0:def_1; set q = |[((p `1) + (r1 / 2)),((p `2) + 0)]|; r1 / 2 < r1 by A3, XREAL_1:216; then |[((p `1) + (r1 / 2)),((p `2) + 0)]| in Ball (u,r1) by A3, A5, Th7; then |[((p `1) + (r1 / 2)),((p `2) + 0)]| in v_strip (G,0) by A4; then ex r2, s2 being Real st ( |[((p `1) + (r1 / 2)),((p `2) + 0)]| = |[r2,s2]| & r2 <= (G * (1,1)) `1 ) by A1; then A6: (p `1) + (r1 / 2) <= (G * (1,1)) `1 by SPPOL_2:1; p `1 < (p `1) + (r1 / 2) by A3, XREAL_1:29, XREAL_1:215; then p `1 < (G * (1,1)) `1 by A6, XXREAL_0:2; hence x in { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } by A5; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } or x in Int (v_strip (G,0)) ) assume x in { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } ; ::_thesis: x in Int (v_strip (G,0)) then consider r, s being Real such that A7: x = |[r,s]| and A8: r < (G * (1,1)) `1 ; reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8; A9: Ball (u,(((G * (1,1)) `1) - r)) c= v_strip (G,0) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (u,(((G * (1,1)) `1) - r)) or y in v_strip (G,0) ) A10: Ball (u,(((G * (1,1)) `1) - r)) = { v where v is Point of (Euclid 2) : dist (u,v) < ((G * (1,1)) `1) - r } by METRIC_1:17; assume y in Ball (u,(((G * (1,1)) `1) - r)) ; ::_thesis: y in v_strip (G,0) then consider v being Point of (Euclid 2) such that A11: v = y and A12: dist (u,v) < ((G * (1,1)) `1) - r by A10; reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:8; ( (r - (q `1)) ^2 >= 0 & ((r - (q `1)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) ) by XREAL_1:6, XREAL_1:63; then A13: sqrt ((r - (q `1)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) by SQUARE_1:26; A14: q = |[(q `1),(q `2)]| by EUCLID:53; then sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < ((G * (1,1)) `1) - r by A12, Th6; then sqrt ((r - (q `1)) ^2) <= ((G * (1,1)) `1) - r by A13, XXREAL_0:2; then A15: abs (r - (q `1)) <= ((G * (1,1)) `1) - r by COMPLEX1:72; percases ( r <= q `1 or r >= q `1 ) ; suppose r <= q `1 ; ::_thesis: y in v_strip (G,0) then A16: (q `1) - r >= 0 by XREAL_1:48; abs (r - (q `1)) = abs (- (r - (q `1))) by COMPLEX1:52 .= (q `1) - r by A16, ABSVALUE:def_1 ; then q `1 <= (G * (1,1)) `1 by A15, XREAL_1:9; hence y in v_strip (G,0) by A1, A11, A14; ::_thesis: verum end; suppose r >= q `1 ; ::_thesis: y in v_strip (G,0) then q `1 <= (G * (1,1)) `1 by A8, XXREAL_0:2; hence y in v_strip (G,0) by A1, A11, A14; ::_thesis: verum end; end; end; reconsider B = Ball (u,(((G * (1,1)) `1) - r)) as Subset of (TOP-REAL 2) by TOPREAL3:8; A17: B is open by Th3; u in Ball (u,(((G * (1,1)) `1) - r)) by A8, Th1, XREAL_1:50; hence x in Int (v_strip (G,0)) by A7, A9, A17, TOPS_1:22; ::_thesis: verum end; theorem Th13: :: GOBOARD6:13 for G being Go-board holds Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } proof let G be Go-board; ::_thesis: Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } 0 <> width G by GOBOARD1:def_3; then 1 <= width G by NAT_1:14; then A1: v_strip (G,(len G)) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 <= r } by GOBOARD5:9; thus Int (v_strip (G,(len G))) c= { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } c= Int (v_strip (G,(len G))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (v_strip (G,(len G))) or x in { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } ) assume A2: x in Int (v_strip (G,(len G))) ; ::_thesis: x in { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } then reconsider u = x as Point of (Euclid 2) by Lm6; consider r1 being real number such that A3: r1 > 0 and A4: Ball (u,r1) c= v_strip (G,(len G)) by A2, Th5; reconsider p = u as Point of (TOP-REAL 2) by Lm6; A5: p = |[(p `1),(p `2)]| by EUCLID:53; reconsider r1 = r1 as Real by XREAL_0:def_1; set q = |[((p `1) - (r1 / 2)),((p `2) + 0)]|; r1 / 2 < r1 by A3, XREAL_1:216; then |[((p `1) - (r1 / 2)),((p `2) + 0)]| in Ball (u,r1) by A3, A5, Th9; then |[((p `1) - (r1 / 2)),((p `2) + 0)]| in v_strip (G,(len G)) by A4; then ex r2, s2 being Real st ( |[((p `1) - (r1 / 2)),((p `2) + 0)]| = |[r2,s2]| & (G * ((len G),1)) `1 <= r2 ) by A1; then (G * ((len G),1)) `1 <= (p `1) - (r1 / 2) by SPPOL_2:1; then A6: ((G * ((len G),1)) `1) + (r1 / 2) <= p `1 by XREAL_1:19; (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + (r1 / 2) by A3, XREAL_1:29, XREAL_1:215; then (G * ((len G),1)) `1 < p `1 by A6, XXREAL_0:2; hence x in { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } by A5; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } or x in Int (v_strip (G,(len G))) ) assume x in { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } ; ::_thesis: x in Int (v_strip (G,(len G))) then consider r, s being Real such that A7: x = |[r,s]| and A8: (G * ((len G),1)) `1 < r ; reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8; A9: Ball (u,(r - ((G * ((len G),1)) `1))) c= v_strip (G,(len G)) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (u,(r - ((G * ((len G),1)) `1))) or y in v_strip (G,(len G)) ) A10: Ball (u,(r - ((G * ((len G),1)) `1))) = { v where v is Point of (Euclid 2) : dist (u,v) < r - ((G * ((len G),1)) `1) } by METRIC_1:17; assume y in Ball (u,(r - ((G * ((len G),1)) `1))) ; ::_thesis: y in v_strip (G,(len G)) then consider v being Point of (Euclid 2) such that A11: v = y and A12: dist (u,v) < r - ((G * ((len G),1)) `1) by A10; reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:8; ( (r - (q `1)) ^2 >= 0 & ((r - (q `1)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) ) by XREAL_1:6, XREAL_1:63; then A13: sqrt ((r - (q `1)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) by SQUARE_1:26; A14: q = |[(q `1),(q `2)]| by EUCLID:53; then sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < r - ((G * ((len G),1)) `1) by A12, Th6; then sqrt ((r - (q `1)) ^2) <= r - ((G * ((len G),1)) `1) by A13, XXREAL_0:2; then A15: abs (r - (q `1)) <= r - ((G * ((len G),1)) `1) by COMPLEX1:72; percases ( r >= q `1 or r <= q `1 ) ; suppose r >= q `1 ; ::_thesis: y in v_strip (G,(len G)) then r - (q `1) >= 0 by XREAL_1:48; then abs (r - (q `1)) = r - (q `1) by ABSVALUE:def_1; then (G * ((len G),1)) `1 <= q `1 by A15, XREAL_1:10; hence y in v_strip (G,(len G)) by A1, A11, A14; ::_thesis: verum end; suppose r <= q `1 ; ::_thesis: y in v_strip (G,(len G)) then (G * ((len G),1)) `1 <= q `1 by A8, XXREAL_0:2; hence y in v_strip (G,(len G)) by A1, A11, A14; ::_thesis: verum end; end; end; reconsider B = Ball (u,(r - ((G * ((len G),1)) `1))) as Subset of (TOP-REAL 2) by TOPREAL3:8; A16: B is open by Th3; u in Ball (u,(r - ((G * ((len G),1)) `1))) by A8, Th1, XREAL_1:50; hence x in Int (v_strip (G,(len G))) by A7, A9, A16, TOPS_1:22; ::_thesis: verum end; theorem Th14: :: GOBOARD6:14 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } ) 0 <> width G by GOBOARD1:def_3; then A1: 1 <= width G by NAT_1:14; assume ( 1 <= i & i < len G ) ; ::_thesis: Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } then A2: v_strip (G,i) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 <= r & r <= (G * ((i + 1),1)) `1 ) } by A1, GOBOARD5:8; thus Int (v_strip (G,i)) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } c= Int (v_strip (G,i)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (v_strip (G,i)) or x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } ) assume A3: x in Int (v_strip (G,i)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } then reconsider u = x as Point of (Euclid 2) by Lm6; consider r1 being real number such that A4: r1 > 0 and A5: Ball (u,r1) c= v_strip (G,i) by A3, Th5; reconsider p = u as Point of (TOP-REAL 2) by Lm6; A6: p = |[(p `1),(p `2)]| by EUCLID:53; reconsider r1 = r1 as Real by XREAL_0:def_1; set q2 = |[((p `1) - (r1 / 2)),((p `2) + 0)]|; A7: r1 / 2 < r1 by A4, XREAL_1:216; then |[((p `1) - (r1 / 2)),((p `2) + 0)]| in Ball (u,r1) by A4, A6, Th9; then |[((p `1) - (r1 / 2)),((p `2) + 0)]| in v_strip (G,i) by A5; then ex r2, s2 being Real st ( |[((p `1) - (r1 / 2)),((p `2) + 0)]| = |[r2,s2]| & (G * (i,1)) `1 <= r2 & r2 <= (G * ((i + 1),1)) `1 ) by A2; then (G * (i,1)) `1 <= (p `1) - (r1 / 2) by SPPOL_2:1; then A8: ((G * (i,1)) `1) + (r1 / 2) <= p `1 by XREAL_1:19; set q1 = |[((p `1) + (r1 / 2)),((p `2) + 0)]|; |[((p `1) + (r1 / 2)),((p `2) + 0)]| in Ball (u,r1) by A4, A6, A7, Th7; then |[((p `1) + (r1 / 2)),((p `2) + 0)]| in v_strip (G,i) by A5; then ex r2, s2 being Real st ( |[((p `1) + (r1 / 2)),((p `2) + 0)]| = |[r2,s2]| & (G * (i,1)) `1 <= r2 & r2 <= (G * ((i + 1),1)) `1 ) by A2; then A9: (p `1) + (r1 / 2) <= (G * ((i + 1),1)) `1 by SPPOL_2:1; (G * (i,1)) `1 < ((G * (i,1)) `1) + (r1 / 2) by A4, XREAL_1:29, XREAL_1:215; then A10: (G * (i,1)) `1 < p `1 by A8, XXREAL_0:2; p `1 < (p `1) + (r1 / 2) by A4, XREAL_1:29, XREAL_1:215; then p `1 < (G * ((i + 1),1)) `1 by A9, XXREAL_0:2; hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by A6, A10; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } or x in Int (v_strip (G,i)) ) assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } ; ::_thesis: x in Int (v_strip (G,i)) then consider r, s being Real such that A11: x = |[r,s]| and A12: (G * (i,1)) `1 < r and A13: r < (G * ((i + 1),1)) `1 ; reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8; ( ((G * ((i + 1),1)) `1) - r > 0 & r - ((G * (i,1)) `1) > 0 ) by A12, A13, XREAL_1:50; then min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)) > 0 by XXREAL_0:15; then A14: u in Ball (u,(min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)))) by Th1; A15: Ball (u,(min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)))) c= v_strip (G,i) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (u,(min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)))) or y in v_strip (G,i) ) A16: Ball (u,(min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)))) = { v where v is Point of (Euclid 2) : dist (u,v) < min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)) } by METRIC_1:17; assume y in Ball (u,(min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)))) ; ::_thesis: y in v_strip (G,i) then consider v being Point of (Euclid 2) such that A17: v = y and A18: dist (u,v) < min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)) by A16; reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:8; ( (r - (q `1)) ^2 >= 0 & ((r - (q `1)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) ) by XREAL_1:6, XREAL_1:63; then A19: sqrt ((r - (q `1)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) by SQUARE_1:26; A20: q = |[(q `1),(q `2)]| by EUCLID:53; then sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)) by A18, Th6; then sqrt ((r - (q `1)) ^2) <= min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)) by A19, XXREAL_0:2; then A21: abs (r - (q `1)) <= min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)) by COMPLEX1:72; then A22: abs (r - (q `1)) <= r - ((G * (i,1)) `1) by XXREAL_0:22; A23: abs (r - (q `1)) <= ((G * ((i + 1),1)) `1) - r by A21, XXREAL_0:22; percases ( r <= q `1 or r >= q `1 ) ; supposeA24: r <= q `1 ; ::_thesis: y in v_strip (G,i) then A25: (q `1) - r >= 0 by XREAL_1:48; abs (r - (q `1)) = abs (- (r - (q `1))) by COMPLEX1:52 .= (q `1) - r by A25, ABSVALUE:def_1 ; then A26: q `1 <= (G * ((i + 1),1)) `1 by A23, XREAL_1:9; (G * (i,1)) `1 <= q `1 by A12, A24, XXREAL_0:2; hence y in v_strip (G,i) by A2, A17, A20, A26; ::_thesis: verum end; supposeA27: r >= q `1 ; ::_thesis: y in v_strip (G,i) then r - (q `1) >= 0 by XREAL_1:48; then abs (r - (q `1)) = r - (q `1) by ABSVALUE:def_1; then A28: (G * (i,1)) `1 <= q `1 by A22, XREAL_1:10; q `1 <= (G * ((i + 1),1)) `1 by A13, A27, XXREAL_0:2; hence y in v_strip (G,i) by A2, A17, A20, A28; ::_thesis: verum end; end; end; reconsider B = Ball (u,(min ((r - ((G * (i,1)) `1)),(((G * ((i + 1),1)) `1) - r)))) as Subset of (TOP-REAL 2) by TOPREAL3:8; B is open by Th3; hence x in Int (v_strip (G,i)) by A11, A14, A15, TOPS_1:22; ::_thesis: verum end; theorem Th15: :: GOBOARD6:15 for G being Go-board holds Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } proof let G be Go-board; ::_thesis: Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } 0 <> len G by GOBOARD1:def_3; then 1 <= len G by NAT_1:14; then A1: h_strip (G,0) = { |[r,s]| where r, s is Real : s <= (G * (1,1)) `2 } by GOBOARD5:7; thus Int (h_strip (G,0)) c= { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } c= Int (h_strip (G,0)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (h_strip (G,0)) or x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } ) assume A2: x in Int (h_strip (G,0)) ; ::_thesis: x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } then reconsider u = x as Point of (Euclid 2) by Lm6; consider s1 being real number such that A3: s1 > 0 and A4: Ball (u,s1) c= h_strip (G,0) by A2, Th5; reconsider p = u as Point of (TOP-REAL 2) by Lm6; A5: p = |[(p `1),(p `2)]| by EUCLID:53; reconsider s1 = s1 as Real by XREAL_0:def_1; set q = |[((p `1) + 0),((p `2) + (s1 / 2))]|; s1 / 2 < s1 by A3, XREAL_1:216; then |[((p `1) + 0),((p `2) + (s1 / 2))]| in Ball (u,s1) by A3, A5, Th8; then |[((p `1) + 0),((p `2) + (s1 / 2))]| in h_strip (G,0) by A4; then ex r2, s2 being Real st ( |[((p `1) + 0),((p `2) + (s1 / 2))]| = |[r2,s2]| & s2 <= (G * (1,1)) `2 ) by A1; then A6: (p `2) + (s1 / 2) <= (G * (1,1)) `2 by SPPOL_2:1; p `2 < (p `2) + (s1 / 2) by A3, XREAL_1:29, XREAL_1:215; then p `2 < (G * (1,1)) `2 by A6, XXREAL_0:2; hence x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } by A5; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } or x in Int (h_strip (G,0)) ) assume x in { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } ; ::_thesis: x in Int (h_strip (G,0)) then consider r, s being Real such that A7: x = |[r,s]| and A8: s < (G * (1,1)) `2 ; reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8; A9: Ball (u,(((G * (1,1)) `2) - s)) c= h_strip (G,0) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (u,(((G * (1,1)) `2) - s)) or y in h_strip (G,0) ) A10: Ball (u,(((G * (1,1)) `2) - s)) = { v where v is Point of (Euclid 2) : dist (u,v) < ((G * (1,1)) `2) - s } by METRIC_1:17; assume y in Ball (u,(((G * (1,1)) `2) - s)) ; ::_thesis: y in h_strip (G,0) then consider v being Point of (Euclid 2) such that A11: v = y and A12: dist (u,v) < ((G * (1,1)) `2) - s by A10; reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:8; ( (s - (q `2)) ^2 >= 0 & ((s - (q `2)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) ) by XREAL_1:6, XREAL_1:63; then A13: sqrt ((s - (q `2)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) by SQUARE_1:26; A14: q = |[(q `1),(q `2)]| by EUCLID:53; then sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < ((G * (1,1)) `2) - s by A12, Th6; then sqrt ((s - (q `2)) ^2) <= ((G * (1,1)) `2) - s by A13, XXREAL_0:2; then A15: abs (s - (q `2)) <= ((G * (1,1)) `2) - s by COMPLEX1:72; percases ( s <= q `2 or s >= q `2 ) ; suppose s <= q `2 ; ::_thesis: y in h_strip (G,0) then A16: (q `2) - s >= 0 by XREAL_1:48; abs (s - (q `2)) = abs (- (s - (q `2))) by COMPLEX1:52 .= (q `2) - s by A16, ABSVALUE:def_1 ; then q `2 <= (G * (1,1)) `2 by A15, XREAL_1:9; hence y in h_strip (G,0) by A1, A11, A14; ::_thesis: verum end; suppose s >= q `2 ; ::_thesis: y in h_strip (G,0) then q `2 <= (G * (1,1)) `2 by A8, XXREAL_0:2; hence y in h_strip (G,0) by A1, A11, A14; ::_thesis: verum end; end; end; reconsider B = Ball (u,(((G * (1,1)) `2) - s)) as Subset of (TOP-REAL 2) by TOPREAL3:8; A17: B is open by Th3; u in Ball (u,(((G * (1,1)) `2) - s)) by A8, Th1, XREAL_1:50; hence x in Int (h_strip (G,0)) by A7, A9, A17, TOPS_1:22; ::_thesis: verum end; theorem Th16: :: GOBOARD6:16 for G being Go-board holds Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } proof let G be Go-board; ::_thesis: Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } 0 <> len G by GOBOARD1:def_3; then 1 <= len G by NAT_1:14; then A1: h_strip (G,(width G)) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 <= s } by GOBOARD5:6; thus Int (h_strip (G,(width G))) c= { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } c= Int (h_strip (G,(width G))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (h_strip (G,(width G))) or x in { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } ) assume A2: x in Int (h_strip (G,(width G))) ; ::_thesis: x in { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } then reconsider u = x as Point of (Euclid 2) by Lm6; consider s1 being real number such that A3: s1 > 0 and A4: Ball (u,s1) c= h_strip (G,(width G)) by A2, Th5; reconsider p = u as Point of (TOP-REAL 2) by Lm6; A5: p = |[(p `1),(p `2)]| by EUCLID:53; reconsider s1 = s1 as Real by XREAL_0:def_1; set q = |[((p `1) + 0),((p `2) - (s1 / 2))]|; s1 / 2 < s1 by A3, XREAL_1:216; then |[((p `1) + 0),((p `2) - (s1 / 2))]| in Ball (u,s1) by A3, A5, Th10; then |[((p `1) + 0),((p `2) - (s1 / 2))]| in h_strip (G,(width G)) by A4; then ex r2, s2 being Real st ( |[((p `1) + 0),((p `2) - (s1 / 2))]| = |[r2,s2]| & (G * (1,(width G))) `2 <= s2 ) by A1; then (G * (1,(width G))) `2 <= (p `2) - (s1 / 2) by SPPOL_2:1; then A6: ((G * (1,(width G))) `2) + (s1 / 2) <= p `2 by XREAL_1:19; (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + (s1 / 2) by A3, XREAL_1:29, XREAL_1:215; then (G * (1,(width G))) `2 < p `2 by A6, XXREAL_0:2; hence x in { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } by A5; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } or x in Int (h_strip (G,(width G))) ) assume x in { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } ; ::_thesis: x in Int (h_strip (G,(width G))) then consider r, s being Real such that A7: x = |[r,s]| and A8: (G * (1,(width G))) `2 < s ; reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8; A9: Ball (u,(s - ((G * (1,(width G))) `2))) c= h_strip (G,(width G)) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (u,(s - ((G * (1,(width G))) `2))) or y in h_strip (G,(width G)) ) A10: Ball (u,(s - ((G * (1,(width G))) `2))) = { v where v is Point of (Euclid 2) : dist (u,v) < s - ((G * (1,(width G))) `2) } by METRIC_1:17; assume y in Ball (u,(s - ((G * (1,(width G))) `2))) ; ::_thesis: y in h_strip (G,(width G)) then consider v being Point of (Euclid 2) such that A11: v = y and A12: dist (u,v) < s - ((G * (1,(width G))) `2) by A10; reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:8; ( (s - (q `2)) ^2 >= 0 & ((s - (q `2)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) ) by XREAL_1:6, XREAL_1:63; then A13: sqrt ((s - (q `2)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) by SQUARE_1:26; A14: q = |[(q `1),(q `2)]| by EUCLID:53; then sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < s - ((G * (1,(width G))) `2) by A12, Th6; then sqrt ((s - (q `2)) ^2) <= s - ((G * (1,(width G))) `2) by A13, XXREAL_0:2; then A15: abs (s - (q `2)) <= s - ((G * (1,(width G))) `2) by COMPLEX1:72; percases ( s >= q `2 or s <= q `2 ) ; suppose s >= q `2 ; ::_thesis: y in h_strip (G,(width G)) then s - (q `2) >= 0 by XREAL_1:48; then abs (s - (q `2)) = s - (q `2) by ABSVALUE:def_1; then (G * (1,(width G))) `2 <= q `2 by A15, XREAL_1:10; hence y in h_strip (G,(width G)) by A1, A11, A14; ::_thesis: verum end; suppose s <= q `2 ; ::_thesis: y in h_strip (G,(width G)) then (G * (1,(width G))) `2 <= q `2 by A8, XXREAL_0:2; hence y in h_strip (G,(width G)) by A1, A11, A14; ::_thesis: verum end; end; end; reconsider B = Ball (u,(s - ((G * (1,(width G))) `2))) as Subset of (TOP-REAL 2) by TOPREAL3:8; A16: B is open by Th3; u in Ball (u,(s - ((G * (1,(width G))) `2))) by A8, Th1, XREAL_1:50; hence x in Int (h_strip (G,(width G))) by A7, A9, A16, TOPS_1:22; ::_thesis: verum end; theorem Th17: :: GOBOARD6:17 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) 0 <> len G by GOBOARD1:def_3; then A1: 1 <= len G by NAT_1:14; assume ( 1 <= j & j < width G ) ; ::_thesis: Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then A2: h_strip (G,j) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 <= s & s <= (G * (1,(j + 1))) `2 ) } by A1, GOBOARD5:5; thus Int (h_strip (G,j)) c= { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } c= Int (h_strip (G,j)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (h_strip (G,j)) or x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) assume A3: x in Int (h_strip (G,j)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then reconsider u = x as Point of (Euclid 2) by Lm6; consider s1 being real number such that A4: s1 > 0 and A5: Ball (u,s1) c= h_strip (G,j) by A3, Th5; reconsider p = u as Point of (TOP-REAL 2) by Lm6; A6: p = |[(p `1),(p `2)]| by EUCLID:53; reconsider s1 = s1 as Real by XREAL_0:def_1; set q2 = |[((p `1) + 0),((p `2) - (s1 / 2))]|; A7: s1 / 2 < s1 by A4, XREAL_1:216; then |[((p `1) + 0),((p `2) - (s1 / 2))]| in Ball (u,s1) by A4, A6, Th10; then |[((p `1) + 0),((p `2) - (s1 / 2))]| in h_strip (G,j) by A5; then ex r2, s2 being Real st ( |[((p `1) + 0),((p `2) - (s1 / 2))]| = |[r2,s2]| & (G * (1,j)) `2 <= s2 & s2 <= (G * (1,(j + 1))) `2 ) by A2; then (G * (1,j)) `2 <= (p `2) - (s1 / 2) by SPPOL_2:1; then A8: ((G * (1,j)) `2) + (s1 / 2) <= p `2 by XREAL_1:19; set q1 = |[((p `1) + 0),((p `2) + (s1 / 2))]|; |[((p `1) + 0),((p `2) + (s1 / 2))]| in Ball (u,s1) by A4, A6, A7, Th8; then |[((p `1) + 0),((p `2) + (s1 / 2))]| in h_strip (G,j) by A5; then ex r2, s2 being Real st ( |[((p `1) + 0),((p `2) + (s1 / 2))]| = |[r2,s2]| & (G * (1,j)) `2 <= s2 & s2 <= (G * (1,(j + 1))) `2 ) by A2; then A9: (p `2) + (s1 / 2) <= (G * (1,(j + 1))) `2 by SPPOL_2:1; (G * (1,j)) `2 < ((G * (1,j)) `2) + (s1 / 2) by A4, XREAL_1:29, XREAL_1:215; then A10: (G * (1,j)) `2 < p `2 by A8, XXREAL_0:2; p `2 < (p `2) + (s1 / 2) by A4, XREAL_1:29, XREAL_1:215; then p `2 < (G * (1,(j + 1))) `2 by A9, XXREAL_0:2; hence x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A6, A10; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } or x in Int (h_strip (G,j)) ) assume x in { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ; ::_thesis: x in Int (h_strip (G,j)) then consider r, s being Real such that A11: x = |[r,s]| and A12: (G * (1,j)) `2 < s and A13: s < (G * (1,(j + 1))) `2 ; reconsider u = |[r,s]| as Point of (Euclid 2) by TOPREAL3:8; ( ((G * (1,(j + 1))) `2) - s > 0 & s - ((G * (1,j)) `2) > 0 ) by A12, A13, XREAL_1:50; then min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)) > 0 by XXREAL_0:15; then A14: u in Ball (u,(min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)))) by Th1; A15: Ball (u,(min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)))) c= h_strip (G,j) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Ball (u,(min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)))) or y in h_strip (G,j) ) A16: Ball (u,(min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)))) = { v where v is Point of (Euclid 2) : dist (u,v) < min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)) } by METRIC_1:17; assume y in Ball (u,(min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)))) ; ::_thesis: y in h_strip (G,j) then consider v being Point of (Euclid 2) such that A17: v = y and A18: dist (u,v) < min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)) by A16; reconsider q = v as Point of (TOP-REAL 2) by TOPREAL3:8; ( (s - (q `2)) ^2 >= 0 & ((s - (q `2)) ^2) + 0 <= ((r - (q `1)) ^2) + ((s - (q `2)) ^2) ) by XREAL_1:6, XREAL_1:63; then A19: sqrt ((s - (q `2)) ^2) <= sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) by SQUARE_1:26; A20: q = |[(q `1),(q `2)]| by EUCLID:53; then sqrt (((r - (q `1)) ^2) + ((s - (q `2)) ^2)) < min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)) by A18, Th6; then sqrt ((s - (q `2)) ^2) <= min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)) by A19, XXREAL_0:2; then A21: abs (s - (q `2)) <= min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)) by COMPLEX1:72; then A22: abs (s - (q `2)) <= s - ((G * (1,j)) `2) by XXREAL_0:22; A23: abs (s - (q `2)) <= ((G * (1,(j + 1))) `2) - s by A21, XXREAL_0:22; percases ( s <= q `2 or s >= q `2 ) ; supposeA24: s <= q `2 ; ::_thesis: y in h_strip (G,j) then A25: (q `2) - s >= 0 by XREAL_1:48; abs (s - (q `2)) = abs (- (s - (q `2))) by COMPLEX1:52 .= (q `2) - s by A25, ABSVALUE:def_1 ; then A26: q `2 <= (G * (1,(j + 1))) `2 by A23, XREAL_1:9; (G * (1,j)) `2 <= q `2 by A12, A24, XXREAL_0:2; hence y in h_strip (G,j) by A2, A17, A20, A26; ::_thesis: verum end; supposeA27: s >= q `2 ; ::_thesis: y in h_strip (G,j) then s - (q `2) >= 0 by XREAL_1:48; then abs (s - (q `2)) = s - (q `2) by ABSVALUE:def_1; then A28: (G * (1,j)) `2 <= q `2 by A22, XREAL_1:10; q `2 <= (G * (1,(j + 1))) `2 by A13, A27, XXREAL_0:2; hence y in h_strip (G,j) by A2, A17, A20, A28; ::_thesis: verum end; end; end; reconsider B = Ball (u,(min ((s - ((G * (1,j)) `2)),(((G * (1,(j + 1))) `2) - s)))) as Subset of (TOP-REAL 2) by TOPREAL3:8; B is open by Th3; hence x in Int (h_strip (G,j)) by A11, A14, A15, TOPS_1:22; ::_thesis: verum end; theorem Th18: :: GOBOARD6:18 for G being Go-board holds Int (cell (G,0,0)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } proof let G be Go-board; ::_thesis: Int (cell (G,0,0)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } cell (G,0,0) = (v_strip (G,0)) /\ (h_strip (G,0)) by GOBOARD5:def_3; then A1: Int (cell (G,0,0)) = (Int (v_strip (G,0))) /\ (Int (h_strip (G,0))) by TOPS_1:17; A2: Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } by Th15; A3: Int (v_strip (G,0)) = { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } by Th12; thus Int (cell (G,0,0)) c= { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } c= Int (cell (G,0,0)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,0,0)) or x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } ) assume A4: x in Int (cell (G,0,0)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } then x in Int (v_strip (G,0)) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: r1 < (G * (1,1)) `1 by A3; x in Int (h_strip (G,0)) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: s2 < (G * (1,1)) `2 by A2; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } or x in Int (cell (G,0,0)) ) assume x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } ; ::_thesis: x in Int (cell (G,0,0)) then A9: ex r, s being Real st ( x = |[r,s]| & r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) ; then A10: x in Int (h_strip (G,0)) by A2; x in Int (v_strip (G,0)) by A3, A9; hence x in Int (cell (G,0,0)) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th19: :: GOBOARD6:19 for G being Go-board holds Int (cell (G,0,(width G))) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } proof let G be Go-board; ::_thesis: Int (cell (G,0,(width G))) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } cell (G,0,(width G)) = (v_strip (G,0)) /\ (h_strip (G,(width G))) by GOBOARD5:def_3; then A1: Int (cell (G,0,(width G))) = (Int (v_strip (G,0))) /\ (Int (h_strip (G,(width G)))) by TOPS_1:17; A2: Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } by Th16; A3: Int (v_strip (G,0)) = { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } by Th12; thus Int (cell (G,0,(width G))) c= { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } c= Int (cell (G,0,(width G))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,0,(width G))) or x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } ) assume A4: x in Int (cell (G,0,(width G))) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } then x in Int (v_strip (G,0)) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: r1 < (G * (1,1)) `1 by A3; x in Int (h_strip (G,(width G))) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: (G * (1,(width G))) `2 < s2 by A2; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } or x in Int (cell (G,0,(width G))) ) assume x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } ; ::_thesis: x in Int (cell (G,0,(width G))) then A9: ex r, s being Real st ( x = |[r,s]| & r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) ; then A10: x in Int (h_strip (G,(width G))) by A2; x in Int (v_strip (G,0)) by A3, A9; hence x in Int (cell (G,0,(width G))) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th20: :: GOBOARD6:20 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds Int (cell (G,0,j)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds Int (cell (G,0,j)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies Int (cell (G,0,j)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) cell (G,0,j) = (v_strip (G,0)) /\ (h_strip (G,j)) by GOBOARD5:def_3; then A1: Int (cell (G,0,j)) = (Int (v_strip (G,0))) /\ (Int (h_strip (G,j))) by TOPS_1:17; assume ( 1 <= j & j < width G ) ; ::_thesis: Int (cell (G,0,j)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then A2: Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by Th17; A3: Int (v_strip (G,0)) = { |[r,s]| where r, s is Real : r < (G * (1,1)) `1 } by Th12; thus Int (cell (G,0,j)) c= { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } c= Int (cell (G,0,j)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,0,j)) or x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) assume A4: x in Int (cell (G,0,j)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then x in Int (v_strip (G,0)) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: r1 < (G * (1,1)) `1 by A3; x in Int (h_strip (G,j)) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: ( (G * (1,j)) `2 < s2 & s2 < (G * (1,(j + 1))) `2 ) by A2; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } or x in Int (cell (G,0,j)) ) assume x in { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ; ::_thesis: x in Int (cell (G,0,j)) then A9: ex r, s being Real st ( x = |[r,s]| & r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) ; then A10: x in Int (h_strip (G,j)) by A2; x in Int (v_strip (G,0)) by A3, A9; hence x in Int (cell (G,0,j)) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th21: :: GOBOARD6:21 for G being Go-board holds Int (cell (G,(len G),0)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } proof let G be Go-board; ::_thesis: Int (cell (G,(len G),0)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } cell (G,(len G),0) = (v_strip (G,(len G))) /\ (h_strip (G,0)) by GOBOARD5:def_3; then A1: Int (cell (G,(len G),0)) = (Int (v_strip (G,(len G)))) /\ (Int (h_strip (G,0))) by TOPS_1:17; A2: Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } by Th15; A3: Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } by Th13; thus Int (cell (G,(len G),0)) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } c= Int (cell (G,(len G),0)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,(len G),0)) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } ) assume A4: x in Int (cell (G,(len G),0)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } then x in Int (v_strip (G,(len G))) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: (G * ((len G),1)) `1 < r1 by A3; x in Int (h_strip (G,0)) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: s2 < (G * (1,1)) `2 by A2; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } or x in Int (cell (G,(len G),0)) ) assume x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } ; ::_thesis: x in Int (cell (G,(len G),0)) then A9: ex r, s being Real st ( x = |[r,s]| & (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) ; then A10: x in Int (h_strip (G,0)) by A2; x in Int (v_strip (G,(len G))) by A3, A9; hence x in Int (cell (G,(len G),0)) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th22: :: GOBOARD6:22 for G being Go-board holds Int (cell (G,(len G),(width G))) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } proof let G be Go-board; ::_thesis: Int (cell (G,(len G),(width G))) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } cell (G,(len G),(width G)) = (v_strip (G,(len G))) /\ (h_strip (G,(width G))) by GOBOARD5:def_3; then A1: Int (cell (G,(len G),(width G))) = (Int (v_strip (G,(len G)))) /\ (Int (h_strip (G,(width G)))) by TOPS_1:17; A2: Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } by Th16; A3: Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } by Th13; thus Int (cell (G,(len G),(width G))) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } c= Int (cell (G,(len G),(width G))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,(len G),(width G))) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } ) assume A4: x in Int (cell (G,(len G),(width G))) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } then x in Int (v_strip (G,(len G))) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: (G * ((len G),1)) `1 < r1 by A3; x in Int (h_strip (G,(width G))) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: (G * (1,(width G))) `2 < s2 by A2; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } or x in Int (cell (G,(len G),(width G))) ) assume x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } ; ::_thesis: x in Int (cell (G,(len G),(width G))) then A9: ex r, s being Real st ( x = |[r,s]| & (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) ; then A10: x in Int (h_strip (G,(width G))) by A2; x in Int (v_strip (G,(len G))) by A3, A9; hence x in Int (cell (G,(len G),(width G))) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th23: :: GOBOARD6:23 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) cell (G,(len G),j) = (v_strip (G,(len G))) /\ (h_strip (G,j)) by GOBOARD5:def_3; then A1: Int (cell (G,(len G),j)) = (Int (v_strip (G,(len G)))) /\ (Int (h_strip (G,j))) by TOPS_1:17; assume ( 1 <= j & j < width G ) ; ::_thesis: Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then A2: Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by Th17; A3: Int (v_strip (G,(len G))) = { |[r,s]| where r, s is Real : (G * ((len G),1)) `1 < r } by Th13; thus Int (cell (G,(len G),j)) c= { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } c= Int (cell (G,(len G),j)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,(len G),j)) or x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) assume A4: x in Int (cell (G,(len G),j)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then x in Int (v_strip (G,(len G))) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: (G * ((len G),1)) `1 < r1 by A3; x in Int (h_strip (G,j)) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: ( (G * (1,j)) `2 < s2 & s2 < (G * (1,(j + 1))) `2 ) by A2; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } or x in Int (cell (G,(len G),j)) ) assume x in { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ; ::_thesis: x in Int (cell (G,(len G),j)) then A9: ex r, s being Real st ( x = |[r,s]| & (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) ; then A10: x in Int (h_strip (G,j)) by A2; x in Int (v_strip (G,(len G))) by A3, A9; hence x in Int (cell (G,(len G),j)) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th24: :: GOBOARD6:24 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds Int (cell (G,i,0)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds Int (cell (G,i,0)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies Int (cell (G,i,0)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } ) cell (G,i,0) = (v_strip (G,i)) /\ (h_strip (G,0)) by GOBOARD5:def_3; then A1: Int (cell (G,i,0)) = (Int (v_strip (G,i))) /\ (Int (h_strip (G,0))) by TOPS_1:17; assume ( 1 <= i & i < len G ) ; ::_thesis: Int (cell (G,i,0)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } then A2: Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by Th14; A3: Int (h_strip (G,0)) = { |[r,s]| where r, s is Real : s < (G * (1,1)) `2 } by Th15; thus Int (cell (G,i,0)) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } c= Int (cell (G,i,0)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,i,0)) or x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } ) assume A4: x in Int (cell (G,i,0)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } then x in Int (v_strip (G,i)) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: ( (G * (i,1)) `1 < r1 & r1 < (G * ((i + 1),1)) `1 ) by A2; x in Int (h_strip (G,0)) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: s2 < (G * (1,1)) `2 by A3; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } or x in Int (cell (G,i,0)) ) assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } ; ::_thesis: x in Int (cell (G,i,0)) then A9: ex r, s being Real st ( x = |[r,s]| & (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) ; then A10: x in Int (h_strip (G,0)) by A3; x in Int (v_strip (G,i)) by A2, A9; hence x in Int (cell (G,i,0)) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th25: :: GOBOARD6:25 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds Int (cell (G,i,(width G))) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds Int (cell (G,i,(width G))) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies Int (cell (G,i,(width G))) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } ) cell (G,i,(width G)) = (v_strip (G,i)) /\ (h_strip (G,(width G))) by GOBOARD5:def_3; then A1: Int (cell (G,i,(width G))) = (Int (v_strip (G,i))) /\ (Int (h_strip (G,(width G)))) by TOPS_1:17; assume ( 1 <= i & i < len G ) ; ::_thesis: Int (cell (G,i,(width G))) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } then A2: Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by Th14; A3: Int (h_strip (G,(width G))) = { |[r,s]| where r, s is Real : (G * (1,(width G))) `2 < s } by Th16; thus Int (cell (G,i,(width G))) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } c= Int (cell (G,i,(width G))) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,i,(width G))) or x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } ) assume A4: x in Int (cell (G,i,(width G))) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } then x in Int (v_strip (G,i)) by A1, XBOOLE_0:def_4; then consider r1, s1 being Real such that A5: x = |[r1,s1]| and A6: ( (G * (i,1)) `1 < r1 & r1 < (G * ((i + 1),1)) `1 ) by A2; x in Int (h_strip (G,(width G))) by A1, A4, XBOOLE_0:def_4; then consider r2, s2 being Real such that A7: x = |[r2,s2]| and A8: (G * (1,(width G))) `2 < s2 by A3; s1 = s2 by A5, A7, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } by A5, A6, A8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } or x in Int (cell (G,i,(width G))) ) assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } ; ::_thesis: x in Int (cell (G,i,(width G))) then A9: ex r, s being Real st ( x = |[r,s]| & (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) ; then A10: x in Int (h_strip (G,(width G))) by A3; x in Int (v_strip (G,i)) by A2, A9; hence x in Int (cell (G,i,(width G))) by A1, A10, XBOOLE_0:def_4; ::_thesis: verum end; theorem Th26: :: GOBOARD6:26 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds Int (cell (G,i,j)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds Int (cell (G,i,j)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies Int (cell (G,i,j)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) assume that A1: ( 1 <= i & i < len G ) and A2: ( 1 <= j & j < width G ) ; ::_thesis: Int (cell (G,i,j)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } A3: Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A2, Th17; cell (G,i,j) = (v_strip (G,i)) /\ (h_strip (G,j)) by GOBOARD5:def_3; then A4: Int (cell (G,i,j)) = (Int (v_strip (G,i))) /\ (Int (h_strip (G,j))) by TOPS_1:17; A5: Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by A1, Th14; thus Int (cell (G,i,j)) c= { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } :: according to XBOOLE_0:def_10 ::_thesis: { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } c= Int (cell (G,i,j)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Int (cell (G,i,j)) or x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ) assume A6: x in Int (cell (G,i,j)) ; ::_thesis: x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } then x in Int (v_strip (G,i)) by A4, XBOOLE_0:def_4; then consider r1, s1 being Real such that A7: x = |[r1,s1]| and A8: ( (G * (i,1)) `1 < r1 & r1 < (G * ((i + 1),1)) `1 ) by A5; x in Int (h_strip (G,j)) by A4, A6, XBOOLE_0:def_4; then consider r2, s2 being Real such that A9: x = |[r2,s2]| and A10: ( (G * (1,j)) `2 < s2 & s2 < (G * (1,(j + 1))) `2 ) by A3; s1 = s2 by A7, A9, SPPOL_2:1; hence x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A7, A8, A10; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } or x in Int (cell (G,i,j)) ) assume x in { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } ; ::_thesis: x in Int (cell (G,i,j)) then A11: ex r, s being Real st ( x = |[r,s]| & (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) ; then A12: x in Int (h_strip (G,j)) by A3; x in Int (v_strip (G,i)) by A5, A11; hence x in Int (cell (G,i,j)) by A4, A12, XBOOLE_0:def_4; ::_thesis: verum end; theorem :: GOBOARD6:27 for j being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= j & j <= width G & p in Int (h_strip (G,j)) holds p `2 > (G * (1,j)) `2 proof let j be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= j & j <= width G & p in Int (h_strip (G,j)) holds p `2 > (G * (1,j)) `2 let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= j & j <= width G & p in Int (h_strip (G,j)) holds p `2 > (G * (1,j)) `2 let G be Go-board; ::_thesis: ( 1 <= j & j <= width G & p in Int (h_strip (G,j)) implies p `2 > (G * (1,j)) `2 ) assume that A1: 1 <= j and A2: j <= width G and A3: p in Int (h_strip (G,j)) ; ::_thesis: p `2 > (G * (1,j)) `2 percases ( j = width G or j < width G ) by A2, XXREAL_0:1; suppose j = width G ; ::_thesis: p `2 > (G * (1,j)) `2 then Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : (G * (1,j)) `2 < s } by Th16; then ex r, s being Real st ( p = |[r,s]| & (G * (1,j)) `2 < s ) by A3; hence p `2 > (G * (1,j)) `2 by EUCLID:52; ::_thesis: verum end; suppose j < width G ; ::_thesis: p `2 > (G * (1,j)) `2 then Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A1, Th17; then ex r, s being Real st ( p = |[r,s]| & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) by A3; hence p `2 > (G * (1,j)) `2 by EUCLID:52; ::_thesis: verum end; end; end; theorem :: GOBOARD6:28 for j being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st j < width G & p in Int (h_strip (G,j)) holds p `2 < (G * (1,(j + 1))) `2 proof let j be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st j < width G & p in Int (h_strip (G,j)) holds p `2 < (G * (1,(j + 1))) `2 let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st j < width G & p in Int (h_strip (G,j)) holds p `2 < (G * (1,(j + 1))) `2 let G be Go-board; ::_thesis: ( j < width G & p in Int (h_strip (G,j)) implies p `2 < (G * (1,(j + 1))) `2 ) assume that A1: j < width G and A2: p in Int (h_strip (G,j)) ; ::_thesis: p `2 < (G * (1,(j + 1))) `2 percases ( j = 0 or j >= 1 ) by NAT_1:14; suppose j = 0 ; ::_thesis: p `2 < (G * (1,(j + 1))) `2 then Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : s < (G * (1,(j + 1))) `2 } by Th15; then ex r, s being Real st ( p = |[r,s]| & (G * (1,(j + 1))) `2 > s ) by A2; hence p `2 < (G * (1,(j + 1))) `2 by EUCLID:52; ::_thesis: verum end; suppose j >= 1 ; ::_thesis: p `2 < (G * (1,(j + 1))) `2 then Int (h_strip (G,j)) = { |[r,s]| where r, s is Real : ( (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A1, Th17; then ex r, s being Real st ( p = |[r,s]| & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) by A2; hence p `2 < (G * (1,(j + 1))) `2 by EUCLID:52; ::_thesis: verum end; end; end; theorem :: GOBOARD6:29 for i being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i <= len G & p in Int (v_strip (G,i)) holds p `1 > (G * (i,1)) `1 proof let i be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i <= len G & p in Int (v_strip (G,i)) holds p `1 > (G * (i,1)) `1 let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= i & i <= len G & p in Int (v_strip (G,i)) holds p `1 > (G * (i,1)) `1 let G be Go-board; ::_thesis: ( 1 <= i & i <= len G & p in Int (v_strip (G,i)) implies p `1 > (G * (i,1)) `1 ) assume that A1: 1 <= i and A2: i <= len G and A3: p in Int (v_strip (G,i)) ; ::_thesis: p `1 > (G * (i,1)) `1 percases ( i = len G or i < len G ) by A2, XXREAL_0:1; suppose i = len G ; ::_thesis: p `1 > (G * (i,1)) `1 then Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : (G * (i,1)) `1 < r } by Th13; then ex r, s being Real st ( p = |[r,s]| & (G * (i,1)) `1 < r ) by A3; hence p `1 > (G * (i,1)) `1 by EUCLID:52; ::_thesis: verum end; suppose i < len G ; ::_thesis: p `1 > (G * (i,1)) `1 then Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by A1, Th14; then ex r, s being Real st ( p = |[r,s]| & (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) by A3; hence p `1 > (G * (i,1)) `1 by EUCLID:52; ::_thesis: verum end; end; end; theorem :: GOBOARD6:30 for i being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st i < len G & p in Int (v_strip (G,i)) holds p `1 < (G * ((i + 1),1)) `1 proof let i be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st i < len G & p in Int (v_strip (G,i)) holds p `1 < (G * ((i + 1),1)) `1 let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st i < len G & p in Int (v_strip (G,i)) holds p `1 < (G * ((i + 1),1)) `1 let G be Go-board; ::_thesis: ( i < len G & p in Int (v_strip (G,i)) implies p `1 < (G * ((i + 1),1)) `1 ) assume that A1: i < len G and A2: p in Int (v_strip (G,i)) ; ::_thesis: p `1 < (G * ((i + 1),1)) `1 percases ( i = 0 or i >= 1 ) by NAT_1:14; suppose i = 0 ; ::_thesis: p `1 < (G * ((i + 1),1)) `1 then Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : r < (G * ((i + 1),1)) `1 } by Th12; then ex r, s being Real st ( p = |[r,s]| & (G * ((i + 1),1)) `1 > r ) by A2; hence p `1 < (G * ((i + 1),1)) `1 by EUCLID:52; ::_thesis: verum end; suppose i >= 1 ; ::_thesis: p `1 < (G * ((i + 1),1)) `1 then Int (v_strip (G,i)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) } by A1, Th14; then ex r, s being Real st ( p = |[r,s]| & (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 ) by A2; hence p `1 < (G * ((i + 1),1)) `1 by EUCLID:52; ::_thesis: verum end; end; end; theorem Th31: :: GOBOARD6:31 for i, j being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (cell (G,i,j)) proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (cell (G,i,j)) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (cell (G,i,j)) ) assume that A1: 1 <= i and A2: i + 1 <= len G and A3: 1 <= j and A4: j + 1 <= width G ; ::_thesis: (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (cell (G,i,j)) A5: j < j + 1 by XREAL_1:29; set r1 = (G * (i,j)) `1 ; set s1 = (G * (i,j)) `2 ; set r2 = (G * ((i + 1),(j + 1))) `1 ; set s2 = (G * ((i + 1),(j + 1))) `2 ; A6: ( 1 <= i + 1 & 1 <= j + 1 ) by NAT_1:11; then A7: (G * (1,(j + 1))) `2 = (G * ((i + 1),(j + 1))) `2 by A2, A4, GOBOARD5:1; ( i < len G & j < width G ) by A2, A4, NAT_1:13; then A8: Int (cell (G,i,j)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A1, A3, Th26; ( G * (i,j) = |[((G * (i,j)) `1),((G * (i,j)) `2)]| & G * ((i + 1),(j + 1)) = |[((G * ((i + 1),(j + 1))) `1),((G * ((i + 1),(j + 1))) `2)]| ) by EUCLID:53; then (G * (i,j)) + (G * ((i + 1),(j + 1))) = |[(((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1)),(((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2))]| by EUCLID:56; then A9: (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) = |[((1 / 2) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1))),((1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)))]| by EUCLID:58; i <= i + 1 by NAT_1:11; then A10: i <= len G by A2, XXREAL_0:2; then A11: 1 <= len G by A1, XXREAL_0:2; j <= j + 1 by NAT_1:11; then A12: j <= width G by A4, XXREAL_0:2; then A13: 1 <= width G by A3, XXREAL_0:2; A14: (G * (i,1)) `1 = (G * (i,j)) `1 by A1, A3, A10, A12, GOBOARD5:2; (G * (1,j)) `2 = (G * (i,j)) `2 by A1, A3, A10, A12, GOBOARD5:1; then A15: (G * (i,j)) `2 < (G * ((i + 1),(j + 1))) `2 by A3, A4, A7, A11, A5, GOBOARD5:4; then ((G * (i,j)) `2) + ((G * (i,j)) `2) < ((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2) by XREAL_1:6; then (1 / 2) * (((G * (i,j)) `2) + ((G * (i,j)) `2)) < (1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)) by XREAL_1:68; then A16: (G * (1,j)) `2 < (1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)) by A1, A3, A10, A12, GOBOARD5:1; A17: i < i + 1 by XREAL_1:29; (G * ((i + 1),1)) `1 = (G * ((i + 1),(j + 1))) `1 by A2, A4, A6, GOBOARD5:2; then A18: (G * (i,j)) `1 < (G * ((i + 1),(j + 1))) `1 by A1, A2, A14, A13, A17, GOBOARD5:3; then ((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1) < ((G * ((i + 1),(j + 1))) `1) + ((G * ((i + 1),(j + 1))) `1) by XREAL_1:6; then (1 / 2) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1)) < (1 / 2) * (((G * ((i + 1),(j + 1))) `1) + ((G * ((i + 1),(j + 1))) `1)) by XREAL_1:68; then A19: (1 / 2) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1)) < (G * ((i + 1),1)) `1 by A2, A4, A6, GOBOARD5:2; ((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2) < ((G * ((i + 1),(j + 1))) `2) + ((G * ((i + 1),(j + 1))) `2) by A15, XREAL_1:6; then (1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)) < (1 / 2) * (((G * ((i + 1),(j + 1))) `2) + ((G * ((i + 1),(j + 1))) `2)) by XREAL_1:68; then A20: (1 / 2) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2)) < (G * (1,(j + 1))) `2 by A2, A4, A6, GOBOARD5:1; ((G * (i,j)) `1) + ((G * (i,j)) `1) < ((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1) by A18, XREAL_1:6; then (1 / 2) * (((G * (i,j)) `1) + ((G * (i,j)) `1)) < (1 / 2) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1)) by XREAL_1:68; hence (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))) in Int (cell (G,i,j)) by A9, A14, A19, A16, A20, A8; ::_thesis: verum end; theorem Th32: :: GOBOARD6:32 for i being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G holds ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]| in Int (cell (G,i,(width G))) proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G holds ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]| in Int (cell (G,i,(width G))) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G implies ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]| in Int (cell (G,i,(width G))) ) assume that A1: 1 <= i and A2: i + 1 <= len G ; ::_thesis: ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]| in Int (cell (G,i,(width G))) set r1 = (G * (i,(width G))) `1 ; set s1 = (G * (i,(width G))) `2 ; set r2 = (G * ((i + 1),(width G))) `1 ; width G <> 0 by GOBOARD1:def_3; then A3: 1 <= width G by NAT_1:14; i < len G by A2, NAT_1:13; then A4: Int (cell (G,i,(width G))) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s ) } by A1, Th25; width G <> 0 by GOBOARD1:def_3; then A5: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A6: (G * (i,(width G))) `1 < (G * ((i + 1),(width G))) `1 by A1, A2, A5, GOBOARD5:3; then ((G * (i,(width G))) `1) + ((G * (i,(width G))) `1) < ((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1) by XREAL_1:6; then A7: (1 / 2) * (((G * (i,(width G))) `1) + ((G * (i,(width G))) `1)) < (1 / 2) * (((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1)) by XREAL_1:68; A8: i < len G by A2, NAT_1:13; then A9: (G * (1,(width G))) `2 = (G * (i,(width G))) `2 by A1, A3, GOBOARD5:1; then A10: (G * (1,(width G))) `2 < ((G * (i,(width G))) `2) + 1 by XREAL_1:29; A11: 1 <= i + 1 by NAT_1:11; then (G * (1,(width G))) `2 = (G * ((i + 1),(width G))) `2 by A2, A3, GOBOARD5:1; then ( G * (i,(width G)) = |[((G * (i,(width G))) `1),((G * (i,(width G))) `2)]| & G * ((i + 1),(width G)) = |[((G * ((i + 1),(width G))) `1),((G * (i,(width G))) `2)]| ) by A9, EUCLID:53; then ( (1 / 2) * (((G * (i,(width G))) `2) + ((G * (i,(width G))) `2)) = (G * (i,(width G))) `2 & (G * (i,(width G))) + (G * ((i + 1),(width G))) = |[(((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1)),(((G * (i,(width G))) `2) + ((G * (i,(width G))) `2))]| ) by EUCLID:56; then (1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))) = |[((1 / 2) * (((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1))),((G * (i,(width G))) `2)]| by EUCLID:58; then A12: ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]| = |[(((1 / 2) * (((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1))) + 0),(((G * (i,(width G))) `2) + 1)]| by EUCLID:56; ((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1) < ((G * ((i + 1),(width G))) `1) + ((G * ((i + 1),(width G))) `1) by A6, XREAL_1:6; then (1 / 2) * (((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1)) < (1 / 2) * (((G * ((i + 1),(width G))) `1) + ((G * ((i + 1),(width G))) `1)) by XREAL_1:68; then A13: (1 / 2) * (((G * (i,(width G))) `1) + ((G * ((i + 1),(width G))) `1)) < (G * ((i + 1),1)) `1 by A2, A11, A3, GOBOARD5:2; (G * (i,1)) `1 = (G * (i,(width G))) `1 by A1, A8, A3, GOBOARD5:2; hence ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]| in Int (cell (G,i,(width G))) by A12, A7, A13, A10, A4; ::_thesis: verum end; theorem Th33: :: GOBOARD6:33 for i being Element of NAT for G being Go-board st 1 <= i & i + 1 <= len G holds ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]| in Int (cell (G,i,0)) proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G holds ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]| in Int (cell (G,i,0)) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G implies ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]| in Int (cell (G,i,0)) ) assume that A1: 1 <= i and A2: i + 1 <= len G ; ::_thesis: ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]| in Int (cell (G,i,0)) set r1 = (G * (i,1)) `1 ; set s1 = (G * (i,1)) `2 ; set r2 = (G * ((i + 1),1)) `1 ; width G <> 0 by GOBOARD1:def_3; then A3: 1 <= width G by NAT_1:14; width G <> 0 by GOBOARD1:def_3; then A4: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A5: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A2, A4, GOBOARD5:3; then ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A6: (1 / 2) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < (1 / 2) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by XREAL_1:68; i < len G by A2, NAT_1:13; then A7: (G * (1,1)) `2 = (G * (i,1)) `2 by A1, A3, GOBOARD5:1; then (G * (i,1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A8: ((G * (i,1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; 1 <= i + 1 by NAT_1:11; then (G * (1,1)) `2 = (G * ((i + 1),1)) `2 by A2, A3, GOBOARD5:1; then ( G * (i,1) = |[((G * (i,1)) `1),((G * (i,1)) `2)]| & G * ((i + 1),1) = |[((G * ((i + 1),1)) `1),((G * (i,1)) `2)]| ) by A7, EUCLID:53; then ( (1 / 2) * (((G * (i,1)) `2) + ((G * (i,1)) `2)) = (G * (i,1)) `2 & (G * (i,1)) + (G * ((i + 1),1)) = |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (i,1)) `2) + ((G * (i,1)) `2))]| ) by EUCLID:56; then (1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))) = |[((1 / 2) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((G * (i,1)) `2)]| by EUCLID:58; then A9: ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]| = |[(((1 / 2) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) - 0),(((G * (i,1)) `2) - 1)]| by EUCLID:62 .= |[((1 / 2) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((G * (i,1)) `2) - 1)]| ; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A5, XREAL_1:6; then A10: (1 / 2) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < (1 / 2) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by XREAL_1:68; i < len G by A2, NAT_1:13; then Int (cell (G,i,0)) = { |[r,s]| where r, s is Real : ( (G * (i,1)) `1 < r & r < (G * ((i + 1),1)) `1 & s < (G * (1,1)) `2 ) } by A1, Th24; hence ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]| in Int (cell (G,i,0)) by A9, A6, A10, A8; ::_thesis: verum end; theorem Th34: :: GOBOARD6:34 for j being Element of NAT for G being Go-board st 1 <= j & j + 1 <= width G holds ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]| in Int (cell (G,(len G),j)) proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j + 1 <= width G holds ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]| in Int (cell (G,(len G),j)) let G be Go-board; ::_thesis: ( 1 <= j & j + 1 <= width G implies ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]| in Int (cell (G,(len G),j)) ) assume that A1: 1 <= j and A2: j + 1 <= width G ; ::_thesis: ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]| in Int (cell (G,(len G),j)) set s1 = (G * ((len G),j)) `2 ; set r1 = (G * ((len G),j)) `1 ; set s2 = (G * ((len G),(j + 1))) `2 ; len G <> 0 by GOBOARD1:def_3; then A3: 1 <= len G by NAT_1:14; j < width G by A2, NAT_1:13; then A4: Int (cell (G,(len G),j)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A1, Th23; len G <> 0 by GOBOARD1:def_3; then A5: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A6: (G * ((len G),j)) `2 < (G * ((len G),(j + 1))) `2 by A1, A2, A5, GOBOARD5:4; then ((G * ((len G),j)) `2) + ((G * ((len G),j)) `2) < ((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2) by XREAL_1:6; then A7: (1 / 2) * (((G * ((len G),j)) `2) + ((G * ((len G),j)) `2)) < (1 / 2) * (((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2)) by XREAL_1:68; A8: j < width G by A2, NAT_1:13; then A9: (G * ((len G),1)) `1 = (G * ((len G),j)) `1 by A1, A3, GOBOARD5:2; then A10: (G * ((len G),1)) `1 < ((G * ((len G),j)) `1) + 1 by XREAL_1:29; A11: 1 <= j + 1 by NAT_1:11; then (G * ((len G),1)) `1 = (G * ((len G),(j + 1))) `1 by A2, A3, GOBOARD5:2; then ( G * ((len G),j) = |[((G * ((len G),j)) `1),((G * ((len G),j)) `2)]| & G * ((len G),(j + 1)) = |[((G * ((len G),j)) `1),((G * ((len G),(j + 1))) `2)]| ) by A9, EUCLID:53; then ( (1 / 2) * (((G * ((len G),j)) `1) + ((G * ((len G),j)) `1)) = (G * ((len G),j)) `1 & (G * ((len G),j)) + (G * ((len G),(j + 1))) = |[(((G * ((len G),j)) `1) + ((G * ((len G),j)) `1)),(((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2))]| ) by EUCLID:56; then (1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))) = |[((G * ((len G),j)) `1),((1 / 2) * (((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2)))]| by EUCLID:58; then A12: ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]| = |[(((G * ((len G),j)) `1) + 1),(((1 / 2) * (((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2))) + 0)]| by EUCLID:56; ((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2) < ((G * ((len G),(j + 1))) `2) + ((G * ((len G),(j + 1))) `2) by A6, XREAL_1:6; then (1 / 2) * (((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2)) < (1 / 2) * (((G * ((len G),(j + 1))) `2) + ((G * ((len G),(j + 1))) `2)) by XREAL_1:68; then A13: (1 / 2) * (((G * ((len G),j)) `2) + ((G * ((len G),(j + 1))) `2)) < (G * (1,(j + 1))) `2 by A2, A11, A3, GOBOARD5:1; (G * (1,j)) `2 = (G * ((len G),j)) `2 by A1, A8, A3, GOBOARD5:1; hence ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]| in Int (cell (G,(len G),j)) by A12, A7, A13, A10, A4; ::_thesis: verum end; theorem Th35: :: GOBOARD6:35 for j being Element of NAT for G being Go-board st 1 <= j & j + 1 <= width G holds ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j)) proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j + 1 <= width G holds ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j)) let G be Go-board; ::_thesis: ( 1 <= j & j + 1 <= width G implies ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j)) ) assume that A1: 1 <= j and A2: j + 1 <= width G ; ::_thesis: ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j)) set s1 = (G * (1,j)) `2 ; set r1 = (G * (1,j)) `1 ; set s2 = (G * (1,(j + 1))) `2 ; len G <> 0 by GOBOARD1:def_3; then A3: 1 <= len G by NAT_1:14; len G <> 0 by GOBOARD1:def_3; then A4: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A5: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A2, A4, GOBOARD5:4; then ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A6: (1 / 2) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < (1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by XREAL_1:68; j < width G by A2, NAT_1:13; then A7: (G * (1,1)) `1 = (G * (1,j)) `1 by A1, A3, GOBOARD5:2; then (G * (1,j)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A8: ((G * (1,j)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19; 1 <= j + 1 by NAT_1:11; then (G * (1,1)) `1 = (G * (1,(j + 1))) `1 by A2, A3, GOBOARD5:2; then ( G * (1,j) = |[((G * (1,j)) `1),((G * (1,j)) `2)]| & G * (1,(j + 1)) = |[((G * (1,j)) `1),((G * (1,(j + 1))) `2)]| ) by A7, EUCLID:53; then ( (1 / 2) * (((G * (1,j)) `1) + ((G * (1,j)) `1)) = (G * (1,j)) `1 & (G * (1,j)) + (G * (1,(j + 1))) = |[(((G * (1,j)) `1) + ((G * (1,j)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]| ) by EUCLID:56; then (1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))) = |[((G * (1,j)) `1),((1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:58; then A9: ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| = |[(((G * (1,j)) `1) - 1),(((1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) - 0)]| by EUCLID:62 .= |[(((G * (1,j)) `1) - 1),((1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| ; ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A5, XREAL_1:6; then A10: (1 / 2) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < (1 / 2) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by XREAL_1:68; j < width G by A2, NAT_1:13; then Int (cell (G,0,j)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,j)) `2 < s & s < (G * (1,(j + 1))) `2 ) } by A1, Th20; hence ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]| in Int (cell (G,0,j)) by A9, A6, A10, A8; ::_thesis: verum end; theorem Th36: :: GOBOARD6:36 for G being Go-board holds (G * (1,1)) - |[1,1]| in Int (cell (G,0,0)) proof let G be Go-board; ::_thesis: (G * (1,1)) - |[1,1]| in Int (cell (G,0,0)) set s1 = (G * (1,1)) `2 ; set r1 = (G * (1,1)) `1 ; G * (1,1) = |[((G * (1,1)) `1),((G * (1,1)) `2)]| by EUCLID:53; then A1: (G * (1,1)) - |[1,1]| = |[(((G * (1,1)) `1) - 1),(((G * (1,1)) `2) - 1)]| by EUCLID:62; (G * (1,1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A2: ((G * (1,1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; (G * (1,1)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A3: ((G * (1,1)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19; Int (cell (G,0,0)) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & s < (G * (1,1)) `2 ) } by Th18; hence (G * (1,1)) - |[1,1]| in Int (cell (G,0,0)) by A1, A2, A3; ::_thesis: verum end; theorem Th37: :: GOBOARD6:37 for G being Go-board holds (G * ((len G),(width G))) + |[1,1]| in Int (cell (G,(len G),(width G))) proof let G be Go-board; ::_thesis: (G * ((len G),(width G))) + |[1,1]| in Int (cell (G,(len G),(width G))) set s1 = (G * ((len G),(width G))) `2 ; set r1 = (G * ((len G),(width G))) `1 ; len G <> 0 by GOBOARD1:def_3; then A1: 1 <= len G by NAT_1:14; width G <> 0 by GOBOARD1:def_3; then A2: 1 <= width G by NAT_1:14; then (G * ((len G),1)) `1 = (G * ((len G),(width G))) `1 by A1, GOBOARD5:2; then A3: ((G * ((len G),(width G))) `1) + 1 > (G * ((len G),1)) `1 by XREAL_1:29; G * ((len G),(width G)) = |[((G * ((len G),(width G))) `1),((G * ((len G),(width G))) `2)]| by EUCLID:53; then A4: (G * ((len G),(width G))) + |[1,1]| = |[(((G * ((len G),(width G))) `1) + 1),(((G * ((len G),(width G))) `2) + 1)]| by EUCLID:56; (G * (1,(width G))) `2 = (G * ((len G),(width G))) `2 by A2, A1, GOBOARD5:1; then A5: ((G * ((len G),(width G))) `2) + 1 > (G * (1,(width G))) `2 by XREAL_1:29; Int (cell (G,(len G),(width G))) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & (G * (1,(width G))) `2 < s ) } by Th22; hence (G * ((len G),(width G))) + |[1,1]| in Int (cell (G,(len G),(width G))) by A4, A5, A3; ::_thesis: verum end; theorem Th38: :: GOBOARD6:38 for G being Go-board holds (G * (1,(width G))) + |[(- 1),1]| in Int (cell (G,0,(width G))) proof let G be Go-board; ::_thesis: (G * (1,(width G))) + |[(- 1),1]| in Int (cell (G,0,(width G))) set s1 = (G * (1,(width G))) `2 ; set r1 = (G * (1,(width G))) `1 ; len G <> 0 by GOBOARD1:def_3; then A1: 1 <= len G by NAT_1:14; width G <> 0 by GOBOARD1:def_3; then 1 <= width G by NAT_1:14; then (G * (1,1)) `1 = (G * (1,(width G))) `1 by A1, GOBOARD5:2; then (G * (1,(width G))) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A2: ( ((G * (1,(width G))) `2) + 1 > (G * (1,(width G))) `2 & ((G * (1,(width G))) `1) - 1 < (G * (1,1)) `1 ) by XREAL_1:19, XREAL_1:29; G * (1,(width G)) = |[((G * (1,(width G))) `1),((G * (1,(width G))) `2)]| by EUCLID:53; then A3: (G * (1,(width G))) + |[(- 1),1]| = |[(((G * (1,(width G))) `1) + (- 1)),(((G * (1,(width G))) `2) + 1)]| by EUCLID:56 .= |[(((G * (1,(width G))) `1) - 1),(((G * (1,(width G))) `2) + 1)]| ; Int (cell (G,0,(width G))) = { |[r,s]| where r, s is Real : ( r < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s ) } by Th19; hence (G * (1,(width G))) + |[(- 1),1]| in Int (cell (G,0,(width G))) by A3, A2; ::_thesis: verum end; theorem Th39: :: GOBOARD6:39 for G being Go-board holds (G * ((len G),1)) + |[1,(- 1)]| in Int (cell (G,(len G),0)) proof let G be Go-board; ::_thesis: (G * ((len G),1)) + |[1,(- 1)]| in Int (cell (G,(len G),0)) set s1 = (G * ((len G),1)) `2 ; set r1 = (G * ((len G),1)) `1 ; A1: ((G * ((len G),1)) `1) + 1 > (G * ((len G),1)) `1 by XREAL_1:29; len G <> 0 by GOBOARD1:def_3; then A2: 1 <= len G by NAT_1:14; width G <> 0 by GOBOARD1:def_3; then 1 <= width G by NAT_1:14; then (G * (1,1)) `2 = (G * ((len G),1)) `2 by A2, GOBOARD5:1; then (G * ((len G),1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A3: ((G * ((len G),1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; G * ((len G),1) = |[((G * ((len G),1)) `1),((G * ((len G),1)) `2)]| by EUCLID:53; then A4: (G * ((len G),1)) + |[1,(- 1)]| = |[(((G * ((len G),1)) `1) + 1),(((G * ((len G),1)) `2) + (- 1))]| by EUCLID:56 .= |[(((G * ((len G),1)) `1) + 1),(((G * ((len G),1)) `2) - 1)]| ; Int (cell (G,(len G),0)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 < r & s < (G * (1,1)) `2 ) } by Th21; hence (G * ((len G),1)) + |[1,(- 1)]| in Int (cell (G,(len G),0)) by A4, A3, A1; ::_thesis: verum end; theorem Th40: :: GOBOARD6:40 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) or x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} ) assume A5: x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) ; ::_thesis: x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A6: p = ((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) and A7: 0 <= r and A8: r <= 1 by A5; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_(i,j))_+_(G_*_(i,(j_+_1)))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,j))_)_) percases ( r = 1 or r < 1 ) by A8, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) by A6, EUCLID:29 .= 1 * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1))))) by EUCLID:27 .= (1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} by TARSKI:def_1; ::_thesis: verum end; caseA9: r < 1 ; ::_thesis: p in Int (cell (G,i,j)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A10: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ; 0 <> len G by GOBOARD1:def_3; then A11: 1 <= len G by NAT_1:14; A12: j + 1 <= width G by A4, NAT_1:13; j < j + 1 by XREAL_1:29; then A13: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A3, A12, A11, GOBOARD5:4; then A14: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A15: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A7, XREAL_1:64; 1 - r > 0 by A9, XREAL_1:50; then A16: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A14, XREAL_1:68; then A17: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) by A15, A10, XREAL_1:8; A18: ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A13, XREAL_1:6; then A19: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A7, XREAL_1:64; 0 <> width G by GOBOARD1:def_3; then A20: 1 <= width G by NAT_1:14; A21: 1 <= i + 1 by A1, NAT_1:13; A22: Int (cell (G,i,j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, A3, A4, Th26; A23: 1 <= j + 1 by A3, NAT_1:13; A24: G * (i,(j + 1)) = |[((G * (i,(j + 1))) `1),((G * (i,(j + 1))) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (i,(j + 1))) `2)]| by A1, A2, A23, A12, GOBOARD5:2 .= |[((G * (i,1)) `1),((G * (1,(j + 1))) `2)]| by A1, A2, A23, A12, GOBOARD5:1 ; A25: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ; ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A16, A18, XREAL_1:68; then A26: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2 by A19, A25, XREAL_1:8; A27: i + 1 <= len G by A2, NAT_1:13; i < i + 1 by XREAL_1:29; then A28: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A27, A20, GOBOARD5:3; then ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:8; then A29: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A7, XREAL_1:64; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A28, XREAL_1:6; then A30: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A16, XREAL_1:68; (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ; then A31: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) < (G * ((i + 1),1)) `1 by A30, A29, XREAL_1:8; A32: G * (i,j) = |[((G * (i,j)) `1),((G * (i,j)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (i,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:2 .= |[((G * (i,1)) `1),((G * (1,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:1 ; A33: G * ((i + 1),(j + 1)) = |[((G * ((i + 1),(j + 1))) `1),((G * ((i + 1),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * ((i + 1),(j + 1))) `2)]| by A23, A12, A21, A27, GOBOARD5:2 .= |[((G * ((i + 1),1)) `1),((G * (1,(j + 1))) `2)]| by A23, A12, A21, A27, GOBOARD5:1 ; A34: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ; ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by A28, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A16, XREAL_1:68; then A35: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) by A34, XREAL_1:6; p = (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))) by A6, EUCLID:30 .= (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((r * (1 / 2)) * ((G * (i,j)) + (G * (i,(j + 1))))) by EUCLID:30 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * ((G * (i,j)) + (G * (i,(j + 1))))) by A32, A33, EUCLID:56 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * (i,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by A32, A24, EUCLID:56 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * (i,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by EUCLID:58 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,i,j)) by A35, A31, A17, A26, A22; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * (i,(j + 1)))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th41: :: GOBOARD6:41 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) or x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} ) assume A5: x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) ; ::_thesis: x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A6: p = ((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) and A7: 0 <= r and A8: r <= 1 by A5; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_(i,(j_+_1)))_+_(G_*_((i_+_1),(j_+_1)))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,j))_)_) percases ( r = 1 or r < 1 ) by A8, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) by A6, EUCLID:29 .= 1 * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))) by EUCLID:27 .= (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} by TARSKI:def_1; ::_thesis: verum end; caseA9: r < 1 ; ::_thesis: p in Int (cell (G,i,j)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A10: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ; 0 <> width G by GOBOARD1:def_3; then A11: 1 <= width G by NAT_1:14; A12: i + 1 <= len G by A2, NAT_1:13; i < i + 1 by XREAL_1:29; then A13: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A12, A11, GOBOARD5:3; then A14: ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A15: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A7, XREAL_1:64; 1 - r > 0 by A9, XREAL_1:50; then A16: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A14, XREAL_1:68; then A17: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) by A15, A10, XREAL_1:8; 0 <> len G by GOBOARD1:def_3; then A18: 1 <= len G by NAT_1:14; A19: 1 <= i + 1 by A1, NAT_1:13; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A13, XREAL_1:8; then A20: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A7, XREAL_1:64; A21: j + 1 <= width G by A4, NAT_1:13; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A13, XREAL_1:6; then A22: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A16, XREAL_1:68; (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ; then A23: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) < (G * ((i + 1),1)) `1 by A22, A20, XREAL_1:8; A24: Int (cell (G,i,j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, A3, A4, Th26; A25: 1 <= j + 1 by A3, NAT_1:13; j < j + 1 by XREAL_1:29; then A26: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A3, A21, A18, GOBOARD5:4; then A27: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; A28: G * ((i + 1),(j + 1)) = |[((G * ((i + 1),(j + 1))) `1),((G * ((i + 1),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * ((i + 1),(j + 1))) `2)]| by A25, A21, A19, A12, GOBOARD5:2 .= |[((G * ((i + 1),1)) `1),((G * (1,(j + 1))) `2)]| by A25, A21, A19, A12, GOBOARD5:1 ; ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A26, XREAL_1:6; then ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A27, XXREAL_0:2; then A29: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A7, XREAL_1:64; A30: G * (i,j) = |[((G * (i,j)) `1),((G * (i,j)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (i,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:2 .= |[((G * (i,1)) `1),((G * (1,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:1 ; A31: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ; A32: G * (i,(j + 1)) = |[((G * (i,(j + 1))) `1),((G * (i,(j + 1))) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (i,(j + 1))) `2)]| by A1, A2, A25, A21, GOBOARD5:2 .= |[((G * (i,1)) `1),((G * (1,(j + 1))) `2)]| by A1, A2, A25, A21, GOBOARD5:1 ; A33: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ; ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A26, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A16, XREAL_1:68; then A34: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2 by A31, XREAL_1:8; ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A16, A27, XREAL_1:68; then A35: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) by A29, A33, XREAL_1:8; p = (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) by A6, EUCLID:30 .= (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((r * (1 / 2)) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))) by EUCLID:30 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))) by A30, A28, EUCLID:56 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))]|) by A28, A32, EUCLID:56 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))]|) by EUCLID:58 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,i,j)) by A17, A23, A35, A34, A24; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th42: :: GOBOARD6:42 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) or x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} ) assume A5: x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) ; ::_thesis: x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A6: p = ((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) and A7: 0 <= r and A8: r <= 1 by A5; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_((i_+_1),j))_+_(G_*_((i_+_1),(j_+_1)))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,j))_)_) percases ( r = 1 or r < 1 ) by A8, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) by A6, EUCLID:29 .= 1 * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))) by EUCLID:27 .= (1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} by TARSKI:def_1; ::_thesis: verum end; caseA9: r < 1 ; ::_thesis: p in Int (cell (G,i,j)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A10: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ; 0 <> width G by GOBOARD1:def_3; then A11: 1 <= width G by NAT_1:14; A12: i + 1 <= len G by A2, NAT_1:13; i < i + 1 by XREAL_1:29; then A13: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A12, A11, GOBOARD5:3; then A14: ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A13, XREAL_1:6; then ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A14, XXREAL_0:2; then A15: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A7, XREAL_1:64; 1 - r > 0 by A9, XREAL_1:50; then A16: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A14, XREAL_1:68; then A17: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) by A15, A10, XREAL_1:8; 0 <> len G by GOBOARD1:def_3; then A18: 1 <= len G by NAT_1:14; A19: 1 <= j + 1 by A3, NAT_1:13; A20: Int (cell (G,i,j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, A3, A4, Th26; A21: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ; A22: G * (i,j) = |[((G * (i,j)) `1),((G * (i,j)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (i,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:2 .= |[((G * (i,1)) `1),((G * (1,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:1 ; A23: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ; A24: 1 <= i + 1 by A1, NAT_1:13; A25: G * ((i + 1),j) = |[((G * ((i + 1),j)) `1),((G * ((i + 1),j)) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * ((i + 1),j)) `2)]| by A3, A4, A24, A12, GOBOARD5:2 .= |[((G * ((i + 1),1)) `1),((G * (1,j)) `2)]| by A3, A4, A24, A12, GOBOARD5:1 ; A26: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A13, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A16, XREAL_1:68; then A27: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) < (G * ((i + 1),1)) `1 by A26, XREAL_1:8; A28: j + 1 <= width G by A4, NAT_1:13; A29: G * ((i + 1),(j + 1)) = |[((G * ((i + 1),(j + 1))) `1),((G * ((i + 1),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * ((i + 1),(j + 1))) `2)]| by A19, A28, A24, A12, GOBOARD5:2 .= |[((G * ((i + 1),1)) `1),((G * (1,(j + 1))) `2)]| by A19, A28, A24, A12, GOBOARD5:1 ; j < j + 1 by XREAL_1:29; then A30: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A3, A28, A18, GOBOARD5:4; then A31: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A32: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A7, XREAL_1:64; ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A16, A31, XREAL_1:68; then A33: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) by A32, A23, XREAL_1:8; A34: ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A30, XREAL_1:6; then A35: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A7, XREAL_1:64; ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A16, A34, XREAL_1:68; then A36: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2 by A35, A21, XREAL_1:8; p = (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) by A6, EUCLID:30 .= (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((r * (1 / 2)) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))) by EUCLID:30 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))) by A22, A29, EUCLID:56 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * |[(((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by A29, A25, EUCLID:56 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + ((r * (1 / 2)) * |[(((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by EUCLID:58 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,i,j)) by A17, A27, A33, A36, A20; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th43: :: GOBOARD6:43 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j < width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j < width G ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) or x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} ) assume A5: x in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) ; ::_thesis: x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A6: p = ((1 - r) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) and A7: 0 <= r and A8: r <= 1 by A5; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_(i,j))_+_(G_*_((i_+_1),j))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,j))_)_) percases ( r = 1 or r < 1 ) by A8, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) by A6, EUCLID:29 .= 1 * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j)))) by EUCLID:27 .= (1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} by TARSKI:def_1; ::_thesis: verum end; caseA9: r < 1 ; ::_thesis: p in Int (cell (G,i,j)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A10: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ; 0 <> width G by GOBOARD1:def_3; then A11: 1 <= width G by NAT_1:14; A12: i + 1 <= len G by A2, NAT_1:13; i < i + 1 by XREAL_1:29; then A13: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A12, A11, GOBOARD5:3; then A14: ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A15: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A7, XREAL_1:64; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A13, XREAL_1:8; then A16: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A7, XREAL_1:64; A17: 1 <= i + 1 by A1, NAT_1:13; 1 - r > 0 by A9, XREAL_1:50; then A18: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A14, XREAL_1:68; then A19: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) by A15, A10, XREAL_1:8; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A13, XREAL_1:6; then A20: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A18, XREAL_1:68; (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ; then A21: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) < (G * ((i + 1),1)) `1 by A20, A16, XREAL_1:8; A22: Int (cell (G,i,j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, A3, A4, Th26; A23: j + 1 <= width G by A4, NAT_1:13; A24: G * ((i + 1),j) = |[((G * ((i + 1),j)) `1),((G * ((i + 1),j)) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * ((i + 1),j)) `2)]| by A3, A4, A17, A12, GOBOARD5:2 .= |[((G * ((i + 1),1)) `1),((G * (1,j)) `2)]| by A3, A4, A17, A12, GOBOARD5:1 ; A25: 1 <= j + 1 by A3, NAT_1:13; 0 <> len G by GOBOARD1:def_3; then A26: 1 <= len G by NAT_1:14; A27: G * (i,j) = |[((G * (i,j)) `1),((G * (i,j)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (i,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:2 .= |[((G * (i,1)) `1),((G * (1,j)) `2)]| by A1, A2, A3, A4, GOBOARD5:1 ; j < j + 1 by XREAL_1:29; then A28: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A3, A23, A26, GOBOARD5:4; then ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A29: ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A18, XREAL_1:68; A30: G * ((i + 1),(j + 1)) = |[((G * ((i + 1),(j + 1))) `1),((G * ((i + 1),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * ((i + 1),(j + 1))) `2)]| by A25, A23, A17, A12, GOBOARD5:2 .= |[((G * ((i + 1),1)) `1),((G * (1,(j + 1))) `2)]| by A25, A23, A17, A12, GOBOARD5:1 ; A31: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ; A32: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by A28, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A18, XREAL_1:68; then A33: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) by A31, XREAL_1:8; ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A28, XREAL_1:6; then ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A32, XXREAL_0:2; then A34: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A7, XREAL_1:64; (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ; then A35: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) < (G * (1,(j + 1))) `2 by A29, A34, XREAL_1:8; p = (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (r * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))) by A6, EUCLID:30 .= (((1 - r) * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((r * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),j)))) by EUCLID:30 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * ((G * (i,j)) + (G * ((i + 1),j)))) by A27, A30, EUCLID:56 .= (((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,j)) `2))]|) by A27, A24, EUCLID:56 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,j)) `2) + ((G * (1,j)) `2))]|) by EUCLID:58 .= |[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,i,j)) by A19, A21, A33, A35, A22; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,j)) + (G * ((i + 1),j))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th44: :: GOBOARD6:44 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} ) assume that A1: 1 <= j and A2: j < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) or x in (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} ) assume A3: x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) ; ::_thesis: x in (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_(1,j))_+_(G_*_(1,(j_+_1)))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,j))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by A4, EUCLID:29 .= 1 * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) by EUCLID:27 .= (1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,0,j)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set r2 = (G * (1,1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A8: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ; A9: j + 1 <= width G by A2, NAT_1:13; 0 <> len G by GOBOARD1:def_3; then A10: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A11: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A9, A10, GOBOARD5:4; then A12: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A13: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A5, XREAL_1:64; A14: 1 - r > 0 by A7, XREAL_1:50; then A15: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A12, XREAL_1:68; then A16: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) by A13, A8, XREAL_1:8; (G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r) by A14, XREAL_1:29; then A17: ((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1 by XREAL_1:19; A18: 1 <= j + 1 by A1, NAT_1:13; A19: G * (1,(j + 1)) = |[((G * (1,(j + 1))) `1),((G * (1,(j + 1))) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]| by A18, A9, A10, GOBOARD5:2 ; A20: ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A11, XREAL_1:6; then A21: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A5, XREAL_1:64; A22: Int (cell (G,0,j)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th20; A23: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ; ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A15, A20, XREAL_1:68; then A24: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2 by A21, A23, XREAL_1:8; A25: G * (1,j) = |[((G * (1,j)) `1),((G * (1,j)) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,j)) `2)]| by A1, A2, A10, GOBOARD5:2 ; p = (((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1,0]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by A4, EUCLID:49 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1,0]|)) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[(1 - r),0]|) + ((r * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + ((r * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) by A19, A25, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + ((r * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by A19, A25, EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| - |[(1 - r),0]|) + ((r * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by EUCLID:58 .= (|[(((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| - |[(1 - r),0]|) + |[((r * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) - 0)]| + |[((r * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:62 .= |[(((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)) + ((r * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,0,j)) by A17, A16, A24, A22; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th45: :: GOBOARD6:45 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) c= (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) c= (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) c= (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} ) assume that A1: 1 <= j and A2: j < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) c= (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) or x in (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} ) assume A3: x in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) ; ::_thesis: x in (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) + (r * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_((len_G),j))_+_(G_*_((len_G),(j_+_1)))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),j))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) by A4, EUCLID:29 .= 1 * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) by EUCLID:27 .= (1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,(len G),j)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set r2 = (G * ((len G),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A8: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) = (G * (1,j)) `2 ; A9: j + 1 <= width G by A2, NAT_1:13; 0 <> len G by GOBOARD1:def_3; then A10: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A11: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A9, A10, GOBOARD5:4; then A12: ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A13: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) <= (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A5, XREAL_1:64; A14: 1 - r > 0 by A7, XREAL_1:50; then A15: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A12, XREAL_1:68; then A16: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) by A13, A8, XREAL_1:8; A17: ((G * ((len G),1)) `1) + (1 - r) > (G * ((len G),1)) `1 by A14, XREAL_1:29; A18: 1 <= j + 1 by A1, NAT_1:13; A19: ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A11, XREAL_1:6; then A20: (r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) <= (r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A5, XREAL_1:64; A21: Int (cell (G,(len G),j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th23; A22: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) = (G * (1,(j + 1))) `2 ; ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A15, A19, XREAL_1:68; then A23: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) < (G * (1,(j + 1))) `2 by A20, A22, XREAL_1:8; A24: G * ((len G),j) = |[((G * ((len G),j)) `1),((G * ((len G),j)) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),j)) `2)]| by A1, A2, A10, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,j)) `2)]| by A1, A2, A10, GOBOARD5:1 ; A25: G * ((len G),(j + 1)) = |[((G * ((len G),(j + 1))) `1),((G * ((len G),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),(j + 1))) `2)]| by A18, A9, A10, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,(j + 1))) `2)]| by A18, A9, A10, GOBOARD5:1 ; p = (((1 - r) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - r) * |[1,0]|)) + (r * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) by A4, EUCLID:32 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + ((1 - r) * |[1,0]|)) + (r * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[(1 - r),0]|) + ((r * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + ((r * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) by A25, A24, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + ((r * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by A25, A24, EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[(1 - r),0]|) + ((r * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) by EUCLID:58 .= (|[(((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[(1 - r),0]|) + |[((r * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))) + (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + 0)]| + |[((r * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))),((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| by EUCLID:56 .= |[(((((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))) + (1 - r)) + ((r * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)))),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + ((r * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,(len G),j)) by A17, A16, A23, A21; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th46: :: GOBOARD6:46 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} ) assume that A1: 1 <= i and A2: i < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) or x in (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} ) assume A3: x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) ; ::_thesis: x in (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_(i,1))_+_(G_*_((i_+_1),1))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,0))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by A4, EUCLID:29 .= 1 * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) by EUCLID:27 .= (1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,i,0)) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set s2 = (G * (1,1)) `2 ; set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; A8: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ; A9: i + 1 <= len G by A2, NAT_1:13; 0 <> width G by GOBOARD1:def_3; then A10: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A11: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A9, A10, GOBOARD5:3; then A12: ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A13: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A5, XREAL_1:64; A14: 1 - r > 0 by A7, XREAL_1:50; then A15: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A12, XREAL_1:68; then A16: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) by A13, A8, XREAL_1:8; (G * (1,1)) `2 < ((G * (1,1)) `2) + (1 - r) by A14, XREAL_1:29; then A17: ((G * (1,1)) `2) - (1 - r) < (G * (1,1)) `2 by XREAL_1:19; A18: 1 <= i + 1 by A1, NAT_1:13; A19: G * ((i + 1),1) = |[((G * ((i + 1),1)) `1),((G * ((i + 1),1)) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]| by A18, A9, A10, GOBOARD5:1 ; A20: ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A11, XREAL_1:6; then A21: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A5, XREAL_1:64; A22: Int (cell (G,i,0)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & s9 < (G * (1,1)) `2 ) } by A1, A2, Th24; A23: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ; ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A15, A20, XREAL_1:68; then A24: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) < (G * ((i + 1),1)) `1 by A21, A23, XREAL_1:8; A25: G * (i,1) = |[((G * (i,1)) `1),((G * (i,1)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (1,1)) `2)]| by A1, A2, A10, GOBOARD5:1 ; p = (((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[0,1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by A4, EUCLID:49 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[0,1]|)) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,(1 - r)]|) + ((r * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + ((r * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) by A19, A25, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) by A19, A25, EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| - |[0,(1 - r)]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) by EUCLID:58 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| - |[0,(1 - r)]|) + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) - 0),((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r))]| + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| by EUCLID:62 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)))),(((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r)) + ((r * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,i,0)) by A17, A16, A24, A22; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th47: :: GOBOARD6:47 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) c= (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) c= (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) c= (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} ) assume that A1: 1 <= i and A2: i < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) c= (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) or x in (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} ) assume A3: x in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) ; ::_thesis: x in (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) + (r * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((1_/_2)_*_((G_*_(i,(width_G)))_+_(G_*_((i_+_1),(width_G)))))}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,(width_G)))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} then p = (0. (TOP-REAL 2)) + (1 * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) by A4, EUCLID:29 .= 1 * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) by EUCLID:27 .= (1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))) by EUCLID:29 ; hence p in {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,i,(width G))) set r3 = (1 - r) * (1 / 2); set s3 = r * (1 / 2); set s2 = (G * (1,(width G))) `2 ; set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; A8: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) = (G * (i,1)) `1 ; A9: i + 1 <= len G by A2, NAT_1:13; 0 <> width G by GOBOARD1:def_3; then A10: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A11: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A9, A10, GOBOARD5:3; then A12: ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A13: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) <= (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A5, XREAL_1:64; A14: 1 - r > 0 by A7, XREAL_1:50; then A15: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A12, XREAL_1:68; then A16: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) by A13, A8, XREAL_1:8; A17: ((G * (1,(width G))) `2) + (1 - r) > (G * (1,(width G))) `2 by A14, XREAL_1:29; A18: 1 <= i + 1 by A1, NAT_1:13; A19: ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A11, XREAL_1:6; then A20: (r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) <= (r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A5, XREAL_1:64; A21: Int (cell (G,i,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s9 ) } by A1, A2, Th25; A22: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) = (G * ((i + 1),1)) `1 ; ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A15, A19, XREAL_1:68; then A23: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) < (G * ((i + 1),1)) `1 by A20, A22, XREAL_1:8; A24: G * (i,(width G)) = |[((G * (i,(width G))) `1),((G * (i,(width G))) `2)]| by EUCLID:53 .= |[((G * (i,(width G))) `1),((G * (1,(width G))) `2)]| by A1, A2, A10, GOBOARD5:1 .= |[((G * (i,1)) `1),((G * (1,(width G))) `2)]| by A1, A2, A10, GOBOARD5:2 ; A25: G * ((i + 1),(width G)) = |[((G * ((i + 1),(width G))) `1),((G * ((i + 1),(width G))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),(width G))) `1),((G * (1,(width G))) `2)]| by A18, A9, A10, GOBOARD5:1 .= |[((G * ((i + 1),1)) `1),((G * (1,(width G))) `2)]| by A18, A9, A10, GOBOARD5:2 ; p = (((1 - r) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - r) * |[0,1]|)) + (r * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) by A4, EUCLID:32 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + ((1 - r) * |[0,1]|)) + (r * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,(1 - r)]|) + ((r * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + ((r * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) by A25, A24, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) by A25, A24, EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2)))]| + |[0,(1 - r)]|) + ((r * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) by EUCLID:58 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2)))]| + |[0,(1 - r)]|) + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2)))]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + 0),((((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))) + (1 - r))]| + |[((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),((r * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2)))]| by EUCLID:56 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + ((r * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)))),(((((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))) + (1 - r)) + ((r * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))))]| by EUCLID:56 ; hence p in Int (cell (G,i,(width G))) by A17, A16, A23, A21; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th48: :: GOBOARD6:48 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,j)) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,j)) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,j)) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} ) assume that A1: 1 <= j and A2: j < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,j)) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,j)) - |[1,0]|)) or x in (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,j)) - |[1,0]|)) ; ::_thesis: x in (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|)) + (r * ((G * (1,j)) - |[1,0]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(1,j))_-_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,j))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (1,j)) - |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,j)) - |[1,0]|)) by A4, EUCLID:29 .= 1 * ((G * (1,j)) - |[1,0]|) by EUCLID:27 .= (G * (1,j)) - |[1,0]| by EUCLID:29 ; hence p in {((G * (1,j)) - |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,0,j)) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set r2 = (G * (1,1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A9: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + (r * ((G * (1,j)) `2)) = (G * (1,j)) `2 ; A10: j + 1 <= width G by A2, NAT_1:13; (G * (1,1)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A11: ((G * (1,1)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19; A12: Int (cell (G,0,j)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th20; 0 <> len G by GOBOARD1:def_3; then A13: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A14: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A10, A13, GOBOARD5:4; then ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A15: ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by A14, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; then A16: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,j)) `2)) by A9, XREAL_1:6; A17: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) = (G * (1,(j + 1))) `2 ; r * ((G * (1,j)) `2) <= r * ((G * (1,(j + 1))) `2) by A5, A14, XREAL_1:64; then A18: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,j)) `2)) < (G * (1,(j + 1))) `2 by A15, A17, XREAL_1:8; A19: G * (1,j) = |[((G * (1,j)) `1),((G * (1,j)) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,j)) `2)]| by A1, A2, A13, GOBOARD5:2 ; A20: 1 <= j + 1 by A1, NAT_1:13; A21: G * (1,(j + 1)) = |[((G * (1,(j + 1))) `1),((G * (1,(j + 1))) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]| by A20, A10, A13, GOBOARD5:2 ; p = (((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1,0]|)) + (r * ((G * (1,j)) - |[1,0]|)) by A4, EUCLID:49 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1,0]|)) + (r * ((G * (1,j)) - |[1,0]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((G * (1,j)) - |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (r * (|[((G * (1,1)) `1),((G * (1,j)) `2)]| - |[1,0]|)) by A21, A19, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + ((r * |[((G * (1,1)) `1),((G * (1,j)) `2)]|) - (r * |[1,0]|)) by EUCLID:49 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (|[(r * ((G * (1,1)) `1)),(r * ((G * (1,j)) `2))]| - (r * |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (|[(r * ((G * (1,1)) `1)),(r * ((G * (1,j)) `2))]| - |[(r * 1),(r * 0)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,j)) `2)) - 0)]| by EUCLID:62 .= (|[(((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| - |[(1 - r),0]|) + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,j)) `2)) - 0)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) - 0)]| + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,j)) `2)) - 0)]| by EUCLID:62 .= |[(((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)) + ((r * ((G * (1,1)) `1)) - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,j)) `2)))]| by EUCLID:56 ; hence p in Int (cell (G,0,j)) by A11, A16, A18, A12; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,j))) \/ {((G * (1,j)) - |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th49: :: GOBOARD6:49 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} ) assume that A1: 1 <= j and A2: j < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) or x in (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) ; ::_thesis: x in (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|)) + (r * ((G * (1,(j + 1))) - |[1,0]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(1,(j_+_1)))_-_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,j))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (1,(j + 1))) - |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,(j + 1))) - |[1,0]|)) by A4, EUCLID:29 .= 1 * ((G * (1,(j + 1))) - |[1,0]|) by EUCLID:27 .= (G * (1,(j + 1))) - |[1,0]| by EUCLID:29 ; hence p in {((G * (1,(j + 1))) - |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,0,j)) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set r2 = (G * (1,1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A9: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + (r * ((G * (1,j)) `2)) = (G * (1,j)) `2 ; A10: j + 1 <= width G by A2, NAT_1:13; 0 <> len G by GOBOARD1:def_3; then A11: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A12: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A10, A11, GOBOARD5:4; then ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A13: ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; r * ((G * (1,j)) `2) <= r * ((G * (1,(j + 1))) `2) by A5, A12, XREAL_1:64; then A14: (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) by A13, A9, XREAL_1:8; A15: 1 <= j + 1 by A1, NAT_1:13; A16: G * (1,j) = |[((G * (1,j)) `1),((G * (1,j)) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,j)) `2)]| by A1, A2, A11, GOBOARD5:2 ; (G * (1,1)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A17: ((G * (1,1)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19; A18: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) = (G * (1,(j + 1))) `2 ; ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A12, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; then A19: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) < (G * (1,(j + 1))) `2 by A18, XREAL_1:8; A20: Int (cell (G,0,j)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th20; A21: G * (1,(j + 1)) = |[((G * (1,(j + 1))) `1),((G * (1,(j + 1))) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]| by A15, A10, A11, GOBOARD5:2 ; p = (((1 - r) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - r) * |[1,0]|)) + (r * ((G * (1,(j + 1))) - |[1,0]|)) by A4, EUCLID:49 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - ((1 - r) * |[1,0]|)) + (r * ((G * (1,(j + 1))) - |[1,0]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((G * (1,(j + 1))) - |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (r * (|[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]| - |[1,0]|)) by A21, A16, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + ((r * |[((G * (1,1)) `1),((G * (1,(j + 1))) `2)]|) - (r * |[1,0]|)) by EUCLID:49 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (|[(r * ((G * (1,1)) `1)),(r * ((G * (1,(j + 1))) `2))]| - (r * |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + (|[(r * ((G * (1,1)) `1)),(r * ((G * (1,(j + 1))) `2))]| - |[(r * 1),(r * 0)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (1,1)) `1) + ((G * (1,1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) - |[(1 - r),0]|) + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,(j + 1))) `2)) - 0)]| by EUCLID:62 .= (|[(((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| - |[(1 - r),0]|) + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,(j + 1))) `2)) - 0)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) - 0)]| + |[((r * ((G * (1,1)) `1)) - r),((r * ((G * (1,(j + 1))) `2)) - 0)]| by EUCLID:62 .= |[(((((1 - r) * (1 / 2)) * (((G * (1,1)) `1) + ((G * (1,1)) `1))) - (1 - r)) + ((r * ((G * (1,1)) `1)) - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)))]| by EUCLID:56 ; hence p in Int (cell (G,0,j)) by A17, A14, A19, A20; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th50: :: GOBOARD6:50 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),j)) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),j)) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),j)) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} ) assume that A1: 1 <= j and A2: j < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),j)) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),j)) + |[1,0]|)) or x in (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),j)) + |[1,0]|)) ; ::_thesis: x in (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) + (r * ((G * ((len G),j)) + |[1,0]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((len_G),j))_+_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),j))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((len G),j)) + |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((len G),j)) + |[1,0]|)) by A4, EUCLID:29 .= 1 * ((G * ((len G),j)) + |[1,0]|) by EUCLID:27 .= (G * ((len G),j)) + |[1,0]| by EUCLID:29 ; hence p in {((G * ((len G),j)) + |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,(len G),j)) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set r2 = (G * ((len G),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A9: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + (r * ((G * (1,j)) `2)) = (G * (1,j)) `2 ; A10: j + 1 <= width G by A2, NAT_1:13; 0 <> len G by GOBOARD1:def_3; then A11: 1 <= len G by NAT_1:14; A12: G * ((len G),j) = |[((G * ((len G),j)) `1),((G * ((len G),j)) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),j)) `2)]| by A1, A2, A11, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,j)) `2)]| by A1, A2, A11, GOBOARD5:1 ; A13: 1 <= j + 1 by A1, NAT_1:13; j < j + 1 by XREAL_1:29; then A14: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A10, A11, GOBOARD5:4; then ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A15: ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by A14, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; then A16: ( (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + 1 & (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,j)) `2)) ) by A9, XREAL_1:6, XREAL_1:29; A17: Int (cell (G,(len G),j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th23; A18: G * ((len G),(j + 1)) = |[((G * ((len G),(j + 1))) `1),((G * ((len G),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),(j + 1))) `2)]| by A13, A10, A11, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,(j + 1))) `2)]| by A13, A10, A11, GOBOARD5:1 ; A19: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) = (G * (1,(j + 1))) `2 ; r * ((G * (1,j)) `2) <= r * ((G * (1,(j + 1))) `2) by A5, A14, XREAL_1:64; then A20: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,j)) `2)) < (G * (1,(j + 1))) `2 by A15, A19, XREAL_1:8; p = (((1 - r) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - r) * |[1,0]|)) + (r * ((G * ((len G),j)) + |[1,0]|)) by A4, EUCLID:32 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + ((1 - r) * |[1,0]|)) + (r * ((G * ((len G),j)) + |[1,0]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((G * ((len G),j)) + |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + (r * (|[((G * ((len G),1)) `1),((G * (1,j)) `2)]| + |[1,0]|)) by A18, A12, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + ((r * |[((G * ((len G),1)) `1),((G * (1,j)) `2)]|) + (r * |[1,0]|)) by EUCLID:32 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + (|[(r * ((G * ((len G),1)) `1)),(r * ((G * (1,j)) `2))]| + (r * |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + (|[(r * ((G * ((len G),1)) `1)),(r * ((G * (1,j)) `2))]| + |[(r * 1),(r * 0)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + |[((r * ((G * ((len G),1)) `1)) + r),((r * ((G * (1,j)) `2)) + 0)]| by EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[(1 - r),0]|) + |[((r * ((G * ((len G),1)) `1)) + r),((r * ((G * (1,j)) `2)) + 0)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))) + (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + 0)]| + |[((r * ((G * ((len G),1)) `1)) + r),((r * ((G * (1,j)) `2)) + 0)]| by EUCLID:56 .= |[(((((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))) + (1 - r)) + ((r * ((G * ((len G),1)) `1)) + r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,j)) `2)))]| by EUCLID:56 ; hence p in Int (cell (G,(len G),j)) by A16, A20, A17; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),j))) \/ {((G * ((len G),j)) + |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th51: :: GOBOARD6:51 for j being Element of NAT for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} let G be Go-board; ::_thesis: ( 1 <= j & j < width G implies LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} ) assume that A1: 1 <= j and A2: j < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) or x in (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) ; ::_thesis: x in (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) + (r * ((G * ((len G),(j + 1))) + |[1,0]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((len_G),(j_+_1)))_+_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),j))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((len G),(j + 1))) + |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((len G),(j + 1))) + |[1,0]|)) by A4, EUCLID:29 .= 1 * ((G * ((len G),(j + 1))) + |[1,0]|) by EUCLID:27 .= (G * ((len G),(j + 1))) + |[1,0]| by EUCLID:29 ; hence p in {((G * ((len G),(j + 1))) + |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,(len G),j)) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set r2 = (G * ((len G),1)) `1 ; set s1 = (G * (1,j)) `2 ; set s2 = (G * (1,(j + 1))) `2 ; A9: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2))) + (r * ((G * (1,j)) `2)) = (G * (1,j)) `2 ; A10: j + 1 <= width G by A2, NAT_1:13; 0 <> len G by GOBOARD1:def_3; then A11: 1 <= len G by NAT_1:14; j < j + 1 by XREAL_1:29; then A12: (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A10, A11, GOBOARD5:4; then ((G * (1,j)) `2) + ((G * (1,j)) `2) < ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) by XREAL_1:6; then A13: ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,j)) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; A14: (((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) = (G * (1,(j + 1))) `2 ; ((G * (1,j)) `2) + ((G * (1,(j + 1))) `2) < ((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2) by A12, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)) < ((1 - r) * (1 / 2)) * (((G * (1,(j + 1))) `2) + ((G * (1,(j + 1))) `2)) by A8, XREAL_1:68; then A15: (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) < (G * (1,(j + 1))) `2 by A14, XREAL_1:8; A16: G * ((len G),j) = |[((G * ((len G),j)) `1),((G * ((len G),j)) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),j)) `2)]| by A1, A2, A11, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,j)) `2)]| by A1, A2, A11, GOBOARD5:1 ; A17: 1 <= j + 1 by A1, NAT_1:13; r * ((G * (1,j)) `2) <= r * ((G * (1,(j + 1))) `2) by A5, A12, XREAL_1:64; then A18: ( ((G * ((len G),1)) `1) + 1 > (G * ((len G),1)) `1 & (G * (1,j)) `2 < (((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)) ) by A13, A9, XREAL_1:8, XREAL_1:29; A19: Int (cell (G,(len G),j)) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & (G * (1,j)) `2 < s9 & s9 < (G * (1,(j + 1))) `2 ) } by A1, A2, Th23; A20: G * ((len G),(j + 1)) = |[((G * ((len G),(j + 1))) `1),((G * ((len G),(j + 1))) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),(j + 1))) `2)]| by A17, A10, A11, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,(j + 1))) `2)]| by A17, A10, A11, GOBOARD5:1 ; p = (((1 - r) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - r) * |[1,0]|)) + (r * ((G * ((len G),(j + 1))) + |[1,0]|)) by A4, EUCLID:32 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + ((1 - r) * |[1,0]|)) + (r * ((G * ((len G),(j + 1))) + |[1,0]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[((1 - r) * 1),((1 - r) * 0)]|) + (r * ((G * ((len G),(j + 1))) + |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + (r * (|[((G * ((len G),1)) `1),((G * (1,(j + 1))) `2)]| + |[1,0]|)) by A20, A16, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + ((r * |[((G * ((len G),1)) `1),((G * (1,(j + 1))) `2)]|) + (r * |[1,0]|)) by EUCLID:32 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + (|[(r * ((G * ((len G),1)) `1)),(r * ((G * (1,(j + 1))) `2))]| + (r * |[1,0]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + (|[(r * ((G * ((len G),1)) `1)),(r * ((G * (1,(j + 1))) `2))]| + |[(r * 1),(r * 0)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * ((len G),1)) `1) + ((G * ((len G),1)) `1)),(((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))]|) + |[(1 - r),0]|) + |[((r * ((G * ((len G),1)) `1)) + r),((r * ((G * (1,(j + 1))) `2)) + 0)]| by EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2)))]| + |[(1 - r),0]|) + |[((r * ((G * ((len G),1)) `1)) + r),((r * ((G * (1,(j + 1))) `2)) + 0)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))) + (1 - r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + 0)]| + |[((r * ((G * ((len G),1)) `1)) + r),((r * ((G * (1,(j + 1))) `2)) + 0)]| by EUCLID:56 .= |[(((((1 - r) * (1 / 2)) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))) + (1 - r)) + ((r * ((G * ((len G),1)) `1)) + r)),((((1 - r) * (1 / 2)) * (((G * (1,j)) `2) + ((G * (1,(j + 1))) `2))) + (r * ((G * (1,(j + 1))) `2)))]| by EUCLID:56 ; hence p in Int (cell (G,(len G),j)) by A18, A15, A19; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th52: :: GOBOARD6:52 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * (i,1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * (i,1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * (i,1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} ) assume that A1: 1 <= i and A2: i < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * (i,1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * (i,1)) - |[0,1]|)) or x in (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * (i,1)) - |[0,1]|)) ; ::_thesis: x in (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|)) + (r * ((G * (i,1)) - |[0,1]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(i,1))_-_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,0))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (i,1)) - |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (i,1)) - |[0,1]|)) by A4, EUCLID:29 .= 1 * ((G * (i,1)) - |[0,1]|) by EUCLID:27 .= (G * (i,1)) - |[0,1]| by EUCLID:29 ; hence p in {((G * (i,1)) - |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,i,0)) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set s1 = (G * (1,1)) `2 ; set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; A9: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + (r * ((G * (i,1)) `1)) = (G * (i,1)) `1 ; A10: i + 1 <= len G by A2, NAT_1:13; (G * (1,1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A11: ((G * (1,1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; A12: Int (cell (G,i,0)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & s9 < (G * (1,1)) `2 ) } by A1, A2, Th24; 0 <> width G by GOBOARD1:def_3; then A13: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A14: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A10, A13, GOBOARD5:3; then ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A15: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by A14, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; then A16: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * (i,1)) `1)) by A9, XREAL_1:6; A17: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) = (G * ((i + 1),1)) `1 ; r * ((G * (i,1)) `1) <= r * ((G * ((i + 1),1)) `1) by A5, A14, XREAL_1:64; then A18: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * (i,1)) `1)) < (G * ((i + 1),1)) `1 by A15, A17, XREAL_1:8; A19: G * (i,1) = |[((G * (i,1)) `1),((G * (i,1)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (1,1)) `2)]| by A1, A2, A13, GOBOARD5:1 ; A20: 1 <= i + 1 by A1, NAT_1:13; A21: G * ((i + 1),1) = |[((G * ((i + 1),1)) `1),((G * ((i + 1),1)) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]| by A20, A10, A13, GOBOARD5:1 ; p = (((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[0,1]|)) + (r * ((G * (i,1)) - |[0,1]|)) by A4, EUCLID:49 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[0,1]|)) + (r * ((G * (i,1)) - |[0,1]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((G * (i,1)) - |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (r * (|[((G * (i,1)) `1),((G * (1,1)) `2)]| - |[0,1]|)) by A21, A19, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + ((r * |[((G * (i,1)) `1),((G * (1,1)) `2)]|) - (r * |[0,1]|)) by EUCLID:49 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (|[(r * ((G * (i,1)) `1)),(r * ((G * (1,1)) `2))]| - (r * |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (|[(r * ((G * (i,1)) `1)),(r * ((G * (1,1)) `2))]| - |[(r * 0),(r * 1)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + |[((r * ((G * (i,1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]| by EUCLID:62 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| - |[0,(1 - r)]|) + |[((r * ((G * (i,1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) - 0),((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r))]| + |[((r * ((G * (i,1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]| by EUCLID:62 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * (i,1)) `1))),(((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r)) + ((r * ((G * (1,1)) `2)) - r))]| by EUCLID:56 ; hence p in Int (cell (G,i,0)) by A11, A16, A18, A12; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,0))) \/ {((G * (i,1)) - |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th53: :: GOBOARD6:53 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} ) assume that A1: 1 <= i and A2: i < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) or x in (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) ; ::_thesis: x in (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|)) + (r * ((G * ((i + 1),1)) - |[0,1]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((i_+_1),1))_-_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,0))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((i + 1),1)) - |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((i + 1),1)) - |[0,1]|)) by A4, EUCLID:29 .= 1 * ((G * ((i + 1),1)) - |[0,1]|) by EUCLID:27 .= (G * ((i + 1),1)) - |[0,1]| by EUCLID:29 ; hence p in {((G * ((i + 1),1)) - |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,i,0)) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set s1 = (G * (1,1)) `2 ; set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; A9: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + (r * ((G * (i,1)) `1)) = (G * (i,1)) `1 ; A10: i + 1 <= len G by A2, NAT_1:13; 0 <> width G by GOBOARD1:def_3; then A11: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A12: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A10, A11, GOBOARD5:3; then ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A13: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; r * ((G * (i,1)) `1) <= r * ((G * ((i + 1),1)) `1) by A5, A12, XREAL_1:64; then A14: (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) by A13, A9, XREAL_1:8; A15: 1 <= i + 1 by A1, NAT_1:13; A16: G * (i,1) = |[((G * (i,1)) `1),((G * (i,1)) `2)]| by EUCLID:53 .= |[((G * (i,1)) `1),((G * (1,1)) `2)]| by A1, A2, A11, GOBOARD5:1 ; (G * (1,1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A17: ((G * (1,1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; A18: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) = (G * ((i + 1),1)) `1 ; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A12, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; then A19: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) < (G * ((i + 1),1)) `1 by A18, XREAL_1:8; A20: Int (cell (G,i,0)) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & s9 < (G * (1,1)) `2 ) } by A1, A2, Th24; A21: G * ((i + 1),1) = |[((G * ((i + 1),1)) `1),((G * ((i + 1),1)) `2)]| by EUCLID:53 .= |[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]| by A15, A10, A11, GOBOARD5:1 ; p = (((1 - r) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - r) * |[0,1]|)) + (r * ((G * ((i + 1),1)) - |[0,1]|)) by A4, EUCLID:49 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - ((1 - r) * |[0,1]|)) + (r * ((G * ((i + 1),1)) - |[0,1]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((G * ((i + 1),1)) - |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (r * (|[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]| - |[0,1]|)) by A21, A16, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + ((r * |[((G * ((i + 1),1)) `1),((G * (1,1)) `2)]|) - (r * |[0,1]|)) by EUCLID:49 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (|[(r * ((G * ((i + 1),1)) `1)),(r * ((G * (1,1)) `2))]| - (r * |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + (|[(r * ((G * ((i + 1),1)) `1)),(r * ((G * (1,1)) `2))]| - |[(r * 0),(r * 1)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,1)) `2) + ((G * (1,1)) `2))]|) - |[0,(1 - r)]|) + |[((r * ((G * ((i + 1),1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]| by EUCLID:62 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2)))]| - |[0,(1 - r)]|) + |[((r * ((G * ((i + 1),1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) - 0),((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r))]| + |[((r * ((G * ((i + 1),1)) `1)) - 0),((r * ((G * (1,1)) `2)) - r)]| by EUCLID:62 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1))),(((((1 - r) * (1 / 2)) * (((G * (1,1)) `2) + ((G * (1,1)) `2))) - (1 - r)) + ((r * ((G * (1,1)) `2)) - r))]| by EUCLID:56 ; hence p in Int (cell (G,i,0)) by A17, A14, A19, A20; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th54: :: GOBOARD6:54 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * (i,(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * (i,(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * (i,(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} ) assume that A1: 1 <= i and A2: i < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * (i,(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * (i,(width G))) + |[0,1]|)) or x in (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * (i,(width G))) + |[0,1]|)) ; ::_thesis: x in (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) + (r * ((G * (i,(width G))) + |[0,1]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(i,(width_G)))_+_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,(width_G)))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (i,(width G))) + |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (i,(width G))) + |[0,1]|)) by A4, EUCLID:29 .= 1 * ((G * (i,(width G))) + |[0,1]|) by EUCLID:27 .= (G * (i,(width G))) + |[0,1]| by EUCLID:29 ; hence p in {((G * (i,(width G))) + |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,i,(width G))) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set s1 = (G * (1,(width G))) `2 ; set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; A9: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + (r * ((G * (i,1)) `1)) = (G * (i,1)) `1 ; A10: i + 1 <= len G by A2, NAT_1:13; 0 <> width G by GOBOARD1:def_3; then A11: 1 <= width G by NAT_1:14; A12: G * (i,(width G)) = |[((G * (i,(width G))) `1),((G * (i,(width G))) `2)]| by EUCLID:53 .= |[((G * (i,(width G))) `1),((G * (1,(width G))) `2)]| by A1, A2, A11, GOBOARD5:1 .= |[((G * (i,1)) `1),((G * (1,(width G))) `2)]| by A1, A2, A11, GOBOARD5:2 ; A13: 1 <= i + 1 by A1, NAT_1:13; i < i + 1 by XREAL_1:29; then A14: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A10, A11, GOBOARD5:3; then ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A15: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by A14, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; then A16: ( (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + 1 & (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * (i,1)) `1)) ) by A9, XREAL_1:6, XREAL_1:29; A17: Int (cell (G,i,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s9 ) } by A1, A2, Th25; A18: G * ((i + 1),(width G)) = |[((G * ((i + 1),(width G))) `1),((G * ((i + 1),(width G))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),(width G))) `1),((G * (1,(width G))) `2)]| by A13, A10, A11, GOBOARD5:1 .= |[((G * ((i + 1),1)) `1),((G * (1,(width G))) `2)]| by A13, A10, A11, GOBOARD5:2 ; A19: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) = (G * ((i + 1),1)) `1 ; r * ((G * (i,1)) `1) <= r * ((G * ((i + 1),1)) `1) by A5, A14, XREAL_1:64; then A20: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * (i,1)) `1)) < (G * ((i + 1),1)) `1 by A15, A19, XREAL_1:8; p = (((1 - r) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - r) * |[0,1]|)) + (r * ((G * (i,(width G))) + |[0,1]|)) by A4, EUCLID:32 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + ((1 - r) * |[0,1]|)) + (r * ((G * (i,(width G))) + |[0,1]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((G * (i,(width G))) + |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + (r * (|[((G * (i,1)) `1),((G * (1,(width G))) `2)]| + |[0,1]|)) by A18, A12, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + ((r * |[((G * (i,1)) `1),((G * (1,(width G))) `2)]|) + (r * |[0,1]|)) by EUCLID:32 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + (|[(r * ((G * (i,1)) `1)),(r * ((G * (1,(width G))) `2))]| + (r * |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + (|[(r * ((G * (i,1)) `1)),(r * ((G * (1,(width G))) `2))]| + |[(r * 0),(r * 1)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + |[((r * ((G * (i,1)) `1)) + 0),((r * ((G * (1,(width G))) `2)) + r)]| by EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2)))]| + |[0,(1 - r)]|) + |[((r * ((G * (i,1)) `1)) + 0),((r * ((G * (1,(width G))) `2)) + r)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + 0),((((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))) + (1 - r))]| + |[((r * ((G * (i,1)) `1)) + 0),((r * ((G * (1,(width G))) `2)) + r)]| by EUCLID:56 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * (i,1)) `1))),(((((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))) + (1 - r)) + ((r * ((G * (1,(width G))) `2)) + r))]| by EUCLID:56 ; hence p in Int (cell (G,i,(width G))) by A16, A20, A17; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,(width G)))) \/ {((G * (i,(width G))) + |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th55: :: GOBOARD6:55 for i being Element of NAT for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} let G be Go-board; ::_thesis: ( 1 <= i & i < len G implies LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} ) assume that A1: 1 <= i and A2: i < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) or x in (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} ) assume A3: x in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) ; ::_thesis: x in (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A4: p = ((1 - r) * (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) + (r * ((G * ((i + 1),(width G))) + |[0,1]|)) and A5: 0 <= r and A6: r <= 1 by A3; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((i_+_1),(width_G)))_+_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,i,(width_G)))_)_) percases ( r = 1 or r < 1 ) by A6, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((i + 1),(width G))) + |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((i + 1),(width G))) + |[0,1]|)) by A4, EUCLID:29 .= 1 * ((G * ((i + 1),(width G))) + |[0,1]|) by EUCLID:27 .= (G * ((i + 1),(width G))) + |[0,1]| by EUCLID:29 ; hence p in {((G * ((i + 1),(width G))) + |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; caseA7: r < 1 ; ::_thesis: p in Int (cell (G,i,(width G))) set r3 = (1 - r) * (1 / 2); 1 - r > 0 by A7, XREAL_1:50; then A8: (1 - r) * (1 / 2) > (1 / 2) * 0 by XREAL_1:68; set s1 = (G * (1,(width G))) `2 ; set r1 = (G * (i,1)) `1 ; set r2 = (G * ((i + 1),1)) `1 ; A9: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1))) + (r * ((G * (i,1)) `1)) = (G * (i,1)) `1 ; A10: i + 1 <= len G by A2, NAT_1:13; 0 <> width G by GOBOARD1:def_3; then A11: 1 <= width G by NAT_1:14; i < i + 1 by XREAL_1:29; then A12: (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A10, A11, GOBOARD5:3; then ((G * (i,1)) `1) + ((G * (i,1)) `1) < ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) by XREAL_1:6; then A13: ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * (i,1)) `1)) < ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; A14: (((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) = (G * ((i + 1),1)) `1 ; ((G * (i,1)) `1) + ((G * ((i + 1),1)) `1) < ((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1) by A12, XREAL_1:6; then ((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)) < ((1 - r) * (1 / 2)) * (((G * ((i + 1),1)) `1) + ((G * ((i + 1),1)) `1)) by A8, XREAL_1:68; then A15: (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) < (G * ((i + 1),1)) `1 by A14, XREAL_1:8; A16: G * (i,(width G)) = |[((G * (i,(width G))) `1),((G * (i,(width G))) `2)]| by EUCLID:53 .= |[((G * (i,(width G))) `1),((G * (1,(width G))) `2)]| by A1, A2, A11, GOBOARD5:1 .= |[((G * (i,1)) `1),((G * (1,(width G))) `2)]| by A1, A2, A11, GOBOARD5:2 ; A17: 1 <= i + 1 by A1, NAT_1:13; r * ((G * (i,1)) `1) <= r * ((G * ((i + 1),1)) `1) by A5, A12, XREAL_1:64; then A18: ( ((G * (1,(width G))) `2) + 1 > (G * (1,(width G))) `2 & (G * (i,1)) `1 < (((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1)) ) by A13, A9, XREAL_1:8, XREAL_1:29; A19: Int (cell (G,i,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (G * (i,1)) `1 < r9 & r9 < (G * ((i + 1),1)) `1 & (G * (1,(width G))) `2 < s9 ) } by A1, A2, Th25; A20: G * ((i + 1),(width G)) = |[((G * ((i + 1),(width G))) `1),((G * ((i + 1),(width G))) `2)]| by EUCLID:53 .= |[((G * ((i + 1),(width G))) `1),((G * (1,(width G))) `2)]| by A17, A10, A11, GOBOARD5:1 .= |[((G * ((i + 1),1)) `1),((G * (1,(width G))) `2)]| by A17, A10, A11, GOBOARD5:2 ; p = (((1 - r) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - r) * |[0,1]|)) + (r * ((G * ((i + 1),(width G))) + |[0,1]|)) by A4, EUCLID:32 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + ((1 - r) * |[0,1]|)) + (r * ((G * ((i + 1),(width G))) + |[0,1]|)) by EUCLID:30 .= ((((1 - r) * (1 / 2)) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[((1 - r) * 0),((1 - r) * 1)]|) + (r * ((G * ((i + 1),(width G))) + |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + (r * (|[((G * ((i + 1),1)) `1),((G * (1,(width G))) `2)]| + |[0,1]|)) by A20, A16, EUCLID:56 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + ((r * |[((G * ((i + 1),1)) `1),((G * (1,(width G))) `2)]|) + (r * |[0,1]|)) by EUCLID:32 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + (|[(r * ((G * ((i + 1),1)) `1)),(r * ((G * (1,(width G))) `2))]| + (r * |[0,1]|)) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + (|[(r * ((G * ((i + 1),1)) `1)),(r * ((G * (1,(width G))) `2))]| + |[(r * 0),(r * 1)]|) by EUCLID:58 .= ((((1 - r) * (1 / 2)) * |[(((G * (i,1)) `1) + ((G * ((i + 1),1)) `1)),(((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))]|) + |[0,(1 - r)]|) + |[((r * ((G * ((i + 1),1)) `1)) + 0),((r * ((G * (1,(width G))) `2)) + r)]| by EUCLID:56 .= (|[(((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))),(((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2)))]| + |[0,(1 - r)]|) + |[((r * ((G * ((i + 1),1)) `1)) + 0),((r * ((G * (1,(width G))) `2)) + r)]| by EUCLID:58 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + 0),((((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))) + (1 - r))]| + |[((r * ((G * ((i + 1),1)) `1)) + 0),((r * ((G * (1,(width G))) `2)) + r)]| by EUCLID:56 .= |[((((1 - r) * (1 / 2)) * (((G * (i,1)) `1) + ((G * ((i + 1),1)) `1))) + (r * ((G * ((i + 1),1)) `1))),(((((1 - r) * (1 / 2)) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))) + (1 - r)) + ((r * ((G * (1,(width G))) `2)) + r))]| by EUCLID:56 ; hence p in Int (cell (G,i,(width G))) by A18, A15, A19; ::_thesis: verum end; end; end; hence x in (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th56: :: GOBOARD6:56 for G being Go-board holds LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)} proof let G be Go-board; ::_thesis: LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|)) or x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)} ) set r1 = (G * (1,1)) `1 ; set s1 = (G * (1,1)) `2 ; assume A1: x in LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|)) ; ::_thesis: x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * (1,1)) - |[1,1]|)) + (r * ((G * (1,1)) - |[1,0]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(1,1))_-_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,0))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (1,1)) - |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,1)) - |[1,0]|)) by A2, EUCLID:29 .= 1 * ((G * (1,1)) - |[1,0]|) by EUCLID:27 .= (G * (1,1)) - |[1,0]| by EUCLID:29 ; hence p in {((G * (1,1)) - |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,0,0)) then 1 - r > 0 by XREAL_1:50; then (G * (1,1)) `2 < ((G * (1,1)) `2) + (1 - r) by XREAL_1:29; then A4: ((G * (1,1)) `2) - (1 - r) < (G * (1,1)) `2 by XREAL_1:19; A5: G * (1,1) = |[((G * (1,1)) `1),((G * (1,1)) `2)]| by EUCLID:53; (G * (1,1)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A6: ((G * (1,1)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19; A7: Int (cell (G,0,0)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & s9 < (G * (1,1)) `2 ) } by Th18; p = (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|)) + (r * ((G * (1,1)) - |[1,0]|)) by A2, EUCLID:49 .= (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|)) + ((r * (G * (1,1))) - (r * |[1,0]|)) by EUCLID:49 .= ((r * (G * (1,1))) + (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|))) - (r * |[1,0]|) by EUCLID:45 .= (((r * (G * (1,1))) + ((1 - r) * (G * (1,1)))) - ((1 - r) * |[1,1]|)) - (r * |[1,0]|) by EUCLID:45 .= (((r + (1 - r)) * (G * (1,1))) - ((1 - r) * |[1,1]|)) - (r * |[1,0]|) by EUCLID:33 .= ((G * (1,1)) - ((1 - r) * |[1,1]|)) - (r * |[1,0]|) by EUCLID:29 .= ((G * (1,1)) - |[((1 - r) * 1),((1 - r) * 1)]|) - (r * |[1,0]|) by EUCLID:58 .= ((G * (1,1)) - |[(1 - r),(1 - r)]|) - |[(r * 1),(r * 0)]| by EUCLID:58 .= |[(((G * (1,1)) `1) - (1 - r)),(((G * (1,1)) `2) - (1 - r))]| - |[r,0]| by A5, EUCLID:62 .= |[((((G * (1,1)) `1) - (1 - r)) - r),((((G * (1,1)) `2) - (1 - r)) - 0)]| by EUCLID:62 .= |[(((G * (1,1)) `1) - 1),(((G * (1,1)) `2) - (1 - r))]| ; hence p in Int (cell (G,0,0)) by A4, A6, A7; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th57: :: GOBOARD6:57 for G being Go-board holds LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) + |[1,0]|)) c= (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)} proof let G be Go-board; ::_thesis: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) + |[1,0]|)) c= (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) + |[1,0]|)) or x in (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)} ) set r1 = (G * ((len G),1)) `1 ; set s1 = (G * (1,1)) `2 ; assume A1: x in LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) + |[1,0]|)) ; ::_thesis: x in (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * ((len G),1)) + |[1,(- 1)]|)) + (r * ((G * ((len G),1)) + |[1,0]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((len_G),1))_+_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),0))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((len G),1)) + |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((len G),1)) + |[1,0]|)) by A2, EUCLID:29 .= 1 * ((G * ((len G),1)) + |[1,0]|) by EUCLID:27 .= (G * ((len G),1)) + |[1,0]| by EUCLID:29 ; hence p in {((G * ((len G),1)) + |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,(len G),0)) then 1 - r > 0 by XREAL_1:50; then A4: (G * (1,1)) `2 < ((G * (1,1)) `2) + (1 - r) by XREAL_1:29; ((G * (1,1)) `2) + (r - 1) = ((G * (1,1)) `2) - (1 - r) ; then A5: ((G * (1,1)) `2) + (r - 1) < (G * (1,1)) `2 by A4, XREAL_1:19; A6: (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + 1 by XREAL_1:29; 0 <> len G by GOBOARD1:def_3; then A7: 1 <= len G by NAT_1:14; 0 <> width G by GOBOARD1:def_3; then A8: 1 <= width G by NAT_1:14; A9: G * ((len G),1) = |[((G * ((len G),1)) `1),((G * ((len G),1)) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * (1,1)) `2)]| by A8, A7, GOBOARD5:1 ; A10: Int (cell (G,(len G),0)) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & s9 < (G * (1,1)) `2 ) } by Th21; p = (((1 - r) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|)) + (r * ((G * ((len G),1)) + |[1,0]|)) by A2, EUCLID:32 .= (((1 - r) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|)) + ((r * (G * ((len G),1))) + (r * |[1,0]|)) by EUCLID:32 .= ((r * (G * ((len G),1))) + (((1 - r) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|))) + (r * |[1,0]|) by EUCLID:26 .= (((r * (G * ((len G),1))) + ((1 - r) * (G * ((len G),1)))) + ((1 - r) * |[1,(- 1)]|)) + (r * |[1,0]|) by EUCLID:26 .= (((r + (1 - r)) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|)) + (r * |[1,0]|) by EUCLID:33 .= ((G * ((len G),1)) + ((1 - r) * |[1,(- 1)]|)) + (r * |[1,0]|) by EUCLID:29 .= ((G * ((len G),1)) + |[((1 - r) * 1),((1 - r) * (- 1))]|) + (r * |[1,0]|) by EUCLID:58 .= ((G * ((len G),1)) + |[(1 - r),(r - 1)]|) + |[(r * 1),(r * 0)]| by EUCLID:58 .= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,1)) `2) + (r - 1))]| + |[r,0]| by A9, EUCLID:56 .= |[((((G * ((len G),1)) `1) + (1 - r)) + r),((((G * (1,1)) `2) + (r - 1)) + 0)]| by EUCLID:56 .= |[(((G * ((len G),1)) `1) + 1),(((G * (1,1)) `2) + (r - 1))]| ; hence p in Int (cell (G,(len G),0)) by A5, A6, A10; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th58: :: GOBOARD6:58 for G being Go-board holds LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) - |[1,0]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)} proof let G be Go-board; ::_thesis: LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) - |[1,0]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) - |[1,0]|)) or x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)} ) set r1 = (G * (1,1)) `1 ; set s1 = (G * (1,(width G))) `2 ; assume A1: x in LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) - |[1,0]|)) ; ::_thesis: x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * (1,(width G))) + |[(- 1),1]|)) + (r * ((G * (1,(width G))) - |[1,0]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(1,(width_G)))_-_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,(width_G)))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (1,(width G))) - |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,(width G))) - |[1,0]|)) by A2, EUCLID:29 .= 1 * ((G * (1,(width G))) - |[1,0]|) by EUCLID:27 .= (G * (1,(width G))) - |[1,0]| by EUCLID:29 ; hence p in {((G * (1,(width G))) - |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,0,(width G))) then 1 - r > 0 by XREAL_1:50; then A4: (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + (1 - r) by XREAL_1:29; 0 <> width G by GOBOARD1:def_3; then A5: 1 <= width G by NAT_1:14; 0 <> len G by GOBOARD1:def_3; then A6: 1 <= len G by NAT_1:14; A7: G * (1,(width G)) = |[((G * (1,(width G))) `1),((G * (1,(width G))) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,(width G))) `2)]| by A5, A6, GOBOARD5:2 ; A8: Int (cell (G,0,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s9 ) } by Th19; (G * (1,1)) `1 < ((G * (1,1)) `1) + 1 by XREAL_1:29; then A9: ((G * (1,1)) `1) - 1 < (G * (1,1)) `1 by XREAL_1:19; p = (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + (r * ((G * (1,(width G))) - |[1,0]|)) by A2, EUCLID:32 .= (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + ((r * (G * (1,(width G)))) - (r * |[1,0]|)) by EUCLID:49 .= ((r * (G * (1,(width G)))) + (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|))) - (r * |[1,0]|) by EUCLID:45 .= (((r * (G * (1,(width G)))) + ((1 - r) * (G * (1,(width G))))) + ((1 - r) * |[(- 1),1]|)) - (r * |[1,0]|) by EUCLID:26 .= (((r + (1 - r)) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) - (r * |[1,0]|) by EUCLID:33 .= ((G * (1,(width G))) + ((1 - r) * |[(- 1),1]|)) - (r * |[1,0]|) by EUCLID:29 .= ((G * (1,(width G))) + |[((1 - r) * (- 1)),((1 - r) * 1)]|) - (r * |[1,0]|) by EUCLID:58 .= ((G * (1,(width G))) + |[(r - 1),(1 - r)]|) - |[(r * 1),(r * 0)]| by EUCLID:58 .= |[(((G * (1,1)) `1) + (r - 1)),(((G * (1,(width G))) `2) + (1 - r))]| - |[r,0]| by A7, EUCLID:56 .= |[((((G * (1,1)) `1) + (r - 1)) - r),((((G * (1,(width G))) `2) + (1 - r)) - 0)]| by EUCLID:62 .= |[(((G * (1,1)) `1) - 1),(((G * (1,(width G))) `2) + (1 - r))]| ; hence p in Int (cell (G,0,(width G))) by A4, A9, A8; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th59: :: GOBOARD6:59 for G being Go-board holds LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} proof let G be Go-board; ::_thesis: LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) or x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} ) set r1 = (G * ((len G),1)) `1 ; set s1 = (G * (1,(width G))) `2 ; assume A1: x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) ; ::_thesis: x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * ((len G),(width G))) + |[1,1]|)) + (r * ((G * ((len G),(width G))) + |[1,0]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((len_G),(width_G)))_+_|[1,0]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),(width_G)))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((len G),(width G))) + |[1,0]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((len G),(width G))) + |[1,0]|)) by A2, EUCLID:29 .= 1 * ((G * ((len G),(width G))) + |[1,0]|) by EUCLID:27 .= (G * ((len G),(width G))) + |[1,0]| by EUCLID:29 ; hence p in {((G * ((len G),(width G))) + |[1,0]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,(len G),(width G))) then 1 - r > 0 by XREAL_1:50; then A4: (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + (1 - r) by XREAL_1:29; A5: (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + 1 by XREAL_1:29; 0 <> width G by GOBOARD1:def_3; then A6: 1 <= width G by NAT_1:14; 0 <> len G by GOBOARD1:def_3; then A7: 1 <= len G by NAT_1:14; A8: G * ((len G),(width G)) = |[((G * ((len G),(width G))) `1),((G * ((len G),(width G))) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),(width G))) `2)]| by A6, A7, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,(width G))) `2)]| by A6, A7, GOBOARD5:1 ; A9: Int (cell (G,(len G),(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & (G * (1,(width G))) `2 < s9 ) } by Th22; p = (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + (r * ((G * ((len G),(width G))) + |[1,0]|)) by A2, EUCLID:32 .= (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + ((r * (G * ((len G),(width G)))) + (r * |[1,0]|)) by EUCLID:32 .= ((r * (G * ((len G),(width G)))) + (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|))) + (r * |[1,0]|) by EUCLID:26 .= (((r * (G * ((len G),(width G)))) + ((1 - r) * (G * ((len G),(width G))))) + ((1 - r) * |[1,1]|)) + (r * |[1,0]|) by EUCLID:26 .= (((r + (1 - r)) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + (r * |[1,0]|) by EUCLID:33 .= ((G * ((len G),(width G))) + ((1 - r) * |[1,1]|)) + (r * |[1,0]|) by EUCLID:29 .= ((G * ((len G),(width G))) + |[((1 - r) * 1),((1 - r) * 1)]|) + (r * |[1,0]|) by EUCLID:58 .= ((G * ((len G),(width G))) + |[(1 - r),(1 - r)]|) + |[(r * 1),(r * 0)]| by EUCLID:58 .= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,(width G))) `2) + (1 - r))]| + |[r,0]| by A8, EUCLID:56 .= |[((((G * ((len G),1)) `1) + (1 - r)) + r),((((G * (1,(width G))) `2) + (1 - r)) + 0)]| by EUCLID:56 .= |[(((G * ((len G),1)) `1) + 1),(((G * (1,(width G))) `2) + (1 - r))]| ; hence p in Int (cell (G,(len G),(width G))) by A4, A5, A9; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th60: :: GOBOARD6:60 for G being Go-board holds LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} proof let G be Go-board; ::_thesis: LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) or x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} ) set r1 = (G * (1,1)) `1 ; set s1 = (G * (1,1)) `2 ; assume A1: x in LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) ; ::_thesis: x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * (1,1)) - |[1,1]|)) + (r * ((G * (1,1)) - |[0,1]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(1,1))_-_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,0))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (1,1)) - |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,1)) - |[0,1]|)) by A2, EUCLID:29 .= 1 * ((G * (1,1)) - |[0,1]|) by EUCLID:27 .= (G * (1,1)) - |[0,1]| by EUCLID:29 ; hence p in {((G * (1,1)) - |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,0,0)) then 1 - r > 0 by XREAL_1:50; then (G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r) by XREAL_1:29; then A4: ((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1 by XREAL_1:19; A5: G * (1,1) = |[((G * (1,1)) `1),((G * (1,1)) `2)]| by EUCLID:53; (G * (1,1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A6: ((G * (1,1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; A7: Int (cell (G,0,0)) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & s9 < (G * (1,1)) `2 ) } by Th18; p = (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|)) + (r * ((G * (1,1)) - |[0,1]|)) by A2, EUCLID:49 .= (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|)) + ((r * (G * (1,1))) - (r * |[0,1]|)) by EUCLID:49 .= ((r * (G * (1,1))) + (((1 - r) * (G * (1,1))) - ((1 - r) * |[1,1]|))) - (r * |[0,1]|) by EUCLID:45 .= (((r * (G * (1,1))) + ((1 - r) * (G * (1,1)))) - ((1 - r) * |[1,1]|)) - (r * |[0,1]|) by EUCLID:45 .= (((r + (1 - r)) * (G * (1,1))) - ((1 - r) * |[1,1]|)) - (r * |[0,1]|) by EUCLID:33 .= ((G * (1,1)) - ((1 - r) * |[1,1]|)) - (r * |[0,1]|) by EUCLID:29 .= ((G * (1,1)) - |[((1 - r) * 1),((1 - r) * 1)]|) - (r * |[0,1]|) by EUCLID:58 .= ((G * (1,1)) - |[(1 - r),(1 - r)]|) - |[(r * 0),(r * 1)]| by EUCLID:58 .= |[(((G * (1,1)) `1) - (1 - r)),(((G * (1,1)) `2) - (1 - r))]| - |[0,r]| by A5, EUCLID:62 .= |[((((G * (1,1)) `1) - (1 - r)) - 0),((((G * (1,1)) `2) - (1 - r)) - r)]| by EUCLID:62 .= |[(((G * (1,1)) `1) - (1 - r)),(((G * (1,1)) `2) - 1)]| ; hence p in Int (cell (G,0,0)) by A6, A4, A7; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th61: :: GOBOARD6:61 for G being Go-board holds LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) - |[0,1]|)) c= (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)} proof let G be Go-board; ::_thesis: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) - |[0,1]|)) c= (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) - |[0,1]|)) or x in (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)} ) set r1 = (G * ((len G),1)) `1 ; set s1 = (G * (1,1)) `2 ; assume A1: x in LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) - |[0,1]|)) ; ::_thesis: x in (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * ((len G),1)) + |[1,(- 1)]|)) + (r * ((G * ((len G),1)) - |[0,1]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((len_G),1))_-_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),0))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((len G),1)) - |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((len G),1)) - |[0,1]|)) by A2, EUCLID:29 .= 1 * ((G * ((len G),1)) - |[0,1]|) by EUCLID:27 .= (G * ((len G),1)) - |[0,1]| by EUCLID:29 ; hence p in {((G * ((len G),1)) - |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,(len G),0)) then 1 - r > 0 by XREAL_1:50; then A4: (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + (1 - r) by XREAL_1:29; (G * (1,1)) `2 < ((G * (1,1)) `2) + 1 by XREAL_1:29; then A5: ((G * (1,1)) `2) - 1 < (G * (1,1)) `2 by XREAL_1:19; 0 <> len G by GOBOARD1:def_3; then A6: 1 <= len G by NAT_1:14; 0 <> width G by GOBOARD1:def_3; then A7: 1 <= width G by NAT_1:14; A8: G * ((len G),1) = |[((G * ((len G),1)) `1),((G * ((len G),1)) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * (1,1)) `2)]| by A7, A6, GOBOARD5:1 ; A9: Int (cell (G,(len G),0)) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & s9 < (G * (1,1)) `2 ) } by Th21; p = (((1 - r) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|)) + (r * ((G * ((len G),1)) - |[0,1]|)) by A2, EUCLID:32 .= (((1 - r) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|)) + ((r * (G * ((len G),1))) - (r * |[0,1]|)) by EUCLID:49 .= ((r * (G * ((len G),1))) + (((1 - r) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|))) - (r * |[0,1]|) by EUCLID:45 .= (((r * (G * ((len G),1))) + ((1 - r) * (G * ((len G),1)))) + ((1 - r) * |[1,(- 1)]|)) - (r * |[0,1]|) by EUCLID:26 .= (((r + (1 - r)) * (G * ((len G),1))) + ((1 - r) * |[1,(- 1)]|)) - (r * |[0,1]|) by EUCLID:33 .= ((G * ((len G),1)) + ((1 - r) * |[1,(- 1)]|)) - (r * |[0,1]|) by EUCLID:29 .= ((G * ((len G),1)) + |[((1 - r) * 1),((1 - r) * (- 1))]|) - (r * |[0,1]|) by EUCLID:58 .= ((G * ((len G),1)) + |[(1 - r),(r - 1)]|) - |[(r * 0),(r * 1)]| by EUCLID:58 .= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,1)) `2) + (r - 1))]| - |[0,r]| by A8, EUCLID:56 .= |[((((G * ((len G),1)) `1) + (1 - r)) - 0),((((G * (1,1)) `2) + (r - 1)) - r)]| by EUCLID:62 .= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,1)) `2) - 1)]| ; hence p in Int (cell (G,(len G),0)) by A5, A4, A9; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th62: :: GOBOARD6:62 for G being Go-board holds LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} proof let G be Go-board; ::_thesis: LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) or x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} ) set r1 = (G * (1,1)) `1 ; set s1 = (G * (1,(width G))) `2 ; assume A1: x in LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) ; ::_thesis: x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * (1,(width G))) + |[(- 1),1]|)) + (r * ((G * (1,(width G))) + |[0,1]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_(1,(width_G)))_+_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,0,(width_G)))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * (1,(width G))) + |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * (1,(width G))) + |[0,1]|)) by A2, EUCLID:29 .= 1 * ((G * (1,(width G))) + |[0,1]|) by EUCLID:27 .= (G * (1,(width G))) + |[0,1]| by EUCLID:29 ; hence p in {((G * (1,(width G))) + |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,0,(width G))) then 1 - r > 0 by XREAL_1:50; then (G * (1,1)) `1 < ((G * (1,1)) `1) + (1 - r) by XREAL_1:29; then A4: ( (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + 1 & ((G * (1,1)) `1) - (1 - r) < (G * (1,1)) `1 ) by XREAL_1:19, XREAL_1:29; 0 <> width G by GOBOARD1:def_3; then A5: 1 <= width G by NAT_1:14; 0 <> len G by GOBOARD1:def_3; then A6: 1 <= len G by NAT_1:14; A7: G * (1,(width G)) = |[((G * (1,(width G))) `1),((G * (1,(width G))) `2)]| by EUCLID:53 .= |[((G * (1,1)) `1),((G * (1,(width G))) `2)]| by A5, A6, GOBOARD5:2 ; A8: Int (cell (G,0,(width G))) = { |[r9,s9]| where r9, s9 is Real : ( r9 < (G * (1,1)) `1 & (G * (1,(width G))) `2 < s9 ) } by Th19; p = (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + (r * ((G * (1,(width G))) + |[0,1]|)) by A2, EUCLID:32 .= (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + ((r * (G * (1,(width G)))) + (r * |[0,1]|)) by EUCLID:32 .= ((r * (G * (1,(width G)))) + (((1 - r) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|))) + (r * |[0,1]|) by EUCLID:26 .= (((r * (G * (1,(width G)))) + ((1 - r) * (G * (1,(width G))))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0,1]|) by EUCLID:26 .= (((r + (1 - r)) * (G * (1,(width G)))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0,1]|) by EUCLID:33 .= ((G * (1,(width G))) + ((1 - r) * |[(- 1),1]|)) + (r * |[0,1]|) by EUCLID:29 .= ((G * (1,(width G))) + |[((1 - r) * (- 1)),((1 - r) * 1)]|) + (r * |[0,1]|) by EUCLID:58 .= ((G * (1,(width G))) + |[(r - 1),(1 - r)]|) + |[(r * 0),(r * 1)]| by EUCLID:58 .= |[(((G * (1,1)) `1) + (r - 1)),(((G * (1,(width G))) `2) + (1 - r))]| + |[0,r]| by A7, EUCLID:56 .= |[((((G * (1,1)) `1) + (r - 1)) + 0),((((G * (1,(width G))) `2) + (1 - r)) + r)]| by EUCLID:56 .= |[(((G * (1,1)) `1) - (1 - r)),(((G * (1,(width G))) `2) + 1)]| ; hence p in Int (cell (G,0,(width G))) by A4, A8; ::_thesis: verum end; end; end; hence x in (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem Th63: :: GOBOARD6:63 for G being Go-board holds LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[0,1]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} proof let G be Go-board; ::_thesis: LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[0,1]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[0,1]|)) or x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} ) set r1 = (G * ((len G),1)) `1 ; set s1 = (G * (1,(width G))) `2 ; assume A1: x in LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[0,1]|)) ; ::_thesis: x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} then reconsider p = x as Point of (TOP-REAL 2) ; consider r being Real such that A2: p = ((1 - r) * ((G * ((len G),(width G))) + |[1,1]|)) + (r * ((G * ((len G),(width G))) + |[0,1]|)) and 0 <= r and A3: r <= 1 by A1; now__::_thesis:_(_(_r_=_1_&_p_in_{((G_*_((len_G),(width_G)))_+_|[0,1]|)}_)_or_(_r_<_1_&_p_in_Int_(cell_(G,(len_G),(width_G)))_)_) percases ( r = 1 or r < 1 ) by A3, XXREAL_0:1; case r = 1 ; ::_thesis: p in {((G * ((len G),(width G))) + |[0,1]|)} then p = (0. (TOP-REAL 2)) + (1 * ((G * ((len G),(width G))) + |[0,1]|)) by A2, EUCLID:29 .= 1 * ((G * ((len G),(width G))) + |[0,1]|) by EUCLID:27 .= (G * ((len G),(width G))) + |[0,1]| by EUCLID:29 ; hence p in {((G * ((len G),(width G))) + |[0,1]|)} by TARSKI:def_1; ::_thesis: verum end; case r < 1 ; ::_thesis: p in Int (cell (G,(len G),(width G))) then 1 - r > 0 by XREAL_1:50; then A4: ( (G * (1,(width G))) `2 < ((G * (1,(width G))) `2) + 1 & (G * ((len G),1)) `1 < ((G * ((len G),1)) `1) + (1 - r) ) by XREAL_1:29; 0 <> width G by GOBOARD1:def_3; then A5: 1 <= width G by NAT_1:14; 0 <> len G by GOBOARD1:def_3; then A6: 1 <= len G by NAT_1:14; A7: G * ((len G),(width G)) = |[((G * ((len G),(width G))) `1),((G * ((len G),(width G))) `2)]| by EUCLID:53 .= |[((G * ((len G),1)) `1),((G * ((len G),(width G))) `2)]| by A5, A6, GOBOARD5:2 .= |[((G * ((len G),1)) `1),((G * (1,(width G))) `2)]| by A5, A6, GOBOARD5:1 ; A8: Int (cell (G,(len G),(width G))) = { |[r9,s9]| where r9, s9 is Real : ( (G * ((len G),1)) `1 < r9 & (G * (1,(width G))) `2 < s9 ) } by Th22; p = (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + (r * ((G * ((len G),(width G))) + |[0,1]|)) by A2, EUCLID:32 .= (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + ((r * (G * ((len G),(width G)))) + (r * |[0,1]|)) by EUCLID:32 .= ((r * (G * ((len G),(width G)))) + (((1 - r) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|))) + (r * |[0,1]|) by EUCLID:26 .= (((r * (G * ((len G),(width G)))) + ((1 - r) * (G * ((len G),(width G))))) + ((1 - r) * |[1,1]|)) + (r * |[0,1]|) by EUCLID:26 .= (((r + (1 - r)) * (G * ((len G),(width G)))) + ((1 - r) * |[1,1]|)) + (r * |[0,1]|) by EUCLID:33 .= ((G * ((len G),(width G))) + ((1 - r) * |[1,1]|)) + (r * |[0,1]|) by EUCLID:29 .= ((G * ((len G),(width G))) + |[((1 - r) * 1),((1 - r) * 1)]|) + (r * |[0,1]|) by EUCLID:58 .= ((G * ((len G),(width G))) + |[(1 - r),(1 - r)]|) + |[(r * 0),(r * 1)]| by EUCLID:58 .= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,(width G))) `2) + (1 - r))]| + |[0,r]| by A7, EUCLID:56 .= |[((((G * ((len G),1)) `1) + (1 - r)) + 0),((((G * (1,(width G))) `2) + (1 - r)) + r)]| by EUCLID:56 .= |[(((G * ((len G),1)) `1) + (1 - r)),(((G * (1,(width G))) `2) + 1)]| ; hence p in Int (cell (G,(len G),(width G))) by A4, A8; ::_thesis: verum end; end; end; hence x in (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} by XBOOLE_0:def_3; ::_thesis: verum end; theorem :: GOBOARD6:64 for i, j being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 <= j & j + 1 < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,i,(j + 1))))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} proof let i, j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 <= j & j + 1 < width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,i,(j + 1))))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 <= j & j + 1 < width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,i,(j + 1))))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 <= j and A4: j + 1 < width G ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,i,(j + 1))))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} set p1 = G * (i,j); set p2 = G * (i,(j + 1)); set q2 = G * ((i + 1),(j + 1)); set q3 = G * ((i + 1),(j + 2)); set r = (((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)); A5: j + 1 >= 1 by NAT_1:11; set I1 = Int (cell (G,i,j)); set I2 = Int (cell (G,i,(j + 1))); j <= j + 1 by NAT_1:11; then A6: j < width G by A4, XXREAL_0:2; then A7: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} by A1, A2, A3, Th41; j < j + 1 by XREAL_1:29; then (G * (i,j)) `2 < (G * (i,(j + 1))) `2 by A1, A2, A3, A4, GOBOARD5:4; then A8: ((G * (i,(j + 1))) `2) - ((G * (i,j)) `2) > 0 by XREAL_1:50; A9: (j + 1) + 1 = j + (1 + 1) ; then A10: j + 2 >= 1 by NAT_1:11; A11: j + (1 + 1) <= width G by A4, A9, NAT_1:13; A12: ( i + 1 >= 1 & i + 1 <= len G ) by A2, NAT_1:11, NAT_1:13; then A13: (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),1)) `1 by A4, A5, GOBOARD5:2 .= (G * ((i + 1),(j + 2))) `1 by A11, A10, A12, GOBOARD5:2 ; A14: (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A4, A5, A12, GOBOARD5:1 .= (G * (i,(j + 1))) `2 by A1, A2, A4, A5, GOBOARD5:1 ; j + 1 < j + 2 by XREAL_1:6; then (G * ((i + 1),(j + 1))) `2 < (G * ((i + 1),(j + 2))) `2 by A5, A11, A12, GOBOARD5:4; then A15: ((G * (i,(j + 1))) `2) - ((G * (i,j)) `2) < ((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2) by A14, XREAL_1:9; then A16: ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)) = ((G * (i,(j + 1))) `2) - ((G * (i,j)) `2) by A8, XCMPLX_1:87; (G * (i,j)) `1 = (G * (i,1)) `1 by A1, A2, A3, A6, GOBOARD5:2 .= (G * (i,(j + 1))) `1 by A1, A2, A4, A5, GOBOARD5:2 ; then A17: ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) `1 = ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * (i,(j + 1))) `1) + ((G * ((i + 1),(j + 2))) `1))) by A13, Lm1 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1)) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * (i,(j + 1))) `1) + ((G * ((i + 1),(j + 2))) `1))) by Lm1 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1)) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))) `1)) by Lm1 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1)) + ((((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))) `1) by Lm3 .= (((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `1) + ((((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))) `1) by Lm3 .= (((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) `1 by Lm1 ; ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) `2 = ((G * (i,(j + 1))) `2) + ((((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) + (1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))))) * ((G * ((i + 1),(j + 1))) `2)) by Lm1 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * (i,(j + 1))) `2) + ((G * ((i + 1),(j + 2))) `2))) by A14, A16 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))) `2)) by Lm1 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2)) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * (((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))) `2)) by Lm1 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2)) + ((((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))) `2) by Lm3 .= (((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2) + ((((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))) `2) by Lm3 .= (((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) `2 by Lm1 ; then ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))) = |[(((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) `1),(((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) `2)]| by A17, EUCLID:53 .= (G * (i,(j + 1))) + (G * ((i + 1),(j + 1))) by EUCLID:53 ; then A18: (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) = ((1 / 2) * ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + ((1 / 2) * (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) by EUCLID:32 .= (((1 / 2) * (1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((1 / 2) * (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) by EUCLID:30 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + ((1 / 2) * (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) by EUCLID:30 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (((1 / 2) * ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))) by EUCLID:30 .= ((1 - ((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)))) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (((((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2))) * ((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) by EUCLID:30 ; (((G * (i,(j + 1))) `2) - ((G * (i,j)) `2)) / (((G * ((i + 1),(j + 2))) `2) - ((G * (i,j)) `2)) < 1 by A15, A8, XREAL_1:189; then (1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))) in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) by A15, A8, A18; then A19: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) = (LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))))) \/ (LSeg (((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2))))))) by TOPREAL1:5; A20: ((Int (cell (G,i,j))) \/ (Int (cell (G,i,(j + 1))))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} = (Int (cell (G,i,j))) \/ ((Int (cell (G,i,(j + 1)))) \/ ({((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))})) by XBOOLE_1:4 .= (Int (cell (G,i,j))) \/ (((Int (cell (G,i,(j + 1)))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}) by XBOOLE_1:4 .= ((Int (cell (G,i,j))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}) \/ ((Int (cell (G,i,(j + 1)))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))}) by XBOOLE_1:4 ; LSeg (((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) c= (Int (cell (G,i,(j + 1)))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} by A1, A2, A4, A5, A9, Th43; hence LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 2)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,i,(j + 1))))) \/ {((1 / 2) * ((G * (i,(j + 1))) + (G * ((i + 1),(j + 1)))))} by A19, A7, A20, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:65 for j, i being Element of NAT for G being Go-board st 1 <= j & j < width G & 1 <= i & i + 1 < len G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,(i + 1),j)))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} proof let j, i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G & 1 <= i & i + 1 < len G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,(i + 1),j)))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= j & j < width G & 1 <= i & i + 1 < len G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,(i + 1),j)))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} ) assume that A1: 1 <= j and A2: j < width G and A3: 1 <= i and A4: i + 1 < len G ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,(i + 1),j)))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} set p1 = G * (i,j); set p2 = G * ((i + 1),j); set q2 = G * ((i + 1),(j + 1)); set q3 = G * ((i + 2),(j + 1)); set r = (((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)); A5: i + 1 >= 1 by NAT_1:11; set I1 = Int (cell (G,i,j)); set I2 = Int (cell (G,(i + 1),j)); i <= i + 1 by NAT_1:11; then A6: i < len G by A4, XXREAL_0:2; then A7: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))) c= (Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} by A1, A2, A3, Th42; i < i + 1 by XREAL_1:29; then (G * (i,j)) `1 < (G * ((i + 1),j)) `1 by A1, A2, A3, A4, GOBOARD5:3; then A8: ((G * ((i + 1),j)) `1) - ((G * (i,j)) `1) > 0 by XREAL_1:50; A9: (i + 1) + 1 = i + (1 + 1) ; then A10: i + 2 >= 1 by NAT_1:11; A11: i + (1 + 1) <= len G by A4, A9, NAT_1:13; A12: ( j + 1 >= 1 & j + 1 <= width G ) by A2, NAT_1:11, NAT_1:13; then A13: (G * ((i + 1),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A4, A5, GOBOARD5:1 .= (G * ((i + 2),(j + 1))) `2 by A11, A10, A12, GOBOARD5:1 ; A14: (G * ((i + 1),(j + 1))) `1 = (G * ((i + 1),1)) `1 by A4, A5, A12, GOBOARD5:2 .= (G * ((i + 1),j)) `1 by A1, A2, A4, A5, GOBOARD5:2 ; i + 1 < i + 2 by XREAL_1:6; then (G * ((i + 1),(j + 1))) `1 < (G * ((i + 2),(j + 1))) `1 by A5, A11, A12, GOBOARD5:3; then A15: ((G * ((i + 1),j)) `1) - ((G * (i,j)) `1) < ((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1) by A14, XREAL_1:9; then A16: ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)) = ((G * ((i + 1),j)) `1) - ((G * (i,j)) `1) by A8, XCMPLX_1:87; (G * (i,j)) `2 = (G * (1,j)) `2 by A1, A2, A3, A6, GOBOARD5:1 .= (G * ((i + 1),j)) `2 by A1, A2, A4, A5, GOBOARD5:1 ; then A17: ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) `2 = ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) `2) + ((G * ((i + 1),(j + 1))) `2))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 1),j)) `2) + ((G * ((i + 2),(j + 1))) `2))) by A13, Lm1 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2)) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 1),j)) `2) + ((G * ((i + 2),(j + 1))) `2))) by Lm1 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2)) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))) `2)) by Lm1 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `2)) + ((((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))) `2) by Lm3 .= (((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `2) + ((((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))) `2) by Lm3 .= (((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) `2 by Lm1 ; ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) `1 = ((G * ((i + 1),j)) `1) + ((((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) + (1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))))) * ((G * ((i + 1),(j + 1))) `1)) by Lm1 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 1),j)) `1) + ((G * ((i + 2),(j + 1))) `1))) by A14, A16 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) `1) + ((G * ((i + 1),(j + 1))) `1))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))) `1)) by Lm1 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1)) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * (((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))) `1)) by Lm1 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * (((G * (i,j)) + (G * ((i + 1),(j + 1)))) `1)) + ((((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))) `1) by Lm3 .= (((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) `1) + ((((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))) `1) by Lm3 .= (((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) `1 by Lm1 ; then ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))) = |[(((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) `1),(((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) `2)]| by A17, EUCLID:53 .= (G * ((i + 1),j)) + (G * ((i + 1),(j + 1))) by EUCLID:53 ; then A18: (1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) = ((1 / 2) * ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + ((1 / 2) * (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) by EUCLID:32 .= (((1 / 2) * (1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))))) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))) + ((1 / 2) * (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + ((1 / 2) * (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (((1 / 2) * ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)))) * ((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))))) + (((((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1))) * ((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) by EUCLID:30 ; (((G * ((i + 1),j)) `1) - ((G * (i,j)) `1)) / (((G * ((i + 2),(j + 1))) `1) - ((G * (i,j)) `1)) < 1 by A15, A8, XREAL_1:189; then (1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))) in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) by A15, A8, A18; then A19: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) = (LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))))) \/ (LSeg (((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1))))))) by TOPREAL1:5; A20: ((Int (cell (G,i,j))) \/ (Int (cell (G,(i + 1),j)))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} = (Int (cell (G,i,j))) \/ ((Int (cell (G,(i + 1),j))) \/ ({((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))})) by XBOOLE_1:4 .= (Int (cell (G,i,j))) \/ (((Int (cell (G,(i + 1),j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))}) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))}) by XBOOLE_1:4 .= ((Int (cell (G,i,j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))}) \/ ((Int (cell (G,(i + 1),j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))}) by XBOOLE_1:4 ; LSeg (((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) c= (Int (cell (G,(i + 1),j))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} by A1, A2, A4, A5, A9, Th40; hence LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 2),(j + 1)))))) c= ((Int (cell (G,i,j))) \/ (Int (cell (G,(i + 1),j)))) \/ {((1 / 2) * ((G * ((i + 1),j)) + (G * ((i + 1),(j + 1)))))} by A19, A7, A20, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:66 for i being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 < width G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 < width G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 < width G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} set p1 = G * (i,1); set q2 = G * ((i + 1),1); set q3 = G * ((i + 1),2); set r = 1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1); A4: ( i + 1 >= 1 & i + 1 <= len G ) by A2, NAT_1:11, NAT_1:13; A5: 0 + (1 + 1) <= width G by A3, NAT_1:13; then A6: (G * ((i + 1),1)) `1 = (G * ((i + 1),2)) `1 by A4, GOBOARD5:2; A7: (G * ((i + 1),1)) `2 = (G * (1,(0 + 1))) `2 by A3, A4, GOBOARD5:1 .= (G * (i,1)) `2 by A1, A2, A3, GOBOARD5:1 ; then (G * (i,1)) `2 < (G * ((i + 1),2)) `2 by A5, A4, GOBOARD5:4; then A8: ((G * ((i + 1),2)) `2) - ((G * (i,1)) `2) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1 by XREAL_1:29, XREAL_1:129; then A9: 1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1) < 1 by XREAL_1:212; set I1 = Int (cell (G,i,0)); set I2 = Int (cell (G,i,1)); A10: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) c= (Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} by A1, A2, Th46; A11: ((Int (cell (G,i,0))) \/ (Int (cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} = (Int (cell (G,i,0))) \/ ((Int (cell (G,i,1))) \/ ({((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))})) by XBOOLE_1:4 .= (Int (cell (G,i,0))) \/ (((Int (cell (G,i,1))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))}) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))}) by XBOOLE_1:4 .= ((Int (cell (G,i,0))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))}) \/ ((Int (cell (G,i,1))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))}) by XBOOLE_1:4 ; A12: ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|)) `1 = ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) `1) - (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) `1) by Lm2 .= ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) `1) - (|[((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * 0),((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * 1)]| `1) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) `1) - 0 by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) `1) + (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))) `1) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),1))) `1)) + (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))) `1) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),1))) `1)) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),2))) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),1)) `1))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),2))) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),1)) `1))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),1)) `1))) by A6, Lm3 .= ((1 / 2) * (G * ((i + 1),1))) `1 by Lm3 ; A13: ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * (i,1))))) + ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))) = ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * (i,1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1))))) + (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * (i,1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (G * (i,1))) + ((1 / 2) * (G * ((i + 1),2))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * (G * (i,1))) + ((1 / 2) * (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (G * (i,1))) + ((1 / 2) * (G * ((i + 1),2))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * (G * (i,1))) + ((1 / 2) * (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) by EUCLID:32 ; A14: (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),2)) `2))) - ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),1)) `2)))) + (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) = (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * ((i + 1),1)) `2))) + 1) .= 1 by A7, A8, XCMPLX_1:106 ; ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|)) `2 = ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) `2) - (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) `2) by Lm2 .= ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) `2) - (|[((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * 0),((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * 1)]| `2) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) `2) - (1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) by EUCLID:52 .= ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) `2) + (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))) `2)) - (1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),1))) `2)) + (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))) `2)) - (1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),1))) `2)) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),2))) `2))) - (1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),1)) `2))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),2))) `2))) - (1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),1)) `2))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),2)) `2)))) - (1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) by Lm3 .= ((1 / 2) * (G * ((i + 1),1))) `2 by A14, Lm3 ; then A15: (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) = |[(((1 / 2) * (G * ((i + 1),1))) `1),(((1 / 2) * (G * ((i + 1),1))) `2)]| by A12, EUCLID:53 .= (1 / 2) * (G * ((i + 1),1)) by EUCLID:53 ; (1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))) = ((1 / 2) * (G * (i,1))) + ((1 / 2) * (G * ((i + 1),1))) by EUCLID:32 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) + (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 / 2) * (G * ((i + 1),1))) by EUCLID:29 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * (i,1))))) + ((1 / 2) * (G * ((i + 1),1))) by EUCLID:33 .= ((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * (i,1))))) + (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2)))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) by A15, EUCLID:45 .= (((((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * (i,1)))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * (i,1))))) + ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),1))))) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),2))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) by EUCLID:26 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) + (((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|)) by A13, EUCLID:45 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) + (- (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) - ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))))) by EUCLID:44 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|) - ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2)))))) by EUCLID:41 .= (((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * |[0,1]|)) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) by EUCLID:47 .= ((1 - (1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1))) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|)) + ((1 / (((1 / 2) * (((G * ((i + 1),2)) `2) - ((G * (i,1)) `2))) + 1)) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) by EUCLID:49 ; then (1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))) in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) by A8, A9; then A16: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) = (LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) \/ (LSeg (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2)))))) by TOPREAL1:5; (0 + 1) + 1 = 0 + (1 + 1) ; then LSeg (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) c= (Int (cell (G,i,1))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} by A1, A2, A3, Th43; hence LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((1 / 2) * ((G * (i,1)) + (G * ((i + 1),2))))) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,i,1)))) \/ {((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))} by A16, A10, A11, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:67 for i being Element of NAT for G being Go-board st 1 <= i & i < len G & 1 < width G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) c= ((Int (cell (G,i,((width G) -' 1)))) \/ (Int (cell (G,i,(width G))))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 <= i & i < len G & 1 < width G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) c= ((Int (cell (G,i,((width G) -' 1)))) \/ (Int (cell (G,i,(width G))))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} let G be Go-board; ::_thesis: ( 1 <= i & i < len G & 1 < width G implies LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) c= ((Int (cell (G,i,((width G) -' 1)))) \/ (Int (cell (G,i,(width G))))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} ) assume that A1: 1 <= i and A2: i < len G and A3: 1 < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) c= ((Int (cell (G,i,((width G) -' 1)))) \/ (Int (cell (G,i,(width G))))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} set I1 = Int (cell (G,i,((width G) -' 1))); set I2 = Int (cell (G,i,(width G))); set p1 = G * (i,(width G)); set q2 = G * ((i + 1),(width G)); set q3 = G * ((i + 1),((width G) -' 1)); set r = 1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1); A4: ((width G) -' 1) + 1 = width G by A3, XREAL_1:235; then A5: 1 <= (width G) -' 1 by A3, NAT_1:13; A6: (width G) -' 1 < width G by A4, NAT_1:13; then (G * (i,(width G))) + (G * ((i + 1),((width G) -' 1))) = (G * (i,((width G) -' 1))) + (G * ((i + 1),(width G))) by A1, A2, A4, A5, Th11; then A7: LSeg (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) c= (Int (cell (G,i,((width G) -' 1)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} by A1, A2, A4, A5, A6, Th41; A8: ( i + 1 >= 1 & i + 1 <= len G ) by A2, NAT_1:11, NAT_1:13; then A9: (G * ((i + 1),(width G))) `1 = (G * ((i + 1),1)) `1 by A3, GOBOARD5:2 .= (G * ((i + 1),((width G) -' 1))) `1 by A5, A6, A8, GOBOARD5:2 ; A10: (G * ((i + 1),(width G))) `2 = (G * (1,(width G))) `2 by A3, A8, GOBOARD5:1 .= (G * (i,(width G))) `2 by A1, A2, A3, GOBOARD5:1 ; then (G * ((i + 1),((width G) -' 1))) `2 < (G * (i,(width G))) `2 by A5, A6, A8, GOBOARD5:4; then A11: ((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1 by XREAL_1:29, XREAL_1:129; then A12: 1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1) < 1 by XREAL_1:212; A13: ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|)) `1 = ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) `1) + (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) `1) + (|[((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * 0),((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * 1)]| `1) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) `1) + 0 by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) `1) + (((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))) `1) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),(width G)))) `1)) + (((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))) `1) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),(width G)))) `1)) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),((width G) -' 1)))) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),(width G))) `1))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),((width G) -' 1)))) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),(width G))) `1))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),(width G))) `1))) by A9, Lm3 .= ((1 / 2) * (G * ((i + 1),(width G)))) `1 by Lm3 ; A14: ((Int (cell (G,i,((width G) -' 1)))) \/ (Int (cell (G,i,(width G))))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} = (Int (cell (G,i,((width G) -' 1)))) \/ ((Int (cell (G,i,(width G)))) \/ ({((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))})) by XBOOLE_1:4 .= (Int (cell (G,i,((width G) -' 1)))) \/ (((Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))}) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))}) by XBOOLE_1:4 .= ((Int (cell (G,i,((width G) -' 1)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))}) \/ ((Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))}) by XBOOLE_1:4 ; A15: ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * (i,(width G)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))) = ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * (i,(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G)))))) + (((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * (i,(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (G * (i,(width G)))) + ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * (G * (i,(width G)))) + ((1 / 2) * (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (G * (i,(width G)))) + ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * (G * (i,(width G)))) + ((1 / 2) * (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) by EUCLID:32 ; A16: (((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),(width G))) `2))) - ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),((width G) -' 1))) `2)))) + (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) = (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (((G * ((i + 1),(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1) .= 1 by A10, A11, XCMPLX_1:106 ; ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|)) `2 = ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) `2) + (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) `2) + (|[((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * 0),((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * 1)]| `2) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) `2) + (1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) by EUCLID:52 .= ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) `2) + (((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))) `2)) + (1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),(width G)))) `2)) + (((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))) `2)) + (1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * (G * ((i + 1),(width G)))) `2)) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),((width G) -' 1)))) `2))) + (1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),(width G))) `2))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (G * ((i + 1),((width G) -' 1)))) `2))) + (1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * ((G * ((i + 1),(width G))) `2))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * ((i + 1),((width G) -' 1))) `2)))) + (1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) by Lm3 .= ((1 / 2) * (G * ((i + 1),(width G)))) `2 by A16, Lm3 ; then A17: (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|) = |[(((1 / 2) * (G * ((i + 1),(width G)))) `1),(((1 / 2) * (G * ((i + 1),(width G)))) `2)]| by A13, EUCLID:53 .= (1 / 2) * (G * ((i + 1),(width G))) by EUCLID:53 ; (1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))) = ((1 / 2) * (G * (i,(width G)))) + ((1 / 2) * (G * ((i + 1),(width G)))) by EUCLID:32 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) + (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 / 2) * (G * ((i + 1),(width G)))) by EUCLID:29 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * (i,(width G)))))) + ((1 / 2) * (G * ((i + 1),(width G)))) by EUCLID:33 .= ((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * (i,(width G)))))) + (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1))))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|) by A17, EUCLID:26 .= (((((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * (i,(width G))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * (i,(width G)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * (G * ((i + 1),(width G)))))) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (G * ((i + 1),((width G) -' 1)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * |[0,1]|)) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) by A15, EUCLID:26 .= ((1 - (1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1))) * (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) + ((1 / (((1 / 2) * (((G * (i,(width G))) `2) - ((G * ((i + 1),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) by EUCLID:32 ; then (1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))) in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) by A11, A12; then A18: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) = (LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))))) \/ (LSeg (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1))))))) by TOPREAL1:5; LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) c= (Int (cell (G,i,(width G)))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} by A1, A2, Th47; hence LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),((width G) -' 1)))))) c= ((Int (cell (G,i,((width G) -' 1)))) \/ (Int (cell (G,i,(width G))))) \/ {((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))} by A18, A7, A14, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:68 for j being Element of NAT for G being Go-board st 1 <= j & j < width G & 1 < len G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G & 1 < len G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= j & j < width G & 1 < len G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} ) assume that A1: 1 <= j and A2: j < width G and A3: 1 < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} set p1 = G * (1,j); set q2 = G * (1,(j + 1)); set q3 = G * (2,(j + 1)); set r = 1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1); A4: ( j + 1 >= 1 & j + 1 <= width G ) by A2, NAT_1:11, NAT_1:13; A5: 0 + (1 + 1) <= len G by A3, NAT_1:13; then A6: (G * (1,(j + 1))) `2 = (G * (2,(j + 1))) `2 by A4, GOBOARD5:1; A7: (G * (1,(j + 1))) `1 = (G * (1,1)) `1 by A3, A4, GOBOARD5:2 .= (G * (1,j)) `1 by A1, A2, A3, GOBOARD5:2 ; then (G * (1,j)) `1 < (G * (2,(j + 1))) `1 by A5, A4, GOBOARD5:3; then A8: ((G * (2,(j + 1))) `1) - ((G * (1,j)) `1) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1 by XREAL_1:29, XREAL_1:129; then A9: 1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1) < 1 by XREAL_1:212; set I1 = Int (cell (G,0,j)); set I2 = Int (cell (G,1,j)); A10: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) c= (Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by A1, A2, Th44; A11: ((Int (cell (G,0,j))) \/ (Int (cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} = (Int (cell (G,0,j))) \/ ((Int (cell (G,1,j))) \/ ({((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))})) by XBOOLE_1:4 .= (Int (cell (G,0,j))) \/ (((Int (cell (G,1,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}) by XBOOLE_1:4 .= ((Int (cell (G,0,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}) \/ ((Int (cell (G,1,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))}) by XBOOLE_1:4 ; A12: ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|)) `2 = ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) `2) - (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) `2) by Lm2 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) `2) - (|[((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * 1),((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * 0)]| `2) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) `2) - 0 by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) `2) + (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))) `2) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * (G * (1,(j + 1)))) `2)) + (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))) `2) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * (G * (1,(j + 1)))) `2)) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (G * (2,(j + 1)))) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,(j + 1))) `2))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (G * (2,(j + 1)))) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,(j + 1))) `2))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,(j + 1))) `2))) by A6, Lm3 .= ((1 / 2) * (G * (1,(j + 1)))) `2 by Lm3 ; A13: ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (1,j))))) + ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))) = ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (1,j))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1)))))) + (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (1,j)))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (G * (1,j))) + ((1 / 2) * (G * (2,(j + 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * (G * (1,j))) + ((1 / 2) * (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (G * (1,j))) + ((1 / 2) * (G * (2,(j + 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * (G * (1,j))) + ((1 / 2) * (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) by EUCLID:32 ; A14: (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (2,(j + 1))) `1))) - ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,(j + 1))) `1)))) + (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) = (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,(j + 1))) `1))) + 1) .= 1 by A7, A8, XCMPLX_1:106 ; ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|)) `1 = ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) `1) - (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) `1) by Lm2 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) `1) - (|[((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * 1),((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * 0)]| `1) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) `1) - (1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) by EUCLID:52 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) `1) + (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * (G * (1,(j + 1)))) `1)) + (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * (G * (1,(j + 1)))) `1)) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (G * (2,(j + 1)))) `1))) - (1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,(j + 1))) `1))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * (((1 / 2) * (G * (2,(j + 1)))) `1))) - (1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,(j + 1))) `1))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (2,(j + 1))) `1)))) - (1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) by Lm3 .= ((1 / 2) * (G * (1,(j + 1)))) `1 by A14, Lm3 ; then A15: (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) = |[(((1 / 2) * (G * (1,(j + 1)))) `1),(((1 / 2) * (G * (1,(j + 1)))) `2)]| by A12, EUCLID:53 .= (1 / 2) * (G * (1,(j + 1))) by EUCLID:53 ; (1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))) = ((1 / 2) * (G * (1,j))) + ((1 / 2) * (G * (1,(j + 1)))) by EUCLID:32 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) + (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 / 2) * (G * (1,(j + 1)))) by EUCLID:29 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (1,j))))) + ((1 / 2) * (G * (1,(j + 1)))) by EUCLID:33 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (1,j))))) + (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1))))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) by A15, EUCLID:45 .= (((((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,j)))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (1,j))))) + ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * (G * (1,(j + 1)))))) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * (G * (2,(j + 1)))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) by EUCLID:26 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) + (((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|)) by A13, EUCLID:45 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) + (- (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) - ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))))) by EUCLID:44 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|) - ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1))))))) by EUCLID:41 .= (((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * |[1,0]|)) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) by EUCLID:47 .= ((1 - (1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1))) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|)) + ((1 / (((1 / 2) * (((G * (2,(j + 1))) `1) - ((G * (1,j)) `1))) + 1)) * ((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) by EUCLID:49 ; then (1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))) in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) by A8, A9; then A16: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) = (LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) \/ (LSeg (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1))))))) by TOPREAL1:5; (0 + 1) + 1 = 0 + (1 + 1) ; then LSeg (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) c= (Int (cell (G,1,j))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by A1, A2, A3, Th40; hence LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((1 / 2) * ((G * (1,j)) + (G * (2,(j + 1)))))) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,1,j)))) \/ {((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))} by A16, A10, A11, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:69 for j being Element of NAT for G being Go-board st 1 <= j & j < width G & 1 < len G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) c= ((Int (cell (G,((len G) -' 1),j))) \/ (Int (cell (G,(len G),j)))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 <= j & j < width G & 1 < len G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) c= ((Int (cell (G,((len G) -' 1),j))) \/ (Int (cell (G,(len G),j)))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} let G be Go-board; ::_thesis: ( 1 <= j & j < width G & 1 < len G implies LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) c= ((Int (cell (G,((len G) -' 1),j))) \/ (Int (cell (G,(len G),j)))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} ) assume that A1: 1 <= j and A2: j < width G and A3: 1 < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) c= ((Int (cell (G,((len G) -' 1),j))) \/ (Int (cell (G,(len G),j)))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} set I1 = Int (cell (G,((len G) -' 1),j)); set I2 = Int (cell (G,(len G),j)); set p1 = G * ((len G),j); set q2 = G * ((len G),(j + 1)); set q3 = G * (((len G) -' 1),(j + 1)); set r = 1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1); A4: ((len G) -' 1) + 1 = len G by A3, XREAL_1:235; then A5: 1 <= (len G) -' 1 by A3, NAT_1:13; A6: (len G) -' 1 < len G by A4, NAT_1:13; then (G * (((len G) -' 1),j)) + (G * ((len G),(j + 1))) = (G * ((len G),j)) + (G * (((len G) -' 1),(j + 1))) by A1, A2, A4, A5, Th11; then A7: LSeg (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) c= (Int (cell (G,((len G) -' 1),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} by A1, A2, A4, A5, A6, Th42; A8: ( j + 1 >= 1 & j + 1 <= width G ) by A2, NAT_1:11, NAT_1:13; then A9: (G * ((len G),(j + 1))) `2 = (G * (1,(j + 1))) `2 by A3, GOBOARD5:1 .= (G * (((len G) -' 1),(j + 1))) `2 by A5, A6, A8, GOBOARD5:1 ; A10: (G * ((len G),(j + 1))) `1 = (G * ((len G),1)) `1 by A3, A8, GOBOARD5:2 .= (G * ((len G),j)) `1 by A1, A2, A3, GOBOARD5:2 ; then (G * (((len G) -' 1),(j + 1))) `1 < (G * ((len G),j)) `1 by A5, A6, A8, GOBOARD5:3; then A11: ((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1 by XREAL_1:29, XREAL_1:129; then A12: 1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1) < 1 by XREAL_1:212; A13: ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|)) `2 = ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) `2) + (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) `2) + (|[((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * 1),((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * 0)]| `2) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) `2) + 0 by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) `2) + (((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))) `2) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * (G * ((len G),(j + 1)))) `2)) + (((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))) `2) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * (G * ((len G),(j + 1)))) `2)) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (G * (((len G) -' 1),(j + 1)))) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * ((G * ((len G),(j + 1))) `2))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (G * (((len G) -' 1),(j + 1)))) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * ((G * ((len G),(j + 1))) `2))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),(j + 1))) `2))) by A9, Lm3 .= ((1 / 2) * (G * ((len G),(j + 1)))) `2 by Lm3 ; A14: ((Int (cell (G,((len G) -' 1),j))) \/ (Int (cell (G,(len G),j)))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} = (Int (cell (G,((len G) -' 1),j))) \/ ((Int (cell (G,(len G),j))) \/ ({((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))})) by XBOOLE_1:4 .= (Int (cell (G,((len G) -' 1),j))) \/ (((Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))}) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))}) by XBOOLE_1:4 .= ((Int (cell (G,((len G) -' 1),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))}) \/ ((Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))}) by XBOOLE_1:4 ; A15: ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * ((len G),j))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))) = ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * ((len G),j))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1)))))) + (((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * ((len G),j)))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (G * ((len G),j))) + ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * (G * ((len G),j))) + ((1 / 2) * (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (G * ((len G),j))) + ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * (G * ((len G),j))) + ((1 / 2) * (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) by EUCLID:32 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) by EUCLID:32 ; A16: (((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),(j + 1))) `1))) - ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * (((len G) -' 1),(j + 1))) `1)))) + (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) = (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (((G * ((len G),(j + 1))) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1) .= 1 by A10, A11, XCMPLX_1:106 ; ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|)) `1 = ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) `1) + (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) `1) + (|[((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * 1),((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * 0)]| `1) by EUCLID:58 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) `1) + (1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) by EUCLID:52 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) `1) + (((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * (G * ((len G),(j + 1)))) `1)) + (((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * (G * ((len G),(j + 1)))) `1)) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (G * (((len G) -' 1),(j + 1)))) `1))) + (1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * ((G * ((len G),(j + 1))) `1))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * (((1 / 2) * (G * (((len G) -' 1),(j + 1)))) `1))) + (1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * ((G * ((len G),(j + 1))) `1))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * (((len G) -' 1),(j + 1))) `1)))) + (1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) by Lm3 .= ((1 / 2) * (G * ((len G),(j + 1)))) `1 by A16, Lm3 ; then A17: (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|) = |[(((1 / 2) * (G * ((len G),(j + 1)))) `1),(((1 / 2) * (G * ((len G),(j + 1)))) `2)]| by A13, EUCLID:53 .= (1 / 2) * (G * ((len G),(j + 1))) by EUCLID:53 ; (1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))) = ((1 / 2) * (G * ((len G),j))) + ((1 / 2) * (G * ((len G),(j + 1)))) by EUCLID:32 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) + (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 / 2) * (G * ((len G),(j + 1)))) by EUCLID:29 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * ((len G),j))))) + ((1 / 2) * (G * ((len G),(j + 1)))) by EUCLID:33 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * ((len G),j))))) + (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1))))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|) by A17, EUCLID:26 .= (((((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),j)))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * ((len G),j))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * (G * ((len G),(j + 1)))))) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * (G * (((len G) -' 1),(j + 1)))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|) by EUCLID:26 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * |[1,0]|)) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) by A15, EUCLID:26 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1))) * (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) + ((1 / (((1 / 2) * (((G * ((len G),j)) `1) - ((G * (((len G) -' 1),(j + 1))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) by EUCLID:32 ; then (1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))) in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) by A11, A12; then A18: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) = (LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))))) \/ (LSeg (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1))))))) by TOPREAL1:5; LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) c= (Int (cell (G,(len G),j))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} by A1, A2, Th45; hence LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((1 / 2) * ((G * ((len G),j)) + (G * (((len G) -' 1),(j + 1)))))) c= ((Int (cell (G,((len G) -' 1),j))) \/ (Int (cell (G,(len G),j)))) \/ {((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))} by A18, A7, A14, XBOOLE_1:13; ::_thesis: verum end; Lm7: (1 / 2) + (1 / 2) = 1 ; theorem :: GOBOARD6:70 for j being Element of NAT for G being Go-board st 1 < len G & 1 <= j & j + 1 < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,0,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1,0]|)} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 < len G & 1 <= j & j + 1 < width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,0,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1,0]|)} let G be Go-board; ::_thesis: ( 1 < len G & 1 <= j & j + 1 < width G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,0,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1,0]|)} ) assume that A1: 1 < len G and A2: 1 <= j and A3: j + 1 < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,0,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1,0]|)} set p1 = G * (1,j); set p2 = G * (1,(j + 1)); set q3 = G * (1,(j + 2)); set r = (((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)); A4: (j + 1) + 1 = j + (1 + 1) ; then A5: j + 2 >= 1 by NAT_1:11; A6: j + (1 + 1) <= width G by A3, A4, NAT_1:13; set I1 = Int (cell (G,0,j)); set I2 = Int (cell (G,0,(j + 1))); A7: ((Int (cell (G,0,j))) \/ (Int (cell (G,0,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1,0]|)} = (Int (cell (G,0,j))) \/ ((Int (cell (G,0,(j + 1)))) \/ ({((G * (1,(j + 1))) - |[1,0]|)} \/ {((G * (1,(j + 1))) - |[1,0]|)})) by XBOOLE_1:4 .= (Int (cell (G,0,j))) \/ (((Int (cell (G,0,(j + 1)))) \/ {((G * (1,(j + 1))) - |[1,0]|)}) \/ {((G * (1,(j + 1))) - |[1,0]|)}) by XBOOLE_1:4 .= ((Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)}) \/ ((Int (cell (G,0,(j + 1)))) \/ {((G * (1,(j + 1))) - |[1,0]|)}) by XBOOLE_1:4 ; A8: LSeg ((((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) c= (Int (cell (G,0,(j + 1)))) \/ {((G * (1,(j + 1))) - |[1,0]|)} by A3, A4, Th48, NAT_1:11; j < j + 1 by XREAL_1:29; then (G * (1,j)) `2 < (G * (1,(j + 1))) `2 by A1, A2, A3, GOBOARD5:4; then A9: ((G * (1,(j + 1))) `2) - ((G * (1,j)) `2) > 0 by XREAL_1:50; A10: j + 1 >= 1 by NAT_1:11; then A11: (G * (1,(j + 1))) `1 = (G * (1,1)) `1 by A1, A3, GOBOARD5:2 .= (G * (1,(j + 2))) `1 by A1, A6, A5, GOBOARD5:2 ; j <= j + 1 by NAT_1:11; then A12: j < width G by A3, XXREAL_0:2; then (G * (1,j)) `1 = (G * (1,1)) `1 by A1, A2, GOBOARD5:2 .= (G * (1,(j + 1))) `1 by A1, A3, A10, GOBOARD5:2 ; then A13: 1 * ((G * (1,(j + 1))) `1) = ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((G * (1,j)) `1)) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((G * (1,(j + 2))) `1)) by A11 .= (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) `1) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((G * (1,(j + 2))) `1)) by Lm3 .= (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) `1) + ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))) `1) by Lm3 .= (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))) `1 by Lm1 ; j + 1 < j + 2 by XREAL_1:6; then (G * (1,(j + 1))) `2 < (G * (1,(j + 2))) `2 by A1, A10, A6, GOBOARD5:4; then A14: ((G * (1,(j + 1))) `2) - ((G * (1,j)) `2) < ((G * (1,(j + 2))) `2) - ((G * (1,j)) `2) by XREAL_1:9; then ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)) = ((G * (1,(j + 1))) `2) - ((G * (1,j)) `2) by A9, XCMPLX_1:87; then (G * (1,(j + 1))) `2 = ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((G * (1,j)) `2)) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((G * (1,(j + 2))) `2)) ; then 1 * ((G * (1,(j + 1))) `2) = (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) `2) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((G * (1,(j + 2))) `2)) by Lm3 .= (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) `2) + ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))) `2) by Lm3 .= (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))) `2 by Lm1 ; then A15: ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))) = |[((G * (1,(j + 1))) `1),((G * (1,(j + 1))) `2)]| by A13, EUCLID:53 .= G * (1,(j + 1)) by EUCLID:53 ; G * (1,(j + 1)) = 1 * (G * (1,(j + 1))) by EUCLID:29 .= ((1 / 2) * (G * (1,(j + 1)))) + ((1 / 2) * (G * (1,(j + 1)))) by Lm7, EUCLID:33 .= ((1 / 2) * (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) + ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + ((1 / 2) * (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by A15, EUCLID:29 .= ((1 / 2) * (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1)))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + ((1 / 2) * (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by EUCLID:33 .= (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + ((1 / 2) * (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j)))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by EUCLID:32 .= ((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + (((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1))))) + (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j)))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))))) by EUCLID:26 .= ((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j)))) + (((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))))) by EUCLID:26 .= (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + ((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))))) + (((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by EUCLID:26 .= ((((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1))))) + ((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))) by EUCLID:26 .= (((1 / 2) * (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,(j + 1)))) + ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (G * (1,j))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((G * (1,j)) + (G * (1,(j + 1)))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))) by EUCLID:32 .= ((((1 / 2) * (1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))))) * ((G * (1,j)) + (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2))))) by EUCLID:30 .= (((1 / 2) * (1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))))) * ((G * (1,j)) + (G * (1,(j + 1))))) + (((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by EUCLID:26 .= (((1 / 2) * (1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))))) * ((G * (1,j)) + (G * (1,(j + 1))))) + ((1 / 2) * ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 1)))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (G * (1,(j + 2)))))) by EUCLID:32 .= (((1 / 2) * (1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))))) * ((G * (1,j)) + (G * (1,(j + 1))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) by EUCLID:32 ; then A16: G * (1,(j + 1)) = ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) + ((1 / 2) * (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) by EUCLID:30 .= ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) + (((1 / 2) * ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) by EUCLID:30 .= ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) by EUCLID:30 ; A17: ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * (((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|)) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) = (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * |[1,0]|)) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * (((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) by EUCLID:49 .= (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * |[1,0]|)) + ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) - (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * |[1,0]|)) by EUCLID:49 .= ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) + (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1)))))) - ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * |[1,0]|))) - (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * |[1,0]|) by EUCLID:45 .= (((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) + ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) - ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * |[1,0]|)) - (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * |[1,0]|) by EUCLID:45 .= ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) + ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) - (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * |[1,0]|) + (((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * |[1,0]|)) by EUCLID:46 .= ((((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2))) * ((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2)))))) + ((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))))) - (((1 - ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) + ((((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)))) * |[1,0]|) by EUCLID:33 .= (G * (1,(j + 1))) - |[1,0]| by A16, EUCLID:29 ; (((G * (1,(j + 1))) `2) - ((G * (1,j)) `2)) / (((G * (1,(j + 2))) `2) - ((G * (1,j)) `2)) < 1 by A14, A9, XREAL_1:189; then (G * (1,(j + 1))) - |[1,0]| in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) by A14, A9, A17; then A18: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) = (LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|))) \/ (LSeg (((G * (1,(j + 1))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|))) by TOPREAL1:5; LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),((G * (1,(j + 1))) - |[1,0]|)) c= (Int (cell (G,0,j))) \/ {((G * (1,(j + 1))) - |[1,0]|)} by A2, A12, Th49; hence LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),(((1 / 2) * ((G * (1,(j + 1))) + (G * (1,(j + 2))))) - |[1,0]|)) c= ((Int (cell (G,0,j))) \/ (Int (cell (G,0,(j + 1))))) \/ {((G * (1,(j + 1))) - |[1,0]|)} by A18, A8, A7, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:71 for j being Element of NAT for G being Go-board st 1 < len G & 1 <= j & j + 1 < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) c= ((Int (cell (G,(len G),j))) \/ (Int (cell (G,(len G),(j + 1))))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} proof let j be Element of NAT ; ::_thesis: for G being Go-board st 1 < len G & 1 <= j & j + 1 < width G holds LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) c= ((Int (cell (G,(len G),j))) \/ (Int (cell (G,(len G),(j + 1))))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} let G be Go-board; ::_thesis: ( 1 < len G & 1 <= j & j + 1 < width G implies LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) c= ((Int (cell (G,(len G),j))) \/ (Int (cell (G,(len G),(j + 1))))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} ) assume that A1: 1 < len G and A2: 1 <= j and A3: j + 1 < width G ; ::_thesis: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) c= ((Int (cell (G,(len G),j))) \/ (Int (cell (G,(len G),(j + 1))))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} set p1 = G * ((len G),j); set p2 = G * ((len G),(j + 1)); set q3 = G * ((len G),(j + 2)); set r = (((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)); A4: (j + 1) + 1 = j + (1 + 1) ; then A5: j + 2 >= 1 by NAT_1:11; A6: j + (1 + 1) <= width G by A3, A4, NAT_1:13; set I1 = Int (cell (G,(len G),j)); set I2 = Int (cell (G,(len G),(j + 1))); A7: ((Int (cell (G,(len G),j))) \/ (Int (cell (G,(len G),(j + 1))))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} = (Int (cell (G,(len G),j))) \/ ((Int (cell (G,(len G),(j + 1)))) \/ ({((G * ((len G),(j + 1))) + |[1,0]|)} \/ {((G * ((len G),(j + 1))) + |[1,0]|)})) by XBOOLE_1:4 .= (Int (cell (G,(len G),j))) \/ (((Int (cell (G,(len G),(j + 1)))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)}) \/ {((G * ((len G),(j + 1))) + |[1,0]|)}) by XBOOLE_1:4 .= ((Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)}) \/ ((Int (cell (G,(len G),(j + 1)))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)}) by XBOOLE_1:4 ; A8: LSeg ((((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) c= (Int (cell (G,(len G),(j + 1)))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} by A3, A4, Th50, NAT_1:11; j < j + 1 by XREAL_1:29; then (G * ((len G),j)) `2 < (G * ((len G),(j + 1))) `2 by A1, A2, A3, GOBOARD5:4; then A9: ((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2) > 0 by XREAL_1:50; A10: j + 1 >= 1 by NAT_1:11; then A11: (G * ((len G),(j + 1))) `1 = (G * ((len G),1)) `1 by A1, A3, GOBOARD5:2 .= (G * ((len G),(j + 2))) `1 by A1, A6, A5, GOBOARD5:2 ; j <= j + 1 by NAT_1:11; then A12: j < width G by A3, XXREAL_0:2; then (G * ((len G),j)) `1 = (G * ((len G),1)) `1 by A1, A2, GOBOARD5:2 .= (G * ((len G),(j + 1))) `1 by A1, A3, A10, GOBOARD5:2 ; then A13: 1 * ((G * ((len G),(j + 1))) `1) = ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((G * ((len G),j)) `1)) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((G * ((len G),(j + 2))) `1)) by A11 .= (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) `1) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((G * ((len G),(j + 2))) `1)) by Lm3 .= (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) `1) + ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))) `1) by Lm3 .= (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))) `1 by Lm1 ; j + 1 < j + 2 by XREAL_1:6; then (G * ((len G),(j + 1))) `2 < (G * ((len G),(j + 2))) `2 by A1, A10, A6, GOBOARD5:4; then A14: ((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2) < ((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2) by XREAL_1:9; then ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)) = ((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2) by A9, XCMPLX_1:87; then (G * ((len G),(j + 1))) `2 = ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((G * ((len G),j)) `2)) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((G * ((len G),(j + 2))) `2)) ; then 1 * ((G * ((len G),(j + 1))) `2) = (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) `2) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((G * ((len G),(j + 2))) `2)) by Lm3 .= (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) `2) + ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))) `2) by Lm3 .= (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))) `2 by Lm1 ; then A15: ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))) = |[((G * ((len G),(j + 1))) `1),((G * ((len G),(j + 1))) `2)]| by A13, EUCLID:53 .= G * ((len G),(j + 1)) by EUCLID:53 ; G * ((len G),(j + 1)) = 1 * (G * ((len G),(j + 1))) by EUCLID:29 .= ((1 / 2) * (G * ((len G),(j + 1)))) + ((1 / 2) * (G * ((len G),(j + 1)))) by Lm7, EUCLID:33 .= ((1 / 2) * (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) + ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + ((1 / 2) * (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by A15, EUCLID:29 .= ((1 / 2) * (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1)))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + ((1 / 2) * (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by EUCLID:33 .= (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + ((1 / 2) * (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j)))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by EUCLID:32 .= ((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + (((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1))))) + (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j)))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))))) by EUCLID:26 .= ((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j)))) + (((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))))) by EUCLID:26 .= (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + ((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))))) + (((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by EUCLID:26 .= ((((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1))))) + ((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))) by EUCLID:26 .= (((1 / 2) * (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),(j + 1)))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (G * ((len G),j))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))) by EUCLID:32 .= ((((1 / 2) * (1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))))) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2))))) by EUCLID:30 ; then A16: G * ((len G),(j + 1)) = (((1 / 2) * (1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))))) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + (((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by EUCLID:26 .= (((1 / 2) * (1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))))) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + ((1 / 2) * ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 1)))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (G * ((len G),(j + 2)))))) by EUCLID:32 .= (((1 / 2) * (1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))))) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) by EUCLID:32 .= ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 / 2) * (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) by EUCLID:30 .= ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + (((1 / 2) * ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) by EUCLID:30 .= ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) by EUCLID:30 ; A17: ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * (((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) = (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * |[1,0]|)) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * (((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) by EUCLID:32 .= (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * |[1,0]|)) + ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * |[1,0]|)) by EUCLID:32 .= ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) + (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1)))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * |[1,0]|))) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * |[1,0]|) by EUCLID:26 .= (((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * |[1,0]|)) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * |[1,0]|) by EUCLID:26 .= ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))))) + (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * |[1,0]|) + (((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * |[1,0]|)) by EUCLID:26 .= ((((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2))) * ((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2)))))) + ((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))))) + (((1 - ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) + ((((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)))) * |[1,0]|) by EUCLID:33 .= (G * ((len G),(j + 1))) + |[1,0]| by A16, EUCLID:29 ; (((G * ((len G),(j + 1))) `2) - ((G * ((len G),j)) `2)) / (((G * ((len G),(j + 2))) `2) - ((G * ((len G),j)) `2)) < 1 by A14, A9, XREAL_1:189; then (G * ((len G),(j + 1))) + |[1,0]| in LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) by A14, A9, A17; then A18: LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) = (LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|))) \/ (LSeg (((G * ((len G),(j + 1))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|))) by TOPREAL1:5; LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),((G * ((len G),(j + 1))) + |[1,0]|)) c= (Int (cell (G,(len G),j))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} by A2, A12, Th51; hence LSeg ((((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(j + 1))) + (G * ((len G),(j + 2))))) + |[1,0]|)) c= ((Int (cell (G,(len G),j))) \/ (Int (cell (G,(len G),(j + 1))))) \/ {((G * ((len G),(j + 1))) + |[1,0]|)} by A18, A8, A7, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:72 for i being Element of NAT for G being Go-board st 1 < width G & 1 <= i & i + 1 < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,(i + 1),0)))) \/ {((G * ((i + 1),1)) - |[0,1]|)} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 < width G & 1 <= i & i + 1 < len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,(i + 1),0)))) \/ {((G * ((i + 1),1)) - |[0,1]|)} let G be Go-board; ::_thesis: ( 1 < width G & 1 <= i & i + 1 < len G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,(i + 1),0)))) \/ {((G * ((i + 1),1)) - |[0,1]|)} ) assume that A1: 1 < width G and A2: 1 <= i and A3: i + 1 < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,(i + 1),0)))) \/ {((G * ((i + 1),1)) - |[0,1]|)} set p1 = G * (i,1); set p2 = G * ((i + 1),1); set q3 = G * ((i + 2),1); set r = (((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)); A4: (i + 1) + 1 = i + (1 + 1) ; then A5: i + 2 >= 1 by NAT_1:11; A6: i + (1 + 1) <= len G by A3, A4, NAT_1:13; set I1 = Int (cell (G,i,0)); set I2 = Int (cell (G,(i + 1),0)); A7: ((Int (cell (G,i,0))) \/ (Int (cell (G,(i + 1),0)))) \/ {((G * ((i + 1),1)) - |[0,1]|)} = (Int (cell (G,i,0))) \/ ((Int (cell (G,(i + 1),0))) \/ ({((G * ((i + 1),1)) - |[0,1]|)} \/ {((G * ((i + 1),1)) - |[0,1]|)})) by XBOOLE_1:4 .= (Int (cell (G,i,0))) \/ (((Int (cell (G,(i + 1),0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}) \/ {((G * ((i + 1),1)) - |[0,1]|)}) by XBOOLE_1:4 .= ((Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}) \/ ((Int (cell (G,(i + 1),0))) \/ {((G * ((i + 1),1)) - |[0,1]|)}) by XBOOLE_1:4 ; A8: LSeg ((((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,(i + 1),0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} by A3, A4, Th52, NAT_1:11; i < i + 1 by XREAL_1:29; then (G * (i,1)) `1 < (G * ((i + 1),1)) `1 by A1, A2, A3, GOBOARD5:3; then A9: ((G * ((i + 1),1)) `1) - ((G * (i,1)) `1) > 0 by XREAL_1:50; A10: i + 1 >= 1 by NAT_1:11; then A11: (G * ((i + 1),1)) `2 = (G * (1,1)) `2 by A1, A3, GOBOARD5:1 .= (G * ((i + 2),1)) `2 by A1, A6, A5, GOBOARD5:1 ; i <= i + 1 by NAT_1:11; then A12: i < len G by A3, XXREAL_0:2; then (G * (i,1)) `2 = (G * (1,1)) `2 by A1, A2, GOBOARD5:1 .= (G * ((i + 1),1)) `2 by A1, A3, A10, GOBOARD5:1 ; then A13: 1 * ((G * ((i + 1),1)) `2) = ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((G * (i,1)) `2)) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((G * ((i + 2),1)) `2)) by A11 .= (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) `2) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((G * ((i + 2),1)) `2)) by Lm3 .= (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) `2) + ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))) `2) by Lm3 .= (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))) `2 by Lm1 ; i + 1 < i + 2 by XREAL_1:6; then (G * ((i + 1),1)) `1 < (G * ((i + 2),1)) `1 by A1, A10, A6, GOBOARD5:3; then A14: ((G * ((i + 1),1)) `1) - ((G * (i,1)) `1) < ((G * ((i + 2),1)) `1) - ((G * (i,1)) `1) by XREAL_1:9; then ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)) = ((G * ((i + 1),1)) `1) - ((G * (i,1)) `1) by A9, XCMPLX_1:87; then (G * ((i + 1),1)) `1 = ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((G * (i,1)) `1)) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((G * ((i + 2),1)) `1)) ; then 1 * ((G * ((i + 1),1)) `1) = (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) `1) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((G * ((i + 2),1)) `1)) by Lm3 .= (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) `1) + ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))) `1) by Lm3 .= (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))) `1 by Lm1 ; then A15: ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))) = |[((G * ((i + 1),1)) `1),((G * ((i + 1),1)) `2)]| by A13, EUCLID:53 .= G * ((i + 1),1) by EUCLID:53 ; G * ((i + 1),1) = 1 * (G * ((i + 1),1)) by EUCLID:29 .= ((1 / 2) * (G * ((i + 1),1))) + ((1 / 2) * (G * ((i + 1),1))) by Lm7, EUCLID:33 .= ((1 / 2) * (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) + ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + ((1 / 2) * (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by A15, EUCLID:29 .= ((1 / 2) * (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + ((1 / 2) * (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by EUCLID:33 .= (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + ((1 / 2) * (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by EUCLID:32 .= ((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + (((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1)))) + (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))))) by EUCLID:26 .= ((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1)))) + (((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))))) by EUCLID:26 .= (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + ((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))))) + (((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by EUCLID:26 .= ((((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1)))) + ((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))) by EUCLID:26 .= (((1 / 2) * (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * ((i + 1),1))) + ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (G * (i,1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((G * (i,1)) + (G * ((i + 1),1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))) by EUCLID:32 .= ((((1 / 2) * (1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))))) * ((G * (i,1)) + (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1)))) by EUCLID:30 .= (((1 / 2) * (1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))))) * ((G * (i,1)) + (G * ((i + 1),1)))) + (((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by EUCLID:26 .= (((1 / 2) * (1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))))) * ((G * (i,1)) + (G * ((i + 1),1)))) + ((1 / 2) * ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 1),1))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (G * ((i + 2),1))))) by EUCLID:32 ; then A16: G * ((i + 1),1) = (((1 / 2) * (1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))))) * ((G * (i,1)) + (G * ((i + 1),1)))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) by EUCLID:32 .= ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) + ((1 / 2) * (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) + (((1 / 2) * ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) by EUCLID:30 ; A17: ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * (((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|)) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) = (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * |[0,1]|)) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * (((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) by EUCLID:49 .= (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * |[0,1]|)) + ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) - (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * |[0,1]|)) by EUCLID:49 .= ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) + (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1))))) - ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * |[0,1]|))) - (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * |[0,1]|) by EUCLID:45 .= (((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) + ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) - ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * |[0,1]|)) - (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * |[0,1]|) by EUCLID:45 .= ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) + ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) - (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * |[0,1]|) + (((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * |[0,1]|)) by EUCLID:46 .= ((((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1))) * ((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1))))) + ((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))))) - (((1 - ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) + ((((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)))) * |[0,1]|) by EUCLID:33 .= (G * ((i + 1),1)) - |[0,1]| by A16, EUCLID:29 ; (((G * ((i + 1),1)) `1) - ((G * (i,1)) `1)) / (((G * ((i + 2),1)) `1) - ((G * (i,1)) `1)) < 1 by A14, A9, XREAL_1:189; then (G * ((i + 1),1)) - |[0,1]| in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) by A14, A9, A17; then A18: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) = (LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|))) \/ (LSeg (((G * ((i + 1),1)) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|))) by TOPREAL1:5; LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),((G * ((i + 1),1)) - |[0,1]|)) c= (Int (cell (G,i,0))) \/ {((G * ((i + 1),1)) - |[0,1]|)} by A2, A12, Th53; hence LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),(((1 / 2) * ((G * ((i + 1),1)) + (G * ((i + 2),1)))) - |[0,1]|)) c= ((Int (cell (G,i,0))) \/ (Int (cell (G,(i + 1),0)))) \/ {((G * ((i + 1),1)) - |[0,1]|)} by A18, A8, A7, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:73 for i being Element of NAT for G being Go-board st 1 < width G & 1 <= i & i + 1 < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) c= ((Int (cell (G,i,(width G)))) \/ (Int (cell (G,(i + 1),(width G))))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} proof let i be Element of NAT ; ::_thesis: for G being Go-board st 1 < width G & 1 <= i & i + 1 < len G holds LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) c= ((Int (cell (G,i,(width G)))) \/ (Int (cell (G,(i + 1),(width G))))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} let G be Go-board; ::_thesis: ( 1 < width G & 1 <= i & i + 1 < len G implies LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) c= ((Int (cell (G,i,(width G)))) \/ (Int (cell (G,(i + 1),(width G))))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} ) assume that A1: 1 < width G and A2: 1 <= i and A3: i + 1 < len G ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) c= ((Int (cell (G,i,(width G)))) \/ (Int (cell (G,(i + 1),(width G))))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} set p1 = G * (i,(width G)); set p2 = G * ((i + 1),(width G)); set q3 = G * ((i + 2),(width G)); set r = (((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)); A4: (i + 1) + 1 = i + (1 + 1) ; then A5: i + 2 >= 1 by NAT_1:11; A6: i + (1 + 1) <= len G by A3, A4, NAT_1:13; set I1 = Int (cell (G,i,(width G))); set I2 = Int (cell (G,(i + 1),(width G))); A7: ((Int (cell (G,i,(width G)))) \/ (Int (cell (G,(i + 1),(width G))))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} = (Int (cell (G,i,(width G)))) \/ ((Int (cell (G,(i + 1),(width G)))) \/ ({((G * ((i + 1),(width G))) + |[0,1]|)} \/ {((G * ((i + 1),(width G))) + |[0,1]|)})) by XBOOLE_1:4 .= (Int (cell (G,i,(width G)))) \/ (((Int (cell (G,(i + 1),(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)}) \/ {((G * ((i + 1),(width G))) + |[0,1]|)}) by XBOOLE_1:4 .= ((Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)}) \/ ((Int (cell (G,(i + 1),(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)}) by XBOOLE_1:4 ; A8: LSeg ((((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) c= (Int (cell (G,(i + 1),(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} by A3, A4, Th54, NAT_1:11; i < i + 1 by XREAL_1:29; then (G * (i,(width G))) `1 < (G * ((i + 1),(width G))) `1 by A1, A2, A3, GOBOARD5:3; then A9: ((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1) > 0 by XREAL_1:50; A10: i + 1 >= 1 by NAT_1:11; then A11: (G * ((i + 1),(width G))) `2 = (G * (1,(width G))) `2 by A1, A3, GOBOARD5:1 .= (G * ((i + 2),(width G))) `2 by A1, A6, A5, GOBOARD5:1 ; i <= i + 1 by NAT_1:11; then A12: i < len G by A3, XXREAL_0:2; then (G * (i,(width G))) `2 = (G * (1,(width G))) `2 by A1, A2, GOBOARD5:1 .= (G * ((i + 1),(width G))) `2 by A1, A3, A10, GOBOARD5:1 ; then A13: 1 * ((G * ((i + 1),(width G))) `2) = ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((G * (i,(width G))) `2)) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((G * ((i + 2),(width G))) `2)) by A11 .= (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) `2) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((G * ((i + 2),(width G))) `2)) by Lm3 .= (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) `2) + ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))) `2) by Lm3 .= (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))) `2 by Lm1 ; i + 1 < i + 2 by XREAL_1:6; then (G * ((i + 1),(width G))) `1 < (G * ((i + 2),(width G))) `1 by A1, A10, A6, GOBOARD5:3; then A14: ((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1) < ((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1) by XREAL_1:9; then ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)) = ((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1) by A9, XCMPLX_1:87; then (G * ((i + 1),(width G))) `1 = ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((G * (i,(width G))) `1)) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((G * ((i + 2),(width G))) `1)) ; then 1 * ((G * ((i + 1),(width G))) `1) = (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) `1) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((G * ((i + 2),(width G))) `1)) by Lm3 .= (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) `1) + ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))) `1) by Lm3 .= (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))) `1 by Lm1 ; then A15: ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))) = |[((G * ((i + 1),(width G))) `1),((G * ((i + 1),(width G))) `2)]| by A13, EUCLID:53 .= G * ((i + 1),(width G)) by EUCLID:53 ; G * ((i + 1),(width G)) = 1 * (G * ((i + 1),(width G))) by EUCLID:29 .= ((1 / 2) * (G * ((i + 1),(width G)))) + ((1 / 2) * (G * ((i + 1),(width G)))) by Lm7, EUCLID:33 .= ((1 / 2) * (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) + ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + ((1 / 2) * (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by A15, EUCLID:29 .= ((1 / 2) * (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + ((1 / 2) * (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by EUCLID:33 .= (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + ((1 / 2) * (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by EUCLID:32 .= ((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + (((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G))))) + (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))))) by EUCLID:26 .= ((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G))))) + (((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))))) by EUCLID:26 .= (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + ((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))))) + (((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by EUCLID:26 .= ((((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G))))) + ((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))) by EUCLID:26 .= (((1 / 2) * (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * ((i + 1),(width G)))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (G * (i,(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))) by EUCLID:32 .= (((1 / 2) * ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))) by EUCLID:32 .= ((((1 / 2) * (1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))))) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G))))) by EUCLID:30 ; then A16: G * ((i + 1),(width G)) = (((1 / 2) * (1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))))) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + (((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by EUCLID:26 .= (((1 / 2) * (1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))))) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + ((1 / 2) * ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 1),(width G)))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (G * ((i + 2),(width G)))))) by EUCLID:32 .= (((1 / 2) * (1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))))) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) by EUCLID:32 .= ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 / 2) * (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + (((1 / 2) * ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) by EUCLID:30 .= ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) by EUCLID:30 ; A17: ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * (((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) = (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * |[0,1]|)) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * (((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) by EUCLID:32 .= (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * |[0,1]|)) + ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * |[0,1]|)) by EUCLID:32 .= ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) + (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G)))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * |[0,1]|))) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * |[0,1]|) by EUCLID:26 .= (((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * |[0,1]|)) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * |[0,1]|) by EUCLID:26 .= ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))))) + (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * |[0,1]|) + (((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * |[0,1]|)) by EUCLID:26 .= ((((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1))) * ((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G)))))) + ((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))))) + (((1 - ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) + ((((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)))) * |[0,1]|) by EUCLID:33 .= (G * ((i + 1),(width G))) + |[0,1]| by A16, EUCLID:29 ; (((G * ((i + 1),(width G))) `1) - ((G * (i,(width G))) `1)) / (((G * ((i + 2),(width G))) `1) - ((G * (i,(width G))) `1)) < 1 by A14, A9, XREAL_1:189; then (G * ((i + 1),(width G))) + |[0,1]| in LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) by A14, A9, A17; then A18: LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) = (LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|))) \/ (LSeg (((G * ((i + 1),(width G))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|))) by TOPREAL1:5; LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),((G * ((i + 1),(width G))) + |[0,1]|)) c= (Int (cell (G,i,(width G)))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} by A2, A12, Th55; hence LSeg ((((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|),(((1 / 2) * ((G * ((i + 1),(width G))) + (G * ((i + 2),(width G))))) + |[0,1]|)) c= ((Int (cell (G,i,(width G)))) \/ (Int (cell (G,(i + 1),(width G))))) \/ {((G * ((i + 1),(width G))) + |[0,1]|)} by A18, A8, A7, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:74 for G being Go-board st 1 < len G & 1 < width G holds LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} proof let G be Go-board; ::_thesis: ( 1 < len G & 1 < width G implies LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} ) assume that A1: 1 < len G and A2: 1 < width G ; ::_thesis: LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} set q2 = G * (1,1); set q3 = G * (1,2); set r = 1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1); A3: 0 + (1 + 1) <= width G by A2, NAT_1:13; then A4: (G * (1,1)) `1 = (G * (1,2)) `1 by A1, GOBOARD5:2; (G * (1,1)) `2 < (G * (1,2)) `2 by A1, A3, GOBOARD5:4; then A5: ((G * (1,2)) `2) - ((G * (1,1)) `2) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1 by XREAL_1:29, XREAL_1:129; then A6: 1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1) < 1 by XREAL_1:212; A7: (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `1) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) - |[1,1]|) `1)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `1) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) - |[1,1]|) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) `1) - (|[1,1]| `1))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|) `1)) by Lm2 .= ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) `1) - (|[1,1]| `1))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1) - (|[1,0]| `1))) by Lm2 .= ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (((G * (1,1)) `1) - 1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1) - (|[1,0]| `1))) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1) - 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `1))) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) + (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) + (G * (1,2))) `1)))) - 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `1)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) `1) + ((G * (1,1)) `1))))) - 1 by A4, Lm1 .= ((G * (1,1)) `1) - (|[1,0]| `1) by EUCLID:52 .= ((G * (1,1)) - |[1,0]|) `1 by Lm2 ; A8: (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * ((G * (1,2)) `2))) - ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * ((G * (1,1)) `2)))) + (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) = (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1) .= 1 by A5, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2) by EUCLID:49 .= ((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) `2) - (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * |[1,1]|) `2)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2) by Lm2 .= ((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) `2) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (|[1,1]| `2))) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * (G * (1,1))) `2) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) `2) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|) `2)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `2) - (|[1,0]| `2))) by Lm2 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (1,2)))) `2) - 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) + (G * (1,2))) `2))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) `2)) - (1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * ((1 / 2) * (((G * (1,1)) `2) + ((G * (1,2)) `2)))) by Lm1 .= ((G * (1,1)) `2) - 0 by A8 .= ((G * (1,1)) `2) - (|[1,0]| `2) by EUCLID:52 .= ((G * (1,1)) - |[1,0]|) `2 by Lm2 ; then ((1 - (1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (1,2)) `2) - ((G * (1,1)) `2))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) = |[(((G * (1,1)) - |[1,0]|) `1),(((G * (1,1)) - |[1,0]|) `2)]| by A7, EUCLID:53 .= (G * (1,1)) - |[1,0]| by EUCLID:53 ; then (G * (1,1)) - |[1,0]| in LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) by A5, A6; then A9: LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) = (LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|))) \/ (LSeg (((G * (1,1)) - |[1,0]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|))) by TOPREAL1:5; set I1 = Int (cell (G,0,0)); set I2 = Int (cell (G,0,1)); (0 + 1) + 1 = 0 + (1 + 1) ; then A10: LSeg (((G * (1,1)) - |[1,0]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= (Int (cell (G,0,1))) \/ {((G * (1,1)) - |[1,0]|)} by A2, Th48; A11: ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} = (Int (cell (G,0,0))) \/ ((Int (cell (G,0,1))) \/ ({((G * (1,1)) - |[1,0]|)} \/ {((G * (1,1)) - |[1,0]|)})) by XBOOLE_1:4 .= (Int (cell (G,0,0))) \/ (((Int (cell (G,0,1))) \/ {((G * (1,1)) - |[1,0]|)}) \/ {((G * (1,1)) - |[1,0]|)}) by XBOOLE_1:4 .= ((Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)}) \/ ((Int (cell (G,0,1))) \/ {((G * (1,1)) - |[1,0]|)}) by XBOOLE_1:4 ; LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[1,0]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[1,0]|)} by Th56; hence LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (1,2)))) - |[1,0]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,0,1)))) \/ {((G * (1,1)) - |[1,0]|)} by A9, A10, A11, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:75 for G being Go-board st 1 < len G & 1 < width G holds LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,(len G),1)))) \/ {((G * ((len G),1)) + |[1,0]|)} proof let G be Go-board; ::_thesis: ( 1 < len G & 1 < width G implies LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,(len G),1)))) \/ {((G * ((len G),1)) + |[1,0]|)} ) assume that A1: 1 < len G and A2: 1 < width G ; ::_thesis: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,(len G),1)))) \/ {((G * ((len G),1)) + |[1,0]|)} set q2 = G * ((len G),1); set q3 = G * ((len G),2); set r = 1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1); A3: 0 + (1 + 1) <= width G by A2, NAT_1:13; then A4: (G * ((len G),1)) `1 = (G * ((len G),2)) `1 by A1, GOBOARD5:2; (G * ((len G),1)) `2 < (G * ((len G),2)) `2 by A1, A3, GOBOARD5:4; then A5: ((G * ((len G),2)) `2) - ((G * ((len G),1)) `2) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1 by XREAL_1:29, XREAL_1:129; then A6: 1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1) < 1 by XREAL_1:212; A7: (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) `1) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `1) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (((G * ((len G),1)) + |[1,(- 1)]|) `1)) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `1) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (((G * ((len G),1)) + |[1,(- 1)]|) `1)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (((G * ((len G),1)) `1) + (|[1,(- 1)]| `1))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|) `1)) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (((G * ((len G),1)) `1) + (|[1,(- 1)]| `1))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) `1) + (|[1,0]| `1))) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (((G * ((len G),1)) `1) + 1)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) `1) + (|[1,0]| `1))) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `1)) + ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) `1) + 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `1)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) `1))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) + (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `1)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((1 / 2) * (((G * ((len G),1)) + (G * ((len G),2))) `1)))) + 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `1)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((1 / 2) * (((G * ((len G),1)) `1) + ((G * ((len G),1)) `1))))) + 1 by A4, Lm1 .= ((G * ((len G),1)) `1) + (|[1,0]| `1) by EUCLID:52 .= ((G * ((len G),1)) + |[1,0]|) `1 by Lm1 ; A8: (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((1 / 2) * ((G * ((len G),2)) `2))) - ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((1 / 2) * ((G * ((len G),1)) `2)))) + (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) = (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1) .= 1 by A5, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) `2) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (G * ((len G),1))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * |[1,(- 1)]|)) `2) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `2) by EUCLID:32 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (G * ((len G),1))) `2) + (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * |[1,(- 1)]|) `2)) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (G * ((len G),1))) `2) + ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (|[1,(- 1)]| `2))) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `2) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (G * ((len G),1))) `2) + ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * (- 1))) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `2) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `2)) + (- (1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))))) + (((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) `2) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `2)) - (1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|) `2)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `2)) - (1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) `2) + (|[1,0]| `2))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `2)) - (1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) `2) + 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `2)) - (1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((1 / 2) * (((G * ((len G),1)) + (G * ((len G),2))) `2))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) `2)) - (1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * ((1 / 2) * (((G * ((len G),1)) `2) + ((G * ((len G),2)) `2)))) by Lm1 .= ((G * ((len G),1)) `2) + 0 by A8 .= ((G * ((len G),1)) `2) + (|[1,0]| `2) by EUCLID:52 .= ((G * ((len G),1)) + |[1,0]|) `2 by Lm1 ; then ((1 - (1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) + ((1 / (((1 / 2) * (((G * ((len G),2)) `2) - ((G * ((len G),1)) `2))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) = |[(((G * ((len G),1)) + |[1,0]|) `1),(((G * ((len G),1)) + |[1,0]|) `2)]| by A7, EUCLID:53 .= (G * ((len G),1)) + |[1,0]| by EUCLID:53 ; then (G * ((len G),1)) + |[1,0]| in LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) by A5, A6; then A9: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) = (LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) + |[1,0]|))) \/ (LSeg (((G * ((len G),1)) + |[1,0]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|))) by TOPREAL1:5; set I1 = Int (cell (G,(len G),0)); set I2 = Int (cell (G,(len G),1)); (0 + 1) + 1 = 0 + (1 + 1) ; then A10: LSeg (((G * ((len G),1)) + |[1,0]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) c= (Int (cell (G,(len G),1))) \/ {((G * ((len G),1)) + |[1,0]|)} by A2, Th50; A11: ((Int (cell (G,(len G),0))) \/ (Int (cell (G,(len G),1)))) \/ {((G * ((len G),1)) + |[1,0]|)} = (Int (cell (G,(len G),0))) \/ ((Int (cell (G,(len G),1))) \/ ({((G * ((len G),1)) + |[1,0]|)} \/ {((G * ((len G),1)) + |[1,0]|)})) by XBOOLE_1:4 .= (Int (cell (G,(len G),0))) \/ (((Int (cell (G,(len G),1))) \/ {((G * ((len G),1)) + |[1,0]|)}) \/ {((G * ((len G),1)) + |[1,0]|)}) by XBOOLE_1:4 .= ((Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)}) \/ ((Int (cell (G,(len G),1))) \/ {((G * ((len G),1)) + |[1,0]|)}) by XBOOLE_1:4 ; LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) + |[1,0]|)) c= (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) + |[1,0]|)} by Th57; hence LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * ((len G),2)))) + |[1,0]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,(len G),1)))) \/ {((G * ((len G),1)) + |[1,0]|)} by A9, A10, A11, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:76 for G being Go-board st 1 < len G & 1 < width G holds LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,0,((width G) -' 1))))) \/ {((G * (1,(width G))) - |[1,0]|)} proof let G be Go-board; ::_thesis: ( 1 < len G & 1 < width G implies LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,0,((width G) -' 1))))) \/ {((G * (1,(width G))) - |[1,0]|)} ) assume that A1: 1 < len G and A2: 1 < width G ; ::_thesis: LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,0,((width G) -' 1))))) \/ {((G * (1,(width G))) - |[1,0]|)} set q2 = G * (1,(width G)); set q3 = G * (1,((width G) -' 1)); set r = 1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1); A3: ((width G) -' 1) + 1 = width G by A2, XREAL_1:235; then A4: (width G) -' 1 >= 1 by A2, NAT_1:13; A5: (width G) -' 1 < width G by A3, NAT_1:13; then (G * (1,((width G) -' 1))) `2 < (G * (1,(width G))) `2 by A1, A4, GOBOARD5:4; then A6: ((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1 by XREAL_1:29, XREAL_1:129; then A7: 1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1) < 1 by XREAL_1:212; A8: (G * (1,(width G))) `1 = (G * (1,1)) `1 by A1, A2, GOBOARD5:2 .= (G * (1,((width G) -' 1))) `1 by A1, A4, A5, GOBOARD5:2 ; A9: (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) `1) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `1) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (((G * (1,(width G))) + |[(- 1),1]|) `1)) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `1) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (((G * (1,(width G))) + |[(- 1),1]|) `1)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (((G * (1,(width G))) `1) + (|[(- 1),1]| `1))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|) `1)) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (((G * (1,(width G))) `1) + (|[(- 1),1]| `1))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) `1) - (|[1,0]| `1))) by Lm2 .= ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (((G * (1,(width G))) `1) + (- 1))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) `1) - (|[1,0]| `1))) by EUCLID:52 .= ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (((G * (1,(width G))) `1) - 1)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) `1) - 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `1)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) `1))) - 1 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `1)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * (1,(width G))) + (G * (1,((width G) -' 1)))) `1)))) - 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `1)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * (1,(width G))) `1) + ((G * (1,(width G))) `1))))) - 1 by A8, Lm1 .= ((G * (1,(width G))) `1) - (|[1,0]| `1) by EUCLID:52 .= ((G * (1,(width G))) - |[1,0]|) `1 by Lm2 ; A10: (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * (1,(width G))) `2))) - ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * (1,((width G) -' 1))) `2)))) + (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) = (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1) .= 1 by A6, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) `2) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (G * (1,(width G)))) + ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * |[(- 1),1]|)) `2) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `2) by EUCLID:32 .= ((((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (G * (1,(width G)))) `2) + (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * |[(- 1),1]|) `2)) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (G * (1,(width G)))) `2) + ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (|[(- 1),1]| `2))) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `2) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * (G * (1,(width G)))) `2) + ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `2) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `2)) + ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) `2) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|) `2)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) `2) - (|[1,0]| `2))) by Lm2 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) `2) - 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * (1,(width G))) + (G * (1,((width G) -' 1)))) `2))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * (1,(width G))) `2) + ((G * (1,((width G) -' 1))) `2)))) by Lm1 .= ((G * (1,(width G))) `2) - 0 by A10 .= ((G * (1,(width G))) `2) - (|[1,0]| `2) by EUCLID:52 .= ((G * (1,(width G))) - |[1,0]|) `2 by Lm2 ; then ((1 - (1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * (1,(width G))) `2) - ((G * (1,((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) = |[(((G * (1,(width G))) - |[1,0]|) `1),(((G * (1,(width G))) - |[1,0]|) `2)]| by A9, EUCLID:53 .= (G * (1,(width G))) - |[1,0]| by EUCLID:53 ; then (G * (1,(width G))) - |[1,0]| in LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) by A6, A7; then A11: LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) = (LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) - |[1,0]|))) \/ (LSeg (((G * (1,(width G))) - |[1,0]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|))) by TOPREAL1:5; set I1 = Int (cell (G,0,(width G))); set I2 = Int (cell (G,0,((width G) -' 1))); A12: ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,0,((width G) -' 1))))) \/ {((G * (1,(width G))) - |[1,0]|)} = (Int (cell (G,0,(width G)))) \/ ((Int (cell (G,0,((width G) -' 1)))) \/ ({((G * (1,(width G))) - |[1,0]|)} \/ {((G * (1,(width G))) - |[1,0]|)})) by XBOOLE_1:4 .= (Int (cell (G,0,(width G)))) \/ (((Int (cell (G,0,((width G) -' 1)))) \/ {((G * (1,(width G))) - |[1,0]|)}) \/ {((G * (1,(width G))) - |[1,0]|)}) by XBOOLE_1:4 .= ((Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)}) \/ ((Int (cell (G,0,((width G) -' 1)))) \/ {((G * (1,(width G))) - |[1,0]|)}) by XBOOLE_1:4 ; A13: LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) - |[1,0]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) - |[1,0]|)} by Th58; LSeg (((G * (1,(width G))) - |[1,0]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) c= (Int (cell (G,0,((width G) -' 1)))) \/ {((G * (1,(width G))) - |[1,0]|)} by A3, A4, A5, Th49; hence LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (1,((width G) -' 1))))) - |[1,0]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,0,((width G) -' 1))))) \/ {((G * (1,(width G))) - |[1,0]|)} by A11, A13, A12, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:77 for G being Go-board st 1 < len G & 1 < width G holds LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,(len G),((width G) -' 1))))) \/ {((G * ((len G),(width G))) + |[1,0]|)} proof let G be Go-board; ::_thesis: ( 1 < len G & 1 < width G implies LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,(len G),((width G) -' 1))))) \/ {((G * ((len G),(width G))) + |[1,0]|)} ) assume that A1: 1 < len G and A2: 1 < width G ; ::_thesis: LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,(len G),((width G) -' 1))))) \/ {((G * ((len G),(width G))) + |[1,0]|)} set q2 = G * ((len G),(width G)); set q3 = G * ((len G),((width G) -' 1)); set r = 1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1); A3: ((width G) -' 1) + 1 = width G by A2, XREAL_1:235; then A4: (width G) -' 1 >= 1 by A2, NAT_1:13; A5: (width G) -' 1 < width G by A3, NAT_1:13; then (G * ((len G),((width G) -' 1))) `2 < (G * ((len G),(width G))) `2 by A1, A4, GOBOARD5:4; then A6: ((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1 by XREAL_1:29, XREAL_1:129; then A7: 1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1) < 1 by XREAL_1:212; A8: (G * ((len G),(width G))) `1 = (G * ((len G),1)) `1 by A1, A2, GOBOARD5:2 .= (G * ((len G),((width G) -' 1))) `1 by A1, A4, A5, GOBOARD5:2 ; A9: (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `1) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (((G * ((len G),(width G))) + |[1,1]|) `1)) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `1) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (((G * ((len G),(width G))) + |[1,1]|) `1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|) `1)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (((G * ((len G),(width G))) `1) + (|[1,1]| `1))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|) `1)) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (((G * ((len G),(width G))) `1) + (|[1,1]| `1))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) `1) + (|[1,0]| `1))) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (((G * ((len G),(width G))) `1) + 1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) `1) + (|[1,0]| `1))) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `1)) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) `1) + 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) `1))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) + (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1)))) `1)))) + 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) `1) + ((G * ((len G),(width G))) `1))))) + 1 by A8, Lm1 .= ((G * ((len G),(width G))) `1) + (|[1,0]| `1) by EUCLID:52 .= ((G * ((len G),(width G))) + |[1,0]|) `1 by Lm1 ; A10: (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * ((len G),(width G))) `2))) - ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * ((G * ((len G),((width G) -' 1))) `2)))) + (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) = (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1) .= 1 by A6, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (G * ((len G),(width G)))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `2) by EUCLID:32 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (G * ((len G),(width G)))) `2) + (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * |[1,1]|) `2)) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `2) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (G * ((len G),(width G)))) `2) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (|[1,1]| `2))) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `2) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * (G * ((len G),(width G)))) `2) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `2) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)))) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) `2) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|) `2)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) `2) + (|[1,0]| `2))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) `2) + 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1)))) `2))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) `2)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) `2) + ((G * ((len G),((width G) -' 1))) `2)))) by Lm1 .= ((G * ((len G),(width G))) `2) + 0 by A10 .= ((G * ((len G),(width G))) `2) + (|[1,0]| `2) by EUCLID:52 .= ((G * ((len G),(width G))) + |[1,0]|) `2 by Lm1 ; then ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `2) - ((G * ((len G),((width G) -' 1))) `2))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) = |[(((G * ((len G),(width G))) + |[1,0]|) `1),(((G * ((len G),(width G))) + |[1,0]|) `2)]| by A9, EUCLID:53 .= (G * ((len G),(width G))) + |[1,0]| by EUCLID:53 ; then (G * ((len G),(width G))) + |[1,0]| in LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) by A6, A7; then A11: LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) = (LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|))) \/ (LSeg (((G * ((len G),(width G))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|))) by TOPREAL1:5; set I1 = Int (cell (G,(len G),(width G))); set I2 = Int (cell (G,(len G),((width G) -' 1))); A12: ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,(len G),((width G) -' 1))))) \/ {((G * ((len G),(width G))) + |[1,0]|)} = (Int (cell (G,(len G),(width G)))) \/ ((Int (cell (G,(len G),((width G) -' 1)))) \/ ({((G * ((len G),(width G))) + |[1,0]|)} \/ {((G * ((len G),(width G))) + |[1,0]|)})) by XBOOLE_1:4 .= (Int (cell (G,(len G),(width G)))) \/ (((Int (cell (G,(len G),((width G) -' 1)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}) \/ {((G * ((len G),(width G))) + |[1,0]|)}) by XBOOLE_1:4 .= ((Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}) \/ ((Int (cell (G,(len G),((width G) -' 1)))) \/ {((G * ((len G),(width G))) + |[1,0]|)}) by XBOOLE_1:4 ; A13: LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[1,0]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} by Th59; LSeg (((G * ((len G),(width G))) + |[1,0]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) c= (Int (cell (G,(len G),((width G) -' 1)))) \/ {((G * ((len G),(width G))) + |[1,0]|)} by A3, A4, A5, Th51; hence LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * ((len G),((width G) -' 1))))) + |[1,0]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,(len G),((width G) -' 1))))) \/ {((G * ((len G),(width G))) + |[1,0]|)} by A11, A13, A12, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:78 for G being Go-board st 1 < width G & 1 < len G holds LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,1,0)))) \/ {((G * (1,1)) - |[0,1]|)} proof let G be Go-board; ::_thesis: ( 1 < width G & 1 < len G implies LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,1,0)))) \/ {((G * (1,1)) - |[0,1]|)} ) assume that A1: 1 < width G and A2: 1 < len G ; ::_thesis: LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,1,0)))) \/ {((G * (1,1)) - |[0,1]|)} set q2 = G * (1,1); set q3 = G * (2,1); set r = 1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1); A3: 0 + (1 + 1) <= len G by A2, NAT_1:13; then A4: (G * (1,1)) `2 = (G * (2,1)) `2 by A1, GOBOARD5:1; (G * (1,1)) `1 < (G * (2,1)) `1 by A1, A3, GOBOARD5:3; then A5: ((G * (2,1)) `1) - ((G * (1,1)) `1) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1 by XREAL_1:29, XREAL_1:129; then A6: 1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1) < 1 by XREAL_1:212; A7: (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) - |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `2) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (((G * (1,1)) - |[1,1]|) `2)) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `2) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (((G * (1,1)) - |[1,1]|) `2)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (((G * (1,1)) `2) - (|[1,1]| `2))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|) `2)) by Lm2 .= ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (((G * (1,1)) `2) - (|[1,1]| `2))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) `2) - (|[0,1]| `2))) by Lm2 .= ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (((G * (1,1)) `2) - 1)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) `2) - (|[0,1]| `2))) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `2)) - ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) `2) - 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `2)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) `2))) - ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) + (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `2)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((1 / 2) * (((G * (1,1)) + (G * (2,1))) `2)))) - 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `2)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((1 / 2) * (((G * (1,1)) `2) + ((G * (1,1)) `2))))) - 1 by A4, Lm1 .= ((G * (1,1)) `2) - (|[0,1]| `2) by EUCLID:52 .= ((G * (1,1)) - |[0,1]|) `2 by Lm2 ; A8: (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((1 / 2) * ((G * (2,1)) `1))) - ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((1 / 2) * ((G * (1,1)) `1)))) + (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) = (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1) .= 1 by A5, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) - |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (G * (1,1))) - ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `1) by EUCLID:49 .= ((((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (G * (1,1))) `1) - (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * |[1,1]|) `1)) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `1) by Lm2 .= ((((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (G * (1,1))) `1) - ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (|[1,1]| `1))) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `1) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * (G * (1,1))) `1) - ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `1) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `1)) - ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) `1) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|) `1)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) `1) - (|[0,1]| `1))) by Lm2 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((((1 / 2) * ((G * (1,1)) + (G * (2,1)))) `1) - 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((1 / 2) * (((G * (1,1)) + (G * (2,1))) `1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * ((1 / 2) * (((G * (1,1)) `1) + ((G * (2,1)) `1)))) by Lm1 .= ((G * (1,1)) `1) - 0 by A8 .= ((G * (1,1)) `1) - (|[0,1]| `1) by EUCLID:52 .= ((G * (1,1)) - |[0,1]|) `1 by Lm2 ; then ((1 - (1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1))) * ((G * (1,1)) - |[1,1]|)) + ((1 / (((1 / 2) * (((G * (2,1)) `1) - ((G * (1,1)) `1))) + 1)) * (((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) = |[(((G * (1,1)) - |[0,1]|) `1),(((G * (1,1)) - |[0,1]|) `2)]| by A7, EUCLID:53 .= (G * (1,1)) - |[0,1]| by EUCLID:53 ; then (G * (1,1)) - |[0,1]| in LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) by A5, A6; then A9: LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) = (LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|))) \/ (LSeg (((G * (1,1)) - |[0,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|))) by TOPREAL1:5; set I1 = Int (cell (G,0,0)); set I2 = Int (cell (G,1,0)); (0 + 1) + 1 = 0 + (1 + 1) ; then A10: LSeg (((G * (1,1)) - |[0,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) c= (Int (cell (G,1,0))) \/ {((G * (1,1)) - |[0,1]|)} by A2, Th52; A11: ((Int (cell (G,0,0))) \/ (Int (cell (G,1,0)))) \/ {((G * (1,1)) - |[0,1]|)} = (Int (cell (G,0,0))) \/ ((Int (cell (G,1,0))) \/ ({((G * (1,1)) - |[0,1]|)} \/ {((G * (1,1)) - |[0,1]|)})) by XBOOLE_1:4 .= (Int (cell (G,0,0))) \/ (((Int (cell (G,1,0))) \/ {((G * (1,1)) - |[0,1]|)}) \/ {((G * (1,1)) - |[0,1]|)}) by XBOOLE_1:4 .= ((Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)}) \/ ((Int (cell (G,1,0))) \/ {((G * (1,1)) - |[0,1]|)}) by XBOOLE_1:4 ; LSeg (((G * (1,1)) - |[1,1]|),((G * (1,1)) - |[0,1]|)) c= (Int (cell (G,0,0))) \/ {((G * (1,1)) - |[0,1]|)} by Th60; hence LSeg (((G * (1,1)) - |[1,1]|),(((1 / 2) * ((G * (1,1)) + (G * (2,1)))) - |[0,1]|)) c= ((Int (cell (G,0,0))) \/ (Int (cell (G,1,0)))) \/ {((G * (1,1)) - |[0,1]|)} by A9, A10, A11, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:79 for G being Go-board st 1 < width G & 1 < len G holds LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,1,(width G))))) \/ {((G * (1,(width G))) + |[0,1]|)} proof let G be Go-board; ::_thesis: ( 1 < width G & 1 < len G implies LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,1,(width G))))) \/ {((G * (1,(width G))) + |[0,1]|)} ) assume that A1: 1 < width G and A2: 1 < len G ; ::_thesis: LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,1,(width G))))) \/ {((G * (1,(width G))) + |[0,1]|)} set q2 = G * (1,(width G)); set q3 = G * (2,(width G)); set r = 1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1); A3: 0 + (1 + 1) <= len G by A2, NAT_1:13; then A4: (G * (1,(width G))) `2 = (G * (2,(width G))) `2 by A1, GOBOARD5:1; (G * (1,(width G))) `1 < (G * (2,(width G))) `1 by A1, A3, GOBOARD5:3; then A5: ((G * (2,(width G))) `1) - ((G * (1,(width G))) `1) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1 by XREAL_1:29, XREAL_1:129; then A6: 1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1) < 1 by XREAL_1:212; A7: (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) `2) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `2) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (((G * (1,(width G))) + |[(- 1),1]|) `2)) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `2) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (((G * (1,(width G))) + |[(- 1),1]|) `2)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (((G * (1,(width G))) `2) + (|[(- 1),1]| `2))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|) `2)) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (((G * (1,(width G))) `2) + (|[(- 1),1]| `2))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) `2) + (|[0,1]| `2))) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (((G * (1,(width G))) `2) + 1)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) `2) + (|[0,1]| `2))) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `2)) + ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) `2) + 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `2)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) `2))) + ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) + (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `2)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((1 / 2) * (((G * (1,(width G))) + (G * (2,(width G)))) `2)))) + 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `2)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((1 / 2) * (((G * (1,(width G))) `2) + ((G * (1,(width G))) `2))))) + 1 by A4, Lm1 .= ((G * (1,(width G))) `2) + (|[0,1]| `2) by EUCLID:52 .= ((G * (1,(width G))) + |[0,1]|) `2 by Lm1 ; A8: (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((1 / 2) * ((G * (2,(width G))) `1))) - ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((1 / 2) * ((G * (1,(width G))) `1)))) + (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) = (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1) .= 1 by A5, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) `1) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (G * (1,(width G)))) + ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * |[(- 1),1]|)) `1) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) by EUCLID:32 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (G * (1,(width G)))) `1) + (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * |[(- 1),1]|) `1)) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (G * (1,(width G)))) `1) + ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (|[(- 1),1]| `1))) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (G * (1,(width G)))) `1) + ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (- 1))) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) by EUCLID:52 .= ((((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * (G * (1,(width G)))) `1) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + (((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) `1) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|) `1)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) `1) + (|[0,1]| `1))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) `1) + 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((1 / 2) * (((G * (1,(width G))) + (G * (2,(width G)))) `1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) `1)) - (1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * ((1 / 2) * (((G * (1,(width G))) `1) + ((G * (2,(width G))) `1)))) by Lm1 .= ((G * (1,(width G))) `1) + 0 by A8 .= ((G * (1,(width G))) `1) + (|[0,1]| `1) by EUCLID:52 .= ((G * (1,(width G))) + |[0,1]|) `1 by Lm1 ; then ((1 - (1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1))) * ((G * (1,(width G))) + |[(- 1),1]|)) + ((1 / (((1 / 2) * (((G * (2,(width G))) `1) - ((G * (1,(width G))) `1))) + 1)) * (((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) = |[(((G * (1,(width G))) + |[0,1]|) `1),(((G * (1,(width G))) + |[0,1]|) `2)]| by A7, EUCLID:53 .= (G * (1,(width G))) + |[0,1]| by EUCLID:53 ; then (G * (1,(width G))) + |[0,1]| in LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) by A5, A6; then A9: LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) = (LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|))) \/ (LSeg (((G * (1,(width G))) + |[0,1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|))) by TOPREAL1:5; set I1 = Int (cell (G,0,(width G))); set I2 = Int (cell (G,1,(width G))); (0 + 1) + 1 = 0 + (1 + 1) ; then A10: LSeg (((G * (1,(width G))) + |[0,1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) c= (Int (cell (G,1,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} by A2, Th54; A11: ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,1,(width G))))) \/ {((G * (1,(width G))) + |[0,1]|)} = (Int (cell (G,0,(width G)))) \/ ((Int (cell (G,1,(width G)))) \/ ({((G * (1,(width G))) + |[0,1]|)} \/ {((G * (1,(width G))) + |[0,1]|)})) by XBOOLE_1:4 .= (Int (cell (G,0,(width G)))) \/ (((Int (cell (G,1,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)}) \/ {((G * (1,(width G))) + |[0,1]|)}) by XBOOLE_1:4 .= ((Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)}) \/ ((Int (cell (G,1,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)}) by XBOOLE_1:4 ; LSeg (((G * (1,(width G))) + |[(- 1),1]|),((G * (1,(width G))) + |[0,1]|)) c= (Int (cell (G,0,(width G)))) \/ {((G * (1,(width G))) + |[0,1]|)} by Th62; hence LSeg (((G * (1,(width G))) + |[(- 1),1]|),(((1 / 2) * ((G * (1,(width G))) + (G * (2,(width G))))) + |[0,1]|)) c= ((Int (cell (G,0,(width G)))) \/ (Int (cell (G,1,(width G))))) \/ {((G * (1,(width G))) + |[0,1]|)} by A9, A10, A11, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:80 for G being Go-board st 1 < width G & 1 < len G holds LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,((len G) -' 1),0)))) \/ {((G * ((len G),1)) - |[0,1]|)} proof let G be Go-board; ::_thesis: ( 1 < width G & 1 < len G implies LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,((len G) -' 1),0)))) \/ {((G * ((len G),1)) - |[0,1]|)} ) assume that A1: 1 < width G and A2: 1 < len G ; ::_thesis: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,((len G) -' 1),0)))) \/ {((G * ((len G),1)) - |[0,1]|)} set q2 = G * ((len G),1); set q3 = G * (((len G) -' 1),1); set r = 1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1); A3: ((len G) -' 1) + 1 = len G by A2, XREAL_1:235; then A4: (len G) -' 1 >= 1 by A2, NAT_1:13; A5: (len G) -' 1 < len G by A3, NAT_1:13; then (G * (((len G) -' 1),1)) `1 < (G * ((len G),1)) `1 by A1, A4, GOBOARD5:3; then A6: ((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1 by XREAL_1:29, XREAL_1:129; then A7: 1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1) < 1 by XREAL_1:212; A8: (G * ((len G),1)) `2 = (G * (1,1)) `2 by A1, A2, GOBOARD5:1 .= (G * (((len G) -' 1),1)) `2 by A1, A4, A5, GOBOARD5:1 ; A9: (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) `2) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `2) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (((G * ((len G),1)) + |[1,(- 1)]|) `2)) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `2) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (((G * ((len G),1)) + |[1,(- 1)]|) `2)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (((G * ((len G),1)) `2) + (|[1,(- 1)]| `2))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|) `2)) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (((G * ((len G),1)) `2) + (|[1,(- 1)]| `2))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) `2) - (|[0,1]| `2))) by Lm2 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (((G * ((len G),1)) `2) + (- 1))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) `2) - (|[0,1]| `2))) by EUCLID:52 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (((G * ((len G),1)) `2) - 1)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) `2) - 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `2)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) `2))) - 1 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `2)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((1 / 2) * (((G * ((len G),1)) + (G * (((len G) -' 1),1))) `2)))) - 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `2)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((1 / 2) * (((G * ((len G),1)) `2) + ((G * ((len G),1)) `2))))) - 1 by A8, Lm1 .= ((G * ((len G),1)) `2) - (|[0,1]| `2) by EUCLID:52 .= ((G * ((len G),1)) - |[0,1]|) `2 by Lm2 ; A10: (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((1 / 2) * ((G * ((len G),1)) `1))) - ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((1 / 2) * ((G * (((len G) -' 1),1)) `1)))) + (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) = (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1) .= 1 by A6, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) `1) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (G * ((len G),1))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * |[1,(- 1)]|)) `1) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `1) by EUCLID:32 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (G * ((len G),1))) `1) + (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * |[1,(- 1)]|) `1)) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (G * ((len G),1))) `1) + ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (|[1,(- 1)]| `1))) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `1) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * (G * ((len G),1))) `1) + ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `1) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `1)) + ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) `1) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|) `1)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) `1) - (|[0,1]| `1))) by Lm2 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) `1) - 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((1 / 2) * (((G * ((len G),1)) + (G * (((len G) -' 1),1))) `1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * ((1 / 2) * (((G * ((len G),1)) `1) + ((G * (((len G) -' 1),1)) `1)))) by Lm1 .= ((G * ((len G),1)) `1) - 0 by A10 .= ((G * ((len G),1)) `1) - (|[0,1]| `1) by EUCLID:52 .= ((G * ((len G),1)) - |[0,1]|) `1 by Lm2 ; then ((1 - (1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1))) * ((G * ((len G),1)) + |[1,(- 1)]|)) + ((1 / (((1 / 2) * (((G * ((len G),1)) `1) - ((G * (((len G) -' 1),1)) `1))) + 1)) * (((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) = |[(((G * ((len G),1)) - |[0,1]|) `1),(((G * ((len G),1)) - |[0,1]|) `2)]| by A9, EUCLID:53 .= (G * ((len G),1)) - |[0,1]| by EUCLID:53 ; then (G * ((len G),1)) - |[0,1]| in LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) by A6, A7; then A11: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) = (LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) - |[0,1]|))) \/ (LSeg (((G * ((len G),1)) - |[0,1]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|))) by TOPREAL1:5; set I1 = Int (cell (G,(len G),0)); set I2 = Int (cell (G,((len G) -' 1),0)); A12: ((Int (cell (G,(len G),0))) \/ (Int (cell (G,((len G) -' 1),0)))) \/ {((G * ((len G),1)) - |[0,1]|)} = (Int (cell (G,(len G),0))) \/ ((Int (cell (G,((len G) -' 1),0))) \/ ({((G * ((len G),1)) - |[0,1]|)} \/ {((G * ((len G),1)) - |[0,1]|)})) by XBOOLE_1:4 .= (Int (cell (G,(len G),0))) \/ (((Int (cell (G,((len G) -' 1),0))) \/ {((G * ((len G),1)) - |[0,1]|)}) \/ {((G * ((len G),1)) - |[0,1]|)}) by XBOOLE_1:4 .= ((Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)}) \/ ((Int (cell (G,((len G) -' 1),0))) \/ {((G * ((len G),1)) - |[0,1]|)}) by XBOOLE_1:4 ; A13: LSeg (((G * ((len G),1)) + |[1,(- 1)]|),((G * ((len G),1)) - |[0,1]|)) c= (Int (cell (G,(len G),0))) \/ {((G * ((len G),1)) - |[0,1]|)} by Th61; LSeg (((G * ((len G),1)) - |[0,1]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) c= (Int (cell (G,((len G) -' 1),0))) \/ {((G * ((len G),1)) - |[0,1]|)} by A3, A4, A5, Th53; hence LSeg (((G * ((len G),1)) + |[1,(- 1)]|),(((1 / 2) * ((G * ((len G),1)) + (G * (((len G) -' 1),1)))) - |[0,1]|)) c= ((Int (cell (G,(len G),0))) \/ (Int (cell (G,((len G) -' 1),0)))) \/ {((G * ((len G),1)) - |[0,1]|)} by A11, A13, A12, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:81 for G being Go-board st 1 < width G & 1 < len G holds LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,((len G) -' 1),(width G))))) \/ {((G * ((len G),(width G))) + |[0,1]|)} proof let G be Go-board; ::_thesis: ( 1 < width G & 1 < len G implies LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,((len G) -' 1),(width G))))) \/ {((G * ((len G),(width G))) + |[0,1]|)} ) assume that A1: 1 < width G and A2: 1 < len G ; ::_thesis: LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,((len G) -' 1),(width G))))) \/ {((G * ((len G),(width G))) + |[0,1]|)} set q2 = G * ((len G),(width G)); set q3 = G * (((len G) -' 1),(width G)); set r = 1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1); A3: ((len G) -' 1) + 1 = len G by A2, XREAL_1:235; then A4: (len G) -' 1 >= 1 by A2, NAT_1:13; A5: (len G) -' 1 < len G by A3, NAT_1:13; then (G * (((len G) -' 1),(width G))) `1 < (G * ((len G),(width G))) `1 by A1, A4, GOBOARD5:3; then A6: ((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1) > 0 by XREAL_1:50; then 1 < ((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1 by XREAL_1:29, XREAL_1:129; then A7: 1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1) < 1 by XREAL_1:212; A8: (G * ((len G),(width G))) `2 = (G * (1,(width G))) `2 by A1, A2, GOBOARD5:1 .= (G * (((len G) -' 1),(width G))) `2 by A1, A4, A5, GOBOARD5:1 ; A9: (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|))) `2 = (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) `2) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `2) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (((G * ((len G),(width G))) + |[1,1]|) `2)) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `2) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (((G * ((len G),(width G))) + |[1,1]|) `2)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|) `2)) by Lm3 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (((G * ((len G),(width G))) `2) + (|[1,1]| `2))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|) `2)) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (((G * ((len G),(width G))) `2) + (|[1,1]| `2))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) `2) + (|[0,1]| `2))) by Lm1 .= ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (((G * ((len G),(width G))) `2) + 1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) `2) + (|[0,1]| `2))) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `2)) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * 1)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) `2) + 1)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `2)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) `2))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) + (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `2)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G)))) `2)))) + 1 by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `2)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) `2) + ((G * ((len G),(width G))) `2))))) + 1 by A8, Lm1 .= ((G * ((len G),(width G))) `2) + (|[0,1]| `2) by EUCLID:52 .= ((G * ((len G),(width G))) + |[0,1]|) `2 by Lm1 ; A10: (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((1 / 2) * ((G * ((len G),(width G))) `1))) - ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((1 / 2) * ((G * (((len G) -' 1),(width G))) `1)))) + (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) = (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1) .= 1 by A6, XCMPLX_1:106 ; (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|))) `1 = (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (G * ((len G),(width G)))) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * |[1,1]|)) `1) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `1) by EUCLID:32 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (G * ((len G),(width G)))) `1) + (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * |[1,1]|) `1)) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `1) by Lm1 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (G * ((len G),(width G)))) `1) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (|[1,1]| `1))) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `1) by Lm3 .= ((((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * (G * ((len G),(width G)))) `1) + ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * 1)) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `1) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)))) + (((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) `1) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|) `1)) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) `1) + (|[0,1]| `1))) by Lm1 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) `1) + 0)) by EUCLID:52 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G)))) `1))) by Lm3 .= (((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) `1)) + (1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)))) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * ((1 / 2) * (((G * ((len G),(width G))) `1) + ((G * (((len G) -' 1),(width G))) `1)))) by Lm1 .= ((G * ((len G),(width G))) `1) + 0 by A10 .= ((G * ((len G),(width G))) `1) + (|[0,1]| `1) by EUCLID:52 .= ((G * ((len G),(width G))) + |[0,1]|) `1 by Lm1 ; then ((1 - (1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1))) * ((G * ((len G),(width G))) + |[1,1]|)) + ((1 / (((1 / 2) * (((G * ((len G),(width G))) `1) - ((G * (((len G) -' 1),(width G))) `1))) + 1)) * (((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) = |[(((G * ((len G),(width G))) + |[0,1]|) `1),(((G * ((len G),(width G))) + |[0,1]|) `2)]| by A9, EUCLID:53 .= (G * ((len G),(width G))) + |[0,1]| by EUCLID:53 ; then (G * ((len G),(width G))) + |[0,1]| in LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) by A6, A7; then A11: LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) = (LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[0,1]|))) \/ (LSeg (((G * ((len G),(width G))) + |[0,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|))) by TOPREAL1:5; set I1 = Int (cell (G,(len G),(width G))); set I2 = Int (cell (G,((len G) -' 1),(width G))); A12: ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,((len G) -' 1),(width G))))) \/ {((G * ((len G),(width G))) + |[0,1]|)} = (Int (cell (G,(len G),(width G)))) \/ ((Int (cell (G,((len G) -' 1),(width G)))) \/ ({((G * ((len G),(width G))) + |[0,1]|)} \/ {((G * ((len G),(width G))) + |[0,1]|)})) by XBOOLE_1:4 .= (Int (cell (G,(len G),(width G)))) \/ (((Int (cell (G,((len G) -' 1),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)}) \/ {((G * ((len G),(width G))) + |[0,1]|)}) by XBOOLE_1:4 .= ((Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)}) \/ ((Int (cell (G,((len G) -' 1),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)}) by XBOOLE_1:4 ; A13: LSeg (((G * ((len G),(width G))) + |[1,1]|),((G * ((len G),(width G))) + |[0,1]|)) c= (Int (cell (G,(len G),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} by Th63; LSeg (((G * ((len G),(width G))) + |[0,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) c= (Int (cell (G,((len G) -' 1),(width G)))) \/ {((G * ((len G),(width G))) + |[0,1]|)} by A3, A4, A5, Th55; hence LSeg (((G * ((len G),(width G))) + |[1,1]|),(((1 / 2) * ((G * ((len G),(width G))) + (G * (((len G) -' 1),(width G))))) + |[0,1]|)) c= ((Int (cell (G,(len G),(width G)))) \/ (Int (cell (G,((len G) -' 1),(width G))))) \/ {((G * ((len G),(width G))) + |[0,1]|)} by A11, A13, A12, XBOOLE_1:13; ::_thesis: verum end; theorem :: GOBOARD6:82 for i, j being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) meets Int (cell (G,i,j)) proof let i, j be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) meets Int (cell (G,i,j)) let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G holds LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) meets Int (cell (G,i,j)) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G implies LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) meets Int (cell (G,i,j)) ) assume A1: ( 1 <= i & i + 1 <= len G & 1 <= j & j + 1 <= width G ) ; ::_thesis: LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) meets Int (cell (G,i,j)) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(((1_/_2)_*_((G_*_(i,j))_+_(G_*_((i_+_1),(j_+_1))))),p)_&_a_in_Int_(cell_(G,i,j))_) take a = (1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1)))); ::_thesis: ( a in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) & a in Int (cell (G,i,j)) ) thus a in LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,i,j)) thus a in Int (cell (G,i,j)) by A1, Th31; ::_thesis: verum end; hence LSeg (((1 / 2) * ((G * (i,j)) + (G * ((i + 1),(j + 1))))),p) meets Int (cell (G,i,j)) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:83 for i being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i + 1 <= len G holds LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) meets Int (cell (G,i,(width G))) proof let i be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i + 1 <= len G holds LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) meets Int (cell (G,i,(width G))) let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G holds LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) meets Int (cell (G,i,(width G))) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G implies LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) meets Int (cell (G,i,(width G))) ) assume A1: ( 1 <= i & i + 1 <= len G ) ; ::_thesis: LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) meets Int (cell (G,i,(width G))) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(p,(((1_/_2)_*_((G_*_(i,(width_G)))_+_(G_*_((i_+_1),(width_G)))))_+_|[0,1]|))_&_a_in_Int_(cell_(G,i,(width_G)))_) take a = ((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|; ::_thesis: ( a in LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) & a in Int (cell (G,i,(width G))) ) thus a in LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,i,(width G))) thus a in Int (cell (G,i,(width G))) by A1, Th32; ::_thesis: verum end; hence LSeg (p,(((1 / 2) * ((G * (i,(width G))) + (G * ((i + 1),(width G))))) + |[0,1]|)) meets Int (cell (G,i,(width G))) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:84 for i being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i + 1 <= len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) meets Int (cell (G,i,0)) proof let i be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= i & i + 1 <= len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) meets Int (cell (G,i,0)) let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= i & i + 1 <= len G holds LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) meets Int (cell (G,i,0)) let G be Go-board; ::_thesis: ( 1 <= i & i + 1 <= len G implies LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) meets Int (cell (G,i,0)) ) assume A1: ( 1 <= i & i + 1 <= len G ) ; ::_thesis: LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) meets Int (cell (G,i,0)) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_((((1_/_2)_*_((G_*_(i,1))_+_(G_*_((i_+_1),1))))_-_|[0,1]|),p)_&_a_in_Int_(cell_(G,i,0))_) take a = ((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|; ::_thesis: ( a in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) & a in Int (cell (G,i,0)) ) thus a in LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,i,0)) thus a in Int (cell (G,i,0)) by A1, Th33; ::_thesis: verum end; hence LSeg ((((1 / 2) * ((G * (i,1)) + (G * ((i + 1),1)))) - |[0,1]|),p) meets Int (cell (G,i,0)) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:85 for j being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= j & j + 1 <= width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) meets Int (cell (G,0,j)) proof let j be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= j & j + 1 <= width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) meets Int (cell (G,0,j)) let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= j & j + 1 <= width G holds LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) meets Int (cell (G,0,j)) let G be Go-board; ::_thesis: ( 1 <= j & j + 1 <= width G implies LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) meets Int (cell (G,0,j)) ) assume A1: ( 1 <= j & j + 1 <= width G ) ; ::_thesis: LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) meets Int (cell (G,0,j)) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_((((1_/_2)_*_((G_*_(1,j))_+_(G_*_(1,(j_+_1)))))_-_|[1,0]|),p)_&_a_in_Int_(cell_(G,0,j))_) take a = ((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|; ::_thesis: ( a in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) & a in Int (cell (G,0,j)) ) thus a in LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,0,j)) thus a in Int (cell (G,0,j)) by A1, Th35; ::_thesis: verum end; hence LSeg ((((1 / 2) * ((G * (1,j)) + (G * (1,(j + 1))))) - |[1,0]|),p) meets Int (cell (G,0,j)) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:86 for j being Element of NAT for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= j & j + 1 <= width G holds LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) meets Int (cell (G,(len G),j)) proof let j be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) for G being Go-board st 1 <= j & j + 1 <= width G holds LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) meets Int (cell (G,(len G),j)) let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board st 1 <= j & j + 1 <= width G holds LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) meets Int (cell (G,(len G),j)) let G be Go-board; ::_thesis: ( 1 <= j & j + 1 <= width G implies LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) meets Int (cell (G,(len G),j)) ) assume A1: ( 1 <= j & j + 1 <= width G ) ; ::_thesis: LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) meets Int (cell (G,(len G),j)) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(p,(((1_/_2)_*_((G_*_((len_G),j))_+_(G_*_((len_G),(j_+_1)))))_+_|[1,0]|))_&_a_in_Int_(cell_(G,(len_G),j))_) take a = ((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|; ::_thesis: ( a in LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) & a in Int (cell (G,(len G),j)) ) thus a in LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,(len G),j)) thus a in Int (cell (G,(len G),j)) by A1, Th34; ::_thesis: verum end; hence LSeg (p,(((1 / 2) * ((G * ((len G),j)) + (G * ((len G),(j + 1))))) + |[1,0]|)) meets Int (cell (G,(len G),j)) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:87 for p being Point of (TOP-REAL 2) for G being Go-board holds LSeg (p,((G * (1,1)) - |[1,1]|)) meets Int (cell (G,0,0)) proof let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board holds LSeg (p,((G * (1,1)) - |[1,1]|)) meets Int (cell (G,0,0)) let G be Go-board; ::_thesis: LSeg (p,((G * (1,1)) - |[1,1]|)) meets Int (cell (G,0,0)) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(p,((G_*_(1,1))_-_|[1,1]|))_&_a_in_Int_(cell_(G,0,0))_) take a = (G * (1,1)) - |[1,1]|; ::_thesis: ( a in LSeg (p,((G * (1,1)) - |[1,1]|)) & a in Int (cell (G,0,0)) ) thus a in LSeg (p,((G * (1,1)) - |[1,1]|)) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,0,0)) thus a in Int (cell (G,0,0)) by Th36; ::_thesis: verum end; hence LSeg (p,((G * (1,1)) - |[1,1]|)) meets Int (cell (G,0,0)) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:88 for p being Point of (TOP-REAL 2) for G being Go-board holds LSeg (p,((G * ((len G),(width G))) + |[1,1]|)) meets Int (cell (G,(len G),(width G))) proof let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board holds LSeg (p,((G * ((len G),(width G))) + |[1,1]|)) meets Int (cell (G,(len G),(width G))) let G be Go-board; ::_thesis: LSeg (p,((G * ((len G),(width G))) + |[1,1]|)) meets Int (cell (G,(len G),(width G))) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(p,((G_*_((len_G),(width_G)))_+_|[1,1]|))_&_a_in_Int_(cell_(G,(len_G),(width_G)))_) take a = (G * ((len G),(width G))) + |[1,1]|; ::_thesis: ( a in LSeg (p,((G * ((len G),(width G))) + |[1,1]|)) & a in Int (cell (G,(len G),(width G))) ) thus a in LSeg (p,((G * ((len G),(width G))) + |[1,1]|)) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,(len G),(width G))) thus a in Int (cell (G,(len G),(width G))) by Th37; ::_thesis: verum end; hence LSeg (p,((G * ((len G),(width G))) + |[1,1]|)) meets Int (cell (G,(len G),(width G))) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:89 for p being Point of (TOP-REAL 2) for G being Go-board holds LSeg (p,((G * (1,(width G))) + |[(- 1),1]|)) meets Int (cell (G,0,(width G))) proof let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board holds LSeg (p,((G * (1,(width G))) + |[(- 1),1]|)) meets Int (cell (G,0,(width G))) let G be Go-board; ::_thesis: LSeg (p,((G * (1,(width G))) + |[(- 1),1]|)) meets Int (cell (G,0,(width G))) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(p,((G_*_(1,(width_G)))_+_|[(-_1),1]|))_&_a_in_Int_(cell_(G,0,(width_G)))_) take a = (G * (1,(width G))) + |[(- 1),1]|; ::_thesis: ( a in LSeg (p,((G * (1,(width G))) + |[(- 1),1]|)) & a in Int (cell (G,0,(width G))) ) thus a in LSeg (p,((G * (1,(width G))) + |[(- 1),1]|)) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,0,(width G))) thus a in Int (cell (G,0,(width G))) by Th38; ::_thesis: verum end; hence LSeg (p,((G * (1,(width G))) + |[(- 1),1]|)) meets Int (cell (G,0,(width G))) by XBOOLE_0:3; ::_thesis: verum end; theorem :: GOBOARD6:90 for p being Point of (TOP-REAL 2) for G being Go-board holds LSeg (p,((G * ((len G),1)) + |[1,(- 1)]|)) meets Int (cell (G,(len G),0)) proof let p be Point of (TOP-REAL 2); ::_thesis: for G being Go-board holds LSeg (p,((G * ((len G),1)) + |[1,(- 1)]|)) meets Int (cell (G,(len G),0)) let G be Go-board; ::_thesis: LSeg (p,((G * ((len G),1)) + |[1,(- 1)]|)) meets Int (cell (G,(len G),0)) now__::_thesis:_ex_a_being_Element_of_the_carrier_of_(TOP-REAL_2)_st_ (_a_in_LSeg_(p,((G_*_((len_G),1))_+_|[1,(-_1)]|))_&_a_in_Int_(cell_(G,(len_G),0))_) take a = (G * ((len G),1)) + |[1,(- 1)]|; ::_thesis: ( a in LSeg (p,((G * ((len G),1)) + |[1,(- 1)]|)) & a in Int (cell (G,(len G),0)) ) thus a in LSeg (p,((G * ((len G),1)) + |[1,(- 1)]|)) by RLTOPSP1:68; ::_thesis: a in Int (cell (G,(len G),0)) thus a in Int (cell (G,(len G),0)) by Th39; ::_thesis: verum end; hence LSeg (p,((G * ((len G),1)) + |[1,(- 1)]|)) meets Int (cell (G,(len G),0)) by XBOOLE_0:3; ::_thesis: verum end; theorem Th91: :: GOBOARD6:91 for M being non empty MetrSpace for p being Point of M for q being Point of (TopSpaceMetr M) for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q proof let M be non empty MetrSpace; ::_thesis: for p being Point of M for q being Point of (TopSpaceMetr M) for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q let p be Point of M; ::_thesis: for q being Point of (TopSpaceMetr M) for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q let q be Point of (TopSpaceMetr M); ::_thesis: for r being real number st p = q & r > 0 holds Ball (p,r) is a_neighborhood of q let r be real number ; ::_thesis: ( p = q & r > 0 implies Ball (p,r) is a_neighborhood of q ) reconsider A = Ball (p,r) as Subset of (TopSpaceMetr M) by TOPMETR:12; assume ( p = q & r > 0 ) ; ::_thesis: Ball (p,r) is a_neighborhood of q then q in A by Th1; hence Ball (p,r) is a_neighborhood of q by CONNSP_2:3, TOPMETR:14; ::_thesis: verum end; theorem :: GOBOARD6:92 for M being non empty MetrSpace for A being Subset of (TopSpaceMetr M) for p being Point of M holds ( p in Cl A iff for r being real number st r > 0 holds Ball (p,r) meets A ) proof let M be non empty MetrSpace; ::_thesis: for A being Subset of (TopSpaceMetr M) for p being Point of M holds ( p in Cl A iff for r being real number st r > 0 holds Ball (p,r) meets A ) let A be Subset of (TopSpaceMetr M); ::_thesis: for p being Point of M holds ( p in Cl A iff for r being real number st r > 0 holds Ball (p,r) meets A ) let p be Point of M; ::_thesis: ( p in Cl A iff for r being real number st r > 0 holds Ball (p,r) meets A ) reconsider p9 = p as Point of (TopSpaceMetr M) by TOPMETR:12; hereby ::_thesis: ( ( for r being real number st r > 0 holds Ball (p,r) meets A ) implies p in Cl A ) assume A1: p in Cl A ; ::_thesis: for r being real number st r > 0 holds Ball (p,r) meets A let r be real number ; ::_thesis: ( r > 0 implies Ball (p,r) meets A ) reconsider B = Ball (p,r) as Subset of (TopSpaceMetr M) by TOPMETR:12; assume r > 0 ; ::_thesis: Ball (p,r) meets A then B is a_neighborhood of p9 by Th91; hence Ball (p,r) meets A by A1, CONNSP_2:27; ::_thesis: verum end; assume A2: for r being real number st r > 0 holds Ball (p,r) meets A ; ::_thesis: p in Cl A for G being a_neighborhood of p9 holds G meets A proof let G be a_neighborhood of p9; ::_thesis: G meets A p in Int G by CONNSP_2:def_1; then ex r being real number st ( r > 0 & Ball (p,r) c= G ) by Th4; hence G meets A by A2, XBOOLE_1:63; ::_thesis: verum end; hence p in Cl A by CONNSP_2:27; ::_thesis: verum end; theorem :: GOBOARD6:93 for n being Nat for A being Subset of (TOP-REAL n) for p being Point of (TOP-REAL n) for p9 being Point of (Euclid n) st p = p9 holds for s being real number st s > 0 holds ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) proof let n be Nat; ::_thesis: for A being Subset of (TOP-REAL n) for p being Point of (TOP-REAL n) for p9 being Point of (Euclid n) st p = p9 holds for s being real number st s > 0 holds ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) let A be Subset of (TOP-REAL n); ::_thesis: for p being Point of (TOP-REAL n) for p9 being Point of (Euclid n) st p = p9 holds for s being real number st s > 0 holds ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) let p be Point of (TOP-REAL n); ::_thesis: for p9 being Point of (Euclid n) st p = p9 holds for s being real number st s > 0 holds ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) let p9 be Point of (Euclid n); ::_thesis: ( p = p9 implies for s being real number st s > 0 holds ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) ) assume A1: p = p9 ; ::_thesis: for s being real number st s > 0 holds ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) let s be real number ; ::_thesis: ( s > 0 implies ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) ) assume A2: s > 0 ; ::_thesis: ( p in Cl A iff for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) hereby ::_thesis: ( ( for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ) implies p in Cl A ) assume A3: p in Cl A ; ::_thesis: for r being real number st 0 < r & r < s holds Ball (p9,r) meets A let r be real number ; ::_thesis: ( 0 < r & r < s implies Ball (p9,r) meets A ) assume that A4: 0 < r and r < s ; ::_thesis: Ball (p9,r) meets A reconsider B = Ball (p9,r) as Subset of (TOP-REAL n) by TOPREAL3:8; B is a_neighborhood of p by A1, A4, Th2; hence Ball (p9,r) meets A by A3, CONNSP_2:27; ::_thesis: verum end; reconsider s1 = s as Real by XREAL_0:def_1; assume A5: for r being real number st 0 < r & r < s holds Ball (p9,r) meets A ; ::_thesis: p in Cl A for G being a_neighborhood of p holds G meets A proof let G be a_neighborhood of p; ::_thesis: G meets A p in Int G by CONNSP_2:def_1; then consider r9 being real number such that A6: r9 > 0 and A7: Ball (p9,r9) c= G by A1, Th5; set r = min (r9,(s1 / 2)); reconsider rr = min (r9,(s1 / 2)), rr9 = r9 as Real by XREAL_0:def_1; Ball (p9,rr) c= Ball (p9,rr9) by PCOMPS_1:1, XXREAL_0:17; then A8: Ball (p9,(min (r9,(s1 / 2)))) c= G by A7, XBOOLE_1:1; ( s1 / 2 < s1 & min (r9,(s1 / 2)) <= s1 / 2 ) by A2, XREAL_1:216, XXREAL_0:17; then A9: min (r9,(s1 / 2)) < s by XXREAL_0:2; s1 / 2 > 0 by A2, XREAL_1:215; then min (r9,(s1 / 2)) > 0 by A6, XXREAL_0:15; hence G meets A by A5, A8, A9, XBOOLE_1:63; ::_thesis: verum end; hence p in Cl A by CONNSP_2:27; ::_thesis: verum end;