:: GOBOARD3 semantic presentation begin Lm1: now__::_thesis:_for_f_being_FinSequence_of_(TOP-REAL_2) for_k_being_Element_of_NAT_st_len_f_=_k_+_1_&_k_<>_0_&_f_is_unfolded_holds_ f_|_k_is_unfolded let f be FinSequence of (TOP-REAL 2); ::_thesis: for k being Element of NAT st len f = k + 1 & k <> 0 & f is unfolded holds f | k is unfolded let k be Element of NAT ; ::_thesis: ( len f = k + 1 & k <> 0 & f is unfolded implies f | k is unfolded ) A1: dom (f | k) = Seg (len (f | k)) by FINSEQ_1:def_3; assume A2: len f = k + 1 ; ::_thesis: ( k <> 0 & f is unfolded implies f | k is unfolded ) then A3: len (f | k) = k by FINSEQ_1:59, NAT_1:11; assume k <> 0 ; ::_thesis: ( f is unfolded implies f | k is unfolded ) then A4: 0 + 1 <= k by NAT_1:13; assume A5: f is unfolded ; ::_thesis: f | k is unfolded A6: k <= k + 1 by NAT_1:11; then A7: k in dom f by A2, A4, FINSEQ_3:25; thus f | k is unfolded ::_thesis: verum proof let n be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= n or not n + 2 <= len (f | k) or (LSeg ((f | k),n)) /\ (LSeg ((f | k),(n + 1))) = {((f | k) /. (n + 1))} ) set f1 = f | k; assume that A8: 1 <= n and A9: n + 2 <= len (f | k) ; ::_thesis: (LSeg ((f | k),n)) /\ (LSeg ((f | k),(n + 1))) = {((f | k) /. (n + 1))} reconsider n = n as Element of NAT by ORDINAL1:def_12; A10: n + 1 in dom (f | k) by A8, A9, SEQ_4:135; n in dom (f | k) by A8, A9, SEQ_4:135; then A11: LSeg ((f | k),n) = LSeg (f,n) by A10, TOPREAL3:17; len (f | k) <= len f by A2, A6, FINSEQ_1:59; then A12: n + 2 <= len f by A9, XXREAL_0:2; A13: (n + 1) + 1 = n + (1 + 1) ; n + 2 in dom (f | k) by A8, A9, SEQ_4:135; then A14: LSeg ((f | k),(n + 1)) = LSeg (f,(n + 1)) by A10, A13, TOPREAL3:17; (f | k) /. (n + 1) = f /. (n + 1) by A7, A3, A1, A10, FINSEQ_4:71; hence (LSeg ((f | k),n)) /\ (LSeg ((f | k),(n + 1))) = {((f | k) /. (n + 1))} by A5, A8, A11, A14, A12, TOPREAL1:def_6; ::_thesis: verum end; end; theorem Th1: :: GOBOARD3:1 for f being FinSequence of (TOP-REAL 2) for G being Go-board st ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special holds ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) proof let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special holds ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) let G be Go-board; ::_thesis: ( ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ) defpred S1[ Element of NAT ] means for f being FinSequence of (TOP-REAL 2) st len f = $1 & ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special holds ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ); A1: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A2: S1[k] ; ::_thesis: S1[k + 1] let f be FinSequence of (TOP-REAL 2); ::_thesis: ( len f = k + 1 & ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ) assume that A3: len f = k + 1 and A4: for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) and A5: f is one-to-one and A6: f is unfolded and A7: f is s.n.c. and A8: f is special ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) percases ( k = 0 or k <> 0 ) ; supposeA9: k = 0 ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) take g = f; ::_thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) A10: dom f = {1} by A3, A9, FINSEQ_1:2, FINSEQ_1:def_3; now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g_&_n_+_1_in_dom_g_holds_ for_i1,_i2,_j1,_j2_being_Element_of_NAT_st_[i1,i2]_in_Indices_G_&_[j1,j2]_in_Indices_G_&_g_/._n_=_G_*_(i1,i2)_&_g_/._(n_+_1)_=_G_*_(j1,j2)_holds_ (abs_(i1_-_j1))_+_(abs_(i2_-_j2))_=_1 let n be Element of NAT ; ::_thesis: ( n in dom g & n + 1 in dom g implies for i1, i2, j1, j2 being Element of NAT st [i1,i2] in Indices G & [j1,j2] in Indices G & g /. n = G * (i1,i2) & g /. (n + 1) = G * (j1,j2) holds (abs (i1 - j1)) + (abs (i2 - j2)) = 1 ) assume that A11: n in dom g and A12: n + 1 in dom g ; ::_thesis: for i1, i2, j1, j2 being Element of NAT st [i1,i2] in Indices G & [j1,j2] in Indices G & g /. n = G * (i1,i2) & g /. (n + 1) = G * (j1,j2) holds (abs (i1 - j1)) + (abs (i2 - j2)) = 1 n = 1 by A10, A11, TARSKI:def_1; hence for i1, i2, j1, j2 being Element of NAT st [i1,i2] in Indices G & [j1,j2] in Indices G & g /. n = G * (i1,i2) & g /. (n + 1) = G * (j1,j2) holds (abs (i1 - j1)) + (abs (i2 - j2)) = 1 by A10, A12, TARSKI:def_1; ::_thesis: verum end; hence g is_sequence_on G by A4, GOBOARD1:def_9; ::_thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) thus ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) by A5, A6, A7, A8; ::_thesis: verum end; supposeA13: k <> 0 ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) A14: len (f | k) = k by A3, FINSEQ_1:59, NAT_1:11; set f1 = f | k; A15: f | k is unfolded by A3, A6, A13, Lm1; A16: f | k is s.n.c. by A7, GOBOARD2:7; f | k = f | (Seg k) by FINSEQ_1:def_15; then A17: f | k is one-to-one by A5, FUNCT_1:52; A18: dom G = Seg (len G) by FINSEQ_1:def_3; 1 <= len f by A3, NAT_1:11; then A19: k + 1 in dom f by A3, FINSEQ_3:25; then consider j1, j2 being Element of NAT such that A20: [j1,j2] in Indices G and A21: f /. (k + 1) = G * (j1,j2) by A4; A22: Indices G = [:(dom G),(Seg (width G)):] by MATRIX_1:def_4; then A23: j1 in dom G by A20, ZFMISC_1:87; A24: 0 + 1 <= k by A13, NAT_1:13; then A25: 1 in Seg k by FINSEQ_1:1; A26: k <= k + 1 by NAT_1:11; then A27: k in dom f by A3, A24, FINSEQ_3:25; then consider i1, i2 being Element of NAT such that A28: [i1,i2] in Indices G and A29: f /. k = G * (i1,i2) by A4; reconsider l1 = Line (G,i1), c1 = Col (G,i2) as FinSequence of (TOP-REAL 2) ; set x1 = X_axis l1; set y1 = Y_axis l1; set x2 = X_axis c1; set y2 = Y_axis c1; A30: ( dom (Y_axis l1) = Seg (len (Y_axis l1)) & len (Y_axis l1) = len l1 ) by FINSEQ_1:def_3, GOBOARD1:def_2; A31: dom (f | k) = Seg (len (f | k)) by FINSEQ_1:def_3; A32: f | k is special proof let n be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= n or not n + 1 <= len (f | k) or ((f | k) /. n) `1 = ((f | k) /. (n + 1)) `1 or ((f | k) /. n) `2 = ((f | k) /. (n + 1)) `2 ) assume that A33: 1 <= n and A34: n + 1 <= len (f | k) ; ::_thesis: ( ((f | k) /. n) `1 = ((f | k) /. (n + 1)) `1 or ((f | k) /. n) `2 = ((f | k) /. (n + 1)) `2 ) n + 1 in dom (f | k) by A33, A34, SEQ_4:134; then A35: (f | k) /. (n + 1) = f /. (n + 1) by A27, A14, A31, FINSEQ_4:71; len (f | k) <= len f by A3, A26, FINSEQ_1:59; then A36: n + 1 <= len f by A34, XXREAL_0:2; n in dom (f | k) by A33, A34, SEQ_4:134; then (f | k) /. n = f /. n by A27, A14, A31, FINSEQ_4:71; hence ( ((f | k) /. n) `1 = ((f | k) /. (n + 1)) `1 or ((f | k) /. n) `2 = ((f | k) /. (n + 1)) `2 ) by A8, A33, A35, A36, TOPREAL1:def_5; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_(f_|_k)_holds_ ex_i,_j_being_Element_of_NAT_st_ (_[i,j]_in_Indices_G_&_(f_|_k)_/._n_=_G_*_(i,j)_) let n be Element of NAT ; ::_thesis: ( n in dom (f | k) implies ex i, j being Element of NAT st ( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) ) assume A37: n in dom (f | k) ; ::_thesis: ex i, j being Element of NAT st ( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) then n in dom f by A27, A14, A31, FINSEQ_4:71; then consider i, j being Element of NAT such that A38: ( [i,j] in Indices G & f /. n = G * (i,j) ) by A4; take i = i; ::_thesis: ex j being Element of NAT st ( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) take j = j; ::_thesis: ( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) thus ( [i,j] in Indices G & (f | k) /. n = G * (i,j) ) by A27, A14, A31, A37, A38, FINSEQ_4:71; ::_thesis: verum end; then consider g1 being FinSequence of (TOP-REAL 2) such that A39: g1 is_sequence_on G and A40: g1 is one-to-one and A41: g1 is unfolded and A42: g1 is s.n.c. and A43: g1 is special and A44: L~ g1 = L~ (f | k) and A45: g1 /. 1 = (f | k) /. 1 and A46: g1 /. (len g1) = (f | k) /. (len (f | k)) and A47: len (f | k) <= len g1 by A2, A14, A17, A15, A16, A32; A48: for n being Element of NAT st n in dom g1 & n + 1 in dom g1 holds for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g1 /. n = G * (m,k) & g1 /. (n + 1) = G * (i,j) holds (abs (m - i)) + (abs (k - j)) = 1 by A39, GOBOARD1:def_9; A49: ( 1 < k implies rng g1 c= L~ (f | k) ) proof assume 1 < k ; ::_thesis: rng g1 c= L~ (f | k) then A50: 1 + 1 <= k by NAT_1:13; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g1 or x in L~ (f | k) ) assume x in rng g1 ; ::_thesis: x in L~ (f | k) then ex n being Element of NAT st ( n in dom g1 & g1 /. n = x ) by PARTFUN2:2; hence x in L~ (f | k) by A14, A44, A47, A50, GOBOARD1:1, XXREAL_0:2; ::_thesis: verum end; A51: k in Seg k by A24, FINSEQ_1:1; A52: ( k = 1 implies ( L~ g1 = {} & rng g1 = {(f /. k)} ) ) proof A53: g1 /. (len g1) = f /. k by A27, A14, A51, A46, FINSEQ_4:71; assume A54: k = 1 ; ::_thesis: ( L~ g1 = {} & rng g1 = {(f /. k)} ) hence L~ g1 = {} by A14, A44, TOPREAL1:22; ::_thesis: rng g1 = {(f /. k)} then A55: ( len g1 = 1 or len g1 = 0 ) by TOPREAL1:22; A56: rng g1 c= {(f /. k)} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g1 or x in {(f /. k)} ) assume x in rng g1 ; ::_thesis: x in {(f /. k)} then consider n being Element of NAT such that A57: n in dom g1 and A58: g1 /. n = x by PARTFUN2:2; n in Seg (len g1) by A57, FINSEQ_1:def_3; then n = len g1 by A55, FINSEQ_1:2, TARSKI:def_1; hence x in {(f /. k)} by A53, A58, TARSKI:def_1; ::_thesis: verum end; 1 <= len g1 by A3, A47, A54, FINSEQ_1:59; then len g1 in dom g1 by FINSEQ_3:25; then f /. k in rng g1 by A53, PARTFUN2:2; then {(f /. k)} c= rng g1 by ZFMISC_1:31; hence rng g1 = {(f /. k)} by A56, XBOOLE_0:def_10; ::_thesis: verum end; A59: len c1 = len G by MATRIX_1:def_8; then A60: dom c1 = Seg (len G) by FINSEQ_1:def_3 .= dom G by FINSEQ_1:def_3 ; A61: ( dom (Y_axis c1) = Seg (len (Y_axis c1)) & len (Y_axis c1) = len c1 ) by FINSEQ_1:def_3, GOBOARD1:def_2; A62: ( dom (X_axis l1) = Seg (len (X_axis l1)) & len (X_axis l1) = len l1 ) by FINSEQ_1:def_3, GOBOARD1:def_1; A63: dom (X_axis c1) = Seg (len (X_axis c1)) by FINSEQ_1:def_3; A64: len (X_axis c1) = len c1 by GOBOARD1:def_1; then A65: dom c1 = Seg (len (X_axis c1)) by FINSEQ_1:def_3 .= dom (X_axis c1) by FINSEQ_1:def_3 ; A66: i1 in dom G by A28, A22, ZFMISC_1:87; then A67: X_axis l1 is constant by GOBOARD1:def_4; A68: i2 in Seg (width G) by A28, A22, ZFMISC_1:87; then A69: X_axis c1 is increasing by GOBOARD1:def_7; A70: Y_axis c1 is constant by A68, GOBOARD1:def_5; A71: Y_axis l1 is increasing by A66, GOBOARD1:def_6; A72: len l1 = width G by MATRIX_1:def_7; then A73: Seg (width G) = dom l1 by FINSEQ_1:def_3; A74: j2 in Seg (width G) by A20, A22, ZFMISC_1:87; A75: for n being Element of NAT st n in dom g1 holds ex m, k being Element of NAT st ( [m,k] in Indices G & g1 /. n = G * (m,k) ) by A39, GOBOARD1:def_9; now__::_thesis:_ex_g_being_FinSequence_of_(TOP-REAL_2)_st_ (_g_is_sequence_on_G_&_g_is_one-to-one_&_g_is_unfolded_&_g_is_s.n.c._&_g_is_special_&_L~_f_=_L~_g_&_f_/._1_=_g_/._1_&_f_/._(len_f)_=_g_/._(len_g)_&_len_f_<=_len_g_) percases ( i1 = j1 or i2 = j2 ) by A8, A27, A28, A29, A19, A20, A21, GOBOARD2:11; supposeA76: i1 = j1 ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) set ppi = G * (i1,i2); set pj = G * (i1,j2); now__::_thesis:_(_(_i2_>_j2_&_ex_g_being_FinSequence_of_the_U1_of_(TOP-REAL_2)_st_ (_g_is_sequence_on_G_&_g_is_one-to-one_&_g_is_unfolded_&_g_is_s.n.c._&_g_is_special_&_L~_g_=_L~_f_&_g_/._1_=_f_/._1_&_g_/._(len_g)_=_f_/._(len_f)_&_len_f_<=_len_g_)_)_or_(_i2_=_j2_&_contradiction_)_or_(_i2_<_j2_&_ex_g_being_FinSequence_of_the_U1_of_(TOP-REAL_2)_st_ (_g_is_sequence_on_G_&_g_is_one-to-one_&_g_is_unfolded_&_g_is_s.n.c._&_g_is_special_&_L~_g_=_L~_f_&_g_/._1_=_f_/._1_&_g_/._(len_g)_=_f_/._(len_f)_&_len_f_<=_len_g_)_)_) percases ( i2 > j2 or i2 = j2 or i2 < j2 ) by XXREAL_0:1; caseA77: i2 > j2 ; ::_thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) l1 /. i2 = l1 . i2 by A68, A73, PARTFUN1:def_6; then A78: l1 /. i2 = G * (i1,i2) by A68, MATRIX_1:def_7; then A79: (Y_axis l1) . i2 = (G * (i1,i2)) `2 by A68, A30, A72, GOBOARD1:def_2; l1 /. j2 = l1 . j2 by A74, A73, PARTFUN1:def_6; then A80: l1 /. j2 = G * (i1,j2) by A74, MATRIX_1:def_7; then A81: (Y_axis l1) . j2 = (G * (i1,j2)) `2 by A74, A30, A72, GOBOARD1:def_2; then A82: (G * (i1,j2)) `2 < (G * (i1,i2)) `2 by A68, A74, A71, A30, A72, A77, A79, SEQM_3:def_1; reconsider l = i2 - j2 as Element of NAT by A77, INT_1:5; defpred S2[ Nat, set ] means for m being Element of NAT st m = i2 - $1 holds $2 = G * (i1,m); set lk = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } ; A83: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:53; A84: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_Seg_l_holds_ (_i2_-_n_is_Element_of_NAT_&_[i1,(i2_-_n)]_in_Indices_G_&_i2_-_n_in_Seg_(width_G)_) let n be Element of NAT ; ::_thesis: ( n in Seg l implies ( i2 - n is Element of NAT & [i1,(i2 - n)] in Indices G & i2 - n in Seg (width G) ) ) assume n in Seg l ; ::_thesis: ( i2 - n is Element of NAT & [i1,(i2 - n)] in Indices G & i2 - n in Seg (width G) ) then A85: n <= l by FINSEQ_1:1; l <= i2 by XREAL_1:43; then reconsider w = i2 - n as Element of NAT by A85, INT_1:5, XXREAL_0:2; ( i2 - n <= i2 & i2 <= width G ) by A68, FINSEQ_1:1, XREAL_1:43; then A86: w <= width G by XXREAL_0:2; A87: 1 <= j2 by A74, FINSEQ_1:1; i2 - l <= i2 - n by A85, XREAL_1:13; then 1 <= w by A87, XXREAL_0:2; then w in Seg (width G) by A86, FINSEQ_1:1; hence ( i2 - n is Element of NAT & [i1,(i2 - n)] in Indices G & i2 - n in Seg (width G) ) by A22, A66, ZFMISC_1:87; ::_thesis: verum end; A88: now__::_thesis:_for_n_being_Nat_st_n_in_Seg_l_holds_ ex_p_being_Element_of_the_U1_of_(TOP-REAL_2)_st_S2[n,p] let n be Nat; ::_thesis: ( n in Seg l implies ex p being Element of the U1 of (TOP-REAL 2) st S2[n,p] ) assume n in Seg l ; ::_thesis: ex p being Element of the U1 of (TOP-REAL 2) st S2[n,p] then reconsider m = i2 - n as Element of NAT by A84; take p = G * (i1,m); ::_thesis: S2[n,p] thus S2[n,p] ; ::_thesis: verum end; consider g2 being FinSequence of (TOP-REAL 2) such that A89: ( len g2 = l & ( for n being Nat st n in Seg l holds S2[n,g2 /. n] ) ) from FINSEQ_4:sch_1(A88); take g = g1 ^ g2; ::_thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A90: dom g2 = Seg l by A89, FINSEQ_1:def_3; now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_holds_ ex_k,_m_being_Element_of_NAT_st_ (_[k,m]_in_Indices_G_&_g2_/._n_=_G_*_(k,m)_) let n be Element of NAT ; ::_thesis: ( n in dom g2 implies ex k, m being Element of NAT st ( [k,m] in Indices G & g2 /. n = G * (k,m) ) ) assume A91: n in dom g2 ; ::_thesis: ex k, m being Element of NAT st ( [k,m] in Indices G & g2 /. n = G * (k,m) ) then reconsider m = i2 - n as Element of NAT by A84, A90; take k = i1; ::_thesis: ex m being Element of NAT st ( [k,m] in Indices G & g2 /. n = G * (k,m) ) take m = m; ::_thesis: ( [k,m] in Indices G & g2 /. n = G * (k,m) ) thus ( [k,m] in Indices G & g2 /. n = G * (k,m) ) by A84, A89, A90, A91; ::_thesis: verum end; then A92: for n being Element of NAT st n in dom g holds ex i, j being Element of NAT st ( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:23; A93: dom g2 = Seg (len g2) by FINSEQ_1:def_3; A94: (X_axis l1) . i2 = (G * (i1,i2)) `1 by A68, A62, A72, A78, GOBOARD1:def_1; A95: now__::_thesis:_for_n_being_Element_of_NAT_ for_p_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_g2_/._n_=_p_holds_ (_p_`1_=_(G_*_(i1,i2))_`1_&_(G_*_(i1,j2))_`2_<=_p_`2_&_p_`2_<=_(G_*_(i1,i2))_`2_&_p_in_rng_l1_&_p_`2_<_(G_*_(i1,i2))_`2_) let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 & p `2 < (G * (i1,i2)) `2 ) let p be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & g2 /. n = p implies ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 & p `2 < (G * (i1,i2)) `2 ) ) assume that A96: n in dom g2 and A97: g2 /. n = p ; ::_thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 & p `2 < (G * (i1,i2)) `2 ) reconsider n1 = i2 - n as Element of NAT by A84, A90, A96; n <= len g2 by A96, FINSEQ_3:25; then A98: i2 - (len g2) <= n1 by XREAL_1:13; set pn = G * (i1,n1); A99: g2 /. n = G * (i1,n1) by A89, A93, A96; A100: i2 - n in Seg (width G) by A84, A89, A93, A96; then A101: (X_axis l1) . n1 = (X_axis l1) . i2 by A68, A67, A62, A72, SEQM_3:def_10; l1 /. n1 = l1 . n1 by A73, A100, PARTFUN1:def_6; then A102: l1 /. n1 = G * (i1,n1) by A100, MATRIX_1:def_7; then A103: (Y_axis l1) . n1 = (G * (i1,n1)) `2 by A30, A72, A100, GOBOARD1:def_2; (X_axis l1) . n1 = (G * (i1,n1)) `1 by A62, A72, A100, A102, GOBOARD1:def_1; hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 ) by A68, A74, A71, A30, A72, A89, A79, A81, A94, A97, A100, A99, A98, A101, A103, SEQ_4:137, XREAL_1:43; ::_thesis: ( p in rng l1 & p `2 < (G * (i1,i2)) `2 ) dom l1 = Seg (len l1) by FINSEQ_1:def_3; hence p in rng l1 by A72, A97, A100, A99, A102, PARTFUN2:2; ::_thesis: p `2 < (G * (i1,i2)) `2 1 <= n by A96, FINSEQ_3:25; then n1 < i2 by XREAL_1:44; hence p `2 < (G * (i1,i2)) `2 by A68, A71, A30, A72, A79, A97, A100, A99, A103, SEQM_3:def_1; ::_thesis: verum end; A104: g2 is special proof let n be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) set p = g2 /. n; assume A105: ( 1 <= n & n + 1 <= len g2 ) ; ::_thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) then n in dom g2 by SEQ_4:134; then A106: (g2 /. n) `1 = (G * (i1,i2)) `1 by A95; n + 1 in dom g2 by A105, SEQ_4:134; hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A95, A106; ::_thesis: verum end; A107: now__::_thesis:_for_n,_m_being_Element_of_NAT_ for_p,_q_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<_m_&_g2_/._n_=_p_&_g2_/._m_=_q_holds_ q_`2_<_p_`2 let n, m be Element of NAT ; ::_thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds q `2 < p `2 let p, q be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies q `2 < p `2 ) assume that A108: n in dom g2 and A109: m in dom g2 and A110: n < m and A111: ( g2 /. n = p & g2 /. m = q ) ; ::_thesis: q `2 < p `2 A112: i2 - n in Seg (width G) by A84, A90, A108; reconsider n1 = i2 - n, m1 = i2 - m as Element of NAT by A84, A90, A108, A109; set pn = G * (i1,n1); set pm = G * (i1,m1); A113: m1 < n1 by A110, XREAL_1:15; l1 /. n1 = l1 . n1 by A73, A84, A90, A108, PARTFUN1:def_6; then l1 /. n1 = G * (i1,n1) by A112, MATRIX_1:def_7; then A114: (Y_axis l1) . n1 = (G * (i1,n1)) `2 by A30, A72, A112, GOBOARD1:def_2; A115: i2 - m in Seg (width G) by A84, A90, A109; l1 /. m1 = l1 . m1 by A73, A84, A90, A109, PARTFUN1:def_6; then l1 /. m1 = G * (i1,m1) by A115, MATRIX_1:def_7; then A116: (Y_axis l1) . m1 = (G * (i1,m1)) `2 by A30, A72, A115, GOBOARD1:def_2; ( g2 /. n = G * (i1,n1) & g2 /. m = G * (i1,m1) ) by A89, A90, A108, A109; hence q `2 < p `2 by A71, A30, A72, A111, A112, A115, A113, A114, A116, SEQM_3:def_1; ::_thesis: verum end; for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds LSeg (g2,n) misses LSeg (g2,m) proof let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) ) assume that A117: m > n + 1 and A118: n in dom g2 and A119: n + 1 in dom g2 and A120: m in dom g2 and A121: m + 1 in dom g2 and A122: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A123: ( p1 `1 = (G * (i1,i2)) `1 & p2 `1 = (G * (i1,i2)) `1 ) by A95, A118, A119; n < n + 1 by NAT_1:13; then A124: p2 `2 < p1 `2 by A107, A118, A119; set lp = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p2 `2 <= w `2 & w `2 <= p1 `2 ) } ; set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } ; A125: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:53; m < m + 1 by NAT_1:13; then A126: q2 `2 < q1 `2 by A107, A120, A121; A127: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; set x = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)); A128: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,n) by A122, XBOOLE_0:def_4; A129: ( q1 `1 = (G * (i1,i2)) `1 & q2 `1 = (G * (i1,i2)) `1 ) by A95, A120, A121; A130: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,m) by A122, XBOOLE_0:def_4; ( 1 <= m & m + 1 <= len g2 ) by A120, A121, FINSEQ_3:25; then LSeg (g2,m) = LSeg (q2,q1) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } by A126, A129, A127, TOPREAL3:9 ; then A131: ex tm being Point of (TOP-REAL 2) st ( tm = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tm `1 = (G * (i1,i2)) `1 & q2 `2 <= tm `2 & tm `2 <= q1 `2 ) by A130; ( 1 <= n & n + 1 <= len g2 ) by A118, A119, FINSEQ_3:25; then LSeg (g2,n) = LSeg (p2,p1) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p2 `2 <= w `2 & w `2 <= p1 `2 ) } by A124, A123, A125, TOPREAL3:9 ; then A132: ex tn being Point of (TOP-REAL 2) st ( tn = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tn `1 = (G * (i1,i2)) `1 & p2 `2 <= tn `2 & tn `2 <= p1 `2 ) by A128; q1 `2 < p2 `2 by A107, A117, A119, A120; hence contradiction by A132, A131, XXREAL_0:2; ::_thesis: verum end; then A133: g2 is s.n.c. by GOBOARD2:1; A134: not f /. k in L~ g2 proof set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume f /. k in L~ g2 ; ::_thesis: contradiction then consider X being set such that A135: f /. k in X and A136: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A137: X = LSeg (g2,m) and A138: ( 1 <= m & m + 1 <= len g2 ) by A136; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A139: m in dom g2 by A138, SEQ_4:134; then A140: q1 `1 = (G * (i1,i2)) `1 by A95; set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } ; A141: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; A142: m + 1 in dom g2 by A138, SEQ_4:134; then A143: q2 `1 = (G * (i1,i2)) `1 by A95; m < m + 1 by NAT_1:13; then A144: q2 `2 < q1 `2 by A107, A139, A142; LSeg (g2,m) = LSeg (q2,q1) by A138, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q2 `2 <= w `2 & w `2 <= q1 `2 ) } by A140, A143, A144, A141, TOPREAL3:9 ; then ex p being Point of (TOP-REAL 2) st ( p = f /. k & p `1 = (G * (i1,i2)) `1 & q2 `2 <= p `2 & p `2 <= q1 `2 ) by A135, A137; hence contradiction by A29, A95, A139; ::_thesis: verum end; (X_axis l1) . j2 = (G * (i1,j2)) `1 by A74, A62, A72, A80, GOBOARD1:def_1; then A145: (G * (i1,i2)) `1 = (G * (i1,j2)) `1 by A68, A74, A67, A62, A72, A94, SEQM_3:def_10; now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<>_m_holds_ not_g2_/._n_=_g2_/._m let n, m be Element of NAT ; ::_thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m ) assume that A146: ( n in dom g2 & m in dom g2 ) and A147: n <> m ; ::_thesis: not g2 /. n = g2 /. m reconsider n1 = i2 - n, m1 = i2 - m as Element of NAT by A84, A90, A146; A148: ( g2 /. n = G * (i1,n1) & g2 /. m = G * (i1,m1) ) by A89, A90, A146; assume A149: g2 /. n = g2 /. m ; ::_thesis: contradiction ( [i1,(i2 - n)] in Indices G & [i1,(i2 - m)] in Indices G ) by A84, A90, A146; then n1 = m1 by A148, A149, GOBOARD1:5; hence contradiction by A147; ::_thesis: verum end; then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds n = m ; then A150: g2 is one-to-one by PARTFUN2:9; reconsider m1 = i2 - l as Element of NAT ; A151: G * (i1,j2) = |[((G * (i1,j2)) `1),((G * (i1,j2)) `2)]| by EUCLID:53; A152: LSeg (f,k) = LSeg ((G * (i1,j2)),(G * (i1,i2))) by A3, A24, A29, A21, A76, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A82, A145, A83, A151, TOPREAL3:9 ; A153: rng g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g2 or x in LSeg (f,k) ) assume x in rng g2 ; ::_thesis: x in LSeg (f,k) then consider n being Element of NAT such that A154: n in dom g2 and A155: g2 /. n = x by PARTFUN2:2; reconsider n1 = i2 - n as Element of NAT by A84, A89, A93, A154; set pn = G * (i1,n1); A156: g2 /. n = G * (i1,n1) by A89, A93, A154; then A157: (G * (i1,n1)) `2 <= (G * (i1,i2)) `2 by A95, A154; ( (G * (i1,n1)) `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= (G * (i1,n1)) `2 ) by A95, A154, A156; hence x in LSeg (f,k) by A152, A155, A156, A157; ::_thesis: verum end; A158: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_&_n_+_1_in_dom_g2_holds_ for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g2_/._n_=_G_*_(l1,l2)_&_g2_/._(n_+_1)_=_G_*_(l3,l4)_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let n be Element of NAT ; ::_thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A159: n in dom g2 and A160: n + 1 in dom g2 ; ::_thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 reconsider m1 = i2 - n, m2 = i2 - (n + 1) as Element of NAT by A84, A90, A159, A160; let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A161: [l1,l2] in Indices G and A162: [l3,l4] in Indices G and A163: g2 /. n = G * (l1,l2) and A164: g2 /. (n + 1) = G * (l3,l4) ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ( [i1,(i2 - (n + 1))] in Indices G & g2 /. (n + 1) = G * (i1,m2) ) by A84, A89, A90, A160; then A165: ( l3 = i1 & l4 = m2 ) by A162, A164, GOBOARD1:5; ( [i1,(i2 - n)] in Indices G & g2 /. n = G * (i1,m1) ) by A84, A89, A90, A159; then ( l1 = i1 & l2 = m1 ) by A161, A163, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs ((i2 - n) - (i2 - (n + 1)))) by A165, ABSVALUE:2 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; now__::_thesis:_for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g1_/._(len_g1)_=_G_*_(l1,l2)_&_g2_/._1_=_G_*_(l3,l4)_&_len_g1_in_dom_g1_&_1_in_dom_g2_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A166: [l1,l2] in Indices G and A167: [l3,l4] in Indices G and A168: g1 /. (len g1) = G * (l1,l2) and A169: g2 /. 1 = G * (l3,l4) and len g1 in dom g1 and A170: 1 in dom g2 ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 reconsider m1 = i2 - 1 as Element of NAT by A84, A90, A170; ( [i1,(i2 - 1)] in Indices G & g2 /. 1 = G * (i1,m1) ) by A84, A89, A90, A170; then A171: ( l3 = i1 & l4 = m1 ) by A167, A169, GOBOARD1:5; (f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:71; then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A166, A168, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs (i2 - (i2 - 1))) by A171, ABSVALUE:2 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; then for n being Element of NAT st n in dom g & n + 1 in dom g holds for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds (abs (m - i)) + (abs (k - j)) = 1 by A48, A158, GOBOARD1:24; hence g is_sequence_on G by A92, GOBOARD1:def_9; ::_thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A172: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (i1,j2))) by A3, A24, A29, A21, A76, TOPREAL1:def_3; A173: not f /. k in rng g2 proof assume f /. k in rng g2 ; ::_thesis: contradiction then consider n being Element of NAT such that A174: n in dom g2 and A175: g2 /. n = f /. k by PARTFUN2:2; reconsider n1 = i2 - n as Element of NAT by A84, A89, A93, A174; ( [i1,(i2 - n)] in Indices G & g2 /. n = G * (i1,n1) ) by A84, A89, A93, A174; then A176: n1 = i2 by A28, A29, A175, GOBOARD1:5; 0 < n by A93, A174, FINSEQ_1:1; hence contradiction by A176; ::_thesis: verum end; (rng g1) /\ (rng g2) = {} proof set x = the Element of (rng g1) /\ (rng g2); assume A177: not (rng g1) /\ (rng g2) = {} ; ::_thesis: contradiction then A178: the Element of (rng g1) /\ (rng g2) in rng g2 by XBOOLE_0:def_4; A179: the Element of (rng g1) /\ (rng g2) in rng g1 by A177, XBOOLE_0:def_4; now__::_thesis:_contradiction percases ( k = 1 or 1 < k ) by A24, XXREAL_0:1; suppose k = 1 ; ::_thesis: contradiction hence contradiction by A52, A173, A179, A178, TARSKI:def_1; ::_thesis: verum end; suppose 1 < k ; ::_thesis: contradiction then ( the Element of (rng g1) /\ (rng g2) in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A153, A179, A178, GOBOARD2:4, XBOOLE_0:def_4; hence contradiction by A173, A178, TARSKI:def_1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; then rng g1 misses rng g2 by XBOOLE_0:def_7; hence g is one-to-one by A40, A150, FINSEQ_3:91; ::_thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A180: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} proof let n be Element of NAT ; ::_thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} ) assume that A181: 1 <= n and A182: n + 2 <= len g2 ; ::_thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} A183: n + 1 in dom g2 by A181, A182, SEQ_4:135; then g2 /. (n + 1) in rng g2 by PARTFUN2:2; then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153; then consider u1 being Point of (TOP-REAL 2) such that A184: g2 /. (n + 1) = u1 and A185: u1 `1 = (G * (i1,i2)) `1 and (G * (i1,j2)) `2 <= u1 `2 and u1 `2 <= (G * (i1,i2)) `2 ; A186: n + 2 in dom g2 by A181, A182, SEQ_4:135; then g2 /. (n + 2) in rng g2 by PARTFUN2:2; then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153; then consider u2 being Point of (TOP-REAL 2) such that A187: g2 /. (n + 2) = u2 and A188: u2 `1 = (G * (i1,i2)) `1 and (G * (i1,j2)) `2 <= u2 `2 and u2 `2 <= (G * (i1,i2)) `2 ; ( n + (1 + 1) = (n + 1) + 1 & 1 <= n + 1 ) by NAT_1:11; then A189: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A182, A184, A187, TOPREAL1:def_3; n + 1 < (n + 1) + 1 by NAT_1:13; then A190: u2 `2 < u1 `2 by A107, A183, A186, A184, A187; A191: n in dom g2 by A181, A182, SEQ_4:135; then g2 /. n in rng g2 by PARTFUN2:2; then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153; then consider u being Point of (TOP-REAL 2) such that A192: g2 /. n = u and A193: u `1 = (G * (i1,i2)) `1 and (G * (i1,j2)) `2 <= u `2 and u `2 <= (G * (i1,i2)) `2 ; n + 1 <= n + 2 by XREAL_1:6; then n + 1 <= len g2 by A182, XXREAL_0:2; then A194: LSeg (g2,n) = LSeg (u,u1) by A181, A192, A184, TOPREAL1:def_3; set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= u `2 ) } ; n < n + 1 by NAT_1:13; then A195: u1 `2 < u `2 by A107, A191, A183, A192, A184; then A196: u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= u `2 ) } by A185, A190; ( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:53; then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= u `2 ) } by A192, A193, A187, A188, A190, A195, TOPREAL3:9, XXREAL_0:2; hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A192, A184, A187, A194, A189, A196, TOPREAL1:8; ::_thesis: verum end; thus g is unfolded ::_thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof let n be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ) assume that A197: 1 <= n and A198: n + 2 <= len g ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A199: (n + 1) + 1 <= len g by A198; A200: n + (1 + 1) = (n + 1) + 1 ; A201: n <= n + 1 by NAT_1:11; n + 1 <= (n + 1) + 1 by NAT_1:11; then A202: n + 1 <= len g by A198, XXREAL_0:2; A203: len g = (len g1) + (len g2) by FINSEQ_1:22; (n + 2) - (len g1) = (n - (len g1)) + 2 ; then A204: (n - (len g1)) + 2 <= len g2 by A198, A203, XREAL_1:20; A205: ( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11; percases ( n + 2 <= len g1 or len g1 < n + 2 ) ; supposeA206: n + 2 <= len g1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A207: n + (1 + 1) = (n + 1) + 1 ; A208: n + 1 in dom g1 by A197, A206, SEQ_4:135; then A209: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:68; n in dom g1 by A197, A206, SEQ_4:135; then A210: LSeg (g1,n) = LSeg (g,n) by A208, TOPREAL3:18; n + 2 in dom g1 by A197, A206, SEQ_4:135; then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A208, A207, TOPREAL3:18; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A197, A206, A210, A209, TOPREAL1:def_6; ::_thesis: verum end; suppose len g1 < n + 2 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then (len g1) + 1 <= n + 2 by NAT_1:13; then A211: len g1 <= (n + 2) - 1 by XREAL_1:19; now__::_thesis:_(LSeg_(g,n))_/\_(LSeg_(g,(n_+_1)))_=_{(g_/._(n_+_1))} percases ( len g1 = n + 1 or len g1 <> n + 1 ) ; supposeA212: len g1 = n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} now__::_thesis:_not_k_=_1 1 < len g1 by A197, A212, NAT_1:13; then A213: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A213, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A214: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; ( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A197, A198, A202, A205, TOPREAL1:21; then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def_4; then A215: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:31; A216: 1 <= (len g) - (len g1) by A199, A212, XREAL_1:19; then 1 in dom g2 by A203, FINSEQ_3:25; then A217: g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153; then consider u1 being Point of (TOP-REAL 2) such that A218: g2 /. 1 = u1 and u1 `1 = (G * (i1,i2)) `1 and (G * (i1,j2)) `2 <= u1 `2 and u1 `2 <= (G * (i1,i2)) `2 ; G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (i1,j2))) by RLTOPSP1:68; then A219: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A172, A153, A217, A218, TOPREAL1:6; 1 <= n + 1 by NAT_1:11; then A220: n + 1 in dom g1 by A212, FINSEQ_3:25; then A221: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A212, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; n in dom g1 by A197, A201, A212, FINSEQ_3:25; then A222: LSeg (g,n) = LSeg (g1,n) by A220, TOPREAL3:18; g /. (n + 2) = g2 /. 1 by A200, A203, A212, A216, SEQ_4:136; then A223: LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A198, A205, A221, A218, TOPREAL1:def_3; LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:19; then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A214, A222, A221, A219, A223, XBOOLE_1:27; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A215, XBOOLE_0:def_10; ::_thesis: verum end; suppose len g1 <> n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then len g1 < n + 1 by A211, XXREAL_0:1; then A224: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_(LSeg_(g,n))_/\_(LSeg_(g,(n_+_1)))_=_{(g_/._(n_+_1))} percases ( len g1 = n or len g1 <> n ) ; supposeA225: len g1 = n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then 1 <= len g2 by A202, A203, XREAL_1:6; then A226: g /. (n + 1) = g2 /. 1 by A225, SEQ_4:136; A227: 0 + 2 <= len g2 by A198, A203, A225, XREAL_1:6; then 1 <= len g2 by XXREAL_0:2; then A228: 1 in dom g2 by FINSEQ_3:25; then g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153; then consider u1 being Point of (TOP-REAL 2) such that A229: g2 /. 1 = u1 and A230: u1 `1 = (G * (i1,i2)) `1 and (G * (i1,j2)) `2 <= u1 `2 and A231: u1 `2 <= (G * (i1,i2)) `2 ; A232: 2 in dom g2 by A227, FINSEQ_3:25; then g2 /. 2 in rng g2 by PARTFUN2:2; then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A152, A153; then consider u2 being Point of (TOP-REAL 2) such that A233: g2 /. 2 = u2 and A234: u2 `1 = (G * (i1,i2)) `1 and (G * (i1,j2)) `2 <= u2 `2 and A235: u2 `2 <= (G * (i1,i2)) `2 ; set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } ; u2 `2 < u1 `2 by A107, A228, A232, A229, A233; then A236: u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A230, A231; u2 = |[(u2 `1),(u2 `2)]| by EUCLID:53; then A237: LSeg ((G * (i1,i2)),(g2 /. 2)) = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u2 `2 <= w `2 & w `2 <= (G * (i1,i2)) `2 ) } by A83, A233, A234, A235, TOPREAL3:9; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then g /. n = (f | k) /. (len (f | k)) by A46, A225, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; then A238: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A197, A202, A226, A229, TOPREAL1:def_3; 2 <= len g2 by A198, A203, A225, XREAL_1:6; then g /. (n + 2) = g2 /. 2 by A225, SEQ_4:136; then LSeg (g,(n + 1)) = LSeg (u1,u2) by A198, A205, A226, A229, A233, TOPREAL1:def_3; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A226, A229, A233, A236, A238, A237, TOPREAL1:8; ::_thesis: verum end; suppose len g1 <> n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A239: len g1 < n by A224, XXREAL_0:1; then (len g1) + 1 <= n by NAT_1:13; then A240: 1 <= n1 by XREAL_1:19; n1 + (len g1) = n ; then A241: LSeg (g,n) = LSeg (g2,n1) by A202, A239, GOBOARD2:5; A242: n + 1 = (n1 + 1) + (len g1) ; (n1 + 1) + (len g1) = n + 1 ; then n1 + 1 <= len g2 by A202, A203, XREAL_1:6; then A243: g /. (n + 1) = g2 /. (n1 + 1) by A242, NAT_1:11, SEQ_4:136; len g1 < n + 1 by A201, A239, XXREAL_0:2; then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A199, A242, GOBOARD2:5; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A180, A204, A241, A243, A240; ::_thesis: verum end; end; end; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; ::_thesis: verum end; end; end; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; ::_thesis: verum end; end; end; A244: L~ g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g2 or x in LSeg (f,k) ) set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume x in L~ g2 ; ::_thesis: x in LSeg (f,k) then consider X being set such that A245: x in X and A246: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A247: X = LSeg (g2,m) and A248: ( 1 <= m & m + 1 <= len g2 ) by A246; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A249: LSeg (g2,m) = LSeg (q1,q2) by A248, TOPREAL1:def_3; m + 1 in dom g2 by A248, SEQ_4:134; then A250: g2 /. (m + 1) in rng g2 by PARTFUN2:2; m in dom g2 by A248, SEQ_4:134; then g2 /. m in rng g2 by PARTFUN2:2; then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (i1,j2))) by A172, A153, A250, TOPREAL1:6; hence x in LSeg (f,k) by A172, A245, A247, A249; ::_thesis: verum end; A251: (L~ g1) /\ (L~ g2) = {} proof percases ( k = 1 or k <> 1 ) ; suppose k = 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} hence (L~ g1) /\ (L~ g2) = {} by A52; ::_thesis: verum end; suppose k <> 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} then 1 < k by A24, XXREAL_0:1; then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:4; then A252: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A244, XBOOLE_1:26; now__::_thesis:_not_(L~_g1)_/\_(L~_g2)_<>_{} set x = the Element of (L~ g1) /\ (L~ g2); assume (L~ g1) /\ (L~ g2) <> {} ; ::_thesis: contradiction then ( the Element of (L~ g1) /\ (L~ g2) in {(f /. k)} & the Element of (L~ g1) /\ (L~ g2) in L~ g2 ) by A252, TARSKI:def_3, XBOOLE_0:def_4; hence contradiction by A134, TARSKI:def_1; ::_thesis: verum end; hence (L~ g1) /\ (L~ g2) = {} ; ::_thesis: verum end; end; end; for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds LSeg (g,n) misses LSeg (g,m) proof A253: 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A254: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:71 ; reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ; set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ; set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ; let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) ) assume that A255: m > n + 1 and A256: n in dom g and A257: n + 1 in dom g and A258: m in dom g and A259: m + 1 in dom g ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A260: 1 <= n by A256, FINSEQ_3:25; j2 + 1 <= i2 by A77, NAT_1:13; then A261: 1 <= l by XREAL_1:19; then A262: 1 in dom g2 by A89, FINSEQ_3:25; then A263: ( qq `1 = (G * (i1,i2)) `1 & qq `2 < (G * (i1,i2)) `2 ) by A95; A264: g /. ((len g1) + 1) = qq by A89, A261, SEQ_4:136; A265: (G * (i1,j2)) `2 <= qq `2 by A95, A262; A266: m + 1 <= len g by A259, FINSEQ_3:25; A267: 1 <= m + 1 by A259, FINSEQ_3:25; A268: 1 <= n + 1 by A257, FINSEQ_3:25; A269: n + 1 <= len g by A257, FINSEQ_3:25; A270: qq = |[(qq `1),(qq `2)]| by EUCLID:53; A271: 1 <= m by A258, FINSEQ_3:25; set ql = { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & qq `2 <= z `2 & z `2 <= (G * (i1,i2)) `2 ) } ; A272: n <= n + 1 by NAT_1:11; A273: len g = (len g1) + (len g2) by FINSEQ_1:22; then (len g1) + 1 <= len g by A89, A261, XREAL_1:7; then A274: LSeg (g,(len g1)) = LSeg (qq,(G * (i1,i2))) by A253, A254, A264, TOPREAL1:def_3 .= { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & qq `2 <= z `2 & z `2 <= (G * (i1,i2)) `2 ) } by A83, A263, A270, TOPREAL3:9 ; A275: m <= m + 1 by NAT_1:11; then A276: n + 1 <= m + 1 by A255, XXREAL_0:2; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m + 1 <= len g1 or len g1 < m + 1 ) ; supposeA277: m + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then m <= len g1 by A275, XXREAL_0:2; then A278: m in dom g1 by A271, FINSEQ_3:25; m + 1 in dom g1 by A267, A277, FINSEQ_3:25; then A279: LSeg (g,m) = LSeg (g1,m) by A278, TOPREAL3:18; A280: n + 1 <= len g1 by A276, A277, XXREAL_0:2; then n <= len g1 by A272, XXREAL_0:2; then A281: n in dom g1 by A260, FINSEQ_3:25; n + 1 in dom g1 by A268, A280, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A281, TOPREAL3:18; hence LSeg (g,n) misses LSeg (g,m) by A42, A255, A279, TOPREAL1:def_7; ::_thesis: verum end; suppose len g1 < m + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A282: len g1 <= m by NAT_1:13; then reconsider m1 = m - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m = len g1 or m <> len g1 ) ; supposeA283: m = len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A284: LSeg (g,m) c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) ) assume x in LSeg (g,m) ; ::_thesis: x in LSeg (f,k) then consider px being Point of (TOP-REAL 2) such that A285: ( px = x & px `1 = (G * (i1,i2)) `1 ) and A286: qq `2 <= px `2 and A287: px `2 <= (G * (i1,i2)) `2 by A274, A283; (G * (i1,j2)) `2 <= px `2 by A265, A286, XXREAL_0:2; hence x in LSeg (f,k) by A152, A285, A287; ::_thesis: verum end; n <= len g1 by A255, A272, A283, XXREAL_0:2; then A288: n in dom g1 by A260, FINSEQ_3:25; now__::_thesis:_not_k_=_1 1 < len g1 by A255, A268, A283, XXREAL_0:2; then A289: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A289, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A290: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; A291: n + 1 in dom g1 by A255, A268, A283, FINSEQ_3:25; then A292: LSeg (g,n) = LSeg (g1,n) by A288, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A255, A260, A283; then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:74; then A293: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A290, A284, XBOOLE_1:27; now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); assume A294: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A295: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in {(f /. k)} by A293, A294, TARSKI:def_3; then A296: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) = f /. k by TARSKI:def_1; f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:71; hence contradiction by A40, A41, A42, A255, A283, A288, A291, A292, A295, A296, GOBOARD2:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose m <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A297: len g1 < m by A282, XXREAL_0:1; then (len g1) + 1 <= m by NAT_1:13; then A298: 1 <= m1 by XREAL_1:19; m + 1 = (m1 + 1) + (len g1) ; then A299: m1 + 1 <= len g2 by A266, A273, XREAL_1:6; m = m1 + (len g1) ; then A300: LSeg (g,m) = LSeg (g2,m1) by A266, A297, GOBOARD2:5; then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A298, A299; then A301: LSeg (g,m) c= L~ g2 by ZFMISC_1:74; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( n + 1 <= len g1 or len g1 < n + 1 ) ; supposeA302: n + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then n <= len g1 by A272, XXREAL_0:2; then A303: n in dom g1 by A260, FINSEQ_3:25; n + 1 in dom g1 by A268, A302, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A303, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A260, A302; then LSeg (g,n) c= L~ g1 by ZFMISC_1:74; then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A251, A301, XBOOLE_1:3, XBOOLE_1:27; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose len g1 < n + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A304: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; A305: (n - (len g1)) + 1 = (n + 1) - (len g1) ; A306: n = n1 + (len g1) ; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( len g1 = n or n <> len g1 ) ; supposeA307: len g1 = n ; ::_thesis: LSeg (g,n) misses LSeg (g,m) now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ; set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); set q1l = { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q2 `2 <= v `2 & v `2 <= q1 `2 ) } ; A308: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; assume A309: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A310: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,m) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by A309, XBOOLE_0:def_4; then A311: ex qx being Point of (TOP-REAL 2) st ( qx = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qx `1 = (G * (i1,i2)) `1 & qq `2 <= qx `2 & qx `2 <= (G * (i1,i2)) `2 ) by A274, A307; A312: m1 in dom g2 by A298, A299, SEQ_4:134; then A313: q1 `1 = (G * (i1,i2)) `1 by A95; A314: m1 + 1 in dom g2 by A298, A299, SEQ_4:134; then A315: q2 `1 = (G * (i1,i2)) `1 by A95; m1 < m1 + 1 by NAT_1:13; then A316: q2 `2 < q1 `2 by A107, A312, A314; LSeg (g2,m1) = LSeg (q2,q1) by A298, A299, TOPREAL1:def_3 .= { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q2 `2 <= v `2 & v `2 <= q1 `2 ) } by A313, A315, A316, A308, TOPREAL3:9 ; then A317: ex qy being Point of (TOP-REAL 2) st ( qy = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qy `1 = (G * (i1,i2)) `1 & q2 `2 <= qy `2 & qy `2 <= q1 `2 ) by A300, A310; ( m1 > n1 + 1 & n1 + 1 >= 1 ) by A255, A305, NAT_1:11, XREAL_1:9; then m1 > 1 by XXREAL_0:2; then q1 `2 < qq `2 by A107, A262, A312; hence contradiction by A311, A317, XXREAL_0:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose n <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then len g1 < n by A304, XXREAL_0:1; then A318: LSeg (g,n) = LSeg (g2,n1) by A269, A306, GOBOARD2:5; m1 > n1 + 1 by A255, A305, XREAL_1:9; hence LSeg (g,n) misses LSeg (g,m) by A133, A300, A318, TOPREAL1:def_7; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; hence g is s.n.c. by GOBOARD2:1; ::_thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) now__::_thesis:_(_(g1_/._(len_g1))_`1_=_(g2_/._1)_`1_or_(g1_/._(len_g1))_`2_=_(g2_/._1)_`2_) set p = g1 /. (len g1); set q = g2 /. 1; j2 + 1 <= i2 by A77, NAT_1:13; then 1 <= l by XREAL_1:19; then 1 in dom g2 by A90, FINSEQ_1:1; then (g2 /. 1) `1 = (G * (i1,i2)) `1 by A95; hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:71; ::_thesis: verum end; hence g is special by A43, A104, GOBOARD2:8; ::_thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) thus L~ g = L~ f ::_thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ; set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ; A319: len g = (len g1) + (len g2) by FINSEQ_1:22; A320: now__::_thesis:_for_j_being_Element_of_NAT_st_len_g1_<=_j_&_j_<=_len_g_holds_ for_p_being_Point_of_(TOP-REAL_2)_st_p_=_g_/._j_holds_ (_p_`1_=_(G_*_(i1,i2))_`1_&_(G_*_(i1,j2))_`2_<=_p_`2_&_p_`2_<=_(G_*_(i1,i2))_`2_&_p_in_rng_l1_) let j be Element of NAT ; ::_thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) ) assume that A321: len g1 <= j and A322: j <= len g ; ::_thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) reconsider w = j - (len g1) as Element of NAT by A321, INT_1:5; let p be Point of (TOP-REAL 2); ::_thesis: ( p = g /. j implies ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) ) assume A323: p = g /. j ; ::_thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) now__::_thesis:_(_p_`1_=_(G_*_(i1,i2))_`1_&_(G_*_(i1,j2))_`2_<=_p_`2_&_p_`2_<=_(G_*_(i1,i2))_`2_&_p_in_rng_l1_) percases ( j = len g1 or j <> len g1 ) ; supposeA324: j = len g1 ; ::_thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A325: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence p `1 = (G * (i1,i2)) `1 by A323, A324; ::_thesis: ( (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) thus ( (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 ) by A68, A74, A71, A30, A72, A77, A79, A81, A323, A324, A325, SEQM_3:def_1; ::_thesis: p in rng l1 dom l1 = Seg (len l1) by FINSEQ_1:def_3; hence p in rng l1 by A68, A72, A78, A323, A324, A325, PARTFUN2:2; ::_thesis: verum end; suppose j <> len g1 ; ::_thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) then len g1 < j by A321, XXREAL_0:1; then (len g1) + 1 <= j by NAT_1:13; then A326: 1 <= w by XREAL_1:19; A327: w <= len g2 by A319, A322, XREAL_1:20; then A328: w in dom g2 by A326, FINSEQ_3:25; w + (len g1) = j ; then g /. j = g2 /. w by A326, A327, SEQ_4:136; hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) by A95, A323, A328; ::_thesis: verum end; end; end; hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= p `2 & p `2 <= (G * (i1,i2)) `2 & p in rng l1 ) ; ::_thesis: verum end; thus L~ g c= L~ f :: according to XBOOLE_0:def_10 ::_thesis: L~ f c= L~ g proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g or x in L~ f ) assume x in L~ g ; ::_thesis: x in L~ f then consider X being set such that A329: x in X and A330: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def_4; consider i being Element of NAT such that A331: X = LSeg (g,i) and A332: 1 <= i and A333: i + 1 <= len g by A330; percases ( i + 1 <= len g1 or i + 1 > len g1 ) ; supposeA334: i + 1 <= len g1 ; ::_thesis: x in L~ f i <= i + 1 by NAT_1:11; then i <= len g1 by A334, XXREAL_0:2; then A335: i in dom g1 by A332, FINSEQ_3:25; 1 <= i + 1 by NAT_1:11; then i + 1 in dom g1 by A334, FINSEQ_3:25; then X = LSeg (g1,i) by A331, A335, TOPREAL3:18; then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A332, A334; then A336: x in L~ (f | k) by A44, A329, TARSKI:def_4; L~ (f | k) c= L~ f by TOPREAL3:20; hence x in L~ f by A336; ::_thesis: verum end; supposeA337: i + 1 > len g1 ; ::_thesis: x in L~ f reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ; A338: i <= len g by A333, NAT_1:13; A339: len g1 <= i by A337, NAT_1:13; then A340: q1 `1 = (G * (i1,i2)) `1 by A320, A338; A341: q1 `2 <= (G * (i1,i2)) `2 by A320, A339, A338; A342: (G * (i1,j2)) `2 <= q1 `2 by A320, A339, A338; q2 `1 = (G * (i1,i2)) `1 by A320, A333, A337; then A343: q2 = |[(q1 `1),(q2 `2)]| by A340, EUCLID:53; A344: q2 `2 <= (G * (i1,i2)) `2 by A320, A333, A337; A345: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A332, A333, EUCLID:53, TOPREAL1:def_3; A346: (G * (i1,j2)) `2 <= q2 `2 by A320, A333, A337; now__::_thesis:_x_in_L~_f percases ( q1 `2 > q2 `2 or q1 `2 = q2 `2 or q1 `2 < q2 `2 ) by XXREAL_0:1; suppose q1 `2 > q2 `2 ; ::_thesis: x in L~ f then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = q1 `1 & q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) } by A343, A345, TOPREAL3:9; then consider p2 being Point of (TOP-REAL 2) such that A347: ( p2 = x & p2 `1 = q1 `1 ) and A348: ( q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) by A329, A331; ( (G * (i1,j2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,i2)) `2 ) by A341, A346, A348, XXREAL_0:2; then A349: x in LSeg (f,k) by A152, A340, A347; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A349, TARSKI:def_4; ::_thesis: verum end; suppose q1 `2 = q2 `2 ; ::_thesis: x in L~ f then LSeg (g,i) = {q1} by A343, A345, RLTOPSP1:70; then x = q1 by A329, A331, TARSKI:def_1; then A350: x in LSeg (f,k) by A152, A340, A342, A341; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A350, TARSKI:def_4; ::_thesis: verum end; suppose q1 `2 < q2 `2 ; ::_thesis: x in L~ f then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = q1 `1 & q1 `2 <= p1 `2 & p1 `2 <= q2 `2 ) } by A343, A345, TOPREAL3:9; then consider p2 being Point of (TOP-REAL 2) such that A351: ( p2 = x & p2 `1 = q1 `1 ) and A352: ( q1 `2 <= p2 `2 & p2 `2 <= q2 `2 ) by A329, A331; ( (G * (i1,j2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,i2)) `2 ) by A342, A344, A352, XXREAL_0:2; then A353: x in LSeg (f,k) by A152, A340, A351; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A353, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ f ; ::_thesis: verum end; end; end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ f or x in L~ g ) assume x in L~ f ; ::_thesis: x in L~ g then A354: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:3; percases ( x in L~ (f | k) or x in LSeg (f,k) ) by A354, XBOOLE_0:def_3; supposeA355: x in L~ (f | k) ; ::_thesis: x in L~ g L~ g1 c= L~ g by GOBOARD2:6; hence x in L~ g by A44, A355; ::_thesis: verum end; suppose x in LSeg (f,k) ; ::_thesis: x in L~ g then consider p1 being Point of (TOP-REAL 2) such that A356: p1 = x and A357: p1 `1 = (G * (i1,i2)) `1 and A358: (G * (i1,j2)) `2 <= p1 `2 and A359: p1 `2 <= (G * (i1,i2)) `2 by A152; defpred S3[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds q `2 >= p1 `2 ) ); A360: now__::_thesis:_ex_n_being_Nat_st_S3[n] reconsider n = len g1 as Nat ; take n = n; ::_thesis: S3[n] thus S3[n] ::_thesis: verum proof thus ( len g1 <= n & n <= len g ) by A319, XREAL_1:31; ::_thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds q `2 >= p1 `2 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A361: len g1 in dom g1 by FINSEQ_3:25; let q be Point of (TOP-REAL 2); ::_thesis: ( q = g /. n implies q `2 >= p1 `2 ) assume q = g /. n ; ::_thesis: q `2 >= p1 `2 then q = (f | k) /. (len (f | k)) by A46, A361, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence q `2 >= p1 `2 by A359; ::_thesis: verum end; end; A362: for n being Nat st S3[n] holds n <= len g ; consider ma being Nat such that A363: ( S3[ma] & ( for n being Nat st S3[n] holds n <= ma ) ) from NAT_1:sch_6(A362, A360); reconsider ma = ma as Element of NAT by ORDINAL1:def_12; now__::_thesis:_x_in_L~_g percases ( ma = len g or ma <> len g ) ; supposeA364: ma = len g ; ::_thesis: x in L~ g j2 + 1 <= i2 by A77, NAT_1:13; then A365: 1 <= l by XREAL_1:19; then (len g1) + 1 <= ma by A89, A319, A364, XREAL_1:7; then A366: len g1 <= ma - 1 by XREAL_1:19; then 0 + 1 <= ma by XREAL_1:19; then reconsider m1 = ma - 1 as Element of NAT by INT_1:5; reconsider q = g /. m1 as Point of (TOP-REAL 2) ; A367: ma - 1 <= len g by A364, XREAL_1:43; then A368: q `1 = (G * (i1,i2)) `1 by A320, A366; A369: (G * (i1,j2)) `2 <= q `2 by A320, A367, A366; set lq = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= e `2 & e `2 <= q `2 ) } ; A370: i2 - l = j2 ; A371: l in dom g2 by A89, A365, FINSEQ_3:25; then A372: g /. ma = g2 /. l by A89, A319, A364, FINSEQ_4:69 .= G * (i1,j2) by A89, A90, A371, A370 ; then p1 `2 <= (G * (i1,j2)) `2 by A363; then A373: p1 `2 = (G * (i1,j2)) `2 by A358, XXREAL_0:1; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A374: 1 <= m1 by A366, XXREAL_0:2; A375: m1 + 1 = ma ; then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg ((G * (i1,j2)),q) ) by A364, A372, A374, EUCLID:53, TOPREAL1:def_3; then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & (G * (i1,j2)) `2 <= e `2 & e `2 <= q `2 ) } by A145, A151, A368, A369, TOPREAL3:9; then A376: p1 in LSeg (g,m1) by A357, A373, A369; LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A364, A374, A375; hence x in L~ g by A356, A376, TARSKI:def_4; ::_thesis: verum end; suppose ma <> len g ; ::_thesis: x in L~ g then ma < len g by A363, XXREAL_0:1; then A377: ma + 1 <= len g by NAT_1:13; reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ; set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa1 `2 <= p2 `2 & p2 `2 <= qa `2 ) } ; A378: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:53; A379: p1 `2 <= qa `2 by A363; A380: len g1 <= ma + 1 by A363, NAT_1:13; then A381: qa1 `1 = (G * (i1,i2)) `1 by A320, A377; A382: now__::_thesis:_not_p1_`2_<=_qa1_`2 assume p1 `2 <= qa1 `2 ; ::_thesis: contradiction then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds p1 `2 <= q `2 ; then ma + 1 <= ma by A363, A377, A380; hence contradiction by XREAL_1:29; ::_thesis: verum end; A383: ( qa `1 = (G * (i1,i2)) `1 & qa = |[(qa `1),(qa `2)]| ) by A320, A363, EUCLID:53; A384: 1 <= ma by A24, A14, A47, A363, NAT_1:13; then LSeg (g,ma) = LSeg (qa1,qa) by A377, TOPREAL1:def_3 .= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa1 `2 <= p2 `2 & p2 `2 <= qa `2 ) } by A379, A382, A381, A383, A378, TOPREAL3:9, XXREAL_0:2 ; then A385: x in LSeg (g,ma) by A356, A357, A379, A382; LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A377, A384; hence x in L~ g by A385, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ g ; ::_thesis: verum end; end; end; A386: len g = (len g1) + (len g2) by FINSEQ_1:22; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then 1 in dom g1 by FINSEQ_3:25; hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:68 .= f /. 1 by A27, A25, FINSEQ_4:71 ; ::_thesis: ( g /. (len g) = f /. (len f) & len f <= len g ) j2 + 1 <= i2 by A77, NAT_1:13; then A387: 1 <= l by XREAL_1:19; then A388: l in dom g2 by A90, FINSEQ_1:1; hence g /. (len g) = g2 /. l by A89, A386, FINSEQ_4:69 .= G * (i1,m1) by A89, A90, A388 .= f /. (len f) by A3, A21, A76 ; ::_thesis: len f <= len g thus len f <= len g by A3, A14, A47, A89, A387, A386, XREAL_1:7; ::_thesis: verum end; caseA389: i2 = j2 ; ::_thesis: contradiction k <> k + 1 ; hence contradiction by A5, A27, A29, A19, A21, A76, A389, PARTFUN2:10; ::_thesis: verum end; caseA390: i2 < j2 ; ::_thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) l1 /. i2 = l1 . i2 by A68, A73, PARTFUN1:def_6; then A391: l1 /. i2 = G * (i1,i2) by A68, MATRIX_1:def_7; then A392: (Y_axis l1) . i2 = (G * (i1,i2)) `2 by A68, A30, A72, GOBOARD1:def_2; l1 /. j2 = l1 . j2 by A74, A73, PARTFUN1:def_6; then A393: l1 /. j2 = G * (i1,j2) by A74, MATRIX_1:def_7; then A394: (Y_axis l1) . j2 = (G * (i1,j2)) `2 by A74, A30, A72, GOBOARD1:def_2; then A395: (G * (i1,i2)) `2 < (G * (i1,j2)) `2 by A68, A74, A71, A30, A72, A390, A392, SEQM_3:def_1; reconsider l = j2 - i2 as Element of NAT by A390, INT_1:5; deffunc H1( Nat) -> Element of the U1 of (TOP-REAL 2) = G * (i1,(i2 + $1)); consider g2 being FinSequence of (TOP-REAL 2) such that A396: ( len g2 = l & ( for n being Nat st n in dom g2 holds g2 /. n = H1(n) ) ) from FINSEQ_4:sch_2(); take g = g1 ^ g2; ::_thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A397: dom g2 = Seg (len g2) by FINSEQ_1:def_3; A398: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_Seg_l_holds_ (_i2_+_n_in_Seg_(width_G)_&_[i1,(i2_+_n)]_in_Indices_G_) let n be Element of NAT ; ::_thesis: ( n in Seg l implies ( i2 + n in Seg (width G) & [i1,(i2 + n)] in Indices G ) ) A399: n <= i2 + n by NAT_1:11; assume A400: n in Seg l ; ::_thesis: ( i2 + n in Seg (width G) & [i1,(i2 + n)] in Indices G ) then n <= l by FINSEQ_1:1; then A401: i2 + n <= l + i2 by XREAL_1:7; j2 <= width G by A74, FINSEQ_1:1; then A402: i2 + n <= width G by A401, XXREAL_0:2; 1 <= n by A400, FINSEQ_1:1; then 1 <= i2 + n by A399, XXREAL_0:2; hence i2 + n in Seg (width G) by A402, FINSEQ_1:1; ::_thesis: [i1,(i2 + n)] in Indices G hence [i1,(i2 + n)] in Indices G by A22, A66, ZFMISC_1:87; ::_thesis: verum end; now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_holds_ ex_m_being_Element_of_NAT_ex_k_being_Element_of_NAT_st_ (_[m,k]_in_Indices_G_&_g2_/._n_=_G_*_(m,k)_) let n be Element of NAT ; ::_thesis: ( n in dom g2 implies ex m being Element of NAT ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) ) assume A403: n in dom g2 ; ::_thesis: ex m being Element of NAT ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) take m = i1; ::_thesis: ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) take k = i2 + n; ::_thesis: ( [m,k] in Indices G & g2 /. n = G * (m,k) ) thus ( [m,k] in Indices G & g2 /. n = G * (m,k) ) by A396, A398, A397, A403; ::_thesis: verum end; then A404: for n being Element of NAT st n in dom g holds ex i, j being Element of NAT st ( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:23; A405: (X_axis l1) . i2 = (G * (i1,i2)) `1 by A68, A62, A72, A391, GOBOARD1:def_1; A406: now__::_thesis:_for_n_being_Element_of_NAT_ for_p_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_g2_/._n_=_p_holds_ (_p_`1_=_(G_*_(i1,i2))_`1_&_(G_*_(i1,i2))_`2_<=_p_`2_&_p_`2_<=_(G_*_(i1,j2))_`2_&_p_in_rng_l1_&_p_`2_>_(G_*_(i1,i2))_`2_) let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 & p `2 > (G * (i1,i2)) `2 ) let p be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & g2 /. n = p implies ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 & p `2 > (G * (i1,i2)) `2 ) ) assume that A407: n in dom g2 and A408: g2 /. n = p ; ::_thesis: ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 & p `2 > (G * (i1,i2)) `2 ) A409: g2 /. n = G * (i1,(i2 + n)) by A396, A407; set n1 = i2 + n; set pn = G * (i1,(i2 + n)); A410: i2 + n in Seg (width G) by A396, A398, A397, A407; then A411: (X_axis l1) . (i2 + n) = (X_axis l1) . i2 by A68, A67, A62, A72, SEQM_3:def_10; l1 /. (i2 + n) = l1 . (i2 + n) by A73, A396, A398, A397, A407, PARTFUN1:def_6; then A412: l1 /. (i2 + n) = G * (i1,(i2 + n)) by A410, MATRIX_1:def_7; then A413: (Y_axis l1) . (i2 + n) = (G * (i1,(i2 + n))) `2 by A30, A72, A410, GOBOARD1:def_2; n <= len g2 by A397, A407, FINSEQ_1:1; then A414: i2 + n <= i2 + (len g2) by XREAL_1:7; (X_axis l1) . (i2 + n) = (G * (i1,(i2 + n))) `1 by A62, A72, A410, A412, GOBOARD1:def_1; hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 ) by A68, A74, A71, A30, A72, A396, A392, A394, A405, A408, A409, A410, A414, A411, A413, SEQ_4:137, XREAL_1:31; ::_thesis: ( p in rng l1 & p `2 > (G * (i1,i2)) `2 ) dom l1 = Seg (len l1) by FINSEQ_1:def_3; hence p in rng l1 by A72, A408, A409, A410, A412, PARTFUN2:2; ::_thesis: p `2 > (G * (i1,i2)) `2 1 <= n by A397, A407, FINSEQ_1:1; then i2 < i2 + n by XREAL_1:29; hence p `2 > (G * (i1,i2)) `2 by A68, A71, A30, A72, A392, A408, A409, A410, A413, SEQM_3:def_1; ::_thesis: verum end; A415: g2 is special proof let n be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) set p = g2 /. n; assume A416: ( 1 <= n & n + 1 <= len g2 ) ; ::_thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) then n in dom g2 by SEQ_4:134; then A417: (g2 /. n) `1 = (G * (i1,i2)) `1 by A406; n + 1 in dom g2 by A416, SEQ_4:134; hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A406, A417; ::_thesis: verum end; now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<>_m_holds_ not_g2_/._n_=_g2_/._m let n, m be Element of NAT ; ::_thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m ) assume that A418: ( n in dom g2 & m in dom g2 ) and A419: n <> m ; ::_thesis: not g2 /. n = g2 /. m A420: ( g2 /. n = G * (i1,(i2 + n)) & g2 /. m = G * (i1,(i2 + m)) ) by A396, A418; assume A421: g2 /. n = g2 /. m ; ::_thesis: contradiction ( [i1,(i2 + n)] in Indices G & [i1,(i2 + m)] in Indices G ) by A396, A398, A397, A418; then i2 + n = i2 + m by A420, A421, GOBOARD1:5; hence contradiction by A419; ::_thesis: verum end; then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds n = m ; then A422: g2 is one-to-one by PARTFUN2:9; set lk = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } ; A423: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:53; A424: now__::_thesis:_for_n,_m_being_Element_of_NAT_ for_p,_q_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<_m_&_g2_/._n_=_p_&_g2_/._m_=_q_holds_ p_`2_<_q_`2 let n, m be Element of NAT ; ::_thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds p `2 < q `2 let p, q be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies p `2 < q `2 ) assume that A425: n in dom g2 and A426: m in dom g2 and A427: n < m and A428: ( g2 /. n = p & g2 /. m = q ) ; ::_thesis: p `2 < q `2 A429: i2 + n in Seg (width G) by A396, A398, A397, A425; set n1 = i2 + n; set m1 = i2 + m; set pn = G * (i1,(i2 + n)); set pm = G * (i1,(i2 + m)); A430: i2 + n < i2 + m by A427, XREAL_1:8; l1 /. (i2 + n) = l1 . (i2 + n) by A73, A396, A398, A397, A425, PARTFUN1:def_6; then l1 /. (i2 + n) = G * (i1,(i2 + n)) by A429, MATRIX_1:def_7; then A431: (Y_axis l1) . (i2 + n) = (G * (i1,(i2 + n))) `2 by A30, A72, A429, GOBOARD1:def_2; A432: i2 + m in Seg (width G) by A396, A398, A397, A426; l1 /. (i2 + m) = l1 . (i2 + m) by A73, A396, A398, A397, A426, PARTFUN1:def_6; then l1 /. (i2 + m) = G * (i1,(i2 + m)) by A432, MATRIX_1:def_7; then A433: (Y_axis l1) . (i2 + m) = (G * (i1,(i2 + m))) `2 by A30, A72, A432, GOBOARD1:def_2; ( g2 /. n = G * (i1,(i2 + n)) & g2 /. m = G * (i1,(i2 + m)) ) by A396, A425, A426; hence p `2 < q `2 by A71, A30, A72, A428, A429, A432, A430, A431, A433, SEQM_3:def_1; ::_thesis: verum end; for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds LSeg (g2,n) misses LSeg (g2,m) proof let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) ) assume that A434: m > n + 1 and A435: n in dom g2 and A436: n + 1 in dom g2 and A437: m in dom g2 and A438: m + 1 in dom g2 and A439: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A440: ( p1 `1 = (G * (i1,i2)) `1 & p2 `1 = (G * (i1,i2)) `1 ) by A406, A435, A436; n < n + 1 by NAT_1:13; then A441: p1 `2 < p2 `2 by A424, A435, A436; set lp = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p1 `2 <= w `2 & w `2 <= p2 `2 ) } ; set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } ; A442: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:53; m < m + 1 by NAT_1:13; then A443: q1 `2 < q2 `2 by A424, A437, A438; A444: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; set x = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)); A445: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,n) by A439, XBOOLE_0:def_4; A446: ( q1 `1 = (G * (i1,i2)) `1 & q2 `1 = (G * (i1,i2)) `1 ) by A406, A437, A438; A447: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,m) by A439, XBOOLE_0:def_4; ( 1 <= m & m + 1 <= len g2 ) by A437, A438, FINSEQ_3:25; then LSeg (g2,m) = LSeg (q1,q2) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } by A443, A446, A444, TOPREAL3:9 ; then A448: ex tm being Point of (TOP-REAL 2) st ( tm = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tm `1 = (G * (i1,i2)) `1 & q1 `2 <= tm `2 & tm `2 <= q2 `2 ) by A447; ( 1 <= n & n + 1 <= len g2 ) by A435, A436, FINSEQ_3:25; then LSeg (g2,n) = LSeg (p1,p2) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & p1 `2 <= w `2 & w `2 <= p2 `2 ) } by A441, A440, A442, TOPREAL3:9 ; then A449: ex tn being Point of (TOP-REAL 2) st ( tn = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tn `1 = (G * (i1,i2)) `1 & p1 `2 <= tn `2 & tn `2 <= p2 `2 ) by A445; p2 `2 < q1 `2 by A424, A434, A436, A437; hence contradiction by A449, A448, XXREAL_0:2; ::_thesis: verum end; then A450: g2 is s.n.c. by GOBOARD2:1; A451: not f /. k in L~ g2 proof set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume f /. k in L~ g2 ; ::_thesis: contradiction then consider X being set such that A452: f /. k in X and A453: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A454: X = LSeg (g2,m) and A455: ( 1 <= m & m + 1 <= len g2 ) by A453; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A456: m in dom g2 by A455, SEQ_4:134; then A457: q1 `1 = (G * (i1,i2)) `1 by A406; set lq = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } ; A458: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; A459: m + 1 in dom g2 by A455, SEQ_4:134; then A460: q2 `1 = (G * (i1,i2)) `1 by A406; m < m + 1 by NAT_1:13; then A461: q1 `2 < q2 `2 by A424, A456, A459; LSeg (g2,m) = LSeg (q1,q2) by A455, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & q1 `2 <= w `2 & w `2 <= q2 `2 ) } by A457, A460, A461, A458, TOPREAL3:9 ; then ex p being Point of (TOP-REAL 2) st ( p = f /. k & p `1 = (G * (i1,i2)) `1 & q1 `2 <= p `2 & p `2 <= q2 `2 ) by A452, A454; hence contradiction by A29, A406, A456; ::_thesis: verum end; (X_axis l1) . j2 = (G * (i1,j2)) `1 by A74, A62, A72, A393, GOBOARD1:def_1; then A462: (G * (i1,i2)) `1 = (G * (i1,j2)) `1 by A68, A74, A67, A62, A72, A405, SEQM_3:def_10; A463: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_&_n_+_1_in_dom_g2_holds_ for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g2_/._n_=_G_*_(l1,l2)_&_g2_/._(n_+_1)_=_G_*_(l3,l4)_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let n be Element of NAT ; ::_thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A464: n in dom g2 and A465: n + 1 in dom g2 ; ::_thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A466: [l1,l2] in Indices G and A467: [l3,l4] in Indices G and A468: g2 /. n = G * (l1,l2) and A469: g2 /. (n + 1) = G * (l3,l4) ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ( g2 /. (n + 1) = G * (i1,(i2 + (n + 1))) & [i1,(i2 + (n + 1))] in Indices G ) by A396, A398, A397, A465; then A470: ( l3 = i1 & l4 = i2 + (n + 1) ) by A467, A469, GOBOARD1:5; ( g2 /. n = G * (i1,(i2 + n)) & [i1,(i2 + n)] in Indices G ) by A396, A398, A397, A464; then ( l1 = i1 & l2 = i2 + n ) by A466, A468, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs ((i2 + n) - (i2 + (n + 1)))) by A470, ABSVALUE:2 .= abs (- 1) .= abs 1 by COMPLEX1:52 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; now__::_thesis:_for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g1_/._(len_g1)_=_G_*_(l1,l2)_&_g2_/._1_=_G_*_(l3,l4)_&_len_g1_in_dom_g1_&_1_in_dom_g2_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A471: [l1,l2] in Indices G and A472: [l3,l4] in Indices G and A473: g1 /. (len g1) = G * (l1,l2) and A474: g2 /. 1 = G * (l3,l4) and len g1 in dom g1 and A475: 1 in dom g2 ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ( g2 /. 1 = G * (i1,(i2 + 1)) & [i1,(i2 + 1)] in Indices G ) by A396, A398, A397, A475; then A476: ( l3 = i1 & l4 = i2 + 1 ) by A472, A474, GOBOARD1:5; (f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:71; then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A471, A473, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = 0 + (abs (i2 - (i2 + 1))) by A476, ABSVALUE:2 .= abs ((i2 - i2) + (- 1)) .= abs 1 by COMPLEX1:52 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; then for n being Element of NAT st n in dom g & n + 1 in dom g holds for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds (abs (m - i)) + (abs (k - j)) = 1 by A48, A463, GOBOARD1:24; hence g is_sequence_on G by A404, GOBOARD1:def_9; ::_thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A477: G * (i1,j2) = |[((G * (i1,j2)) `1),((G * (i1,j2)) `2)]| by EUCLID:53; A478: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (i1,j2))) by A3, A24, A29, A21, A76, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A395, A462, A423, A477, TOPREAL3:9 ; A479: rng g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g2 or x in LSeg (f,k) ) assume x in rng g2 ; ::_thesis: x in LSeg (f,k) then consider n being Element of NAT such that A480: n in dom g2 and A481: g2 /. n = x by PARTFUN2:2; set pn = G * (i1,(i2 + n)); A482: g2 /. n = G * (i1,(i2 + n)) by A396, A480; then A483: (G * (i1,(i2 + n))) `2 <= (G * (i1,j2)) `2 by A406, A480; ( (G * (i1,(i2 + n))) `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= (G * (i1,(i2 + n))) `2 ) by A406, A480, A482; hence x in LSeg (f,k) by A478, A481, A482, A483; ::_thesis: verum end; A484: not f /. k in rng g2 proof assume f /. k in rng g2 ; ::_thesis: contradiction then consider n being Element of NAT such that A485: n in dom g2 and A486: g2 /. n = f /. k by PARTFUN2:2; A487: 0 < n by A485, FINSEQ_3:25; A488: g2 /. n = G * (i1,(i2 + n)) by A396, A485; dom g2 = Seg (len g2) by FINSEQ_1:def_3; then [i1,(i2 + n)] in Indices G by A396, A398, A485; then i2 + n = i2 by A28, A29, A486, A488, GOBOARD1:5; hence contradiction by A487; ::_thesis: verum end; (rng g1) /\ (rng g2) = {} proof set x = the Element of (rng g1) /\ (rng g2); assume A489: not (rng g1) /\ (rng g2) = {} ; ::_thesis: contradiction then A490: the Element of (rng g1) /\ (rng g2) in rng g2 by XBOOLE_0:def_4; A491: the Element of (rng g1) /\ (rng g2) in rng g1 by A489, XBOOLE_0:def_4; now__::_thesis:_contradiction percases ( k = 1 or 1 < k ) by A24, XXREAL_0:1; suppose k = 1 ; ::_thesis: contradiction hence contradiction by A52, A484, A491, A490, TARSKI:def_1; ::_thesis: verum end; suppose 1 < k ; ::_thesis: contradiction then ( the Element of (rng g1) /\ (rng g2) in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A479, A491, A490, GOBOARD2:4, XBOOLE_0:def_4; hence contradiction by A484, A490, TARSKI:def_1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; then rng g1 misses rng g2 by XBOOLE_0:def_7; hence g is one-to-one by A40, A422, FINSEQ_3:91; ::_thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A492: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (i1,j2))) by A3, A24, A29, A21, A76, TOPREAL1:def_3; A493: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} proof let n be Element of NAT ; ::_thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} ) assume that A494: 1 <= n and A495: n + 2 <= len g2 ; ::_thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} A496: n + 1 in dom g2 by A494, A495, SEQ_4:135; then g2 /. (n + 1) in rng g2 by PARTFUN2:2; then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479; then consider u1 being Point of (TOP-REAL 2) such that A497: g2 /. (n + 1) = u1 and A498: u1 `1 = (G * (i1,i2)) `1 and (G * (i1,i2)) `2 <= u1 `2 and u1 `2 <= (G * (i1,j2)) `2 ; A499: n + 2 in dom g2 by A494, A495, SEQ_4:135; then g2 /. (n + 2) in rng g2 by PARTFUN2:2; then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479; then consider u2 being Point of (TOP-REAL 2) such that A500: g2 /. (n + 2) = u2 and A501: u2 `1 = (G * (i1,i2)) `1 and (G * (i1,i2)) `2 <= u2 `2 and u2 `2 <= (G * (i1,j2)) `2 ; ( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11; then A502: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A495, A497, A500, TOPREAL1:def_3; n + 1 < (n + 1) + 1 by NAT_1:13; then A503: u1 `2 < u2 `2 by A424, A496, A499, A497, A500; A504: n in dom g2 by A494, A495, SEQ_4:135; then g2 /. n in rng g2 by PARTFUN2:2; then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479; then consider u being Point of (TOP-REAL 2) such that A505: g2 /. n = u and A506: u `1 = (G * (i1,i2)) `1 and (G * (i1,i2)) `2 <= u `2 and u `2 <= (G * (i1,j2)) `2 ; n + 1 <= n + 2 by XREAL_1:6; then n + 1 <= len g2 by A495, XXREAL_0:2; then A507: LSeg (g2,n) = LSeg (u,u1) by A494, A505, A497, TOPREAL1:def_3; set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u `2 <= w `2 & w `2 <= u2 `2 ) } ; n < n + 1 by NAT_1:13; then A508: u `2 < u1 `2 by A424, A504, A496, A505, A497; then A509: u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u `2 <= w `2 & w `2 <= u2 `2 ) } by A498, A503; ( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:53; then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & u `2 <= w `2 & w `2 <= u2 `2 ) } by A505, A506, A500, A501, A503, A508, TOPREAL3:9, XXREAL_0:2; hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A505, A497, A500, A507, A502, A509, TOPREAL1:8; ::_thesis: verum end; thus g is unfolded ::_thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof let n be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ) assume that A510: 1 <= n and A511: n + 2 <= len g ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A512: (n + 1) + 1 <= len g by A511; n + 1 <= (n + 1) + 1 by NAT_1:11; then A513: n + 1 <= len g by A511, XXREAL_0:2; A514: len g = (len g1) + (len g2) by FINSEQ_1:22; (n + 2) - (len g1) = (n - (len g1)) + 2 ; then A515: (n - (len g1)) + 2 <= len g2 by A511, A514, XREAL_1:20; A516: 1 <= n + 1 by NAT_1:11; A517: n <= n + 1 by NAT_1:11; A518: n + (1 + 1) = (n + 1) + 1 ; percases ( n + 2 <= len g1 or len g1 < n + 2 ) ; supposeA519: n + 2 <= len g1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A520: n + (1 + 1) = (n + 1) + 1 ; A521: n + 1 in dom g1 by A510, A519, SEQ_4:135; then A522: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:68; n in dom g1 by A510, A519, SEQ_4:135; then A523: LSeg (g1,n) = LSeg (g,n) by A521, TOPREAL3:18; n + 2 in dom g1 by A510, A519, SEQ_4:135; then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A521, A520, TOPREAL3:18; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A510, A519, A523, A522, TOPREAL1:def_6; ::_thesis: verum end; suppose len g1 < n + 2 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then (len g1) + 1 <= n + 2 by NAT_1:13; then A524: len g1 <= (n + 2) - 1 by XREAL_1:19; now__::_thesis:_(LSeg_(g,n))_/\_(LSeg_(g,(n_+_1)))_=_{(g_/._(n_+_1))} percases ( len g1 = n + 1 or len g1 <> n + 1 ) ; supposeA525: len g1 = n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} now__::_thesis:_not_k_=_1 1 < len g1 by A510, A525, NAT_1:13; then A526: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A526, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A527: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; ( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A510, A511, A516, A513, A518, TOPREAL1:21; then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def_4; then A528: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:31; A529: 1 <= (len g) - (len g1) by A512, A525, XREAL_1:19; then 1 in dom g2 by A514, FINSEQ_3:25; then A530: g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479; then consider u1 being Point of (TOP-REAL 2) such that A531: g2 /. 1 = u1 and u1 `1 = (G * (i1,i2)) `1 and (G * (i1,i2)) `2 <= u1 `2 and u1 `2 <= (G * (i1,j2)) `2 ; G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (i1,j2))) by RLTOPSP1:68; then A532: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A492, A479, A530, A531, TOPREAL1:6; 1 <= n + 1 by NAT_1:11; then A533: n + 1 in dom g1 by A525, FINSEQ_3:25; then A534: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A525, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; n in dom g1 by A510, A517, A525, FINSEQ_3:25; then A535: LSeg (g,n) = LSeg (g1,n) by A533, TOPREAL3:18; g /. (n + 2) = g2 /. 1 by A518, A514, A525, A529, SEQ_4:136; then A536: LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A511, A516, A518, A534, A531, TOPREAL1:def_3; LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:19; then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A527, A535, A534, A532, A536, XBOOLE_1:27; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A528, XBOOLE_0:def_10; ::_thesis: verum end; suppose len g1 <> n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then len g1 < n + 1 by A524, XXREAL_0:1; then A537: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_(LSeg_(g,n))_/\_(LSeg_(g,(n_+_1)))_=_{(g_/._(n_+_1))} percases ( len g1 = n or len g1 <> n ) ; supposeA538: len g1 = n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A539: 2 <= len g2 by A511, A514, XREAL_1:6; then 1 <= len g2 by XXREAL_0:2; then A540: g /. (n + 1) = g2 /. 1 by A538, SEQ_4:136; 1 <= len g2 by A539, XXREAL_0:2; then A541: 1 in dom g2 by FINSEQ_3:25; then g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479; then consider u1 being Point of (TOP-REAL 2) such that A542: g2 /. 1 = u1 and A543: ( u1 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= u1 `2 ) and u1 `2 <= (G * (i1,j2)) `2 ; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then g /. n = (f | k) /. (len (f | k)) by A46, A538, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; then A544: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A510, A513, A540, A542, TOPREAL1:def_3; A545: 2 in dom g2 by A539, FINSEQ_3:25; then g2 /. 2 in rng g2 by PARTFUN2:2; then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= (G * (i1,j2)) `2 ) } by A478, A479; then consider u2 being Point of (TOP-REAL 2) such that A546: g2 /. 2 = u2 and A547: ( u2 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= u2 `2 ) and u2 `2 <= (G * (i1,j2)) `2 ; set lg = { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= u2 `2 ) } ; u1 `2 < u2 `2 by A424, A541, A545, A542, A546; then ( u2 = |[(u2 `1),(u2 `2)]| & u1 in { w where w is Point of (TOP-REAL 2) : ( w `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= w `2 & w `2 <= u2 `2 ) } ) by A543, EUCLID:53; then A548: u1 in LSeg ((G * (i1,i2)),u2) by A423, A547, TOPREAL3:9; g /. (n + 2) = g2 /. 2 by A538, A539, SEQ_4:136; then LSeg (g,(n + 1)) = LSeg (u1,u2) by A511, A516, A518, A540, A542, A546, TOPREAL1:def_3; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A540, A542, A544, A548, TOPREAL1:8; ::_thesis: verum end; suppose len g1 <> n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A549: len g1 < n by A537, XXREAL_0:1; then (len g1) + 1 <= n by NAT_1:13; then A550: 1 <= n1 by XREAL_1:19; n1 + (len g1) = n ; then A551: LSeg (g,n) = LSeg (g2,n1) by A513, A549, GOBOARD2:5; A552: (n1 + 1) + (len g1) = n + 1 ; then n1 + 1 <= len g2 by A513, A514, XREAL_1:6; then A553: g /. (n + 1) = g2 /. (n1 + 1) by A552, NAT_1:11, SEQ_4:136; len g1 < n + 1 by A517, A549, XXREAL_0:2; then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A512, A552, GOBOARD2:5; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A493, A515, A551, A553, A550; ::_thesis: verum end; end; end; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; ::_thesis: verum end; end; end; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; ::_thesis: verum end; end; end; A554: L~ g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g2 or x in LSeg (f,k) ) set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume x in L~ g2 ; ::_thesis: x in LSeg (f,k) then consider X being set such that A555: x in X and A556: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A557: X = LSeg (g2,m) and A558: ( 1 <= m & m + 1 <= len g2 ) by A556; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A559: LSeg (g2,m) = LSeg (q1,q2) by A558, TOPREAL1:def_3; m + 1 in dom g2 by A558, SEQ_4:134; then A560: g2 /. (m + 1) in rng g2 by PARTFUN2:2; m in dom g2 by A558, SEQ_4:134; then g2 /. m in rng g2 by PARTFUN2:2; then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (i1,j2))) by A492, A479, A560, TOPREAL1:6; hence x in LSeg (f,k) by A492, A555, A557, A559; ::_thesis: verum end; A561: (L~ g1) /\ (L~ g2) = {} proof percases ( k = 1 or k <> 1 ) ; suppose k = 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} hence (L~ g1) /\ (L~ g2) = {} by A52; ::_thesis: verum end; suppose k <> 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} then 1 < k by A24, XXREAL_0:1; then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:4; then A562: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A554, XBOOLE_1:26; now__::_thesis:_not_(L~_g1)_/\_(L~_g2)_<>_{} set x = the Element of (L~ g1) /\ (L~ g2); assume (L~ g1) /\ (L~ g2) <> {} ; ::_thesis: contradiction then ( the Element of (L~ g1) /\ (L~ g2) in {(f /. k)} & the Element of (L~ g1) /\ (L~ g2) in L~ g2 ) by A562, TARSKI:def_3, XBOOLE_0:def_4; hence contradiction by A451, TARSKI:def_1; ::_thesis: verum end; hence (L~ g1) /\ (L~ g2) = {} ; ::_thesis: verum end; end; end; for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds LSeg (g,n) misses LSeg (g,m) proof A563: 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A564: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:71 ; reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ; set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ; set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ; let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) ) assume that A565: m > n + 1 and A566: n in dom g and A567: n + 1 in dom g and A568: m in dom g and A569: m + 1 in dom g ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A570: 1 <= n by A566, FINSEQ_3:25; i2 + 1 <= j2 by A390, NAT_1:13; then A571: 1 <= l by XREAL_1:19; then A572: 1 in dom g2 by A396, FINSEQ_3:25; then A573: ( qq `1 = (G * (i1,i2)) `1 & qq `2 > (G * (i1,i2)) `2 ) by A406; A574: g /. ((len g1) + 1) = qq by A396, A571, SEQ_4:136; A575: qq `2 <= (G * (i1,j2)) `2 by A406, A572; A576: m + 1 <= len g by A569, FINSEQ_3:25; A577: 1 <= m + 1 by A569, FINSEQ_3:25; A578: 1 <= n + 1 by A567, FINSEQ_3:25; A579: n + 1 <= len g by A567, FINSEQ_3:25; A580: qq = |[(qq `1),(qq `2)]| by EUCLID:53; A581: 1 <= m by A568, FINSEQ_3:25; set ql = { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= z `2 & z `2 <= qq `2 ) } ; A582: n <= n + 1 by NAT_1:11; A583: len g = (len g1) + (len g2) by FINSEQ_1:22; then (len g1) + 1 <= len g by A396, A571, XREAL_1:7; then A584: LSeg (g,(len g1)) = LSeg ((G * (i1,i2)),qq) by A563, A564, A574, TOPREAL1:def_3 .= { z where z is Point of (TOP-REAL 2) : ( z `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= z `2 & z `2 <= qq `2 ) } by A423, A573, A580, TOPREAL3:9 ; A585: m <= m + 1 by NAT_1:11; then A586: n + 1 <= m + 1 by A565, XXREAL_0:2; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m + 1 <= len g1 or len g1 < m + 1 ) ; supposeA587: m + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then m <= len g1 by A585, XXREAL_0:2; then A588: m in dom g1 by A581, FINSEQ_3:25; m + 1 in dom g1 by A577, A587, FINSEQ_3:25; then A589: LSeg (g,m) = LSeg (g1,m) by A588, TOPREAL3:18; A590: n + 1 <= len g1 by A586, A587, XXREAL_0:2; then n <= len g1 by A582, XXREAL_0:2; then A591: n in dom g1 by A570, FINSEQ_3:25; n + 1 in dom g1 by A578, A590, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A591, TOPREAL3:18; hence LSeg (g,n) misses LSeg (g,m) by A42, A565, A589, TOPREAL1:def_7; ::_thesis: verum end; suppose len g1 < m + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A592: len g1 <= m by NAT_1:13; then reconsider m1 = m - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m = len g1 or m <> len g1 ) ; supposeA593: m = len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A594: LSeg (g,m) c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) ) assume x in LSeg (g,m) ; ::_thesis: x in LSeg (f,k) then consider px being Point of (TOP-REAL 2) such that A595: ( px = x & px `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= px `2 ) and A596: px `2 <= qq `2 by A584, A593; (G * (i1,j2)) `2 >= px `2 by A575, A596, XXREAL_0:2; hence x in LSeg (f,k) by A478, A595; ::_thesis: verum end; n <= len g1 by A565, A582, A593, XXREAL_0:2; then A597: n in dom g1 by A570, FINSEQ_3:25; now__::_thesis:_not_k_=_1 1 < len g1 by A565, A578, A593, XXREAL_0:2; then A598: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A598, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A599: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; A600: n + 1 in dom g1 by A565, A578, A593, FINSEQ_3:25; then A601: LSeg (g,n) = LSeg (g1,n) by A597, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A565, A570, A593; then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:74; then A602: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A599, A594, XBOOLE_1:27; now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); assume A603: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A604: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in {(f /. k)} by A602, A603, TARSKI:def_3; then A605: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) = f /. k by TARSKI:def_1; f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:71; hence contradiction by A40, A41, A42, A565, A593, A597, A600, A601, A604, A605, GOBOARD2:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose m <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A606: len g1 < m by A592, XXREAL_0:1; then (len g1) + 1 <= m by NAT_1:13; then A607: 1 <= m1 by XREAL_1:19; m + 1 = (m1 + 1) + (len g1) ; then A608: m1 + 1 <= len g2 by A576, A583, XREAL_1:6; m = m1 + (len g1) ; then A609: LSeg (g,m) = LSeg (g2,m1) by A576, A606, GOBOARD2:5; then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A607, A608; then A610: LSeg (g,m) c= L~ g2 by ZFMISC_1:74; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( n + 1 <= len g1 or len g1 < n + 1 ) ; supposeA611: n + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then n <= len g1 by A582, XXREAL_0:2; then A612: n in dom g1 by A570, FINSEQ_3:25; n + 1 in dom g1 by A578, A611, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A612, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A570, A611; then LSeg (g,n) c= L~ g1 by ZFMISC_1:74; then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A561, A610, XBOOLE_1:3, XBOOLE_1:27; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose len g1 < n + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A613: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; A614: (n - (len g1)) + 1 = (n + 1) - (len g1) ; A615: n = n1 + (len g1) ; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( len g1 = n or n <> len g1 ) ; supposeA616: len g1 = n ; ::_thesis: LSeg (g,n) misses LSeg (g,m) now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ; set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); set q1l = { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q1 `2 <= v `2 & v `2 <= q2 `2 ) } ; A617: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; assume A618: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A619: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,m) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by A618, XBOOLE_0:def_4; then A620: ex qx being Point of (TOP-REAL 2) st ( qx = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qx `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= qx `2 & qx `2 <= qq `2 ) by A584, A616; A621: m1 in dom g2 by A607, A608, SEQ_4:134; then A622: q1 `1 = (G * (i1,i2)) `1 by A406; A623: m1 + 1 in dom g2 by A607, A608, SEQ_4:134; then A624: q2 `1 = (G * (i1,i2)) `1 by A406; m1 < m1 + 1 by NAT_1:13; then A625: q1 `2 < q2 `2 by A424, A621, A623; LSeg (g2,m1) = LSeg (q1,q2) by A607, A608, TOPREAL1:def_3 .= { v where v is Point of (TOP-REAL 2) : ( v `1 = (G * (i1,i2)) `1 & q1 `2 <= v `2 & v `2 <= q2 `2 ) } by A622, A624, A625, A617, TOPREAL3:9 ; then A626: ex qy being Point of (TOP-REAL 2) st ( qy = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qy `1 = (G * (i1,i2)) `1 & q1 `2 <= qy `2 & qy `2 <= q2 `2 ) by A609, A619; ( m1 > n1 + 1 & n1 + 1 >= 1 ) by A565, A614, NAT_1:11, XREAL_1:9; then m1 > 1 by XXREAL_0:2; then qq `2 < q1 `2 by A424, A572, A621; hence contradiction by A620, A626, XXREAL_0:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose n <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then len g1 < n by A613, XXREAL_0:1; then A627: LSeg (g,n) = LSeg (g2,n1) by A579, A615, GOBOARD2:5; m1 > n1 + 1 by A565, A614, XREAL_1:9; hence LSeg (g,n) misses LSeg (g,m) by A450, A609, A627, TOPREAL1:def_7; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; hence g is s.n.c. by GOBOARD2:1; ::_thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) now__::_thesis:_(_(g1_/._(len_g1))_`1_=_(g2_/._1)_`1_or_(g1_/._(len_g1))_`2_=_(g2_/._1)_`2_) set p = g1 /. (len g1); set q = g2 /. 1; i2 + 1 <= j2 by A390, NAT_1:13; then 1 <= l by XREAL_1:19; then 1 in dom g2 by A396, FINSEQ_3:25; then (g2 /. 1) `1 = (G * (i1,i2)) `1 by A406; hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:71; ::_thesis: verum end; hence g is special by A43, A415, GOBOARD2:8; ::_thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) thus L~ g = L~ f ::_thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ; set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ; A628: len g = (len g1) + (len g2) by FINSEQ_1:22; A629: now__::_thesis:_for_j_being_Element_of_NAT_st_len_g1_<=_j_&_j_<=_len_g_holds_ for_p_being_Point_of_(TOP-REAL_2)_st_p_=_g_/._j_holds_ (_p_`1_=_(G_*_(i1,i2))_`1_&_(G_*_(i1,i2))_`2_<=_p_`2_&_p_`2_<=_(G_*_(i1,j2))_`2_&_p_in_rng_l1_) let j be Element of NAT ; ::_thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds ( b3 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b3 `2 & b3 `2 <= (G * (i1,j2)) `2 & b3 in rng l1 ) ) assume that A630: len g1 <= j and A631: j <= len g ; ::_thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds ( b3 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b3 `2 & b3 `2 <= (G * (i1,j2)) `2 & b3 in rng l1 ) reconsider w = j - (len g1) as Element of NAT by A630, INT_1:5; let p be Point of (TOP-REAL 2); ::_thesis: ( p = g /. j implies ( b2 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b2 `2 & b2 `2 <= (G * (i1,j2)) `2 & b2 in rng l1 ) ) assume A632: p = g /. j ; ::_thesis: ( b2 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b2 `2 & b2 `2 <= (G * (i1,j2)) `2 & b2 in rng l1 ) percases ( j = len g1 or j <> len g1 ) ; supposeA633: j = len g1 ; ::_thesis: ( b2 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b2 `2 & b2 `2 <= (G * (i1,j2)) `2 & b2 in rng l1 ) 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A634: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence p `1 = (G * (i1,i2)) `1 by A632, A633; ::_thesis: ( (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 ) thus ( (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 ) by A68, A74, A71, A30, A72, A390, A392, A394, A632, A633, A634, SEQM_3:def_1; ::_thesis: p in rng l1 dom l1 = Seg (len l1) by FINSEQ_1:def_3; hence p in rng l1 by A68, A72, A391, A632, A633, A634, PARTFUN2:2; ::_thesis: verum end; suppose j <> len g1 ; ::_thesis: ( b2 `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= b2 `2 & b2 `2 <= (G * (i1,j2)) `2 & b2 in rng l1 ) then len g1 < j by A630, XXREAL_0:1; then (len g1) + 1 <= j by NAT_1:13; then A635: 1 <= w by XREAL_1:19; A636: w <= len g2 by A628, A631, XREAL_1:20; then A637: w in dom g2 by A635, FINSEQ_3:25; j = w + (len g1) ; then g /. j = g2 /. w by A635, A636, SEQ_4:136; hence ( p `1 = (G * (i1,i2)) `1 & (G * (i1,i2)) `2 <= p `2 & p `2 <= (G * (i1,j2)) `2 & p in rng l1 ) by A406, A632, A637; ::_thesis: verum end; end; end; thus L~ g c= L~ f :: according to XBOOLE_0:def_10 ::_thesis: L~ f c= L~ g proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g or x in L~ f ) assume x in L~ g ; ::_thesis: x in L~ f then consider X being set such that A638: x in X and A639: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def_4; consider i being Element of NAT such that A640: X = LSeg (g,i) and A641: 1 <= i and A642: i + 1 <= len g by A639; percases ( i + 1 <= len g1 or i + 1 > len g1 ) ; supposeA643: i + 1 <= len g1 ; ::_thesis: x in L~ f i <= i + 1 by NAT_1:11; then i <= len g1 by A643, XXREAL_0:2; then A644: i in dom g1 by A641, FINSEQ_3:25; 1 <= i + 1 by NAT_1:11; then i + 1 in dom g1 by A643, FINSEQ_3:25; then X = LSeg (g1,i) by A640, A644, TOPREAL3:18; then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A641, A643; then A645: x in L~ (f | k) by A44, A638, TARSKI:def_4; L~ (f | k) c= L~ f by TOPREAL3:20; hence x in L~ f by A645; ::_thesis: verum end; supposeA646: i + 1 > len g1 ; ::_thesis: x in L~ f reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ; A647: i <= len g by A642, NAT_1:13; A648: len g1 <= i by A646, NAT_1:13; then A649: q1 `1 = (G * (i1,i2)) `1 by A629, A647; A650: q1 `2 <= (G * (i1,j2)) `2 by A629, A648, A647; A651: (G * (i1,i2)) `2 <= q1 `2 by A629, A648, A647; q2 `1 = (G * (i1,i2)) `1 by A629, A642, A646; then A652: q2 = |[(q1 `1),(q2 `2)]| by A649, EUCLID:53; A653: q2 `2 <= (G * (i1,j2)) `2 by A629, A642, A646; A654: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A641, A642, EUCLID:53, TOPREAL1:def_3; A655: (G * (i1,i2)) `2 <= q2 `2 by A629, A642, A646; now__::_thesis:_x_in_L~_f percases ( q1 `2 > q2 `2 or q1 `2 = q2 `2 or q1 `2 < q2 `2 ) by XXREAL_0:1; suppose q1 `2 > q2 `2 ; ::_thesis: x in L~ f then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = q1 `1 & q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) } by A652, A654, TOPREAL3:9; then consider p2 being Point of (TOP-REAL 2) such that A656: ( p2 = x & p2 `1 = q1 `1 ) and A657: ( q2 `2 <= p2 `2 & p2 `2 <= q1 `2 ) by A638, A640; ( (G * (i1,i2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,j2)) `2 ) by A650, A655, A657, XXREAL_0:2; then A658: x in LSeg (f,k) by A478, A649, A656; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A658, TARSKI:def_4; ::_thesis: verum end; suppose q1 `2 = q2 `2 ; ::_thesis: x in L~ f then LSeg (g,i) = {q1} by A652, A654, RLTOPSP1:70; then x = q1 by A638, A640, TARSKI:def_1; then A659: x in LSeg (f,k) by A478, A649, A651, A650; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A659, TARSKI:def_4; ::_thesis: verum end; suppose q1 `2 < q2 `2 ; ::_thesis: x in L~ f then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `1 = q1 `1 & q1 `2 <= p1 `2 & p1 `2 <= q2 `2 ) } by A652, A654, TOPREAL3:9; then consider p2 being Point of (TOP-REAL 2) such that A660: ( p2 = x & p2 `1 = q1 `1 ) and A661: ( q1 `2 <= p2 `2 & p2 `2 <= q2 `2 ) by A638, A640; ( (G * (i1,i2)) `2 <= p2 `2 & p2 `2 <= (G * (i1,j2)) `2 ) by A651, A653, A661, XXREAL_0:2; then A662: x in LSeg (f,k) by A478, A649, A660; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A662, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ f ; ::_thesis: verum end; end; end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ f or x in L~ g ) assume x in L~ f ; ::_thesis: x in L~ g then A663: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:3; now__::_thesis:_x_in_L~_g percases ( x in L~ (f | k) or x in LSeg (f,k) ) by A663, XBOOLE_0:def_3; supposeA664: x in L~ (f | k) ; ::_thesis: x in L~ g L~ g1 c= L~ g by GOBOARD2:6; hence x in L~ g by A44, A664; ::_thesis: verum end; suppose x in LSeg (f,k) ; ::_thesis: x in L~ g then consider p1 being Point of (TOP-REAL 2) such that A665: p1 = x and A666: p1 `1 = (G * (i1,i2)) `1 and A667: (G * (i1,i2)) `2 <= p1 `2 and A668: p1 `2 <= (G * (i1,j2)) `2 by A478; defpred S2[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds q `2 <= p1 `2 ) ); A669: now__::_thesis:_ex_n_being_Nat_st_S2[n] reconsider n = len g1 as Nat ; take n = n; ::_thesis: S2[n] thus S2[n] ::_thesis: verum proof thus ( len g1 <= n & n <= len g ) by A628, XREAL_1:31; ::_thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds q `2 <= p1 `2 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A670: len g1 in dom g1 by FINSEQ_3:25; let q be Point of (TOP-REAL 2); ::_thesis: ( q = g /. n implies q `2 <= p1 `2 ) assume q = g /. n ; ::_thesis: q `2 <= p1 `2 then q = (f | k) /. (len (f | k)) by A46, A670, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence q `2 <= p1 `2 by A667; ::_thesis: verum end; end; A671: for n being Nat st S2[n] holds n <= len g ; consider ma being Nat such that A672: ( S2[ma] & ( for n being Nat st S2[n] holds n <= ma ) ) from NAT_1:sch_6(A671, A669); reconsider ma = ma as Element of NAT by ORDINAL1:def_12; now__::_thesis:_x_in_L~_g percases ( ma = len g or ma <> len g ) ; supposeA673: ma = len g ; ::_thesis: x in L~ g i2 + 1 <= j2 by A390, NAT_1:13; then A674: 1 <= l by XREAL_1:19; then (len g1) + 1 <= ma by A396, A628, A673, XREAL_1:7; then A675: len g1 <= ma - 1 by XREAL_1:19; then 0 + 1 <= ma by XREAL_1:19; then reconsider m1 = ma - 1 as Element of NAT by INT_1:5; reconsider q = g /. m1 as Point of (TOP-REAL 2) ; A676: ma - 1 <= len g by A673, XREAL_1:43; then A677: q `1 = (G * (i1,i2)) `1 by A629, A675; A678: q `2 <= (G * (i1,j2)) `2 by A629, A676, A675; set lq = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & q `2 <= e `2 & e `2 <= (G * (i1,j2)) `2 ) } ; A679: i2 + l = j2 ; A680: l in dom g2 by A396, A674, FINSEQ_3:25; then A681: g /. ma = g2 /. l by A396, A628, A673, FINSEQ_4:69 .= G * (i1,j2) by A396, A680, A679 ; then (G * (i1,j2)) `2 <= p1 `2 by A672; then A682: p1 `2 = (G * (i1,j2)) `2 by A668, XXREAL_0:1; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A683: 1 <= m1 by A675, XXREAL_0:2; A684: m1 + 1 = ma ; then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg (q,(G * (i1,j2))) ) by A673, A681, A683, EUCLID:53, TOPREAL1:def_3; then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `1 = (G * (i1,i2)) `1 & q `2 <= e `2 & e `2 <= (G * (i1,j2)) `2 ) } by A462, A477, A677, A678, TOPREAL3:9; then A685: p1 in LSeg (g,m1) by A666, A682, A678; LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A673, A683, A684; hence x in L~ g by A665, A685, TARSKI:def_4; ::_thesis: verum end; suppose ma <> len g ; ::_thesis: x in L~ g then ma < len g by A672, XXREAL_0:1; then A686: ma + 1 <= len g by NAT_1:13; reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ; set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa `2 <= p2 `2 & p2 `2 <= qa1 `2 ) } ; A687: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:53; A688: qa `2 <= p1 `2 by A672; A689: len g1 <= ma + 1 by A672, NAT_1:13; then A690: qa1 `1 = (G * (i1,i2)) `1 by A629, A686; A691: now__::_thesis:_not_qa1_`2_<=_p1_`2 assume qa1 `2 <= p1 `2 ; ::_thesis: contradiction then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds q `2 <= p1 `2 ; then ma + 1 <= ma by A672, A686, A689; hence contradiction by XREAL_1:29; ::_thesis: verum end; A692: ( qa `1 = (G * (i1,i2)) `1 & qa = |[(qa `1),(qa `2)]| ) by A629, A672, EUCLID:53; A693: 1 <= ma by A24, A14, A47, A672, NAT_1:13; then LSeg (g,ma) = LSeg (qa,qa1) by A686, TOPREAL1:def_3 .= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `1 = (G * (i1,i2)) `1 & qa `2 <= p2 `2 & p2 `2 <= qa1 `2 ) } by A688, A691, A690, A692, A687, TOPREAL3:9, XXREAL_0:2 ; then A694: x in LSeg (g,ma) by A665, A666, A688, A691; LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A686, A693; hence x in L~ g by A694, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ g ; ::_thesis: verum end; end; end; hence x in L~ g ; ::_thesis: verum end; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then 1 in dom g1 by FINSEQ_3:25; hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:68 .= f /. 1 by A27, A25, FINSEQ_4:71 ; ::_thesis: ( g /. (len g) = f /. (len f) & len f <= len g ) A695: len g = (len g1) + l by A396, FINSEQ_1:22; i2 + 1 <= j2 by A390, NAT_1:13; then A696: 1 <= l by XREAL_1:19; then A697: l in dom g2 by A396, FINSEQ_3:25; hence g /. (len g) = g2 /. l by A695, FINSEQ_4:69 .= G * (i1,(i2 + l)) by A396, A697 .= f /. (len f) by A3, A21, A76 ; ::_thesis: len f <= len g thus len f <= len g by A3, A14, A47, A696, A695, XREAL_1:7; ::_thesis: verum end; end; end; hence ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ; ::_thesis: verum end; supposeA698: i2 = j2 ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) set ppi = G * (i1,i2); set pj = G * (j1,i2); now__::_thesis:_(_(_i1_>_j1_&_ex_g_being_FinSequence_of_the_U1_of_(TOP-REAL_2)_st_ (_g_is_sequence_on_G_&_g_is_one-to-one_&_g_is_unfolded_&_g_is_s.n.c._&_g_is_special_&_L~_g_=_L~_f_&_g_/._1_=_f_/._1_&_g_/._(len_g)_=_f_/._(len_f)_&_len_f_<=_len_g_)_)_or_(_i1_=_j1_&_contradiction_)_or_(_i1_<_j1_&_ex_g_being_FinSequence_of_the_U1_of_(TOP-REAL_2)_st_ (_g_is_sequence_on_G_&_g_is_one-to-one_&_g_is_unfolded_&_g_is_s.n.c._&_g_is_special_&_L~_g_=_L~_f_&_g_/._1_=_f_/._1_&_g_/._(len_g)_=_f_/._(len_f)_&_len_f_<=_len_g_)_)_) percases ( i1 > j1 or i1 = j1 or i1 < j1 ) by XXREAL_0:1; caseA699: i1 > j1 ; ::_thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) c1 /. i1 = c1 . i1 by A66, A60, PARTFUN1:def_6; then A700: c1 /. i1 = G * (i1,i2) by A66, MATRIX_1:def_8; then A701: (X_axis c1) . i1 = (G * (i1,i2)) `1 by A66, A18, A63, A64, A59, GOBOARD1:def_1; c1 /. j1 = c1 . j1 by A23, A60, PARTFUN1:def_6; then A702: c1 /. j1 = G * (j1,i2) by A23, MATRIX_1:def_8; then A703: (X_axis c1) . j1 = (G * (j1,i2)) `1 by A23, A18, A63, A64, A59, GOBOARD1:def_1; then A704: (G * (j1,i2)) `1 < (G * (i1,i2)) `1 by A66, A23, A18, A69, A63, A64, A59, A699, A701, SEQM_3:def_1; reconsider l = i1 - j1 as Element of NAT by A699, INT_1:5; defpred S2[ Nat, set ] means for m being Element of NAT st m = i1 - $1 holds $2 = G * (m,i2); set lk = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } ; A705: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:53; A706: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_Seg_l_holds_ (_i1_-_n_is_Element_of_NAT_&_[(i1_-_n),i2]_in_Indices_G_&_i1_-_n_in_dom_G_) let n be Element of NAT ; ::_thesis: ( n in Seg l implies ( i1 - n is Element of NAT & [(i1 - n),i2] in Indices G & i1 - n in dom G ) ) assume n in Seg l ; ::_thesis: ( i1 - n is Element of NAT & [(i1 - n),i2] in Indices G & i1 - n in dom G ) then A707: n <= l by FINSEQ_1:1; l <= i1 by XREAL_1:43; then reconsider w = i1 - n as Element of NAT by A707, INT_1:5, XXREAL_0:2; ( i1 - n <= i1 & i1 <= len G ) by A66, FINSEQ_3:25, XREAL_1:43; then A708: w <= len G by XXREAL_0:2; A709: 1 <= j1 by A23, FINSEQ_3:25; i1 - l <= i1 - n by A707, XREAL_1:13; then 1 <= w by A709, XXREAL_0:2; then w in dom G by A708, FINSEQ_3:25; hence ( i1 - n is Element of NAT & [(i1 - n),i2] in Indices G & i1 - n in dom G ) by A22, A68, ZFMISC_1:87; ::_thesis: verum end; A710: now__::_thesis:_for_n_being_Nat_st_n_in_Seg_l_holds_ ex_p_being_Element_of_the_U1_of_(TOP-REAL_2)_st_S2[n,p] let n be Nat; ::_thesis: ( n in Seg l implies ex p being Element of the U1 of (TOP-REAL 2) st S2[n,p] ) assume n in Seg l ; ::_thesis: ex p being Element of the U1 of (TOP-REAL 2) st S2[n,p] then reconsider m = i1 - n as Element of NAT by A706; take p = G * (m,i2); ::_thesis: S2[n,p] thus S2[n,p] ; ::_thesis: verum end; consider g2 being FinSequence of (TOP-REAL 2) such that A711: ( len g2 = l & ( for n being Nat st n in Seg l holds S2[n,g2 /. n] ) ) from FINSEQ_4:sch_1(A710); take g = g1 ^ g2; ::_thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A712: dom g2 = Seg l by A711, FINSEQ_1:def_3; now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_holds_ ex_m,_k_being_Element_of_NAT_st_ (_[m,k]_in_Indices_G_&_g2_/._n_=_G_*_(m,k)_) let n be Element of NAT ; ::_thesis: ( n in dom g2 implies ex m, k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) ) assume A713: n in dom g2 ; ::_thesis: ex m, k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) then reconsider m = i1 - n as Element of NAT by A706, A712; take m = m; ::_thesis: ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) take k = i2; ::_thesis: ( [m,k] in Indices G & g2 /. n = G * (m,k) ) thus ( [m,k] in Indices G & g2 /. n = G * (m,k) ) by A706, A711, A712, A713; ::_thesis: verum end; then A714: for n being Element of NAT st n in dom g holds ex i, j being Element of NAT st ( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:23; A715: Seg (len g2) = dom g2 by FINSEQ_1:def_3; A716: (Y_axis c1) . i1 = (G * (i1,i2)) `2 by A66, A18, A61, A59, A700, GOBOARD1:def_2; A717: now__::_thesis:_for_n_being_Element_of_NAT_ for_p_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_g2_/._n_=_p_holds_ (_p_`2_=_(G_*_(i1,i2))_`2_&_(G_*_(j1,i2))_`1_<=_p_`1_&_p_`1_<=_(G_*_(i1,i2))_`1_&_p_in_rng_c1_&_p_`1_<_(G_*_(i1,i2))_`1_) let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 & p `1 < (G * (i1,i2)) `1 ) let p be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & g2 /. n = p implies ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 & p `1 < (G * (i1,i2)) `1 ) ) assume that A718: n in dom g2 and A719: g2 /. n = p ; ::_thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 & p `1 < (G * (i1,i2)) `1 ) reconsider n1 = i1 - n as Element of NAT by A706, A712, A718; n <= len g2 by A715, A718, FINSEQ_1:1; then A720: i1 - (len g2) <= n1 by XREAL_1:13; set pn = G * (n1,i2); A721: g2 /. n = G * (n1,i2) by A711, A715, A718; A722: i1 - n in dom G by A706, A711, A715, A718; then A723: (Y_axis c1) . n1 = (Y_axis c1) . i1 by A66, A18, A70, A61, A59, SEQM_3:def_10; c1 /. n1 = c1 . n1 by A60, A722, PARTFUN1:def_6; then A724: c1 /. n1 = G * (n1,i2) by A722, MATRIX_1:def_8; then A725: (X_axis c1) . n1 = (G * (n1,i2)) `1 by A18, A63, A64, A59, A722, GOBOARD1:def_1; (Y_axis c1) . n1 = (G * (n1,i2)) `2 by A18, A61, A59, A722, A724, GOBOARD1:def_2; hence ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 ) by A66, A23, A18, A69, A63, A64, A59, A711, A716, A701, A703, A719, A722, A721, A720, A723, A725, SEQ_4:137, XREAL_1:43; ::_thesis: ( p in rng c1 & p `1 < (G * (i1,i2)) `1 ) thus p in rng c1 by A60, A719, A722, A721, A724, PARTFUN2:2; ::_thesis: p `1 < (G * (i1,i2)) `1 1 <= n by A715, A718, FINSEQ_1:1; then n1 < i1 by XREAL_1:44; hence p `1 < (G * (i1,i2)) `1 by A66, A18, A69, A63, A64, A59, A701, A719, A722, A721, A725, SEQM_3:def_1; ::_thesis: verum end; A726: g2 is special proof let n be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) set p = g2 /. n; assume A727: ( 1 <= n & n + 1 <= len g2 ) ; ::_thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) then n in dom g2 by SEQ_4:134; then A728: (g2 /. n) `2 = (G * (i1,i2)) `2 by A717; n + 1 in dom g2 by A727, SEQ_4:134; hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A717, A728; ::_thesis: verum end; A729: now__::_thesis:_for_n,_m_being_Element_of_NAT_ for_p,_q_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<_m_&_g2_/._n_=_p_&_g2_/._m_=_q_holds_ q_`1_<_p_`1 let n, m be Element of NAT ; ::_thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds q `1 < p `1 let p, q be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies q `1 < p `1 ) assume that A730: n in dom g2 and A731: m in dom g2 and A732: n < m and A733: ( g2 /. n = p & g2 /. m = q ) ; ::_thesis: q `1 < p `1 A734: i1 - n in dom G by A706, A712, A730; reconsider n1 = i1 - n, m1 = i1 - m as Element of NAT by A706, A712, A730, A731; set pn = G * (n1,i2); set pm = G * (m1,i2); A735: m1 < n1 by A732, XREAL_1:15; c1 /. n1 = c1 . n1 by A60, A706, A712, A730, PARTFUN1:def_6; then c1 /. n1 = G * (n1,i2) by A734, MATRIX_1:def_8; then A736: (X_axis c1) . n1 = (G * (n1,i2)) `1 by A65, A60, A734, GOBOARD1:def_1; A737: i1 - m in dom G by A706, A712, A731; c1 /. m1 = c1 . m1 by A60, A706, A712, A731, PARTFUN1:def_6; then c1 /. m1 = G * (m1,i2) by A737, MATRIX_1:def_8; then A738: (X_axis c1) . m1 = (G * (m1,i2)) `1 by A65, A60, A737, GOBOARD1:def_1; ( g2 /. n = G * (n1,i2) & g2 /. m = G * (m1,i2) ) by A711, A712, A730, A731; hence q `1 < p `1 by A69, A65, A60, A733, A734, A737, A735, A736, A738, SEQM_3:def_1; ::_thesis: verum end; for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds LSeg (g2,n) misses LSeg (g2,m) proof let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) ) assume that A739: m > n + 1 and A740: n in dom g2 and A741: n + 1 in dom g2 and A742: m in dom g2 and A743: m + 1 in dom g2 and A744: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A745: ( p1 `2 = (G * (i1,i2)) `2 & p2 `2 = (G * (i1,i2)) `2 ) by A717, A740, A741; n < n + 1 by NAT_1:13; then A746: p2 `1 < p1 `1 by A729, A740, A741; set lp = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p2 `1 <= w `1 & w `1 <= p1 `1 ) } ; set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } ; A747: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:53; m < m + 1 by NAT_1:13; then A748: q2 `1 < q1 `1 by A729, A742, A743; A749: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; set x = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)); A750: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,n) by A744, XBOOLE_0:def_4; A751: ( q1 `2 = (G * (i1,i2)) `2 & q2 `2 = (G * (i1,i2)) `2 ) by A717, A742, A743; A752: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,m) by A744, XBOOLE_0:def_4; ( 1 <= m & m + 1 <= len g2 ) by A742, A743, FINSEQ_3:25; then LSeg (g2,m) = LSeg (q2,q1) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } by A748, A751, A749, TOPREAL3:10 ; then A753: ex tm being Point of (TOP-REAL 2) st ( tm = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tm `2 = (G * (i1,i2)) `2 & q2 `1 <= tm `1 & tm `1 <= q1 `1 ) by A752; ( 1 <= n & n + 1 <= len g2 ) by A740, A741, FINSEQ_3:25; then LSeg (g2,n) = LSeg (p2,p1) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p2 `1 <= w `1 & w `1 <= p1 `1 ) } by A746, A745, A747, TOPREAL3:10 ; then A754: ex tn being Point of (TOP-REAL 2) st ( tn = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tn `2 = (G * (i1,i2)) `2 & p2 `1 <= tn `1 & tn `1 <= p1 `1 ) by A750; q1 `1 < p2 `1 by A729, A739, A741, A742; hence contradiction by A754, A753, XXREAL_0:2; ::_thesis: verum end; then A755: g2 is s.n.c. by GOBOARD2:1; A756: not f /. k in L~ g2 proof set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume f /. k in L~ g2 ; ::_thesis: contradiction then consider X being set such that A757: f /. k in X and A758: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A759: X = LSeg (g2,m) and A760: ( 1 <= m & m + 1 <= len g2 ) by A758; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A761: m in dom g2 by A760, SEQ_4:134; then A762: q1 `2 = (G * (i1,i2)) `2 by A717; set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } ; A763: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; A764: m + 1 in dom g2 by A760, SEQ_4:134; then A765: q2 `2 = (G * (i1,i2)) `2 by A717; m < m + 1 by NAT_1:13; then A766: q2 `1 < q1 `1 by A729, A761, A764; LSeg (g2,m) = LSeg (q2,q1) by A760, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q2 `1 <= w `1 & w `1 <= q1 `1 ) } by A762, A765, A766, A763, TOPREAL3:10 ; then ex p being Point of (TOP-REAL 2) st ( p = f /. k & p `2 = (G * (i1,i2)) `2 & q2 `1 <= p `1 & p `1 <= q1 `1 ) by A757, A759; hence contradiction by A29, A717, A761; ::_thesis: verum end; (Y_axis c1) . j1 = (G * (j1,i2)) `2 by A23, A18, A61, A59, A702, GOBOARD1:def_2; then A767: (G * (i1,i2)) `2 = (G * (j1,i2)) `2 by A66, A23, A18, A70, A61, A59, A716, SEQM_3:def_10; now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<>_m_holds_ not_g2_/._n_=_g2_/._m let n, m be Element of NAT ; ::_thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m ) assume that A768: ( n in dom g2 & m in dom g2 ) and A769: n <> m ; ::_thesis: not g2 /. n = g2 /. m reconsider n1 = i1 - n, m1 = i1 - m as Element of NAT by A706, A712, A768; A770: ( g2 /. n = G * (n1,i2) & g2 /. m = G * (m1,i2) ) by A711, A712, A768; assume A771: g2 /. n = g2 /. m ; ::_thesis: contradiction ( [(i1 - n),i2] in Indices G & [(i1 - m),i2] in Indices G ) by A706, A712, A768; then n1 = m1 by A770, A771, GOBOARD1:5; hence contradiction by A769; ::_thesis: verum end; then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds n = m ; then A772: g2 is one-to-one by PARTFUN2:9; reconsider m1 = i1 - l as Element of NAT ; A773: G * (j1,i2) = |[((G * (j1,i2)) `1),((G * (j1,i2)) `2)]| by EUCLID:53; A774: LSeg (f,k) = LSeg ((G * (j1,i2)),(G * (i1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A704, A767, A705, A773, TOPREAL3:10 ; A775: rng g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g2 or x in LSeg (f,k) ) assume x in rng g2 ; ::_thesis: x in LSeg (f,k) then consider n being Element of NAT such that A776: n in dom g2 and A777: g2 /. n = x by PARTFUN2:2; reconsider n1 = i1 - n as Element of NAT by A706, A711, A715, A776; set pn = G * (n1,i2); A778: g2 /. n = G * (n1,i2) by A711, A715, A776; then A779: (G * (n1,i2)) `1 <= (G * (i1,i2)) `1 by A717, A776; ( (G * (n1,i2)) `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= (G * (n1,i2)) `1 ) by A717, A776, A778; hence x in LSeg (f,k) by A774, A777, A778, A779; ::_thesis: verum end; A780: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_&_n_+_1_in_dom_g2_holds_ for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g2_/._n_=_G_*_(l1,l2)_&_g2_/._(n_+_1)_=_G_*_(l3,l4)_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let n be Element of NAT ; ::_thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A781: n in dom g2 and A782: n + 1 in dom g2 ; ::_thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 reconsider m1 = i1 - n, m2 = i1 - (n + 1) as Element of NAT by A706, A712, A781, A782; let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A783: [l1,l2] in Indices G and A784: [l3,l4] in Indices G and A785: g2 /. n = G * (l1,l2) and A786: g2 /. (n + 1) = G * (l3,l4) ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ( [(i1 - (n + 1)),i2] in Indices G & g2 /. (n + 1) = G * (m2,i2) ) by A706, A711, A712, A782; then A787: ( l3 = m2 & l4 = i2 ) by A784, A786, GOBOARD1:5; ( [(i1 - n),i2] in Indices G & g2 /. n = G * (m1,i2) ) by A706, A711, A712, A781; then ( l1 = m1 & l2 = i2 ) by A783, A785, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs ((i1 - n) - (i1 - (n + 1)))) + 0 by A787, ABSVALUE:2 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; now__::_thesis:_for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g1_/._(len_g1)_=_G_*_(l1,l2)_&_g2_/._1_=_G_*_(l3,l4)_&_len_g1_in_dom_g1_&_1_in_dom_g2_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A788: [l1,l2] in Indices G and A789: [l3,l4] in Indices G and A790: g1 /. (len g1) = G * (l1,l2) and A791: g2 /. 1 = G * (l3,l4) and len g1 in dom g1 and A792: 1 in dom g2 ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 reconsider m1 = i1 - 1 as Element of NAT by A706, A712, A792; ( [(i1 - 1),i2] in Indices G & g2 /. 1 = G * (m1,i2) ) by A706, A711, A712, A792; then A793: ( l3 = m1 & l4 = i2 ) by A789, A791, GOBOARD1:5; (f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:71; then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A788, A790, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs (i1 - (i1 - 1))) + 0 by A793, ABSVALUE:2 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; then for n being Element of NAT st n in dom g & n + 1 in dom g holds for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds (abs (m - i)) + (abs (k - j)) = 1 by A48, A780, GOBOARD1:24; hence g is_sequence_on G by A714, GOBOARD1:def_9; ::_thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A794: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (j1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def_3; A795: not f /. k in rng g2 proof assume f /. k in rng g2 ; ::_thesis: contradiction then consider n being Element of NAT such that A796: n in dom g2 and A797: g2 /. n = f /. k by PARTFUN2:2; reconsider n1 = i1 - n as Element of NAT by A706, A711, A715, A796; ( [(i1 - n),i2] in Indices G & g2 /. n = G * (n1,i2) ) by A706, A711, A715, A796; then A798: n1 = i1 by A28, A29, A797, GOBOARD1:5; 0 < n by A715, A796, FINSEQ_1:1; hence contradiction by A798; ::_thesis: verum end; (rng g1) /\ (rng g2) = {} proof set x = the Element of (rng g1) /\ (rng g2); assume A799: not (rng g1) /\ (rng g2) = {} ; ::_thesis: contradiction then A800: the Element of (rng g1) /\ (rng g2) in rng g2 by XBOOLE_0:def_4; A801: the Element of (rng g1) /\ (rng g2) in rng g1 by A799, XBOOLE_0:def_4; now__::_thesis:_contradiction percases ( k = 1 or 1 < k ) by A24, XXREAL_0:1; suppose k = 1 ; ::_thesis: contradiction hence contradiction by A52, A795, A801, A800, TARSKI:def_1; ::_thesis: verum end; suppose 1 < k ; ::_thesis: contradiction then ( the Element of (rng g1) /\ (rng g2) in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A775, A801, A800, GOBOARD2:4, XBOOLE_0:def_4; hence contradiction by A795, A800, TARSKI:def_1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; then rng g1 misses rng g2 by XBOOLE_0:def_7; hence g is one-to-one by A40, A772, FINSEQ_3:91; ::_thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A802: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} proof let n be Element of NAT ; ::_thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} ) assume that A803: 1 <= n and A804: n + 2 <= len g2 ; ::_thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} A805: n + 1 in dom g2 by A803, A804, SEQ_4:135; then g2 /. (n + 1) in rng g2 by PARTFUN2:2; then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775; then consider u1 being Point of (TOP-REAL 2) such that A806: g2 /. (n + 1) = u1 and A807: u1 `2 = (G * (i1,i2)) `2 and (G * (j1,i2)) `1 <= u1 `1 and u1 `1 <= (G * (i1,i2)) `1 ; A808: n + 2 in dom g2 by A803, A804, SEQ_4:135; then g2 /. (n + 2) in rng g2 by PARTFUN2:2; then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775; then consider u2 being Point of (TOP-REAL 2) such that A809: g2 /. (n + 2) = u2 and A810: u2 `2 = (G * (i1,i2)) `2 and (G * (j1,i2)) `1 <= u2 `1 and u2 `1 <= (G * (i1,i2)) `1 ; ( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11; then A811: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A804, A806, A809, TOPREAL1:def_3; n + 1 < (n + 1) + 1 by NAT_1:13; then A812: u2 `1 < u1 `1 by A729, A805, A808, A806, A809; A813: n in dom g2 by A803, A804, SEQ_4:135; then g2 /. n in rng g2 by PARTFUN2:2; then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775; then consider u being Point of (TOP-REAL 2) such that A814: g2 /. n = u and A815: u `2 = (G * (i1,i2)) `2 and (G * (j1,i2)) `1 <= u `1 and u `1 <= (G * (i1,i2)) `1 ; n + 1 <= n + 2 by XREAL_1:6; then n + 1 <= len g2 by A804, XXREAL_0:2; then A816: LSeg (g2,n) = LSeg (u,u1) by A803, A814, A806, TOPREAL1:def_3; set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= u `1 ) } ; n < n + 1 by NAT_1:13; then A817: u1 `1 < u `1 by A729, A813, A805, A814, A806; then A818: u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= u `1 ) } by A807, A812; ( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:53; then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= u `1 ) } by A814, A815, A809, A810, A812, A817, TOPREAL3:10, XXREAL_0:2; hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A814, A806, A809, A816, A811, A818, TOPREAL1:8; ::_thesis: verum end; thus g is unfolded ::_thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof let n be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ) assume that A819: 1 <= n and A820: n + 2 <= len g ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A821: (n + 1) + 1 <= len g by A820; n + 1 <= (n + 1) + 1 by NAT_1:11; then A822: n + 1 <= len g by A820, XXREAL_0:2; A823: len g = (len g1) + (len g2) by FINSEQ_1:22; (n + 2) - (len g1) = (n - (len g1)) + 2 ; then A824: (n - (len g1)) + 2 <= len g2 by A820, A823, XREAL_1:20; A825: 1 <= n + 1 by NAT_1:11; A826: n <= n + 1 by NAT_1:11; A827: n + (1 + 1) = (n + 1) + 1 ; percases ( n + 2 <= len g1 or len g1 < n + 2 ) ; supposeA828: n + 2 <= len g1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A829: n + (1 + 1) = (n + 1) + 1 ; A830: n + 1 in dom g1 by A819, A828, SEQ_4:135; then A831: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:68; n in dom g1 by A819, A828, SEQ_4:135; then A832: LSeg (g1,n) = LSeg (g,n) by A830, TOPREAL3:18; n + 2 in dom g1 by A819, A828, SEQ_4:135; then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A830, A829, TOPREAL3:18; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A819, A828, A832, A831, TOPREAL1:def_6; ::_thesis: verum end; suppose len g1 < n + 2 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then (len g1) + 1 <= n + 2 by NAT_1:13; then A833: len g1 <= (n + 2) - 1 by XREAL_1:19; thus (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ::_thesis: verum proof percases ( len g1 = n + 1 or len g1 <> n + 1 ) ; supposeA834: len g1 = n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then 1 <= (len g) - (len g1) by A821, XREAL_1:19; then 1 in dom g2 by A823, FINSEQ_3:25; then A835: g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775; then consider u1 being Point of (TOP-REAL 2) such that A836: g2 /. 1 = u1 and u1 `2 = (G * (i1,i2)) `2 and (G * (j1,i2)) `1 <= u1 `1 and u1 `1 <= (G * (i1,i2)) `1 ; G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (j1,i2))) by RLTOPSP1:68; then A837: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A794, A775, A835, A836, TOPREAL1:6; 1 <= n + 1 by NAT_1:11; then A838: n + 1 in dom g1 by A834, FINSEQ_3:25; then A839: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A834, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; now__::_thesis:_not_k_=_1 1 < len g1 by A819, A834, NAT_1:13; then A840: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A840, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A841: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; A842: LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:19; n in dom g1 by A819, A826, A834, FINSEQ_3:25; then A843: LSeg (g,n) = LSeg (g1,n) by A838, TOPREAL3:18; ( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A819, A820, A825, A822, A827, TOPREAL1:21; then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def_4; then A844: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:31; 1 <= len g2 by A820, A827, A823, A834, XREAL_1:6; then g /. (n + 2) = g2 /. 1 by A827, A834, SEQ_4:136; then LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A820, A825, A827, A839, A836, TOPREAL1:def_3; then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A842, A841, A843, A839, A837, XBOOLE_1:27; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A844, XBOOLE_0:def_10; ::_thesis: verum end; suppose len g1 <> n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then len g1 < n + 1 by A833, XXREAL_0:1; then A845: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; thus (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ::_thesis: verum proof percases ( len g1 = n or len g1 <> n ) ; supposeA846: len g1 = n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A847: 2 <= len g2 by A820, A823, XREAL_1:6; then 1 <= len g2 by XXREAL_0:2; then A848: g /. (n + 1) = g2 /. 1 by A846, SEQ_4:136; 1 <= len g2 by A847, XXREAL_0:2; then A849: 1 in dom g2 by FINSEQ_3:25; then g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775; then consider u1 being Point of (TOP-REAL 2) such that A850: g2 /. 1 = u1 and A851: u1 `2 = (G * (i1,i2)) `2 and (G * (j1,i2)) `1 <= u1 `1 and A852: u1 `1 <= (G * (i1,i2)) `1 ; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then g /. n = (f | k) /. (len (f | k)) by A46, A846, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; then A853: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A819, A822, A848, A850, TOPREAL1:def_3; A854: 2 in dom g2 by A847, FINSEQ_3:25; then g2 /. 2 in rng g2 by PARTFUN2:2; then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A774, A775; then consider u2 being Point of (TOP-REAL 2) such that A855: g2 /. 2 = u2 and A856: u2 `2 = (G * (i1,i2)) `2 and (G * (j1,i2)) `1 <= u2 `1 and A857: u2 `1 <= (G * (i1,i2)) `1 ; set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } ; u2 = |[(u2 `1),(u2 `2)]| by EUCLID:53; then A858: LSeg ((G * (i1,i2)),(g2 /. 2)) = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A705, A855, A856, A857, TOPREAL3:10; u2 `1 < u1 `1 by A729, A849, A854, A850, A855; then A859: u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u2 `1 <= w `1 & w `1 <= (G * (i1,i2)) `1 ) } by A851, A852; g /. (n + 2) = g2 /. 2 by A846, A847, SEQ_4:136; then LSeg (g,(n + 1)) = LSeg (u1,u2) by A820, A825, A827, A848, A850, A855, TOPREAL1:def_3; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A848, A850, A855, A859, A853, A858, TOPREAL1:8; ::_thesis: verum end; suppose len g1 <> n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A860: len g1 < n by A845, XXREAL_0:1; then (len g1) + 1 <= n by NAT_1:13; then A861: 1 <= n1 by XREAL_1:19; n1 + (len g1) = n ; then A862: LSeg (g,n) = LSeg (g2,n1) by A822, A860, GOBOARD2:5; A863: n + 1 = (n1 + 1) + (len g1) ; (n1 + 1) + (len g1) = n + 1 ; then n1 + 1 <= len g2 by A822, A823, XREAL_1:6; then A864: g /. (n + 1) = g2 /. (n1 + 1) by A863, NAT_1:11, SEQ_4:136; len g1 < n + 1 by A826, A860, XXREAL_0:2; then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A821, A863, GOBOARD2:5; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A802, A824, A862, A864, A861; ::_thesis: verum end; end; end; end; end; end; end; end; end; A865: L~ g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g2 or x in LSeg (f,k) ) set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume x in L~ g2 ; ::_thesis: x in LSeg (f,k) then consider X being set such that A866: x in X and A867: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A868: X = LSeg (g2,m) and A869: ( 1 <= m & m + 1 <= len g2 ) by A867; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A870: LSeg (g2,m) = LSeg (q1,q2) by A869, TOPREAL1:def_3; m + 1 in dom g2 by A869, SEQ_4:134; then A871: g2 /. (m + 1) in rng g2 by PARTFUN2:2; m in dom g2 by A869, SEQ_4:134; then g2 /. m in rng g2 by PARTFUN2:2; then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (j1,i2))) by A794, A775, A871, TOPREAL1:6; hence x in LSeg (f,k) by A794, A866, A868, A870; ::_thesis: verum end; A872: (L~ g1) /\ (L~ g2) = {} proof percases ( k = 1 or k <> 1 ) ; suppose k = 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} hence (L~ g1) /\ (L~ g2) = {} by A52; ::_thesis: verum end; suppose k <> 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} then 1 < k by A24, XXREAL_0:1; then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:4; then A873: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A865, XBOOLE_1:26; now__::_thesis:_not_(L~_g1)_/\_(L~_g2)_<>_{} set x = the Element of (L~ g1) /\ (L~ g2); assume (L~ g1) /\ (L~ g2) <> {} ; ::_thesis: contradiction then ( the Element of (L~ g1) /\ (L~ g2) in {(f /. k)} & the Element of (L~ g1) /\ (L~ g2) in L~ g2 ) by A873, TARSKI:def_3, XBOOLE_0:def_4; hence contradiction by A756, TARSKI:def_1; ::_thesis: verum end; hence (L~ g1) /\ (L~ g2) = {} ; ::_thesis: verum end; end; end; for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds LSeg (g,n) misses LSeg (g,m) proof A874: 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A875: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:71 ; reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ; set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ; set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ; let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) ) assume that A876: m > n + 1 and A877: n in dom g and A878: n + 1 in dom g and A879: m in dom g and A880: m + 1 in dom g ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A881: 1 <= n by A877, FINSEQ_3:25; j1 + 1 <= i1 by A699, NAT_1:13; then A882: 1 <= l by XREAL_1:19; then A883: 1 in dom g2 by A711, FINSEQ_3:25; then A884: ( qq `2 = (G * (i1,i2)) `2 & qq `1 < (G * (i1,i2)) `1 ) by A717; A885: g /. ((len g1) + 1) = qq by A711, A882, SEQ_4:136; A886: (G * (j1,i2)) `1 <= qq `1 by A717, A883; A887: m + 1 <= len g by A880, FINSEQ_3:25; A888: 1 <= m + 1 by A880, FINSEQ_3:25; A889: 1 <= n + 1 by A878, FINSEQ_3:25; A890: n + 1 <= len g by A878, FINSEQ_3:25; A891: qq = |[(qq `1),(qq `2)]| by EUCLID:53; A892: 1 <= m by A879, FINSEQ_3:25; set ql = { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & qq `1 <= z `1 & z `1 <= (G * (i1,i2)) `1 ) } ; A893: n <= n + 1 by NAT_1:11; A894: len g = (len g1) + (len g2) by FINSEQ_1:22; then (len g1) + 1 <= len g by A711, A882, XREAL_1:7; then A895: LSeg (g,(len g1)) = LSeg (qq,(G * (i1,i2))) by A874, A875, A885, TOPREAL1:def_3 .= { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & qq `1 <= z `1 & z `1 <= (G * (i1,i2)) `1 ) } by A705, A884, A891, TOPREAL3:10 ; A896: m <= m + 1 by NAT_1:11; then A897: n + 1 <= m + 1 by A876, XXREAL_0:2; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m + 1 <= len g1 or len g1 < m + 1 ) ; supposeA898: m + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then m <= len g1 by A896, XXREAL_0:2; then A899: m in dom g1 by A892, FINSEQ_3:25; m + 1 in dom g1 by A888, A898, FINSEQ_3:25; then A900: LSeg (g,m) = LSeg (g1,m) by A899, TOPREAL3:18; A901: n + 1 <= len g1 by A897, A898, XXREAL_0:2; then n <= len g1 by A893, XXREAL_0:2; then A902: n in dom g1 by A881, FINSEQ_3:25; n + 1 in dom g1 by A889, A901, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A902, TOPREAL3:18; hence LSeg (g,n) misses LSeg (g,m) by A42, A876, A900, TOPREAL1:def_7; ::_thesis: verum end; suppose len g1 < m + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A903: len g1 <= m by NAT_1:13; then reconsider m1 = m - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m = len g1 or m <> len g1 ) ; supposeA904: m = len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A905: LSeg (g,m) c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) ) assume x in LSeg (g,m) ; ::_thesis: x in LSeg (f,k) then consider px being Point of (TOP-REAL 2) such that A906: ( px = x & px `2 = (G * (i1,i2)) `2 ) and A907: qq `1 <= px `1 and A908: px `1 <= (G * (i1,i2)) `1 by A895, A904; (G * (j1,i2)) `1 <= px `1 by A886, A907, XXREAL_0:2; hence x in LSeg (f,k) by A774, A906, A908; ::_thesis: verum end; n <= len g1 by A876, A893, A904, XXREAL_0:2; then A909: n in dom g1 by A881, FINSEQ_3:25; now__::_thesis:_not_k_=_1 1 < len g1 by A876, A889, A904, XXREAL_0:2; then A910: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A910, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A911: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; A912: n + 1 in dom g1 by A876, A889, A904, FINSEQ_3:25; then A913: LSeg (g,n) = LSeg (g1,n) by A909, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A876, A881, A904; then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:74; then A914: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A911, A905, XBOOLE_1:27; now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); assume A915: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A916: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in {(f /. k)} by A914, A915, TARSKI:def_3; then A917: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) = f /. k by TARSKI:def_1; f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:71; hence contradiction by A40, A41, A42, A876, A904, A909, A912, A913, A916, A917, GOBOARD2:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose m <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A918: len g1 < m by A903, XXREAL_0:1; then (len g1) + 1 <= m by NAT_1:13; then A919: 1 <= m1 by XREAL_1:19; m + 1 = (m1 + 1) + (len g1) ; then A920: m1 + 1 <= len g2 by A887, A894, XREAL_1:6; m = m1 + (len g1) ; then A921: LSeg (g,m) = LSeg (g2,m1) by A887, A918, GOBOARD2:5; then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A919, A920; then A922: LSeg (g,m) c= L~ g2 by ZFMISC_1:74; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( n + 1 <= len g1 or len g1 < n + 1 ) ; supposeA923: n + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then n <= len g1 by A893, XXREAL_0:2; then A924: n in dom g1 by A881, FINSEQ_3:25; n + 1 in dom g1 by A889, A923, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A924, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A881, A923; then LSeg (g,n) c= L~ g1 by ZFMISC_1:74; then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A872, A922, XBOOLE_1:3, XBOOLE_1:27; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose len g1 < n + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A925: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; A926: (n - (len g1)) + 1 = (n + 1) - (len g1) ; A927: n = n1 + (len g1) ; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( len g1 = n or n <> len g1 ) ; supposeA928: len g1 = n ; ::_thesis: LSeg (g,n) misses LSeg (g,m) now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ; set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); set q1l = { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q2 `1 <= v `1 & v `1 <= q1 `1 ) } ; A929: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; assume A930: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A931: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,m) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by A930, XBOOLE_0:def_4; then A932: ex qx being Point of (TOP-REAL 2) st ( qx = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qx `2 = (G * (i1,i2)) `2 & qq `1 <= qx `1 & qx `1 <= (G * (i1,i2)) `1 ) by A895, A928; A933: m1 in dom g2 by A919, A920, SEQ_4:134; then A934: q1 `2 = (G * (i1,i2)) `2 by A717; A935: m1 + 1 in dom g2 by A919, A920, SEQ_4:134; then A936: q2 `2 = (G * (i1,i2)) `2 by A717; m1 < m1 + 1 by NAT_1:13; then A937: q2 `1 < q1 `1 by A729, A933, A935; LSeg (g2,m1) = LSeg (q2,q1) by A919, A920, TOPREAL1:def_3 .= { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q2 `1 <= v `1 & v `1 <= q1 `1 ) } by A934, A936, A937, A929, TOPREAL3:10 ; then A938: ex qy being Point of (TOP-REAL 2) st ( qy = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qy `2 = (G * (i1,i2)) `2 & q2 `1 <= qy `1 & qy `1 <= q1 `1 ) by A921, A931; ( m1 > n1 + 1 & n1 + 1 >= 1 ) by A876, A926, NAT_1:11, XREAL_1:9; then m1 > 1 by XXREAL_0:2; then q1 `1 < qq `1 by A729, A883, A933; hence contradiction by A932, A938, XXREAL_0:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose n <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then len g1 < n by A925, XXREAL_0:1; then A939: LSeg (g,n) = LSeg (g2,n1) by A890, A927, GOBOARD2:5; m1 > n1 + 1 by A876, A926, XREAL_1:9; hence LSeg (g,n) misses LSeg (g,m) by A755, A921, A939, TOPREAL1:def_7; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; hence g is s.n.c. by GOBOARD2:1; ::_thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) now__::_thesis:_(_(g1_/._(len_g1))_`1_=_(g2_/._1)_`1_or_(g1_/._(len_g1))_`2_=_(g2_/._1)_`2_) set p = g1 /. (len g1); set q = g2 /. 1; j1 + 1 <= i1 by A699, NAT_1:13; then 1 <= l by XREAL_1:19; then 1 in dom g2 by A712, FINSEQ_1:1; then (g2 /. 1) `2 = (G * (i1,i2)) `2 by A717; hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:71; ::_thesis: verum end; hence g is special by A43, A726, GOBOARD2:8; ::_thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) thus L~ g = L~ f ::_thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ; set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ; A940: len g = (len g1) + (len g2) by FINSEQ_1:22; A941: now__::_thesis:_for_j_being_Element_of_NAT_st_len_g1_<=_j_&_j_<=_len_g_holds_ for_p_being_Point_of_(TOP-REAL_2)_st_p_=_g_/._j_holds_ (_p_`2_=_(G_*_(i1,i2))_`2_&_(G_*_(j1,i2))_`1_<=_p_`1_&_p_`1_<=_(G_*_(i1,i2))_`1_&_p_in_rng_c1_) let j be Element of NAT ; ::_thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds ( b3 `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b3 `1 & b3 `1 <= (G * (i1,i2)) `1 & b3 in rng c1 ) ) assume that A942: len g1 <= j and A943: j <= len g ; ::_thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds ( b3 `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b3 `1 & b3 `1 <= (G * (i1,i2)) `1 & b3 in rng c1 ) reconsider w = j - (len g1) as Element of NAT by A942, INT_1:5; let p be Point of (TOP-REAL 2); ::_thesis: ( p = g /. j implies ( b2 `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b2 `1 & b2 `1 <= (G * (i1,i2)) `1 & b2 in rng c1 ) ) assume A944: p = g /. j ; ::_thesis: ( b2 `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b2 `1 & b2 `1 <= (G * (i1,i2)) `1 & b2 in rng c1 ) percases ( j = len g1 or j <> len g1 ) ; supposeA945: j = len g1 ; ::_thesis: ( b2 `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b2 `1 & b2 `1 <= (G * (i1,i2)) `1 & b2 in rng c1 ) 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A946: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence p `2 = (G * (i1,i2)) `2 by A944, A945; ::_thesis: ( (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 ) thus ( (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 ) by A66, A23, A18, A69, A63, A64, A59, A699, A701, A703, A944, A945, A946, SEQM_3:def_1; ::_thesis: p in rng c1 Seg (len c1) = dom c1 by FINSEQ_1:def_3; hence p in rng c1 by A66, A18, A59, A700, A944, A945, A946, PARTFUN2:2; ::_thesis: verum end; suppose j <> len g1 ; ::_thesis: ( b2 `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= b2 `1 & b2 `1 <= (G * (i1,i2)) `1 & b2 in rng c1 ) then len g1 < j by A942, XXREAL_0:1; then (len g1) + 1 <= j by NAT_1:13; then A947: 1 <= w by XREAL_1:19; A948: w <= len g2 by A940, A943, XREAL_1:20; then A949: w in dom g2 by A947, FINSEQ_3:25; j = w + (len g1) ; then g /. j = g2 /. w by A947, A948, SEQ_4:136; hence ( p `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= p `1 & p `1 <= (G * (i1,i2)) `1 & p in rng c1 ) by A717, A944, A949; ::_thesis: verum end; end; end; thus L~ g c= L~ f :: according to XBOOLE_0:def_10 ::_thesis: L~ f c= L~ g proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g or x in L~ f ) assume x in L~ g ; ::_thesis: x in L~ f then consider X being set such that A950: x in X and A951: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def_4; consider i being Element of NAT such that A952: X = LSeg (g,i) and A953: 1 <= i and A954: i + 1 <= len g by A951; percases ( i + 1 <= len g1 or i + 1 > len g1 ) ; supposeA955: i + 1 <= len g1 ; ::_thesis: x in L~ f i <= i + 1 by NAT_1:11; then i <= len g1 by A955, XXREAL_0:2; then A956: i in dom g1 by A953, FINSEQ_3:25; 1 <= i + 1 by NAT_1:11; then i + 1 in dom g1 by A955, FINSEQ_3:25; then X = LSeg (g1,i) by A952, A956, TOPREAL3:18; then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A953, A955; then A957: x in L~ (f | k) by A44, A950, TARSKI:def_4; L~ (f | k) c= L~ f by TOPREAL3:20; hence x in L~ f by A957; ::_thesis: verum end; supposeA958: i + 1 > len g1 ; ::_thesis: x in L~ f reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ; A959: i <= len g by A954, NAT_1:13; A960: len g1 <= i by A958, NAT_1:13; then A961: q1 `2 = (G * (i1,i2)) `2 by A941, A959; A962: q1 `1 <= (G * (i1,i2)) `1 by A941, A960, A959; A963: (G * (j1,i2)) `1 <= q1 `1 by A941, A960, A959; q2 `2 = (G * (i1,i2)) `2 by A941, A954, A958; then A964: q2 = |[(q2 `1),(q1 `2)]| by A961, EUCLID:53; A965: q2 `1 <= (G * (i1,i2)) `1 by A941, A954, A958; A966: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A953, A954, EUCLID:53, TOPREAL1:def_3; A967: (G * (j1,i2)) `1 <= q2 `1 by A941, A954, A958; now__::_thesis:_x_in_L~_f percases ( q1 `1 > q2 `1 or q1 `1 = q2 `1 or q1 `1 < q2 `1 ) by XXREAL_0:1; suppose q1 `1 > q2 `1 ; ::_thesis: x in L~ f then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = q1 `2 & q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) } by A964, A966, TOPREAL3:10; then consider p2 being Point of (TOP-REAL 2) such that A968: ( p2 = x & p2 `2 = q1 `2 ) and A969: ( q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) by A950, A952; ( (G * (j1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (i1,i2)) `1 ) by A962, A967, A969, XXREAL_0:2; then A970: x in LSeg (f,k) by A774, A961, A968; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A970, TARSKI:def_4; ::_thesis: verum end; suppose q1 `1 = q2 `1 ; ::_thesis: x in L~ f then LSeg (g,i) = {q1} by A964, A966, RLTOPSP1:70; then x = q1 by A950, A952, TARSKI:def_1; then A971: x in LSeg (f,k) by A774, A961, A963, A962; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A971, TARSKI:def_4; ::_thesis: verum end; suppose q1 `1 < q2 `1 ; ::_thesis: x in L~ f then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `2 = q1 `2 & q1 `1 <= p1 `1 & p1 `1 <= q2 `1 ) } by A964, A966, TOPREAL3:10; then consider p2 being Point of (TOP-REAL 2) such that A972: ( p2 = x & p2 `2 = q1 `2 ) and A973: ( q1 `1 <= p2 `1 & p2 `1 <= q2 `1 ) by A950, A952; ( (G * (j1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (i1,i2)) `1 ) by A963, A965, A973, XXREAL_0:2; then A974: x in LSeg (f,k) by A774, A961, A972; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A974, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ f ; ::_thesis: verum end; end; end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ f or x in L~ g ) assume x in L~ f ; ::_thesis: x in L~ g then A975: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:3; percases ( x in L~ (f | k) or x in LSeg (f,k) ) by A975, XBOOLE_0:def_3; supposeA976: x in L~ (f | k) ; ::_thesis: x in L~ g L~ g1 c= L~ g by GOBOARD2:6; hence x in L~ g by A44, A976; ::_thesis: verum end; suppose x in LSeg (f,k) ; ::_thesis: x in L~ g then consider p1 being Point of (TOP-REAL 2) such that A977: p1 = x and A978: p1 `2 = (G * (i1,i2)) `2 and A979: (G * (j1,i2)) `1 <= p1 `1 and A980: p1 `1 <= (G * (i1,i2)) `1 by A774; defpred S3[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds q `1 >= p1 `1 ) ); A981: now__::_thesis:_ex_n_being_Nat_st_S3[n] reconsider n = len g1 as Nat ; take n = n; ::_thesis: S3[n] thus S3[n] ::_thesis: verum proof thus ( len g1 <= n & n <= len g ) by A940, XREAL_1:31; ::_thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds q `1 >= p1 `1 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A982: len g1 in dom g1 by FINSEQ_3:25; let q be Point of (TOP-REAL 2); ::_thesis: ( q = g /. n implies q `1 >= p1 `1 ) assume q = g /. n ; ::_thesis: q `1 >= p1 `1 then q = (f | k) /. (len (f | k)) by A46, A982, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence q `1 >= p1 `1 by A980; ::_thesis: verum end; end; A983: for n being Nat st S3[n] holds n <= len g ; consider ma being Nat such that A984: ( S3[ma] & ( for n being Nat st S3[n] holds n <= ma ) ) from NAT_1:sch_6(A983, A981); reconsider ma = ma as Element of NAT by ORDINAL1:def_12; now__::_thesis:_x_in_L~_g percases ( ma = len g or ma <> len g ) ; supposeA985: ma = len g ; ::_thesis: x in L~ g j1 + 1 <= i1 by A699, NAT_1:13; then A986: 1 <= l by XREAL_1:19; then (len g1) + 1 <= ma by A711, A940, A985, XREAL_1:7; then A987: len g1 <= ma - 1 by XREAL_1:19; then 0 + 1 <= ma by XREAL_1:19; then reconsider m1 = ma - 1 as Element of NAT by INT_1:5; reconsider q = g /. m1 as Point of (TOP-REAL 2) ; A988: ma - 1 <= len g by A985, XREAL_1:43; then A989: q `2 = (G * (i1,i2)) `2 by A941, A987; A990: (G * (j1,i2)) `1 <= q `1 by A941, A988, A987; set lq = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= e `1 & e `1 <= q `1 ) } ; A991: i1 - l = j1 ; A992: l in dom g2 by A711, A986, FINSEQ_3:25; then A993: g /. ma = g2 /. l by A711, A940, A985, FINSEQ_4:69 .= G * (j1,i2) by A711, A712, A992, A991 ; then p1 `1 <= (G * (j1,i2)) `1 by A984; then A994: p1 `1 = (G * (j1,i2)) `1 by A979, XXREAL_0:1; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A995: 1 <= m1 by A987, XXREAL_0:2; A996: m1 + 1 = ma ; then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg ((G * (j1,i2)),q) ) by A985, A993, A995, EUCLID:53, TOPREAL1:def_3; then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & (G * (j1,i2)) `1 <= e `1 & e `1 <= q `1 ) } by A767, A773, A989, A990, TOPREAL3:10; then A997: p1 in LSeg (g,m1) by A978, A994, A990; LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A985, A995, A996; hence x in L~ g by A977, A997, TARSKI:def_4; ::_thesis: verum end; suppose ma <> len g ; ::_thesis: x in L~ g then ma < len g by A984, XXREAL_0:1; then A998: ma + 1 <= len g by NAT_1:13; reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ; set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa1 `1 <= p2 `1 & p2 `1 <= qa `1 ) } ; A999: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:53; A1000: p1 `1 <= qa `1 by A984; A1001: len g1 <= ma + 1 by A984, NAT_1:13; then A1002: qa1 `2 = (G * (i1,i2)) `2 by A941, A998; A1003: now__::_thesis:_not_p1_`1_<=_qa1_`1 assume p1 `1 <= qa1 `1 ; ::_thesis: contradiction then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds p1 `1 <= q `1 ; then ma + 1 <= ma by A984, A998, A1001; hence contradiction by XREAL_1:29; ::_thesis: verum end; A1004: ( qa `2 = (G * (i1,i2)) `2 & qa = |[(qa `1),(qa `2)]| ) by A941, A984, EUCLID:53; A1005: 1 <= ma by A24, A14, A47, A984, NAT_1:13; then LSeg (g,ma) = LSeg (qa1,qa) by A998, TOPREAL1:def_3 .= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa1 `1 <= p2 `1 & p2 `1 <= qa `1 ) } by A1000, A1003, A1002, A1004, A999, TOPREAL3:10, XXREAL_0:2 ; then A1006: x in LSeg (g,ma) by A977, A978, A1000, A1003; LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A998, A1005; hence x in L~ g by A1006, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ g ; ::_thesis: verum end; end; end; A1007: len g = (len g1) + (len g2) by FINSEQ_1:22; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then 1 in dom g1 by FINSEQ_3:25; hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:68 .= f /. 1 by A27, A25, FINSEQ_4:71 ; ::_thesis: ( g /. (len g) = f /. (len f) & len f <= len g ) j1 + 1 <= i1 by A699, NAT_1:13; then A1008: 1 <= l by XREAL_1:19; then A1009: l in dom g2 by A712, FINSEQ_1:1; hence g /. (len g) = g2 /. l by A711, A1007, FINSEQ_4:69 .= G * (m1,i2) by A711, A712, A1009 .= f /. (len f) by A3, A21, A698 ; ::_thesis: len f <= len g thus len f <= len g by A3, A14, A47, A711, A1008, A1007, XREAL_1:7; ::_thesis: verum end; caseA1010: i1 = j1 ; ::_thesis: contradiction k <> k + 1 ; hence contradiction by A5, A27, A29, A19, A21, A698, A1010, PARTFUN2:10; ::_thesis: verum end; caseA1011: i1 < j1 ; ::_thesis: ex g being FinSequence of the U1 of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) c1 /. i1 = c1 . i1 by A66, A60, PARTFUN1:def_6; then A1012: c1 /. i1 = G * (i1,i2) by A66, MATRIX_1:def_8; then A1013: (X_axis c1) . i1 = (G * (i1,i2)) `1 by A66, A65, A60, GOBOARD1:def_1; c1 /. j1 = c1 . j1 by A23, A60, PARTFUN1:def_6; then A1014: c1 /. j1 = G * (j1,i2) by A23, MATRIX_1:def_8; then A1015: (X_axis c1) . j1 = (G * (j1,i2)) `1 by A23, A65, A60, GOBOARD1:def_1; then A1016: (G * (i1,i2)) `1 < (G * (j1,i2)) `1 by A66, A23, A69, A65, A60, A1011, A1013, SEQM_3:def_1; reconsider l = j1 - i1 as Element of NAT by A1011, INT_1:5; deffunc H1( Nat) -> Element of the U1 of (TOP-REAL 2) = G * ((i1 + $1),i2); consider g2 being FinSequence of (TOP-REAL 2) such that A1017: ( len g2 = l & ( for n being Nat st n in dom g2 holds g2 /. n = H1(n) ) ) from FINSEQ_4:sch_2(); take g = g1 ^ g2; ::_thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A1018: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_Seg_l_holds_ (_i1_+_n_in_dom_G_&_[(i1_+_n),i2]_in_Indices_G_) let n be Element of NAT ; ::_thesis: ( n in Seg l implies ( i1 + n in dom G & [(i1 + n),i2] in Indices G ) ) A1019: n <= i1 + n by NAT_1:11; assume A1020: n in Seg l ; ::_thesis: ( i1 + n in dom G & [(i1 + n),i2] in Indices G ) then n <= l by FINSEQ_1:1; then A1021: i1 + n <= l + i1 by XREAL_1:7; j1 <= len G by A23, FINSEQ_3:25; then A1022: i1 + n <= len G by A1021, XXREAL_0:2; 1 <= n by A1020, FINSEQ_1:1; then 1 <= i1 + n by A1019, XXREAL_0:2; hence i1 + n in dom G by A1022, FINSEQ_3:25; ::_thesis: [(i1 + n),i2] in Indices G hence [(i1 + n),i2] in Indices G by A22, A68, ZFMISC_1:87; ::_thesis: verum end; A1023: Seg (len g2) = dom g2 by FINSEQ_1:def_3; now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_holds_ ex_m_being_Element_of_NAT_ex_k_being_Element_of_NAT_st_ (_[m,k]_in_Indices_G_&_g2_/._n_=_G_*_(m,k)_) let n be Element of NAT ; ::_thesis: ( n in dom g2 implies ex m being Element of NAT ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) ) assume A1024: n in dom g2 ; ::_thesis: ex m being Element of NAT ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) take m = i1 + n; ::_thesis: ex k being Element of NAT st ( [m,k] in Indices G & g2 /. n = G * (m,k) ) take k = i2; ::_thesis: ( [m,k] in Indices G & g2 /. n = G * (m,k) ) thus ( [m,k] in Indices G & g2 /. n = G * (m,k) ) by A1017, A1018, A1023, A1024; ::_thesis: verum end; then A1025: for n being Element of NAT st n in dom g holds ex i, j being Element of NAT st ( [i,j] in Indices G & g /. n = G * (i,j) ) by A75, GOBOARD1:23; A1026: (Y_axis c1) . i1 = (G * (i1,i2)) `2 by A66, A63, A64, A65, A61, A60, A1012, GOBOARD1:def_2; A1027: now__::_thesis:_for_n_being_Element_of_NAT_ for_p_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_g2_/._n_=_p_holds_ (_p_`2_=_(G_*_(i1,i2))_`2_&_(G_*_(i1,i2))_`1_<=_p_`1_&_p_`1_<=_(G_*_(j1,i2))_`1_&_p_in_rng_c1_&_p_`1_>_(G_*_(i1,i2))_`1_) let n be Element of NAT ; ::_thesis: for p being Point of (TOP-REAL 2) st n in dom g2 & g2 /. n = p holds ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 & p `1 > (G * (i1,i2)) `1 ) let p be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & g2 /. n = p implies ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 & p `1 > (G * (i1,i2)) `1 ) ) assume that A1028: n in dom g2 and A1029: g2 /. n = p ; ::_thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 & p `1 > (G * (i1,i2)) `1 ) A1030: g2 /. n = G * ((i1 + n),i2) by A1017, A1028; set n1 = i1 + n; set pn = G * ((i1 + n),i2); A1031: i1 + n in dom G by A1017, A1018, A1023, A1028; then A1032: (Y_axis c1) . (i1 + n) = (Y_axis c1) . i1 by A66, A70, A63, A64, A65, A61, A60, SEQM_3:def_10; c1 /. (i1 + n) = c1 . (i1 + n) by A60, A1017, A1018, A1023, A1028, PARTFUN1:def_6; then A1033: c1 /. (i1 + n) = G * ((i1 + n),i2) by A1031, MATRIX_1:def_8; then A1034: (X_axis c1) . (i1 + n) = (G * ((i1 + n),i2)) `1 by A65, A60, A1031, GOBOARD1:def_1; n <= len g2 by A1028, FINSEQ_3:25; then A1035: i1 + n <= i1 + (len g2) by XREAL_1:7; (Y_axis c1) . (i1 + n) = (G * ((i1 + n),i2)) `2 by A63, A64, A65, A61, A60, A1031, A1033, GOBOARD1:def_2; hence ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 ) by A66, A23, A69, A65, A60, A1017, A1026, A1013, A1015, A1029, A1030, A1031, A1035, A1032, A1034, SEQ_4:137, XREAL_1:31; ::_thesis: ( p in rng c1 & p `1 > (G * (i1,i2)) `1 ) thus p in rng c1 by A60, A1029, A1030, A1031, A1033, PARTFUN2:2; ::_thesis: p `1 > (G * (i1,i2)) `1 1 <= n by A1028, FINSEQ_3:25; then i1 < i1 + n by XREAL_1:29; hence p `1 > (G * (i1,i2)) `1 by A66, A69, A65, A60, A1013, A1029, A1030, A1031, A1034, SEQM_3:def_1; ::_thesis: verum end; A1036: g2 is special proof let n be Nat; :: according to TOPREAL1:def_5 ::_thesis: ( not 1 <= n or not n + 1 <= len g2 or (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) set p = g2 /. n; assume A1037: ( 1 <= n & n + 1 <= len g2 ) ; ::_thesis: ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) then n in dom g2 by SEQ_4:134; then A1038: (g2 /. n) `2 = (G * (i1,i2)) `2 by A1027; n + 1 in dom g2 by A1037, SEQ_4:134; hence ( (g2 /. n) `1 = (g2 /. (n + 1)) `1 or (g2 /. n) `2 = (g2 /. (n + 1)) `2 ) by A1027, A1038; ::_thesis: verum end; now__::_thesis:_for_n,_m_being_Element_of_NAT_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<>_m_holds_ not_g2_/._n_=_g2_/._m let n, m be Element of NAT ; ::_thesis: ( n in dom g2 & m in dom g2 & n <> m implies not g2 /. n = g2 /. m ) assume that A1039: ( n in dom g2 & m in dom g2 ) and A1040: n <> m ; ::_thesis: not g2 /. n = g2 /. m A1041: ( g2 /. n = G * ((i1 + n),i2) & g2 /. m = G * ((i1 + m),i2) ) by A1017, A1039; assume A1042: g2 /. n = g2 /. m ; ::_thesis: contradiction ( [(i1 + n),i2] in Indices G & [(i1 + m),i2] in Indices G ) by A1017, A1018, A1023, A1039; then i1 + n = i1 + m by A1041, A1042, GOBOARD1:5; hence contradiction by A1040; ::_thesis: verum end; then for n, m being Element of NAT st n in dom g2 & m in dom g2 & g2 /. n = g2 /. m holds n = m ; then A1043: g2 is one-to-one by PARTFUN2:9; set lk = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } ; A1044: G * (i1,i2) = |[((G * (i1,i2)) `1),((G * (i1,i2)) `2)]| by EUCLID:53; A1045: now__::_thesis:_for_n,_m_being_Element_of_NAT_ for_p,_q_being_Point_of_(TOP-REAL_2)_st_n_in_dom_g2_&_m_in_dom_g2_&_n_<_m_&_g2_/._n_=_p_&_g2_/._m_=_q_holds_ p_`1_<_q_`1 let n, m be Element of NAT ; ::_thesis: for p, q being Point of (TOP-REAL 2) st n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q holds p `1 < q `1 let p, q be Point of (TOP-REAL 2); ::_thesis: ( n in dom g2 & m in dom g2 & n < m & g2 /. n = p & g2 /. m = q implies p `1 < q `1 ) assume that A1046: n in dom g2 and A1047: m in dom g2 and A1048: n < m and A1049: ( g2 /. n = p & g2 /. m = q ) ; ::_thesis: p `1 < q `1 A1050: i1 + n in dom G by A1017, A1018, A1023, A1046; set n1 = i1 + n; set m1 = i1 + m; set pn = G * ((i1 + n),i2); set pm = G * ((i1 + m),i2); A1051: i1 + n < i1 + m by A1048, XREAL_1:8; c1 /. (i1 + n) = c1 . (i1 + n) by A60, A1017, A1018, A1023, A1046, PARTFUN1:def_6; then c1 /. (i1 + n) = G * ((i1 + n),i2) by A1050, MATRIX_1:def_8; then A1052: (X_axis c1) . (i1 + n) = (G * ((i1 + n),i2)) `1 by A65, A60, A1050, GOBOARD1:def_1; A1053: i1 + m in dom G by A1017, A1018, A1023, A1047; c1 /. (i1 + m) = c1 . (i1 + m) by A60, A1017, A1018, A1023, A1047, PARTFUN1:def_6; then c1 /. (i1 + m) = G * ((i1 + m),i2) by A1053, MATRIX_1:def_8; then A1054: (X_axis c1) . (i1 + m) = (G * ((i1 + m),i2)) `1 by A65, A60, A1053, GOBOARD1:def_1; ( g2 /. n = G * ((i1 + n),i2) & g2 /. m = G * ((i1 + m),i2) ) by A1017, A1046, A1047; hence p `1 < q `1 by A69, A65, A60, A1049, A1050, A1053, A1051, A1052, A1054, SEQM_3:def_1; ::_thesis: verum end; for n, m being Element of NAT st m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 holds LSeg (g2,n) misses LSeg (g2,m) proof let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g2 & n + 1 in dom g2 & m in dom g2 & m + 1 in dom g2 implies LSeg (g2,n) misses LSeg (g2,m) ) assume that A1055: m > n + 1 and A1056: n in dom g2 and A1057: n + 1 in dom g2 and A1058: m in dom g2 and A1059: m + 1 in dom g2 and A1060: (LSeg (g2,n)) /\ (LSeg (g2,m)) <> {} ; :: according to XBOOLE_0:def_7 ::_thesis: contradiction reconsider p1 = g2 /. n, p2 = g2 /. (n + 1), q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A1061: ( p1 `2 = (G * (i1,i2)) `2 & p2 `2 = (G * (i1,i2)) `2 ) by A1027, A1056, A1057; n < n + 1 by NAT_1:13; then A1062: p1 `1 < p2 `1 by A1045, A1056, A1057; set lp = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p1 `1 <= w `1 & w `1 <= p2 `1 ) } ; set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } ; A1063: ( p1 = |[(p1 `1),(p1 `2)]| & p2 = |[(p2 `1),(p2 `2)]| ) by EUCLID:53; m < m + 1 by NAT_1:13; then A1064: q1 `1 < q2 `1 by A1045, A1058, A1059; A1065: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; set x = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)); A1066: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,n) by A1060, XBOOLE_0:def_4; A1067: ( q1 `2 = (G * (i1,i2)) `2 & q2 `2 = (G * (i1,i2)) `2 ) by A1027, A1058, A1059; A1068: the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) in LSeg (g2,m) by A1060, XBOOLE_0:def_4; ( 1 <= m & m + 1 <= len g2 ) by A1058, A1059, FINSEQ_3:25; then LSeg (g2,m) = LSeg (q1,q2) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } by A1064, A1067, A1065, TOPREAL3:10 ; then A1069: ex tm being Point of (TOP-REAL 2) st ( tm = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tm `2 = (G * (i1,i2)) `2 & q1 `1 <= tm `1 & tm `1 <= q2 `1 ) by A1068; ( 1 <= n & n + 1 <= len g2 ) by A1056, A1057, FINSEQ_3:25; then LSeg (g2,n) = LSeg (p1,p2) by TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & p1 `1 <= w `1 & w `1 <= p2 `1 ) } by A1062, A1061, A1063, TOPREAL3:10 ; then A1070: ex tn being Point of (TOP-REAL 2) st ( tn = the Element of (LSeg (g2,n)) /\ (LSeg (g2,m)) & tn `2 = (G * (i1,i2)) `2 & p1 `1 <= tn `1 & tn `1 <= p2 `1 ) by A1066; p2 `1 < q1 `1 by A1045, A1055, A1057, A1058; hence contradiction by A1070, A1069, XXREAL_0:2; ::_thesis: verum end; then A1071: g2 is s.n.c. by GOBOARD2:1; A1072: not f /. k in L~ g2 proof set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume f /. k in L~ g2 ; ::_thesis: contradiction then consider X being set such that A1073: f /. k in X and A1074: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A1075: X = LSeg (g2,m) and A1076: ( 1 <= m & m + 1 <= len g2 ) by A1074; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A1077: m in dom g2 by A1076, SEQ_4:134; then A1078: q1 `2 = (G * (i1,i2)) `2 by A1027; set lq = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } ; A1079: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; A1080: m + 1 in dom g2 by A1076, SEQ_4:134; then A1081: q2 `2 = (G * (i1,i2)) `2 by A1027; m < m + 1 by NAT_1:13; then A1082: q1 `1 < q2 `1 by A1045, A1077, A1080; LSeg (g2,m) = LSeg (q1,q2) by A1076, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & q1 `1 <= w `1 & w `1 <= q2 `1 ) } by A1078, A1081, A1082, A1079, TOPREAL3:10 ; then ex p being Point of (TOP-REAL 2) st ( p = f /. k & p `2 = (G * (i1,i2)) `2 & q1 `1 <= p `1 & p `1 <= q2 `1 ) by A1073, A1075; hence contradiction by A29, A1027, A1077; ::_thesis: verum end; (Y_axis c1) . j1 = (G * (j1,i2)) `2 by A23, A63, A64, A65, A61, A60, A1014, GOBOARD1:def_2; then A1083: (G * (i1,i2)) `2 = (G * (j1,i2)) `2 by A66, A23, A70, A63, A64, A65, A61, A60, A1026, SEQM_3:def_10; A1084: now__::_thesis:_for_n_being_Element_of_NAT_st_n_in_dom_g2_&_n_+_1_in_dom_g2_holds_ for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g2_/._n_=_G_*_(l1,l2)_&_g2_/._(n_+_1)_=_G_*_(l3,l4)_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let n be Element of NAT ; ::_thesis: ( n in dom g2 & n + 1 in dom g2 implies for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A1085: n in dom g2 and A1086: n + 1 in dom g2 ; ::_thesis: for l1, l2, l3, l4 being Element of NAT st [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) holds (abs (l1 - l3)) + (abs (l2 - l4)) = 1 let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g2 /. n = G * (l1,l2) & g2 /. (n + 1) = G * (l3,l4) implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A1087: [l1,l2] in Indices G and A1088: [l3,l4] in Indices G and A1089: g2 /. n = G * (l1,l2) and A1090: g2 /. (n + 1) = G * (l3,l4) ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ( g2 /. (n + 1) = G * ((i1 + (n + 1)),i2) & [(i1 + (n + 1)),i2] in Indices G ) by A1017, A1018, A1023, A1086; then A1091: ( l3 = i1 + (n + 1) & l4 = i2 ) by A1088, A1090, GOBOARD1:5; ( g2 /. n = G * ((i1 + n),i2) & [(i1 + n),i2] in Indices G ) by A1017, A1018, A1023, A1085; then ( l1 = i1 + n & l2 = i2 ) by A1087, A1089, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs ((i1 + n) - (i1 + (n + 1)))) + 0 by A1091, ABSVALUE:2 .= abs (- 1) .= abs 1 by COMPLEX1:52 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; now__::_thesis:_for_l1,_l2,_l3,_l4_being_Element_of_NAT_st_[l1,l2]_in_Indices_G_&_[l3,l4]_in_Indices_G_&_g1_/._(len_g1)_=_G_*_(l1,l2)_&_g2_/._1_=_G_*_(l3,l4)_&_len_g1_in_dom_g1_&_1_in_dom_g2_holds_ (abs_(l1_-_l3))_+_(abs_(l2_-_l4))_=_1 let l1, l2, l3, l4 be Element of NAT ; ::_thesis: ( [l1,l2] in Indices G & [l3,l4] in Indices G & g1 /. (len g1) = G * (l1,l2) & g2 /. 1 = G * (l3,l4) & len g1 in dom g1 & 1 in dom g2 implies (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ) assume that A1092: [l1,l2] in Indices G and A1093: [l3,l4] in Indices G and A1094: g1 /. (len g1) = G * (l1,l2) and A1095: g2 /. 1 = G * (l3,l4) and len g1 in dom g1 and A1096: 1 in dom g2 ; ::_thesis: (abs (l1 - l3)) + (abs (l2 - l4)) = 1 ( g2 /. 1 = G * ((i1 + 1),i2) & [(i1 + 1),i2] in Indices G ) by A1017, A1018, A1023, A1096; then A1097: ( l3 = i1 + 1 & l4 = i2 ) by A1093, A1095, GOBOARD1:5; (f | k) /. (len (f | k)) = f /. k by A27, A14, A51, FINSEQ_4:71; then ( l1 = i1 & l2 = i2 ) by A46, A28, A29, A1092, A1094, GOBOARD1:5; hence (abs (l1 - l3)) + (abs (l2 - l4)) = (abs (i1 - (i1 + 1))) + 0 by A1097, ABSVALUE:2 .= abs ((i1 - i1) + (- 1)) .= abs 1 by COMPLEX1:52 .= 1 by ABSVALUE:def_1 ; ::_thesis: verum end; then for n being Element of NAT st n in dom g & n + 1 in dom g holds for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds (abs (m - i)) + (abs (k - j)) = 1 by A48, A1084, GOBOARD1:24; hence g is_sequence_on G by A1025, GOBOARD1:def_9; ::_thesis: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A1098: G * (j1,i2) = |[((G * (j1,i2)) `1),((G * (j1,i2)) `2)]| by EUCLID:53; A1099: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (j1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def_3 .= { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1016, A1083, A1044, A1098, TOPREAL3:10 ; A1100: rng g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g2 or x in LSeg (f,k) ) assume x in rng g2 ; ::_thesis: x in LSeg (f,k) then consider n being Element of NAT such that A1101: n in dom g2 and A1102: g2 /. n = x by PARTFUN2:2; set pn = G * ((i1 + n),i2); A1103: g2 /. n = G * ((i1 + n),i2) by A1017, A1101; then A1104: (G * ((i1 + n),i2)) `1 <= (G * (j1,i2)) `1 by A1027, A1101; ( (G * ((i1 + n),i2)) `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= (G * ((i1 + n),i2)) `1 ) by A1027, A1101, A1103; hence x in LSeg (f,k) by A1099, A1102, A1103, A1104; ::_thesis: verum end; A1105: Seg l = dom g2 by A1017, FINSEQ_1:def_3; A1106: not f /. k in rng g2 proof assume f /. k in rng g2 ; ::_thesis: contradiction then consider n being Element of NAT such that A1107: n in dom g2 and A1108: g2 /. n = f /. k by PARTFUN2:2; ( g2 /. n = G * ((i1 + n),i2) & [(i1 + n),i2] in Indices G ) by A1017, A1105, A1018, A1107; then A1109: i1 + n = i1 by A28, A29, A1108, GOBOARD1:5; 0 < n by A1107, FINSEQ_3:25; hence contradiction by A1109; ::_thesis: verum end; (rng g1) /\ (rng g2) = {} proof set x = the Element of (rng g1) /\ (rng g2); assume A1110: not (rng g1) /\ (rng g2) = {} ; ::_thesis: contradiction then A1111: the Element of (rng g1) /\ (rng g2) in rng g2 by XBOOLE_0:def_4; A1112: the Element of (rng g1) /\ (rng g2) in rng g1 by A1110, XBOOLE_0:def_4; now__::_thesis:_contradiction percases ( k = 1 or 1 < k ) by A24, XXREAL_0:1; suppose k = 1 ; ::_thesis: contradiction hence contradiction by A52, A1106, A1112, A1111, TARSKI:def_1; ::_thesis: verum end; suppose 1 < k ; ::_thesis: contradiction then ( the Element of (rng g1) /\ (rng g2) in (L~ (f | k)) /\ (LSeg (f,k)) & (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} ) by A3, A6, A7, A49, A1100, A1112, A1111, GOBOARD2:4, XBOOLE_0:def_4; hence contradiction by A1106, A1111, TARSKI:def_1; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; then rng g1 misses rng g2 by XBOOLE_0:def_7; hence g is one-to-one by A40, A1043, FINSEQ_3:91; ::_thesis: ( g is unfolded & g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) A1113: LSeg (f,k) = LSeg ((G * (i1,i2)),(G * (j1,i2))) by A3, A24, A29, A21, A698, TOPREAL1:def_3; A1114: for n being Element of NAT st 1 <= n & n + 2 <= len g2 holds (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} proof let n be Element of NAT ; ::_thesis: ( 1 <= n & n + 2 <= len g2 implies (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} ) assume that A1115: 1 <= n and A1116: n + 2 <= len g2 ; ::_thesis: (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} A1117: n + 1 in dom g2 by A1115, A1116, SEQ_4:135; then g2 /. (n + 1) in rng g2 by PARTFUN2:2; then g2 /. (n + 1) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100; then consider u1 being Point of (TOP-REAL 2) such that A1118: g2 /. (n + 1) = u1 and A1119: u1 `2 = (G * (i1,i2)) `2 and (G * (i1,i2)) `1 <= u1 `1 and u1 `1 <= (G * (j1,i2)) `1 ; A1120: n + 2 in dom g2 by A1115, A1116, SEQ_4:135; then g2 /. (n + 2) in rng g2 by PARTFUN2:2; then g2 /. (n + 2) in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100; then consider u2 being Point of (TOP-REAL 2) such that A1121: g2 /. (n + 2) = u2 and A1122: u2 `2 = (G * (i1,i2)) `2 and (G * (i1,i2)) `1 <= u2 `1 and u2 `1 <= (G * (j1,i2)) `1 ; ( 1 <= n + 1 & (n + 1) + 1 = n + (1 + 1) ) by NAT_1:11; then A1123: LSeg (g2,(n + 1)) = LSeg (u1,u2) by A1116, A1118, A1121, TOPREAL1:def_3; n + 1 < (n + 1) + 1 by NAT_1:13; then A1124: u1 `1 < u2 `1 by A1045, A1117, A1120, A1118, A1121; A1125: n in dom g2 by A1115, A1116, SEQ_4:135; then g2 /. n in rng g2 by PARTFUN2:2; then g2 /. n in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100; then consider u being Point of (TOP-REAL 2) such that A1126: g2 /. n = u and A1127: u `2 = (G * (i1,i2)) `2 and (G * (i1,i2)) `1 <= u `1 and u `1 <= (G * (j1,i2)) `1 ; n + 1 <= n + 2 by XREAL_1:6; then n + 1 <= len g2 by A1116, XXREAL_0:2; then A1128: LSeg (g2,n) = LSeg (u,u1) by A1115, A1126, A1118, TOPREAL1:def_3; set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u `1 <= w `1 & w `1 <= u2 `1 ) } ; n < n + 1 by NAT_1:13; then A1129: u `1 < u1 `1 by A1045, A1125, A1117, A1126, A1118; then A1130: u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u `1 <= w `1 & w `1 <= u2 `1 ) } by A1119, A1124; ( u = |[(u `1),(u `2)]| & u2 = |[(u2 `1),(u2 `2)]| ) by EUCLID:53; then LSeg ((g2 /. n),(g2 /. (n + 2))) = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & u `1 <= w `1 & w `1 <= u2 `1 ) } by A1126, A1127, A1121, A1122, A1124, A1129, TOPREAL3:10, XXREAL_0:2; hence (LSeg (g2,n)) /\ (LSeg (g2,(n + 1))) = {(g2 /. (n + 1))} by A1126, A1118, A1121, A1128, A1123, A1130, TOPREAL1:8; ::_thesis: verum end; thus g is unfolded ::_thesis: ( g is s.n.c. & g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof let n be Nat; :: according to TOPREAL1:def_6 ::_thesis: ( not 1 <= n or not n + 2 <= len g or (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ) assume that A1131: 1 <= n and A1132: n + 2 <= len g ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A1133: (n + 1) + 1 <= len g by A1132; n + 1 <= (n + 1) + 1 by NAT_1:11; then A1134: n + 1 <= len g by A1132, XXREAL_0:2; A1135: len g = (len g1) + (len g2) by FINSEQ_1:22; (n + 2) - (len g1) = (n - (len g1)) + 2 ; then A1136: (n - (len g1)) + 2 <= len g2 by A1132, A1135, XREAL_1:20; A1137: 1 <= n + 1 by NAT_1:11; A1138: n <= n + 1 by NAT_1:11; A1139: n + (1 + 1) = (n + 1) + 1 ; percases ( n + 2 <= len g1 or len g1 < n + 2 ) ; supposeA1140: n + 2 <= len g1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} A1141: n + (1 + 1) = (n + 1) + 1 ; A1142: n + 1 in dom g1 by A1131, A1140, SEQ_4:135; then A1143: g /. (n + 1) = g1 /. (n + 1) by FINSEQ_4:68; n in dom g1 by A1131, A1140, SEQ_4:135; then A1144: LSeg (g1,n) = LSeg (g,n) by A1142, TOPREAL3:18; n + 2 in dom g1 by A1131, A1140, SEQ_4:135; then LSeg (g1,(n + 1)) = LSeg (g,(n + 1)) by A1142, A1141, TOPREAL3:18; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A41, A1131, A1140, A1144, A1143, TOPREAL1:def_6; ::_thesis: verum end; suppose len g1 < n + 2 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then (len g1) + 1 <= n + 2 by NAT_1:13; then A1145: len g1 <= (n + 2) - 1 by XREAL_1:19; now__::_thesis:_(LSeg_(g,n))_/\_(LSeg_(g,(n_+_1)))_=_{(g_/._(n_+_1))} percases ( len g1 = n + 1 or len g1 <> n + 1 ) ; supposeA1146: len g1 = n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then 1 <= (len g) - (len g1) by A1133, XREAL_1:19; then 1 in dom g2 by A1135, FINSEQ_3:25; then A1147: g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100; then consider u1 being Point of (TOP-REAL 2) such that A1148: g2 /. 1 = u1 and u1 `2 = (G * (i1,i2)) `2 and (G * (i1,i2)) `1 <= u1 `1 and u1 `1 <= (G * (j1,i2)) `1 ; G * (i1,i2) in LSeg ((G * (i1,i2)),(G * (j1,i2))) by RLTOPSP1:68; then A1149: LSeg ((G * (i1,i2)),u1) c= LSeg (f,k) by A1113, A1100, A1147, A1148, TOPREAL1:6; 1 <= n + 1 by NAT_1:11; then A1150: n + 1 in dom g1 by A1146, FINSEQ_3:25; then A1151: g /. (n + 1) = (f | k) /. (len (f | k)) by A46, A1146, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; now__::_thesis:_not_k_=_1 1 < len g1 by A1131, A1146, NAT_1:13; then A1152: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A1152, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A1153: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; A1154: LSeg (g1,n) c= L~ (f | k) by A44, TOPREAL3:19; n in dom g1 by A1131, A1138, A1146, FINSEQ_3:25; then A1155: LSeg (g,n) = LSeg (g1,n) by A1150, TOPREAL3:18; ( g /. (n + 1) in LSeg (g,n) & g /. (n + 1) in LSeg (g,(n + 1)) ) by A1131, A1132, A1137, A1134, A1139, TOPREAL1:21; then g /. (n + 1) in (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by XBOOLE_0:def_4; then A1156: {(g /. (n + 1))} c= (LSeg (g,n)) /\ (LSeg (g,(n + 1))) by ZFMISC_1:31; ( n + 2 = 1 + (len g1) & 1 <= len g2 ) by A1132, A1139, A1135, A1146, XREAL_1:6; then g /. (n + 2) = g2 /. 1 by SEQ_4:136; then LSeg (g,(n + 1)) = LSeg ((G * (i1,i2)),u1) by A1132, A1137, A1139, A1151, A1148, TOPREAL1:def_3; then (LSeg (g,n)) /\ (LSeg (g,(n + 1))) c= {(g /. (n + 1))} by A29, A1154, A1153, A1155, A1151, A1149, XBOOLE_1:27; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A1156, XBOOLE_0:def_10; ::_thesis: verum end; suppose len g1 <> n + 1 ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then len g1 < n + 1 by A1145, XXREAL_0:1; then A1157: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_(LSeg_(g,n))_/\_(LSeg_(g,(n_+_1)))_=_{(g_/._(n_+_1))} percases ( len g1 = n or len g1 <> n ) ; supposeA1158: len g1 = n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A1159: 2 <= len g2 by A1132, A1135, XREAL_1:6; then 1 <= len g2 by XXREAL_0:2; then A1160: g /. (n + 1) = g2 /. 1 by A1158, SEQ_4:136; 1 <= len g2 by A1159, XXREAL_0:2; then A1161: 1 in dom g2 by FINSEQ_3:25; then g2 /. 1 in rng g2 by PARTFUN2:2; then g2 /. 1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100; then consider u1 being Point of (TOP-REAL 2) such that A1162: g2 /. 1 = u1 and A1163: ( u1 `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= u1 `1 ) and u1 `1 <= (G * (j1,i2)) `1 ; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then g /. n = (f | k) /. (len (f | k)) by A46, A1158, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; then A1164: LSeg (g,n) = LSeg ((G * (i1,i2)),u1) by A1131, A1134, A1160, A1162, TOPREAL1:def_3; A1165: 2 in dom g2 by A1159, FINSEQ_3:25; then g2 /. 2 in rng g2 by PARTFUN2:2; then g2 /. 2 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= (G * (j1,i2)) `1 ) } by A1099, A1100; then consider u2 being Point of (TOP-REAL 2) such that A1166: g2 /. 2 = u2 and A1167: ( u2 `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= u2 `1 ) and u2 `1 <= (G * (j1,i2)) `1 ; set lg = { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= u2 `1 ) } ; u1 `1 < u2 `1 by A1045, A1161, A1165, A1162, A1166; then ( u2 = |[(u2 `1),(u2 `2)]| & u1 in { w where w is Point of (TOP-REAL 2) : ( w `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= w `1 & w `1 <= u2 `1 ) } ) by A1163, EUCLID:53; then A1168: u1 in LSeg ((G * (i1,i2)),u2) by A1044, A1167, TOPREAL3:10; g /. (n + 2) = g2 /. 2 by A1158, A1159, SEQ_4:136; then LSeg (g,(n + 1)) = LSeg (u1,u2) by A1132, A1137, A1139, A1160, A1162, A1166, TOPREAL1:def_3; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A1160, A1162, A1164, A1168, TOPREAL1:8; ::_thesis: verum end; suppose len g1 <> n ; ::_thesis: (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} then A1169: len g1 < n by A1157, XXREAL_0:1; then (len g1) + 1 <= n by NAT_1:13; then A1170: 1 <= n1 by XREAL_1:19; n1 + (len g1) = n ; then A1171: LSeg (g,n) = LSeg (g2,n1) by A1134, A1169, GOBOARD2:5; A1172: n + 1 = (n1 + 1) + (len g1) ; then n1 + 1 <= len g2 by A1134, A1135, XREAL_1:6; then A1173: g /. (n + 1) = g2 /. (n1 + 1) by A1172, NAT_1:11, SEQ_4:136; len g1 < n + 1 by A1138, A1169, XXREAL_0:2; then LSeg (g,(n + 1)) = LSeg (g2,(n1 + 1)) by A1133, A1172, GOBOARD2:5; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} by A1114, A1136, A1171, A1173, A1170; ::_thesis: verum end; end; end; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; ::_thesis: verum end; end; end; hence (LSeg (g,n)) /\ (LSeg (g,(n + 1))) = {(g /. (n + 1))} ; ::_thesis: verum end; end; end; A1174: L~ g2 c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g2 or x in LSeg (f,k) ) set ls = { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } ; assume x in L~ g2 ; ::_thesis: x in LSeg (f,k) then consider X being set such that A1175: x in X and A1176: X in { (LSeg (g2,m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len g2 ) } by TARSKI:def_4; consider m being Element of NAT such that A1177: X = LSeg (g2,m) and A1178: ( 1 <= m & m + 1 <= len g2 ) by A1176; reconsider q1 = g2 /. m, q2 = g2 /. (m + 1) as Point of (TOP-REAL 2) ; A1179: LSeg (g2,m) = LSeg (q1,q2) by A1178, TOPREAL1:def_3; m + 1 in dom g2 by A1178, SEQ_4:134; then A1180: g2 /. (m + 1) in rng g2 by PARTFUN2:2; m in dom g2 by A1178, SEQ_4:134; then g2 /. m in rng g2 by PARTFUN2:2; then LSeg (q1,q2) c= LSeg ((G * (i1,i2)),(G * (j1,i2))) by A1113, A1100, A1180, TOPREAL1:6; hence x in LSeg (f,k) by A1113, A1175, A1177, A1179; ::_thesis: verum end; A1181: (L~ g1) /\ (L~ g2) = {} proof percases ( k = 1 or k <> 1 ) ; suppose k = 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} hence (L~ g1) /\ (L~ g2) = {} by A52; ::_thesis: verum end; suppose k <> 1 ; ::_thesis: (L~ g1) /\ (L~ g2) = {} then 1 < k by A24, XXREAL_0:1; then (L~ g1) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, A44, GOBOARD2:4; then A1182: (L~ g1) /\ (L~ g2) c= {(f /. k)} by A1174, XBOOLE_1:26; now__::_thesis:_not_(L~_g1)_/\_(L~_g2)_<>_{} set x = the Element of (L~ g1) /\ (L~ g2); assume (L~ g1) /\ (L~ g2) <> {} ; ::_thesis: contradiction then ( the Element of (L~ g1) /\ (L~ g2) in {(f /. k)} & the Element of (L~ g1) /\ (L~ g2) in L~ g2 ) by A1182, TARSKI:def_3, XBOOLE_0:def_4; hence contradiction by A1072, TARSKI:def_1; ::_thesis: verum end; hence (L~ g1) /\ (L~ g2) = {} ; ::_thesis: verum end; end; end; for n, m being Element of NAT st m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g holds LSeg (g,n) misses LSeg (g,m) proof A1183: 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A1184: g /. (len g1) = g1 /. (len g1) by FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A46, A29, FINSEQ_4:71 ; reconsider qq = g2 /. 1 as Point of (TOP-REAL 2) ; set l1 = { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } ; set l2 = { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } ; let n, m be Element of NAT ; ::_thesis: ( m > n + 1 & n in dom g & n + 1 in dom g & m in dom g & m + 1 in dom g implies LSeg (g,n) misses LSeg (g,m) ) assume that A1185: m > n + 1 and A1186: n in dom g and A1187: n + 1 in dom g and A1188: m in dom g and A1189: m + 1 in dom g ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A1190: 1 <= n by A1186, FINSEQ_3:25; i1 + 1 <= j1 by A1011, NAT_1:13; then A1191: 1 <= l by XREAL_1:19; then A1192: 1 in dom g2 by A1017, FINSEQ_3:25; then A1193: ( qq `2 = (G * (i1,i2)) `2 & qq `1 > (G * (i1,i2)) `1 ) by A1027; A1194: g /. ((len g1) + 1) = qq by A1017, A1191, SEQ_4:136; A1195: qq `1 <= (G * (j1,i2)) `1 by A1027, A1192; A1196: m + 1 <= len g by A1189, FINSEQ_3:25; A1197: 1 <= m + 1 by A1189, FINSEQ_3:25; A1198: 1 <= n + 1 by A1187, FINSEQ_3:25; A1199: n + 1 <= len g by A1187, FINSEQ_3:25; A1200: qq = |[(qq `1),(qq `2)]| by EUCLID:53; A1201: 1 <= m by A1188, FINSEQ_3:25; set ql = { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= z `1 & z `1 <= qq `1 ) } ; A1202: n <= n + 1 by NAT_1:11; A1203: len g = (len g1) + (len g2) by FINSEQ_1:22; then (len g1) + 1 <= len g by A1017, A1191, XREAL_1:7; then A1204: LSeg (g,(len g1)) = LSeg ((G * (i1,i2)),qq) by A1183, A1184, A1194, TOPREAL1:def_3 .= { z where z is Point of (TOP-REAL 2) : ( z `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= z `1 & z `1 <= qq `1 ) } by A1044, A1193, A1200, TOPREAL3:10 ; A1205: m <= m + 1 by NAT_1:11; then A1206: n + 1 <= m + 1 by A1185, XXREAL_0:2; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m + 1 <= len g1 or len g1 < m + 1 ) ; supposeA1207: m + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then m <= len g1 by A1205, XXREAL_0:2; then A1208: m in dom g1 by A1201, FINSEQ_3:25; m + 1 in dom g1 by A1197, A1207, FINSEQ_3:25; then A1209: LSeg (g,m) = LSeg (g1,m) by A1208, TOPREAL3:18; A1210: n + 1 <= len g1 by A1206, A1207, XXREAL_0:2; then n <= len g1 by A1202, XXREAL_0:2; then A1211: n in dom g1 by A1190, FINSEQ_3:25; n + 1 in dom g1 by A1198, A1210, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A1211, TOPREAL3:18; hence LSeg (g,n) misses LSeg (g,m) by A42, A1185, A1209, TOPREAL1:def_7; ::_thesis: verum end; suppose len g1 < m + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A1212: len g1 <= m by NAT_1:13; then reconsider m1 = m - (len g1) as Element of NAT by INT_1:5; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( m = len g1 or m <> len g1 ) ; supposeA1213: m = len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) A1214: LSeg (g,m) c= LSeg (f,k) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (g,m) or x in LSeg (f,k) ) assume x in LSeg (g,m) ; ::_thesis: x in LSeg (f,k) then consider px being Point of (TOP-REAL 2) such that A1215: ( px = x & px `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= px `1 ) and A1216: px `1 <= qq `1 by A1204, A1213; (G * (j1,i2)) `1 >= px `1 by A1195, A1216, XXREAL_0:2; hence x in LSeg (f,k) by A1099, A1215; ::_thesis: verum end; n <= len g1 by A1185, A1202, A1213, XXREAL_0:2; then A1217: n in dom g1 by A1190, FINSEQ_3:25; now__::_thesis:_not_k_=_1 1 < len g1 by A1185, A1198, A1213, XXREAL_0:2; then A1218: 1 + 1 <= len g1 by NAT_1:13; assume k = 1 ; ::_thesis: contradiction hence contradiction by A52, A1218, TOPREAL1:23; ::_thesis: verum end; then 1 < k by A24, XXREAL_0:1; then A1219: (L~ (f | k)) /\ (LSeg (f,k)) = {(f /. k)} by A3, A6, A7, GOBOARD2:4; A1220: n + 1 in dom g1 by A1185, A1198, A1213, FINSEQ_3:25; then A1221: LSeg (g,n) = LSeg (g1,n) by A1217, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A1185, A1190, A1213; then LSeg (g,n) c= L~ (f | k) by A44, ZFMISC_1:74; then A1222: (LSeg (g,n)) /\ (LSeg (g,m)) c= {(f /. k)} by A1219, A1214, XBOOLE_1:27; now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); assume A1223: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A1224: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in {(f /. k)} by A1222, A1223, TARSKI:def_3; then A1225: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) = f /. k by TARSKI:def_1; f /. k = g1 /. (len g1) by A27, A14, A51, A46, FINSEQ_4:71; hence contradiction by A40, A41, A42, A1185, A1213, A1217, A1220, A1221, A1224, A1225, GOBOARD2:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose m <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A1226: len g1 < m by A1212, XXREAL_0:1; then (len g1) + 1 <= m by NAT_1:13; then A1227: 1 <= m1 by XREAL_1:19; m + 1 = (m1 + 1) + (len g1) ; then A1228: m1 + 1 <= len g2 by A1196, A1203, XREAL_1:6; m = m1 + (len g1) ; then A1229: LSeg (g,m) = LSeg (g2,m1) by A1196, A1226, GOBOARD2:5; then LSeg (g,m) in { (LSeg (g2,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g2 ) } by A1227, A1228; then A1230: LSeg (g,m) c= L~ g2 by ZFMISC_1:74; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( n + 1 <= len g1 or len g1 < n + 1 ) ; supposeA1231: n + 1 <= len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then n <= len g1 by A1202, XXREAL_0:2; then A1232: n in dom g1 by A1190, FINSEQ_3:25; n + 1 in dom g1 by A1198, A1231, FINSEQ_3:25; then LSeg (g,n) = LSeg (g1,n) by A1232, TOPREAL3:18; then LSeg (g,n) in { (LSeg (g1,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g1 ) } by A1190, A1231; then LSeg (g,n) c= L~ g1 by ZFMISC_1:74; then (LSeg (g,n)) /\ (LSeg (g,m)) = {} by A1181, A1230, XBOOLE_1:3, XBOOLE_1:27; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose len g1 < n + 1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then A1233: len g1 <= n by NAT_1:13; then reconsider n1 = n - (len g1) as Element of NAT by INT_1:5; A1234: (n - (len g1)) + 1 = (n + 1) - (len g1) ; A1235: n = n1 + (len g1) ; now__::_thesis:_LSeg_(g,n)_misses_LSeg_(g,m) percases ( len g1 = n or n <> len g1 ) ; supposeA1236: len g1 = n ; ::_thesis: LSeg (g,n) misses LSeg (g,m) now__::_thesis:_not_(LSeg_(g,n))_/\_(LSeg_(g,m))_<>_{} reconsider q1 = g2 /. m1, q2 = g2 /. (m1 + 1) as Point of (TOP-REAL 2) ; set x = the Element of (LSeg (g,n)) /\ (LSeg (g,m)); set q1l = { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q1 `1 <= v `1 & v `1 <= q2 `1 ) } ; A1237: ( q1 = |[(q1 `1),(q1 `2)]| & q2 = |[(q2 `1),(q2 `2)]| ) by EUCLID:53; assume A1238: (LSeg (g,n)) /\ (LSeg (g,m)) <> {} ; ::_thesis: contradiction then A1239: the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,m) by XBOOLE_0:def_4; the Element of (LSeg (g,n)) /\ (LSeg (g,m)) in LSeg (g,n) by A1238, XBOOLE_0:def_4; then A1240: ex qx being Point of (TOP-REAL 2) st ( qx = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qx `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= qx `1 & qx `1 <= qq `1 ) by A1204, A1236; A1241: m1 in dom g2 by A1227, A1228, SEQ_4:134; then A1242: q1 `2 = (G * (i1,i2)) `2 by A1027; A1243: m1 + 1 in dom g2 by A1227, A1228, SEQ_4:134; then A1244: q2 `2 = (G * (i1,i2)) `2 by A1027; m1 < m1 + 1 by NAT_1:13; then A1245: q1 `1 < q2 `1 by A1045, A1241, A1243; LSeg (g2,m1) = LSeg (q1,q2) by A1227, A1228, TOPREAL1:def_3 .= { v where v is Point of (TOP-REAL 2) : ( v `2 = (G * (i1,i2)) `2 & q1 `1 <= v `1 & v `1 <= q2 `1 ) } by A1242, A1244, A1245, A1237, TOPREAL3:10 ; then A1246: ex qy being Point of (TOP-REAL 2) st ( qy = the Element of (LSeg (g,n)) /\ (LSeg (g,m)) & qy `2 = (G * (i1,i2)) `2 & q1 `1 <= qy `1 & qy `1 <= q2 `1 ) by A1229, A1239; ( m1 > n1 + 1 & n1 + 1 >= 1 ) by A1185, A1234, NAT_1:11, XREAL_1:9; then m1 > 1 by XXREAL_0:2; then qq `1 < q1 `1 by A1045, A1192, A1241; hence contradiction by A1240, A1246, XXREAL_0:2; ::_thesis: verum end; hence LSeg (g,n) misses LSeg (g,m) by XBOOLE_0:def_7; ::_thesis: verum end; suppose n <> len g1 ; ::_thesis: LSeg (g,n) misses LSeg (g,m) then len g1 < n by A1233, XXREAL_0:1; then A1247: LSeg (g,n) = LSeg (g2,n1) by A1199, A1235, GOBOARD2:5; m1 > n1 + 1 by A1185, A1234, XREAL_1:9; hence LSeg (g,n) misses LSeg (g,m) by A1071, A1229, A1247, TOPREAL1:def_7; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; end; end; hence LSeg (g,n) misses LSeg (g,m) ; ::_thesis: verum end; hence g is s.n.c. by GOBOARD2:1; ::_thesis: ( g is special & L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) now__::_thesis:_(_(g1_/._(len_g1))_`1_=_(g2_/._1)_`1_or_(g1_/._(len_g1))_`2_=_(g2_/._1)_`2_) set p = g1 /. (len g1); set q = g2 /. 1; i1 + 1 <= j1 by A1011, NAT_1:13; then 1 <= l by XREAL_1:19; then 1 in dom g2 by A1017, FINSEQ_3:25; then (g2 /. 1) `2 = (G * (i1,i2)) `2 by A1027; hence ( (g1 /. (len g1)) `1 = (g2 /. 1) `1 or (g1 /. (len g1)) `2 = (g2 /. 1) `2 ) by A27, A14, A51, A46, A29, FINSEQ_4:71; ::_thesis: verum end; hence g is special by A43, A1036, GOBOARD2:8; ::_thesis: ( L~ g = L~ f & g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) thus L~ g = L~ f ::_thesis: ( g /. 1 = f /. 1 & g /. (len g) = f /. (len f) & len f <= len g ) proof set lg = { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } ; set lf = { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } ; A1248: len g = (len g1) + (len g2) by FINSEQ_1:22; A1249: now__::_thesis:_for_j_being_Element_of_NAT_st_len_g1_<=_j_&_j_<=_len_g_holds_ for_p_being_Point_of_(TOP-REAL_2)_st_p_=_g_/._j_holds_ (_p_`2_=_(G_*_(i1,i2))_`2_&_(G_*_(i1,i2))_`1_<=_p_`1_&_p_`1_<=_(G_*_(j1,i2))_`1_&_p_in_rng_c1_) let j be Element of NAT ; ::_thesis: ( len g1 <= j & j <= len g implies for p being Point of (TOP-REAL 2) st p = g /. j holds ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) ) assume that A1250: len g1 <= j and A1251: j <= len g ; ::_thesis: for p being Point of (TOP-REAL 2) st p = g /. j holds ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) reconsider w = j - (len g1) as Element of NAT by A1250, INT_1:5; let p be Point of (TOP-REAL 2); ::_thesis: ( p = g /. j implies ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) ) assume A1252: p = g /. j ; ::_thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) now__::_thesis:_(_p_`2_=_(G_*_(i1,i2))_`2_&_(G_*_(i1,i2))_`1_<=_p_`1_&_p_`1_<=_(G_*_(j1,i2))_`1_&_p_in_rng_c1_) percases ( j = len g1 or j <> len g1 ) ; supposeA1253: j = len g1 ; ::_thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then len g1 in dom g1 by FINSEQ_3:25; then A1254: g /. (len g1) = (f | k) /. (len (f | k)) by A46, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence p `2 = (G * (i1,i2)) `2 by A1252, A1253; ::_thesis: ( (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) thus ( (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 ) by A66, A23, A69, A65, A60, A1011, A1013, A1015, A1252, A1253, A1254, SEQM_3:def_1; ::_thesis: p in rng c1 thus p in rng c1 by A66, A60, A1012, A1252, A1253, A1254, PARTFUN2:2; ::_thesis: verum end; suppose j <> len g1 ; ::_thesis: ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) then len g1 < j by A1250, XXREAL_0:1; then (len g1) + 1 <= j by NAT_1:13; then A1255: 1 <= w by XREAL_1:19; A1256: w <= len g2 by A1248, A1251, XREAL_1:20; then A1257: w in dom g2 by A1255, FINSEQ_3:25; j = w + (len g1) ; then g /. j = g2 /. w by A1255, A1256, SEQ_4:136; hence ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) by A1027, A1252, A1257; ::_thesis: verum end; end; end; hence ( p `2 = (G * (i1,i2)) `2 & (G * (i1,i2)) `1 <= p `1 & p `1 <= (G * (j1,i2)) `1 & p in rng c1 ) ; ::_thesis: verum end; thus L~ g c= L~ f :: according to XBOOLE_0:def_10 ::_thesis: L~ f c= L~ g proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ g or x in L~ f ) assume x in L~ g ; ::_thesis: x in L~ f then consider X being set such that A1258: x in X and A1259: X in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by TARSKI:def_4; consider i being Element of NAT such that A1260: X = LSeg (g,i) and A1261: 1 <= i and A1262: i + 1 <= len g by A1259; now__::_thesis:_x_in_L~_f percases ( i + 1 <= len g1 or i + 1 > len g1 ) ; supposeA1263: i + 1 <= len g1 ; ::_thesis: x in L~ f i <= i + 1 by NAT_1:11; then i <= len g1 by A1263, XXREAL_0:2; then A1264: i in dom g1 by A1261, FINSEQ_3:25; 1 <= i + 1 by NAT_1:11; then i + 1 in dom g1 by A1263, FINSEQ_3:25; then X = LSeg (g1,i) by A1260, A1264, TOPREAL3:18; then X in { (LSeg (g1,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len g1 ) } by A1261, A1263; then A1265: x in L~ (f | k) by A44, A1258, TARSKI:def_4; L~ (f | k) c= L~ f by TOPREAL3:20; hence x in L~ f by A1265; ::_thesis: verum end; supposeA1266: i + 1 > len g1 ; ::_thesis: x in L~ f reconsider q1 = g /. i, q2 = g /. (i + 1) as Point of (TOP-REAL 2) ; A1267: i <= len g by A1262, NAT_1:13; A1268: len g1 <= i by A1266, NAT_1:13; then A1269: q1 `2 = (G * (i1,i2)) `2 by A1249, A1267; A1270: q1 `1 <= (G * (j1,i2)) `1 by A1249, A1268, A1267; A1271: (G * (i1,i2)) `1 <= q1 `1 by A1249, A1268, A1267; q2 `2 = (G * (i1,i2)) `2 by A1249, A1262, A1266; then A1272: q2 = |[(q2 `1),(q1 `2)]| by A1269, EUCLID:53; A1273: q2 `1 <= (G * (j1,i2)) `1 by A1249, A1262, A1266; A1274: ( q1 = |[(q1 `1),(q1 `2)]| & LSeg (g,i) = LSeg (q2,q1) ) by A1261, A1262, EUCLID:53, TOPREAL1:def_3; A1275: (G * (i1,i2)) `1 <= q2 `1 by A1249, A1262, A1266; now__::_thesis:_x_in_L~_f percases ( q1 `1 > q2 `1 or q1 `1 = q2 `1 or q1 `1 < q2 `1 ) by XXREAL_0:1; suppose q1 `1 > q2 `1 ; ::_thesis: x in L~ f then LSeg (g,i) = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = q1 `2 & q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) } by A1272, A1274, TOPREAL3:10; then consider p2 being Point of (TOP-REAL 2) such that A1276: ( p2 = x & p2 `2 = q1 `2 ) and A1277: ( q2 `1 <= p2 `1 & p2 `1 <= q1 `1 ) by A1258, A1260; ( (G * (i1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (j1,i2)) `1 ) by A1270, A1275, A1277, XXREAL_0:2; then A1278: x in LSeg (f,k) by A1099, A1269, A1276; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A1278, TARSKI:def_4; ::_thesis: verum end; suppose q1 `1 = q2 `1 ; ::_thesis: x in L~ f then LSeg (g,i) = {q1} by A1272, A1274, RLTOPSP1:70; then x = q1 by A1258, A1260, TARSKI:def_1; then A1279: x in LSeg (f,k) by A1099, A1269, A1271, A1270; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A1279, TARSKI:def_4; ::_thesis: verum end; suppose q1 `1 < q2 `1 ; ::_thesis: x in L~ f then LSeg (g,i) = { p1 where p1 is Point of (TOP-REAL 2) : ( p1 `2 = q1 `2 & q1 `1 <= p1 `1 & p1 `1 <= q2 `1 ) } by A1272, A1274, TOPREAL3:10; then consider p2 being Point of (TOP-REAL 2) such that A1280: ( p2 = x & p2 `2 = q1 `2 ) and A1281: ( q1 `1 <= p2 `1 & p2 `1 <= q2 `1 ) by A1258, A1260; ( (G * (i1,i2)) `1 <= p2 `1 & p2 `1 <= (G * (j1,i2)) `1 ) by A1271, A1273, A1281, XXREAL_0:2; then A1282: x in LSeg (f,k) by A1099, A1269, A1280; LSeg (f,k) in { (LSeg (f,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len f ) } by A3, A24; hence x in L~ f by A1282, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ f ; ::_thesis: verum end; end; end; hence x in L~ f ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ f or x in L~ g ) assume x in L~ f ; ::_thesis: x in L~ g then A1283: x in (L~ (f | k)) \/ (LSeg (f,k)) by A3, A13, GOBOARD2:3; now__::_thesis:_x_in_L~_g percases ( x in L~ (f | k) or x in LSeg (f,k) ) by A1283, XBOOLE_0:def_3; supposeA1284: x in L~ (f | k) ; ::_thesis: x in L~ g L~ g1 c= L~ g by GOBOARD2:6; hence x in L~ g by A44, A1284; ::_thesis: verum end; suppose x in LSeg (f,k) ; ::_thesis: x in L~ g then consider p1 being Point of (TOP-REAL 2) such that A1285: p1 = x and A1286: p1 `2 = (G * (i1,i2)) `2 and A1287: (G * (i1,i2)) `1 <= p1 `1 and A1288: p1 `1 <= (G * (j1,i2)) `1 by A1099; defpred S2[ Nat] means ( len g1 <= $1 & $1 <= len g & ( for q being Point of (TOP-REAL 2) st q = g /. $1 holds q `1 <= p1 `1 ) ); A1289: now__::_thesis:_ex_n_being_Nat_st_S2[n] reconsider n = len g1 as Nat ; take n = n; ::_thesis: S2[n] thus S2[n] ::_thesis: verum proof thus ( len g1 <= n & n <= len g ) by A1248, XREAL_1:31; ::_thesis: for q being Point of (TOP-REAL 2) st q = g /. n holds q `1 <= p1 `1 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A1290: len g1 in dom g1 by FINSEQ_3:25; let q be Point of (TOP-REAL 2); ::_thesis: ( q = g /. n implies q `1 <= p1 `1 ) assume q = g /. n ; ::_thesis: q `1 <= p1 `1 then q = (f | k) /. (len (f | k)) by A46, A1290, FINSEQ_4:68 .= G * (i1,i2) by A27, A14, A51, A29, FINSEQ_4:71 ; hence q `1 <= p1 `1 by A1287; ::_thesis: verum end; end; A1291: for n being Nat st S2[n] holds n <= len g ; consider ma being Nat such that A1292: ( S2[ma] & ( for n being Nat st S2[n] holds n <= ma ) ) from NAT_1:sch_6(A1291, A1289); reconsider ma = ma as Element of NAT by ORDINAL1:def_12; now__::_thesis:_x_in_L~_g percases ( ma = len g or ma <> len g ) ; supposeA1293: ma = len g ; ::_thesis: x in L~ g i1 + 1 <= j1 by A1011, NAT_1:13; then A1294: 1 <= l by XREAL_1:19; then (len g1) + 1 <= ma by A1017, A1248, A1293, XREAL_1:7; then A1295: len g1 <= ma - 1 by XREAL_1:19; then 0 + 1 <= ma by XREAL_1:19; then reconsider m1 = ma - 1 as Element of NAT by INT_1:5; reconsider q = g /. m1 as Point of (TOP-REAL 2) ; A1296: ma - 1 <= len g by A1293, XREAL_1:43; then A1297: q `2 = (G * (i1,i2)) `2 by A1249, A1295; A1298: q `1 <= (G * (j1,i2)) `1 by A1249, A1296, A1295; set lq = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & q `1 <= e `1 & e `1 <= (G * (j1,i2)) `1 ) } ; A1299: i1 + l = j1 ; A1300: l in dom g2 by A1017, A1294, FINSEQ_3:25; then A1301: g /. ma = g2 /. l by A1017, A1248, A1293, FINSEQ_4:69 .= G * (j1,i2) by A1017, A1300, A1299 ; then (G * (j1,i2)) `1 <= p1 `1 by A1292; then A1302: p1 `1 = (G * (j1,i2)) `1 by A1288, XXREAL_0:1; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then A1303: 1 <= m1 by A1295, XXREAL_0:2; A1304: m1 + 1 = ma ; then ( q = |[(q `1),(q `2)]| & LSeg (g,m1) = LSeg (q,(G * (j1,i2))) ) by A1293, A1301, A1303, EUCLID:53, TOPREAL1:def_3; then LSeg (g,m1) = { e where e is Point of (TOP-REAL 2) : ( e `2 = (G * (i1,i2)) `2 & q `1 <= e `1 & e `1 <= (G * (j1,i2)) `1 ) } by A1083, A1098, A1297, A1298, TOPREAL3:10; then A1305: p1 in LSeg (g,m1) by A1286, A1302, A1298; LSeg (g,m1) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A1293, A1303, A1304; hence x in L~ g by A1285, A1305, TARSKI:def_4; ::_thesis: verum end; suppose ma <> len g ; ::_thesis: x in L~ g then ma < len g by A1292, XXREAL_0:1; then A1306: ma + 1 <= len g by NAT_1:13; reconsider qa = g /. ma, qa1 = g /. (ma + 1) as Point of (TOP-REAL 2) ; set lma = { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa `1 <= p2 `1 & p2 `1 <= qa1 `1 ) } ; A1307: qa1 = |[(qa1 `1),(qa1 `2)]| by EUCLID:53; A1308: qa `1 <= p1 `1 by A1292; A1309: len g1 <= ma + 1 by A1292, NAT_1:13; then A1310: qa1 `2 = (G * (i1,i2)) `2 by A1249, A1306; A1311: now__::_thesis:_not_qa1_`1_<=_p1_`1 assume qa1 `1 <= p1 `1 ; ::_thesis: contradiction then for q being Point of (TOP-REAL 2) st q = g /. (ma + 1) holds q `1 <= p1 `1 ; then ma + 1 <= ma by A1292, A1306, A1309; hence contradiction by XREAL_1:29; ::_thesis: verum end; A1312: ( qa `2 = (G * (i1,i2)) `2 & qa = |[(qa `1),(qa `2)]| ) by A1249, A1292, EUCLID:53; A1313: 1 <= ma by A24, A14, A47, A1292, NAT_1:13; then LSeg (g,ma) = LSeg (qa,qa1) by A1306, TOPREAL1:def_3 .= { p2 where p2 is Point of (TOP-REAL 2) : ( p2 `2 = (G * (i1,i2)) `2 & qa `1 <= p2 `1 & p2 `1 <= qa1 `1 ) } by A1308, A1311, A1310, A1312, A1307, TOPREAL3:10, XXREAL_0:2 ; then A1314: x in LSeg (g,ma) by A1285, A1286, A1308, A1311; LSeg (g,ma) in { (LSeg (g,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len g ) } by A1306, A1313; hence x in L~ g by A1314, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in L~ g ; ::_thesis: verum end; end; end; hence x in L~ g ; ::_thesis: verum end; 1 <= len g1 by A24, A14, A47, XXREAL_0:2; then 1 in dom g1 by FINSEQ_3:25; hence g /. 1 = (f | k) /. 1 by A45, FINSEQ_4:68 .= f /. 1 by A27, A25, FINSEQ_4:71 ; ::_thesis: ( g /. (len g) = f /. (len f) & len f <= len g ) A1315: len g = (len g1) + l by A1017, FINSEQ_1:22; i1 + 1 <= j1 by A1011, NAT_1:13; then A1316: 1 <= l by XREAL_1:19; then A1317: l in dom g2 by A1017, FINSEQ_3:25; hence g /. (len g) = g2 /. l by A1315, FINSEQ_4:69 .= G * ((i1 + l),i2) by A1017, A1317 .= f /. (len f) by A3, A21, A698 ; ::_thesis: len f <= len g thus len f <= len g by A3, A14, A47, A1316, A1315, XREAL_1:7; ::_thesis: verum end; end; end; hence ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ; ::_thesis: verum end; end; end; hence ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ; ::_thesis: verum end; end; end; A1318: S1[ 0 ] proof let f be FinSequence of (TOP-REAL 2); ::_thesis: ( len f = 0 & ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ) assume that A1319: len f = 0 and A1320: ( ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special ) ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) take g = f; ::_thesis: ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) f = {} by A1319; then for n being Element of NAT st n in dom g & n + 1 in dom g holds for m, k, i, j being Element of NAT st [m,k] in Indices G & [i,j] in Indices G & g /. n = G * (m,k) & g /. (n + 1) = G * (i,j) holds (abs (m - i)) + (abs (k - j)) = 1 ; hence ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) by A1320, GOBOARD1:def_9; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A1318, A1); hence ( ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is one-to-one & f is unfolded & f is s.n.c. & f is special implies ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is one-to-one & g is unfolded & g is s.n.c. & g is special & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ) ; ::_thesis: verum end; theorem :: GOBOARD3:2 for f being FinSequence of (TOP-REAL 2) for G being Go-board st ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is being_S-Seq holds ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is being_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) proof let f be FinSequence of (TOP-REAL 2); ::_thesis: for G being Go-board st ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is being_S-Seq holds ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is being_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) let G be Go-board; ::_thesis: ( ( for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) ) & f is being_S-Seq implies ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is being_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ) assume that A1: for n being Element of NAT st n in dom f holds ex i, j being Element of NAT st ( [i,j] in Indices G & f /. n = G * (i,j) ) and A2: f is being_S-Seq ; ::_thesis: ex g being FinSequence of (TOP-REAL 2) st ( g is_sequence_on G & g is being_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) ( f is one-to-one & f is unfolded & f is s.n.c. & f is special ) by A2, TOPREAL1:def_8; then consider g being FinSequence of (TOP-REAL 2) such that A3: g is_sequence_on G and A4: ( g is one-to-one & g is unfolded & g is s.n.c. & g is special ) and A5: ( L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) ) and A6: len f <= len g by A1, Th1; take g ; ::_thesis: ( g is_sequence_on G & g is being_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) thus g is_sequence_on G by A3; ::_thesis: ( g is being_S-Seq & L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) 2 <= len f by A2, TOPREAL1:def_8; then 2 <= len g by A6, XXREAL_0:2; hence g is being_S-Seq by A4, TOPREAL1:def_8; ::_thesis: ( L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) thus ( L~ f = L~ g & f /. 1 = g /. 1 & f /. (len f) = g /. (len g) & len f <= len g ) by A5, A6; ::_thesis: verum end;