:: Reconstructions of Special Sequences :: by Yatsuka Nakamura and Roman Matuszewski :: :: Received December 10, 1996 :: Copyright (c) 1996-2012 Association of Mizar Users begin theorem :: JORDAN3:1 for n being Element of NAT for f being FinSequence of (TOP-REAL n) st 2 <= len f holds ( f . 1 in L~ f & f /. 1 in L~ f & f . (len f) in L~ f & f /. (len f) in L~ f ) proofend; theorem Th2: :: JORDAN3:2 for p1, p2, q1, q2 being Point of (TOP-REAL 2) st ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & not q1 `1 = q2 `1 holds q1 `2 = q2 `2 proofend; theorem Th3: :: JORDAN3:3 for p1, p2, q1, q2 being Point of (TOP-REAL 2) st ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & LSeg (q1,q2) c= LSeg (p1,p2) & not q1 `1 = q2 `1 holds q1 `2 = q2 `2 proofend; theorem Th4: :: JORDAN3:4 for f being FinSequence of (TOP-REAL 2) for n being Element of NAT st 2 <= n & f is being_S-Seq holds f | n is being_S-Seq proofend; theorem Th5: :: JORDAN3:5 for f being FinSequence of (TOP-REAL 2) for n being Element of NAT st n <= len f & 2 <= (len f) -' n & f is being_S-Seq holds f /^ n is being_S-Seq proofend; theorem :: JORDAN3:6 for f being FinSequence of (TOP-REAL 2) for k1, k2 being Element of NAT st f is being_S-Seq & 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & k1 <> k2 holds mid (f,k1,k2) is being_S-Seq proofend; begin ::---------------------------------------------------------: :: A Concept of Index for Finite Sequences in TOP-REAL 2 : ::---------------------------------------------------------: definition let f be FinSequence of (TOP-REAL 2); let p be Point of (TOP-REAL 2); assume A1: p in L~ f ; func Index (p,f) -> Element of NAT means :Def1: :: JORDAN3:def 1 ex S being non empty Subset of NAT st ( it = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ); existence ex b1 being Element of NAT ex S being non empty Subset of NAT st ( b1 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) proofend; uniqueness for b1, b2 being Element of NAT st ex S being non empty Subset of NAT st ( b1 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) & ex S being non empty Subset of NAT st ( b2 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) holds b1 = b2 ; end; :: deftheorem Def1 defines Index JORDAN3:def_1_:_ for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds for b3 being Element of NAT holds ( b3 = Index (p,f) iff ex S being non empty Subset of NAT st ( b3 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) ); theorem Th7: :: JORDAN3:7 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for i being Element of NAT st p in LSeg (f,i) holds Index (p,f) <= i proofend; theorem Th8: :: JORDAN3:8 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds ( 1 <= Index (p,f) & Index (p,f) < len f ) proofend; theorem Th9: :: JORDAN3:9 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds p in LSeg (f,(Index (p,f))) proofend; theorem Th10: :: JORDAN3:10 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in LSeg (f,1) holds Index (p,f) = 1 proofend; theorem Th11: :: JORDAN3:11 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st len f >= 2 holds Index ((f /. 1),f) = 1 proofend; theorem Th12: :: JORDAN3:12 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for i1 being Nat st f is being_S-Seq & 1 < i1 & i1 <= len f & p = f . i1 holds (Index (p,f)) + 1 = i1 proofend; theorem Th13: :: JORDAN3:13 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for i1 being Element of NAT st f is s.n.c. & p in LSeg (f,i1) & not i1 = Index (p,f) holds i1 = (Index (p,f)) + 1 proofend; theorem Th14: :: JORDAN3:14 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for i1 being Element of NAT st f is unfolded & f is s.n.c. & i1 + 1 <= len f & p in LSeg (f,i1) & p <> f . i1 holds i1 = Index (p,f) proofend; definition let g be FinSequence of (TOP-REAL 2); let p1, p2 be Point of (TOP-REAL 2); predg is_S-Seq_joining p1,p2 means :Def2: :: JORDAN3:def 2 ( g is being_S-Seq & g . 1 = p1 & g . (len g) = p2 ); end; :: deftheorem Def2 defines is_S-Seq_joining JORDAN3:def_2_:_ for g being FinSequence of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) holds ( g is_S-Seq_joining p1,p2 iff ( g is being_S-Seq & g . 1 = p1 & g . (len g) = p2 ) ); theorem Th15: :: JORDAN3:15 for g being FinSequence of (TOP-REAL 2) for p1, p2 being Point of (TOP-REAL 2) st g is_S-Seq_joining p1,p2 holds Rev g is_S-Seq_joining p2,p1 proofend; theorem Th16: :: JORDAN3:16 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for j being Nat st p in L~ f & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) & 1 <= j & j + 1 <= len g holds LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) proofend; theorem :: JORDAN3:17 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . ((Index (p,f)) + 1) & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) holds g is_S-Seq_joining p,f /. (len f) proofend; theorem Th18: :: JORDAN3:18 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) for j being Nat st p in L~ f & 1 <= j & j + 1 <= len g & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds LSeg (g,j) c= LSeg (f,j) proofend; theorem Th19: :: JORDAN3:19 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds g is_S-Seq_joining f /. 1,p proofend; begin ::----------------------------------------------------------------------: :: Left and Right Cutting Functions for Finite Sequences in TOP-REAL 2 : ::----------------------------------------------------------------------: definition let f be FinSequence of (TOP-REAL 2); let p be Point of (TOP-REAL 2); func L_Cut (f,p) -> FinSequence of (TOP-REAL 2) equals :Def3: :: JORDAN3:def 3 <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) if p <> f . ((Index (p,f)) + 1) otherwise mid (f,((Index (p,f)) + 1),(len f)); correctness coherence ( ( p <> f . ((Index (p,f)) + 1) implies <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) is FinSequence of (TOP-REAL 2) ) & ( not p <> f . ((Index (p,f)) + 1) implies mid (f,((Index (p,f)) + 1),(len f)) is FinSequence of (TOP-REAL 2) ) ); consistency for b1 being FinSequence of (TOP-REAL 2) holds verum; ; func R_Cut (f,p) -> FinSequence of (TOP-REAL 2) equals :Def4: :: JORDAN3:def 4 (mid (f,1,(Index (p,f)))) ^ <*p*> if p <> f . 1 otherwise <*p*>; correctness coherence ( ( p <> f . 1 implies (mid (f,1,(Index (p,f)))) ^ <*p*> is FinSequence of (TOP-REAL 2) ) & ( not p <> f . 1 implies <*p*> is FinSequence of (TOP-REAL 2) ) ); consistency for b1 being FinSequence of (TOP-REAL 2) holds verum; ; end; :: deftheorem Def3 defines L_Cut JORDAN3:def_3_:_ for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) holds ( ( p <> f . ((Index (p,f)) + 1) implies L_Cut (f,p) = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) ) & ( not p <> f . ((Index (p,f)) + 1) implies L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) ) ); :: deftheorem Def4 defines R_Cut JORDAN3:def_4_:_ for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) holds ( ( p <> f . 1 implies R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> ) & ( not p <> f . 1 implies R_Cut (f,p) = <*p*> ) ); theorem Th20: :: JORDAN3:20 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p = f . ((Index (p,f)) + 1) & p <> f . (len f) holds ((Index (p,(Rev f))) + (Index (p,f))) + 1 = len f proofend; theorem Th21: :: JORDAN3:21 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is unfolded & f is s.n.c. & p in L~ f & p <> f . ((Index (p,f)) + 1) holds (Index (p,(Rev f))) + (Index (p,f)) = len f proofend; theorem Th22: :: JORDAN3:22 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f holds L_Cut ((Rev f),p) = Rev (R_Cut (f,p)) proofend; theorem Th23: :: JORDAN3:23 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds ( (L_Cut (f,p)) . 1 = p & ( for i being Element of NAT st 1 < i & i <= len (L_Cut (f,p)) holds ( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) ) ) ) proofend; theorem Th24: :: JORDAN3:24 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds ( (R_Cut (f,p)) . (len (R_Cut (f,p))) = p & ( for i being Element of NAT st 1 <= i & i <= Index (p,f) holds (R_Cut (f,p)) . i = f . i ) ) proofend; theorem :: JORDAN3:25 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds ( ( p <> f . 1 implies len (R_Cut (f,p)) = (Index (p,f)) + 1 ) & ( p = f . 1 implies len (R_Cut (f,p)) = Index (p,f) ) ) proofend; theorem Th26: :: JORDAN3:26 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds ( ( p = f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = (len f) - (Index (p,f)) ) & ( p <> f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 ) ) proofend; definition let p1, p2, q1, q2 be Point of (TOP-REAL 2); pred LE q1,q2,p1,p2 means :Def5: :: JORDAN3:def 5 ( q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & ( for r1, r2 being Real st 0 <= r1 & r1 <= 1 & q1 = ((1 - r1) * p1) + (r1 * p2) & 0 <= r2 & r2 <= 1 & q2 = ((1 - r2) * p1) + (r2 * p2) holds r1 <= r2 ) ); end; :: deftheorem Def5 defines LE JORDAN3:def_5_:_ for p1, p2, q1, q2 being Point of (TOP-REAL 2) holds ( LE q1,q2,p1,p2 iff ( q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & ( for r1, r2 being Real st 0 <= r1 & r1 <= 1 & q1 = ((1 - r1) * p1) + (r1 * p2) & 0 <= r2 & r2 <= 1 & q2 = ((1 - r2) * p1) + (r2 * p2) holds r1 <= r2 ) ) ); definition let p1, p2, q1, q2 be Point of (TOP-REAL 2); pred LT q1,q2,p1,p2 means :Def6: :: JORDAN3:def 6 ( LE q1,q2,p1,p2 & q1 <> q2 ); end; :: deftheorem Def6 defines LT JORDAN3:def_6_:_ for p1, p2, q1, q2 being Point of (TOP-REAL 2) holds ( LT q1,q2,p1,p2 iff ( LE q1,q2,p1,p2 & q1 <> q2 ) ); theorem :: JORDAN3:27 for p1, p2, q1, q2 being Point of (TOP-REAL 2) st LE q1,q2,p1,p2 & LE q2,q1,p1,p2 holds q1 = q2 proofend; theorem Th28: :: JORDAN3:28 for p1, p2, q1, q2 being Point of (TOP-REAL 2) st q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & p1 <> p2 holds ( ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) & ( not LE q1,q2,p1,p2 or not LT q2,q1,p1,p2 ) ) proofend; theorem Th29: :: JORDAN3:29 for f being FinSequence of (TOP-REAL 2) for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & Index (p,f) < Index (q,f) holds q in L~ (L_Cut (f,p)) proofend; theorem Th30: :: JORDAN3:30 for p, q, p1, p2 being Point of (TOP-REAL 2) st LE p,q,p1,p2 holds ( q in LSeg (p,p2) & p in LSeg (p1,q) ) proofend; theorem Th31: :: JORDAN3:31 for f being FinSequence of (TOP-REAL 2) for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & p <> q & Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) holds q in L~ (L_Cut (f,p)) proofend; begin ::--------------------------------------------------------: :: Cutting Both Sides of a Finite Sequence and : :: a Discussion of Speciality of Sequences in TOP-REAL 2 : ::--------------------------------------------------------: definition let f be FinSequence of (TOP-REAL 2); let p, q be Point of (TOP-REAL 2); func B_Cut (f,p,q) -> FinSequence of (TOP-REAL 2) equals :Def7: :: JORDAN3:def 7 R_Cut ((L_Cut (f,p)),q) if ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) otherwise Rev (R_Cut ((L_Cut (f,q)),p)); correctness coherence ( ( ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) implies R_Cut ((L_Cut (f,p)),q) is FinSequence of (TOP-REAL 2) ) & ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) or Rev (R_Cut ((L_Cut (f,q)),p)) is FinSequence of (TOP-REAL 2) ) ); consistency for b1 being FinSequence of (TOP-REAL 2) holds verum; ; end; :: deftheorem Def7 defines B_Cut JORDAN3:def_7_:_ for f being FinSequence of (TOP-REAL 2) for p, q being Point of (TOP-REAL 2) holds ( ( ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) implies B_Cut (f,p,q) = R_Cut ((L_Cut (f,p)),q) ) & ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) or B_Cut (f,p,q) = Rev (R_Cut ((L_Cut (f,q)),p)) ) ); theorem Th32: :: JORDAN3:32 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 holds R_Cut (f,p) is_S-Seq_joining f /. 1,p proofend; theorem Th33: :: JORDAN3:33 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . (len f) holds L_Cut (f,p) is_S-Seq_joining p,f /. (len f) proofend; theorem Th34: :: JORDAN3:34 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . (len f) holds L_Cut (f,p) is being_S-Seq proofend; theorem Th35: :: JORDAN3:35 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 holds R_Cut (f,p) is being_S-Seq proofend; Lm1: for f being FinSequence of (TOP-REAL 2) for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) holds B_Cut (f,p,q) is_S-Seq_joining p,q proofend; theorem Th36: :: JORDAN3:36 for f being FinSequence of (TOP-REAL 2) for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds B_Cut (f,p,q) is_S-Seq_joining p,q proofend; theorem :: JORDAN3:37 for f being FinSequence of (TOP-REAL 2) for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds B_Cut (f,p,q) is being_S-Seq proofend; theorem Th38: :: JORDAN3:38 for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds f ^ (mid (g,2,(len g))) is being_S-Seq proofend; theorem Th39: :: JORDAN3:39 for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds f ^ (mid (g,2,(len g))) is_S-Seq_joining f /. 1,g /. (len g) proofend; theorem :: JORDAN3:40 for f being FinSequence of (TOP-REAL 2) for n being Element of NAT holds L~ (f /^ n) c= L~ f proofend; theorem Th41: :: JORDAN3:41 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds L~ (R_Cut (f,p)) c= L~ f proofend; Lm2: for i being Element of NAT for D being non empty set holds (<*> D) | i = <*> D proofend; Lm3: for D being non empty set for f1 being FinSequence of D for k being Element of NAT st 1 <= k & k <= len f1 holds ( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 ) proofend; Lm4: for D being non empty set for f1 being FinSequence of D holds mid (f1,0,0) = f1 | 1 proofend; Lm5: for D being non empty set for f1 being FinSequence of D for k being Element of NAT st len f1 < k holds mid (f1,k,k) = <*> D proofend; Lm6: for D being non empty set for f1 being FinSequence of D for i1, i2 being Element of NAT holds mid (f1,i1,i2) = Rev (mid (f1,i2,i1)) proofend; Lm7: for h being FinSequence of (TOP-REAL 2) for i1, i2 being Element of NAT st 1 <= i1 & i1 <= len h & 1 <= i2 & i2 <= len h holds L~ (mid (h,i1,i2)) c= L~ h proofend; Lm8: for i, j being Element of NAT for D being non empty set for f being FinSequence of D st i in dom f & j in dom f holds len (mid (f,i,j)) >= 1 proofend; Lm9: for i, j being Element of NAT for D being non empty set for f being FinSequence of D st i in dom f & j in dom f holds not mid (f,i,j) is empty proofend; Lm10: for i, j being Element of NAT for D being non empty set for f being FinSequence of D st i in dom f & j in dom f holds (mid (f,i,j)) /. 1 = f /. i proofend; theorem Th42: :: JORDAN3:42 for f being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st p in L~ f holds L~ (L_Cut (f,p)) c= L~ f proofend; theorem Th43: :: JORDAN3:43 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds (L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g) proofend; theorem :: JORDAN3:44 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq proofend; theorem Th45: :: JORDAN3:45 for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds (mid (f,1,((len f) -' 1))) ^ g is being_S-Seq proofend; theorem Th46: :: JORDAN3:46 for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds (mid (f,1,((len f) -' 1))) ^ g is_S-Seq_joining f /. 1,g /. (len g) proofend; theorem Th47: :: JORDAN3:47 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 holds (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p proofend; theorem :: JORDAN3:48 for f, g being FinSequence of (TOP-REAL 2) for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 holds (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is being_S-Seq proofend;