:: TOPALG_6 semantic presentation begin registration let S be TopSpace; let T be non empty TopSpace; cluster Function-like constant quasi_total -> continuous for Element of bool [: the carrier of S, the carrier of T:]; correctness coherence for b1 being Function of S,T st b1 is constant holds b1 is continuous ; proof let f be Function of S,T; ::_thesis: ( f is constant implies f is continuous ) assume A1: f is constant ; ::_thesis: f is continuous percases ( S is empty or not S is empty ) ; supposeA2: S is empty ; ::_thesis: f is continuous for P1 being Subset of T st P1 is closed holds f " P1 is closed by A2; hence f is continuous by PRE_TOPC:def_6; ::_thesis: verum end; suppose not S is empty ; ::_thesis: f is continuous then consider y being Element of T such that A3: rng f = {y} by A1, FUNCT_2:111; y in rng f by A3, TARSKI:def_1; then ex x being set st ( x in dom f & y = f . x ) by FUNCT_1:def_3; then A4: the_value_of f = y by A1, FUNCT_1:def_12; f = (dom f) --> (the_value_of f) by A1, FUNCOP_1:80; then f = S --> y by A4, FUNCT_2:def_1; hence f is continuous ; ::_thesis: verum end; end; end; end; theorem Th1: :: TOPALG_6:1 L[01] (0,1,0,1) = id (Closed-Interval-TSpace (0,1)) proof L[01] (0,1,0,1) = (id (Closed-Interval-TSpace (0,1))) * (id (Closed-Interval-TSpace (0,1))) by BORSUK_6:def_1, TREAL_1:10, TREAL_1:14; hence L[01] (0,1,0,1) = id (Closed-Interval-TSpace (0,1)) by SYSREL:12; ::_thesis: verum end; theorem Th2: :: TOPALG_6:2 for r1, r2, r3, r4, r5, r6 being real number st r1 < r2 & r3 <= r4 & r5 < r6 holds (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4) proof let r1, r2, r3, r4, r5, r6 be real number ; ::_thesis: ( r1 < r2 & r3 <= r4 & r5 < r6 implies (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4) ) set f1 = L[01] (r1,r2,r3,r4); set f2 = L[01] (r5,r6,r1,r2); set f3 = L[01] (r5,r6,r3,r4); assume A1: ( r1 < r2 & r3 <= r4 & r5 < r6 ) ; ::_thesis: (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4) A2: dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) = [#] (Closed-Interval-TSpace (r5,r6)) by FUNCT_2:def_1 .= dom (L[01] (r5,r6,r3,r4)) by FUNCT_2:def_1 ; for x being set st x in dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) holds ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r5,r6,r3,r4)) . x proof let x be set ; ::_thesis: ( x in dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) implies ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r5,r6,r3,r4)) . x ) assume A3: x in dom ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) ; ::_thesis: ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r5,r6,r3,r4)) . x then A4: x in [#] (Closed-Interval-TSpace (r5,r6)) ; then A5: x in [.r5,r6.] by A1, TOPMETR:18; reconsider r = x as real number by A3; A6: ( r5 <= r & r <= r6 ) by A5, XXREAL_1:1; A7: rng (L[01] (r5,r6,r1,r2)) c= [#] (Closed-Interval-TSpace (r1,r2)) by RELAT_1:def_19; reconsider s = (L[01] (r5,r6,r1,r2)) . x as real number ; x in dom (L[01] (r5,r6,r1,r2)) by A4, FUNCT_2:def_1; then s in rng (L[01] (r5,r6,r1,r2)) by FUNCT_1:3; then s in [#] (Closed-Interval-TSpace (r1,r2)) by A7; then s in [.r1,r2.] by A1, TOPMETR:18; then ( r1 <= s & s <= r2 ) by XXREAL_1:1; then A8: (L[01] (r1,r2,r3,r4)) . s = (((r4 - r3) / (r2 - r1)) * (s - r1)) + r3 by A1, BORSUK_6:35; A9: r2 - r1 <> 0 by A1; A10: ((r4 - r3) / (r2 - r1)) * s = ((r4 - r3) / (r2 - r1)) * ((((r2 - r1) / (r6 - r5)) * (r - r5)) + r1) by A1, A6, BORSUK_6:35 .= ((((r4 - r3) / (r2 - r1)) * ((r2 - r1) / (r6 - r5))) * (r - r5)) + (((r4 - r3) / (r2 - r1)) * r1) .= ((((r4 - r3) / (r6 - r5)) * ((r2 - r1) / (r2 - r1))) * (r - r5)) + (((r4 - r3) / (r2 - r1)) * r1) by XCMPLX_1:85 .= ((((r4 - r3) / (r6 - r5)) * 1) * (r - r5)) + (((r4 - r3) / (r2 - r1)) * r1) by A9, XCMPLX_1:60 .= (((r4 - r3) / (r6 - r5)) * (r - r5)) + (((r4 - r3) / (r2 - r1)) * r1) ; thus ((L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2))) . x = (L[01] (r1,r2,r3,r4)) . ((L[01] (r5,r6,r1,r2)) . x) by A3, FUNCT_1:12 .= (L[01] (r5,r6,r3,r4)) . x by A10, A8, A1, A6, BORSUK_6:35 ; ::_thesis: verum end; hence (L[01] (r1,r2,r3,r4)) * (L[01] (r5,r6,r1,r2)) = L[01] (r5,r6,r3,r4) by A2, FUNCT_1:2; ::_thesis: verum end; registration let n be positive Nat; cluster TOP-REAL n -> infinite ; correctness coherence not TOP-REAL n is finite ; proof A1: the carrier of (TOP-REAL n) = REAL n by EUCLID:22 .= n -tuples_on REAL by EUCLID:def_1 ; deffunc H1( Element of n -tuples_on REAL) -> set = n . 1; consider f being Function such that A2: ( dom f = n -tuples_on REAL & ( for d being Element of n -tuples_on REAL holds f . d = H1(d) ) ) from FUNCT_1:sch_4(); for y being set holds ( y in f .: (n -tuples_on REAL) iff y in REAL ) proof let y be set ; ::_thesis: ( y in f .: (n -tuples_on REAL) iff y in REAL ) 0 + 1 < n + 1 by XREAL_1:6; then 1 <= n by NAT_1:13; then A3: 1 in Seg n by FINSEQ_1:1; hereby ::_thesis: ( y in REAL implies y in f .: (n -tuples_on REAL) ) assume y in f .: (n -tuples_on REAL) ; ::_thesis: y in REAL then consider x being set such that A4: ( x in dom f & x in n -tuples_on REAL & y = f . x ) by FUNCT_1:def_6; reconsider x = x as Element of n -tuples_on REAL by A4; A5: y = x . 1 by A2, A4; x in Funcs ((Seg n),REAL) by A4, FINSEQ_2:93; then ex f1 being Function st ( x = f1 & dom f1 = Seg n & rng f1 c= REAL ) by FUNCT_2:def_2; then y in rng x by A3, A5, FUNCT_1:3; hence y in REAL ; ::_thesis: verum end; assume y in REAL ; ::_thesis: y in f .: (n -tuples_on REAL) then A6: {y} c= REAL by ZFMISC_1:31; set x = (Seg n) --> y; A7: ( dom ((Seg n) --> y) = Seg n & rng ((Seg n) --> y) c= {y} ) by FUNCOP_1:13; rng ((Seg n) --> y) c= REAL by A6, XBOOLE_1:1; then (Seg n) --> y in Funcs ((Seg n),REAL) by A7, FUNCT_2:def_2; then reconsider x = (Seg n) --> y as Element of n -tuples_on REAL by FINSEQ_2:93; f . x = x . 1 by A2 .= y by A3, FUNCOP_1:7 ; hence y in f .: (n -tuples_on REAL) by A2, FUNCT_1:def_6; ::_thesis: verum end; hence not TOP-REAL n is finite by A1, TARSKI:1; ::_thesis: verum end; cluster non empty TopSpace-like n -locally_euclidean -> non empty infinite for TopStruct ; correctness coherence for b1 being non empty TopSpace st b1 is n -locally_euclidean holds b1 is infinite ; proof let M be non empty TopSpace; ::_thesis: ( M is n -locally_euclidean implies M is infinite ) assume A8: M is n -locally_euclidean ; ::_thesis: M is infinite consider p being set such that A9: p in the carrier of M by XBOOLE_0:def_1; reconsider p = p as Point of M by A9; consider U being a_neighborhood of p such that A10: U, [#] (TOP-REAL n) are_homeomorphic by A8, MFOLD_1:13; M | U,(TOP-REAL n) | ([#] (TOP-REAL n)) are_homeomorphic by A10, METRIZTS:def_1; then consider f being Function of ((TOP-REAL n) | ([#] (TOP-REAL n))),(M | U) such that A11: f is being_homeomorphism by T_0TOPSP:def_1; A12: ( dom f = [#] ((TOP-REAL n) | ([#] (TOP-REAL n))) & rng f = [#] (M | U) & f is one-to-one ) by A11, TOPS_2:58; [#] ((TOP-REAL n) | ([#] (TOP-REAL n))) is infinite by PRE_TOPC:def_5; hence M is infinite by A12, CARD_1:59; ::_thesis: verum end; end; theorem Th3: :: TOPALG_6:3 for n being Nat for p being Point of (TOP-REAL n) st p in Sphere ((0. (TOP-REAL n)),1) holds - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p} proof let n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st p in Sphere ((0. (TOP-REAL n)),1) holds - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p} let p be Point of (TOP-REAL n); ::_thesis: ( p in Sphere ((0. (TOP-REAL n)),1) implies - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p} ) reconsider n1 = n as Element of NAT by ORDINAL1:def_12; reconsider p1 = p as Point of (TOP-REAL n1) ; assume p in Sphere ((0. (TOP-REAL n)),1) ; ::_thesis: - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p} then |.(p1 - (0. (TOP-REAL n1))).| = 1 by TOPREAL9:9; then |.(p1 + (- (0. (TOP-REAL n1)))).| = 1 by EUCLID:41; then |.(p + ((- 1) * (0. (TOP-REAL n)))).| = 1 by EUCLID:39; then |.(p + (0. (TOP-REAL n))).| = 1 by EUCLID:28; then A1: |.p.| = 1 by EUCLID:27; reconsider r1 = 1 as real number ; |.(0. (TOP-REAL n)).| <> |.p.| by A1, EUCLID_2:39; then 0. (TOP-REAL n) <> (1 + 1) * p by EUCLID:31; then 0. (TOP-REAL n) <> (r1 * p) + (r1 * p) by EUCLID:33; then 0. (TOP-REAL n) <> (r1 * p) + p by EUCLID:29; then 0. (TOP-REAL n) <> p + p by EUCLID:29; then p + (- p) <> p + p by EUCLID:36; then A2: not - p in {p} by TARSKI:def_1; |.(- p).| = 1 by A1, EUCLID:71; then |.((- p) + (0. (TOP-REAL n))).| = 1 by EUCLID:27; then |.((- p) + ((- 1) * (0. (TOP-REAL n)))).| = 1 by EUCLID:28; then |.((- p) + (- (0. (TOP-REAL n)))).| = 1 by EUCLID:39; then |.((- p1) - (0. (TOP-REAL n1))).| = 1 by EUCLID:41; then - p1 in Sphere ((0. (TOP-REAL n1)),1) by TOPREAL9:9; hence - p in (Sphere ((0. (TOP-REAL n)),1)) \ {p} by A2, XBOOLE_0:def_5; ::_thesis: verum end; theorem Th4: :: TOPALG_6:4 for T being non empty TopStruct for t1, t2 being Point of T for p being Path of t1,t2 holds ( inf (dom p) = 0 & sup (dom p) = 1 ) proof let T be non empty TopStruct ; ::_thesis: for t1, t2 being Point of T for p being Path of t1,t2 holds ( inf (dom p) = 0 & sup (dom p) = 1 ) let t1, t2 be Point of T; ::_thesis: for p being Path of t1,t2 holds ( inf (dom p) = 0 & sup (dom p) = 1 ) let p be Path of t1,t2; ::_thesis: ( inf (dom p) = 0 & sup (dom p) = 1 ) [.0,1.] = dom p by BORSUK_1:40, FUNCT_2:def_1; hence ( inf (dom p) = 0 & sup (dom p) = 1 ) by XXREAL_2:25, XXREAL_2:29; ::_thesis: verum end; theorem Th5: :: TOPALG_6:5 for T being non empty TopSpace for t being Point of T for C1, C2 being constant Loop of t holds C1,C2 are_homotopic proof let T be non empty TopSpace; ::_thesis: for t being Point of T for C1, C2 being constant Loop of t holds C1,C2 are_homotopic let t be Point of T; ::_thesis: for C1, C2 being constant Loop of t holds C1,C2 are_homotopic let C1, C2 be constant Loop of t; ::_thesis: C1,C2 are_homotopic C1 = I[01] --> t by BORSUK_2:5 .= C2 by BORSUK_2:5 ; hence C1,C2 are_homotopic by BORSUK_2:12; ::_thesis: verum end; theorem Th6: :: TOPALG_6:6 for T being non empty TopSpace for S being non empty SubSpace of T for t1, t2 being Point of T for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic holds A,B are_homotopic proof let T be non empty TopSpace; ::_thesis: for S being non empty SubSpace of T for t1, t2 being Point of T for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic holds A,B are_homotopic let S be non empty SubSpace of T; ::_thesis: for t1, t2 being Point of T for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic holds A,B are_homotopic let t1, t2 be Point of T; ::_thesis: for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic holds A,B are_homotopic let s1, s2 be Point of S; ::_thesis: for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic holds A,B are_homotopic let A, B be Path of t1,t2; ::_thesis: for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic holds A,B are_homotopic let C, D be Path of s1,s2; ::_thesis: ( s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & C,D are_homotopic implies A,B are_homotopic ) assume that A1: ( s1,s2 are_connected & t1,t2 are_connected ) and A2: ( A = C & B = D ) ; ::_thesis: ( not C,D are_homotopic or A,B are_homotopic ) given f being Function of [:I[01],I[01]:],S such that A3: f is continuous and A4: for t being Point of I[01] holds ( f . (t,0) = C . t & f . (t,1) = D . t & f . (0,t) = s1 & f . (1,t) = s2 ) ; :: according to BORSUK_2:def_7 ::_thesis: A,B are_homotopic reconsider g = f as Function of [:I[01],I[01]:],T by TOPREALA:7; take g ; :: according to BORSUK_2:def_7 ::_thesis: ( g is continuous & ( for b1 being Element of the carrier of I[01] holds ( g . (b1,0) = A . b1 & g . (b1,1) = B . b1 & g . (0,b1) = t1 & g . (1,b1) = t2 ) ) ) thus g is continuous by A3, PRE_TOPC:26; ::_thesis: for b1 being Element of the carrier of I[01] holds ( g . (b1,0) = A . b1 & g . (b1,1) = B . b1 & g . (0,b1) = t1 & g . (1,b1) = t2 ) ( s1 = C . 0 & s2 = C . 1 & t1 = A . 0 & t2 = A . 1 ) by A1, BORSUK_2:def_2; hence for b1 being Element of the carrier of I[01] holds ( g . (b1,0) = A . b1 & g . (b1,1) = B . b1 & g . (0,b1) = t1 & g . (1,b1) = t2 ) by A2, A4; ::_thesis: verum end; theorem :: TOPALG_6:7 for T being non empty TopSpace for S being non empty SubSpace of T for t1, t2 being Point of T for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) holds Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) proof let T be non empty TopSpace; ::_thesis: for S being non empty SubSpace of T for t1, t2 being Point of T for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) holds Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) let S be non empty SubSpace of T; ::_thesis: for t1, t2 being Point of T for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) holds Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) let t1, t2 be Point of T; ::_thesis: for s1, s2 being Point of S for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) holds Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) let s1, s2 be Point of S; ::_thesis: for A, B being Path of t1,t2 for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) holds Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) let A, B be Path of t1,t2; ::_thesis: for C, D being Path of s1,s2 st s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) holds Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) let C, D be Path of s1,s2; ::_thesis: ( s1,s2 are_connected & t1,t2 are_connected & A = C & B = D & Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) implies Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) ) assume that A1: s1,s2 are_connected and A2: t1,t2 are_connected and A3: ( A = C & B = D ) ; ::_thesis: ( not Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) or Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) ) assume Class ((EqRel (S,s1,s2)),C) = Class ((EqRel (S,s1,s2)),D) ; ::_thesis: Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) then C,D are_homotopic by A1, TOPALG_1:46; then A,B are_homotopic by A1, A2, A3, Th6; hence Class ((EqRel (T,t1,t2)),A) = Class ((EqRel (T,t1,t2)),B) by A2, TOPALG_1:46; ::_thesis: verum end; theorem Th8: :: TOPALG_6:8 for T being non empty trivial TopSpace for t being Point of T for L being Loop of t holds the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))} proof let T be non empty trivial TopSpace; ::_thesis: for t being Point of T for L being Loop of t holds the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))} let t be Point of T; ::_thesis: for L being Loop of t holds the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))} set E = EqRel (T,t); let L be Loop of t; ::_thesis: the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))} thus the carrier of (pi_1 (T,t)) c= {(Class ((EqRel (T,t)),L))} :: according to XBOOLE_0:def_10 ::_thesis: {(Class ((EqRel (T,t)),L))} c= the carrier of (pi_1 (T,t)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (pi_1 (T,t)) or x in {(Class ((EqRel (T,t)),L))} ) assume x in the carrier of (pi_1 (T,t)) ; ::_thesis: x in {(Class ((EqRel (T,t)),L))} then consider P being Loop of t such that A1: x = Class ((EqRel (T,t)),P) by TOPALG_1:47; P = I[01] --> t by TOPREALC:22 .= L by TOPREALC:22 ; hence x in {(Class ((EqRel (T,t)),L))} by A1, TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(Class ((EqRel (T,t)),L))} or x in the carrier of (pi_1 (T,t)) ) assume x in {(Class ((EqRel (T,t)),L))} ; ::_thesis: x in the carrier of (pi_1 (T,t)) then A2: x = Class ((EqRel (T,t)),L) by TARSKI:def_1; L in Loops t by TOPALG_1:def_1; then x in Class (EqRel (T,t)) by A2, EQREL_1:def_3; hence x in the carrier of (pi_1 (T,t)) by TOPALG_1:def_5; ::_thesis: verum end; theorem Th9: :: TOPALG_6:9 for n being Nat for p being Point of (TOP-REAL n) for S being Subset of (TOP-REAL n) st n >= 2 & S = ([#] (TOP-REAL n)) \ {p} holds (TOP-REAL n) | S is pathwise_connected proof let n be Nat; ::_thesis: for p being Point of (TOP-REAL n) for S being Subset of (TOP-REAL n) st n >= 2 & S = ([#] (TOP-REAL n)) \ {p} holds (TOP-REAL n) | S is pathwise_connected let p be Point of (TOP-REAL n); ::_thesis: for S being Subset of (TOP-REAL n) st n >= 2 & S = ([#] (TOP-REAL n)) \ {p} holds (TOP-REAL n) | S is pathwise_connected let S be Subset of (TOP-REAL n); ::_thesis: ( n >= 2 & S = ([#] (TOP-REAL n)) \ {p} implies (TOP-REAL n) | S is pathwise_connected ) assume A1: n >= 2 ; ::_thesis: ( not S = ([#] (TOP-REAL n)) \ {p} or (TOP-REAL n) | S is pathwise_connected ) assume A2: S = ([#] (TOP-REAL n)) \ {p} ; ::_thesis: (TOP-REAL n) | S is pathwise_connected then S is infinite by A1, RAMSEY_1:4; then reconsider T = (TOP-REAL n) | S as non empty SubSpace of TOP-REAL n ; A3: [#] T = ([#] (TOP-REAL n)) \ {p} by A2, PRE_TOPC:def_5; A4: for a, b being Point of T for a1, b1 being Point of (TOP-REAL n) st not p in LSeg (a1,b1) & a1 = a & b1 = b holds a,b are_connected proof let a, b be Point of T; ::_thesis: for a1, b1 being Point of (TOP-REAL n) st not p in LSeg (a1,b1) & a1 = a & b1 = b holds a,b are_connected let a1, b1 be Point of (TOP-REAL n); ::_thesis: ( not p in LSeg (a1,b1) & a1 = a & b1 = b implies a,b are_connected ) assume A5: not p in LSeg (a1,b1) ; ::_thesis: ( not a1 = a or not b1 = b or a,b are_connected ) assume A6: ( a1 = a & b1 = b ) ; ::_thesis: a,b are_connected percases ( a1 <> b1 or a1 = b1 ) ; supposeA7: a1 <> b1 ; ::_thesis: a,b are_connected A8: [#] ((TOP-REAL n) | (LSeg (a1,b1))) = LSeg (a1,b1) by PRE_TOPC:def_5; A9: LSeg (a1,b1) c= ([#] (TOP-REAL n)) \ {p} by A5, ZFMISC_1:34; reconsider Y = (TOP-REAL n) | (LSeg (a1,b1)) as non empty SubSpace of T by A3, A9, A8, RLTOPSP1:68, TSEP_1:4; LSeg (a1,b1) is_an_arc_of a1,b1 by A7, TOPREAL1:9; then consider h being Function of I[01],Y such that A10: h is being_homeomorphism and A11: ( h . 0 = a1 & h . 1 = b1 ) by TOPREAL1:def_1; reconsider f = h as Function of I[01],T by A3, A9, A8, FUNCT_2:7; take f ; :: according to BORSUK_2:def_1 ::_thesis: ( f is continuous & f . 0 = a & f . 1 = b ) thus f is continuous by A10, PRE_TOPC:26; ::_thesis: ( f . 0 = a & f . 1 = b ) thus ( f . 0 = a & f . 1 = b ) by A6, A11; ::_thesis: verum end; suppose a1 = b1 ; ::_thesis: a,b are_connected hence a,b are_connected by A6; ::_thesis: verum end; end; end; for a, b being Point of T holds a,b are_connected proof let a, b be Point of T; ::_thesis: a,b are_connected A12: the carrier of T is Subset of (TOP-REAL n) by TSEP_1:1; ( a in the carrier of T & b in the carrier of T ) ; then reconsider a1 = a, b1 = b as Point of (TOP-REAL n) by A12; percases ( a1 <> b1 or a1 = b1 ) ; supposeA13: a1 <> b1 ; ::_thesis: a,b are_connected percases ( p in LSeg (a1,b1) or not p in LSeg (a1,b1) ) ; supposeA14: p in LSeg (a1,b1) ; ::_thesis: a,b are_connected reconsider n1 = n as Element of NAT by ORDINAL1:def_12; reconsider aa1 = a1, bb1 = b1 as Point of (TOP-REAL n1) ; consider s being Real such that A15: ( 0 <= s & s <= 1 & p = ((1 - s) * aa1) + (s * bb1) ) by A14, JGRAPH_1:35; set q1 = b1 - a1; reconsider k = n - 1 as Nat by A1, CHORD:1; k + 1 > 1 by A1, XXREAL_0:2; then A16: k >= 1 by NAT_1:13; b1 - a1 <> 0. (TOP-REAL (k + 1)) by A13, EUCLID:43; then TPlane ((b1 - a1),p) is k -manifold by MFOLD_2:30; then [#] (TPlane ((b1 - a1),p)) is infinite by A16; then [#] ((TOP-REAL n) | (Plane ((b1 - a1),p))) is infinite by MFOLD_2:def_3; then A17: Plane ((b1 - a1),p) is infinite ; reconsider X = Plane ((b1 - a1),p) as set ; X \ {p} is infinite by A17, RAMSEY_1:4; then consider x being set such that A18: x in X \ {p} by XBOOLE_0:def_1; A19: ( x in X & not x in {p} ) by A18, XBOOLE_0:def_5; then x in { y where y is Point of (TOP-REAL n) : |((b1 - a1),(y - p))| = 0 } by MFOLD_2:def_2; then consider c1 being Point of (TOP-REAL n) such that A20: ( c1 = x & |((b1 - a1),(c1 - p))| = 0 ) ; A21: |((b1 - a1),(b1 - a1))| <> 0 proof assume |((b1 - a1),(b1 - a1))| = 0 ; ::_thesis: contradiction then b1 - a1 = 0. (TOP-REAL n) by EUCLID_2:41; hence contradiction by A13, EUCLID:43; ::_thesis: verum end; A22: not p in LSeg (a1,c1) proof assume A23: p in LSeg (a1,c1) ; ::_thesis: contradiction reconsider cc1 = c1 as Point of (TOP-REAL n1) ; consider r being Real such that A24: ( 0 <= r & r <= 1 & p = ((1 - r) * aa1) + (r * cc1) ) by A23, JGRAPH_1:35; A25: 1 - r <> 0 proof assume 1 - r = 0 ; ::_thesis: contradiction then p = (0. (TOP-REAL n)) + (1 * c1) by A24, EUCLID:29 .= (0. (TOP-REAL n)) + c1 by EUCLID:29 .= c1 by EUCLID:27 ; hence contradiction by A19, A20, TARSKI:def_1; ::_thesis: verum end; set q2 = c1 - a1; c1 - p = (c1 - ((1 - r) * a1)) - (r * c1) by A24, EUCLID:46 .= (c1 + (- ((1 - r) * a1))) - (r * c1) by EUCLID:41 .= (c1 + ((- (1 - r)) * a1)) - (r * c1) by EUCLID:40 .= (c1 + (((- 1) + r) * a1)) - (r * c1) .= (c1 + (((- 1) * a1) + (r * a1))) - (r * c1) by EUCLID:33 .= (c1 + ((- a1) + (r * a1))) - (r * c1) by EUCLID:39 .= ((c1 + (- a1)) + (r * a1)) - (r * c1) by EUCLID:26 .= ((c1 - a1) + (r * a1)) - (r * c1) by EUCLID:41 .= ((c1 - a1) + (r * a1)) + (- (r * c1)) by EUCLID:41 .= ((c1 - a1) + (r * a1)) + (r * (- c1)) by EUCLID:40 .= (c1 - a1) + ((r * a1) + (r * (- c1))) by EUCLID:26 .= (c1 - a1) + (r * (a1 + (- c1))) by EUCLID:32 .= (c1 - a1) + (r * (- (c1 - a1))) by EUCLID:44 .= (c1 - a1) + (- (r * (c1 - a1))) by EUCLID:40 .= (c1 - a1) + ((- r) * (c1 - a1)) by EUCLID:40 .= (1 * (c1 - a1)) + ((- r) * (c1 - a1)) by EUCLID:29 .= (1 + (- r)) * (c1 - a1) by EUCLID:33 .= (1 - r) * (c1 - a1) ; then (1 - r) * |((b1 - a1),(c1 - a1))| = 0 by A20, EUCLID_2:20; then A26: |((b1 - a1),(c1 - a1))| = 0 by A25, XCMPLX_1:6; 0. (TOP-REAL n) = (((1 - r) * a1) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by A15, A24, EUCLID:36 .= (((1 - r) * a1) + (- (((1 - s) * a1) + (s * b1)))) + (r * c1) by EUCLID:26 .= (((1 - r) * a1) + ((- ((1 - s) * a1)) - (s * b1))) + (r * c1) by EUCLID:38 .= (((1 - r) * a1) + ((- ((1 - s) * a1)) + (- (s * b1)))) + (r * c1) by EUCLID:41 .= ((((1 - r) * a1) + (- ((1 - s) * a1))) + (- (s * b1))) + (r * c1) by EUCLID:26 .= ((((1 - r) * a1) + ((- (1 - s)) * a1)) + (- (s * b1))) + (r * c1) by EUCLID:40 .= ((((1 - r) + (- (1 - s))) * a1) + (- (s * b1))) + (r * c1) by EUCLID:33 .= (((s + (- r)) * a1) + (- (s * b1))) + (r * c1) .= (((s * a1) + ((- r) * a1)) + (- (s * b1))) + (r * c1) by EUCLID:33 .= (((s * a1) + (- (s * b1))) + ((- r) * a1)) + (r * c1) by EUCLID:26 .= (((s * a1) + (s * (- b1))) + ((- r) * a1)) + (r * c1) by EUCLID:40 .= ((s * (a1 + (- b1))) + ((- r) * a1)) + (r * c1) by EUCLID:32 .= ((s * (- (b1 - a1))) + ((- r) * a1)) + (r * c1) by EUCLID:44 .= (s * (- (b1 - a1))) + (((- r) * a1) + (r * c1)) by EUCLID:26 .= (s * (- (b1 - a1))) + ((r * c1) + (- (r * a1))) by EUCLID:40 .= (s * (- (b1 - a1))) + ((r * c1) + (r * (- a1))) by EUCLID:40 .= (s * (- (b1 - a1))) + (r * (c1 + (- a1))) by EUCLID:32 .= (s * (- (b1 - a1))) + (r * (c1 - a1)) by EUCLID:41 ; then A27: 0 = |(((s * (- (b1 - a1))) + (r * (c1 - a1))),((s * (- (b1 - a1))) + (r * (c1 - a1))))| by EUCLID_2:34 .= (|((s * (- (b1 - a1))),(s * (- (b1 - a1))))| + (2 * |((s * (- (b1 - a1))),(r * (c1 - a1)))|)) + |((r * (c1 - a1)),(r * (c1 - a1)))| by EUCLID_2:30 ; A28: |((s * (- (b1 - a1))),(s * (- (b1 - a1))))| = s * |((- (b1 - a1)),(s * (- (b1 - a1))))| by EUCLID_2:19 .= s * (s * |((- (b1 - a1)),(- (b1 - a1)))|) by EUCLID_2:20 .= s * (s * |((b1 - a1),(b1 - a1))|) by EUCLID_2:23 .= (s * s) * |((b1 - a1),(b1 - a1))| ; A29: |((r * (c1 - a1)),(r * (c1 - a1)))| = r * |((c1 - a1),(r * (c1 - a1)))| by EUCLID_2:19 .= r * (r * |((c1 - a1),(c1 - a1))|) by EUCLID_2:20 .= (r * r) * |((c1 - a1),(c1 - a1))| ; A30: |((s * (- (b1 - a1))),(r * (c1 - a1)))| = s * |((- (b1 - a1)),(r * (c1 - a1)))| by EUCLID_2:19 .= s * (r * |((- (b1 - a1)),(c1 - a1))|) by EUCLID_2:20 .= s * (r * (- |((b1 - a1),(c1 - a1))|)) by EUCLID_2:21 .= 0 by A26 ; A31: s * s >= 0 by XREAL_1:63; A32: r * r >= 0 by XREAL_1:63; A33: |((b1 - a1),(b1 - a1))| >= 0 by EUCLID_2:35; A34: |((c1 - a1),(c1 - a1))| >= 0 by EUCLID_2:35; A35: s * s <> 0 proof assume s * s = 0 ; ::_thesis: contradiction then s = 0 by XCMPLX_1:6; then p = (1 * a1) + (0. (TOP-REAL n)) by A15, EUCLID:29 .= 1 * a1 by EUCLID:27 .= a1 by EUCLID:29 ; then not p in {p} by A3, XBOOLE_0:def_5; hence contradiction by TARSKI:def_1; ::_thesis: verum end; thus contradiction by A28, A29, A27, A30, A31, A32, A33, A34, A35, A21, XREAL_1:71; ::_thesis: verum end; A36: not p in LSeg (c1,b1) proof assume A37: p in LSeg (c1,b1) ; ::_thesis: contradiction reconsider cc1 = c1 as Point of (TOP-REAL n1) ; consider r being Real such that A38: ( 0 <= r & r <= 1 & p = ((1 - r) * bb1) + (r * cc1) ) by A37, JGRAPH_1:35; A39: 1 - r <> 0 proof assume 1 - r = 0 ; ::_thesis: contradiction then p = (0. (TOP-REAL n)) + (1 * c1) by A38, EUCLID:29 .= (0. (TOP-REAL n)) + c1 by EUCLID:29 .= c1 by EUCLID:27 ; hence contradiction by A19, A20, TARSKI:def_1; ::_thesis: verum end; set q2 = c1 - b1; c1 - p = (c1 - ((1 - r) * b1)) - (r * c1) by A38, EUCLID:46 .= (c1 + (- ((1 - r) * b1))) - (r * c1) by EUCLID:41 .= (c1 + ((- (1 - r)) * b1)) - (r * c1) by EUCLID:40 .= (c1 + (((- 1) + r) * b1)) - (r * c1) .= (c1 + (((- 1) * b1) + (r * b1))) - (r * c1) by EUCLID:33 .= (c1 + ((- b1) + (r * b1))) - (r * c1) by EUCLID:39 .= ((c1 + (- b1)) + (r * b1)) - (r * c1) by EUCLID:26 .= ((c1 - b1) + (r * b1)) - (r * c1) by EUCLID:41 .= ((c1 - b1) + (r * b1)) + (- (r * c1)) by EUCLID:41 .= ((c1 - b1) + (r * b1)) + (r * (- c1)) by EUCLID:40 .= (c1 - b1) + ((r * b1) + (r * (- c1))) by EUCLID:26 .= (c1 - b1) + (r * (b1 + (- c1))) by EUCLID:32 .= (c1 - b1) + (r * (- (c1 - b1))) by EUCLID:44 .= (c1 - b1) + (- (r * (c1 - b1))) by EUCLID:40 .= (c1 - b1) + ((- r) * (c1 - b1)) by EUCLID:40 .= (1 * (c1 - b1)) + ((- r) * (c1 - b1)) by EUCLID:29 .= (1 + (- r)) * (c1 - b1) by EUCLID:33 .= (1 - r) * (c1 - b1) ; then (1 - r) * |((b1 - a1),(c1 - b1))| = 0 by A20, EUCLID_2:20; then A40: |((b1 - a1),(c1 - b1))| = 0 by A39, XCMPLX_1:6; A41: 0. (TOP-REAL n) = (((1 + (- r)) * b1) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by A38, A15, EUCLID:36 .= (((1 * b1) + ((- r) * b1)) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:33 .= ((b1 + ((- r) * b1)) + (r * c1)) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:29 .= (b1 + (((- r) * b1) + (r * c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:26 .= (b1 + ((- (r * b1)) + (r * c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:40 .= (b1 + ((r * (- b1)) + (r * c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:40 .= (b1 + (r * ((- b1) + c1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:32 .= (b1 + (r * (c1 - b1))) + (- (((1 - s) * a1) + (s * b1))) by EUCLID:41 .= (b1 + (- (((1 - s) * a1) + (s * b1)))) + (r * (c1 - b1)) by EUCLID:26 .= (b1 + ((- 1) * ((s * b1) + ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:39 .= (b1 + (((- 1) * (s * b1)) + ((- 1) * ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:32 .= (b1 + ((((- 1) * s) * b1) + ((- 1) * ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:30 .= (b1 + (((- s) * b1) + (- ((1 - s) * a1)))) + (r * (c1 - b1)) by EUCLID:39 .= ((b1 + ((- s) * b1)) + (- ((1 - s) * a1))) + (r * (c1 - b1)) by EUCLID:26 .= (((1 * b1) + ((- s) * b1)) + (- ((1 - s) * a1))) + (r * (c1 - b1)) by EUCLID:29 .= (((1 + (- s)) * b1) + (- ((1 - s) * a1))) + (r * (c1 - b1)) by EUCLID:33 .= (((1 - s) * b1) + ((1 - s) * (- a1))) + (r * (c1 - b1)) by EUCLID:40 .= ((1 - s) * (b1 + (- a1))) + (r * (c1 - b1)) by EUCLID:32 .= ((1 - s) * (b1 - a1)) + (r * (c1 - b1)) by EUCLID:41 ; set t = 1 - s; A42: 0 = |((((1 - s) * (b1 - a1)) + (r * (c1 - b1))),(((1 - s) * (b1 - a1)) + (r * (c1 - b1))))| by A41, EUCLID_2:34 .= (|(((1 - s) * (b1 - a1)),((1 - s) * (b1 - a1)))| + (2 * |(((1 - s) * (b1 - a1)),(r * (c1 - b1)))|)) + |((r * (c1 - b1)),(r * (c1 - b1)))| by EUCLID_2:30 ; A43: |(((1 - s) * (b1 - a1)),((1 - s) * (b1 - a1)))| = (1 - s) * |((b1 - a1),((1 - s) * (b1 - a1)))| by EUCLID_2:19 .= (1 - s) * ((1 - s) * |((b1 - a1),(b1 - a1))|) by EUCLID_2:20 .= ((1 - s) * (1 - s)) * |((b1 - a1),(b1 - a1))| ; A44: |((r * (c1 - b1)),(r * (c1 - b1)))| = r * |((c1 - b1),(r * (c1 - b1)))| by EUCLID_2:19 .= r * (r * |((c1 - b1),(c1 - b1))|) by EUCLID_2:20 .= (r * r) * |((c1 - b1),(c1 - b1))| ; A45: |(((1 - s) * (b1 - a1)),(r * (c1 - b1)))| = (1 - s) * |((b1 - a1),(r * (c1 - b1)))| by EUCLID_2:19 .= (1 - s) * (r * |((b1 - a1),(c1 - b1))|) by EUCLID_2:20 .= 0 by A40 ; A46: (1 - s) * (1 - s) >= 0 by XREAL_1:63; A47: r * r >= 0 by XREAL_1:63; A48: |((b1 - a1),(b1 - a1))| >= 0 by EUCLID_2:35; A49: |((c1 - b1),(c1 - b1))| >= 0 by EUCLID_2:35; A50: (1 - s) * (1 - s) <> 0 proof assume (1 - s) * (1 - s) = 0 ; ::_thesis: contradiction then 1 - s = 0 by XCMPLX_1:6; then p = (0. (TOP-REAL n)) + (1 * b1) by A15, EUCLID:29 .= 1 * b1 by EUCLID:27 .= b1 by EUCLID:29 ; then not p in {p} by A3, XBOOLE_0:def_5; hence contradiction by TARSKI:def_1; ::_thesis: verum end; thus contradiction by A50, A21, A43, A44, A42, A45, A46, A47, A48, A49, XREAL_1:71; ::_thesis: verum end; reconsider c = c1 as Point of T by A20, A19, A3, XBOOLE_0:def_5; ( a,c are_connected & c,b are_connected ) by A22, A36, A4; hence a,b are_connected by BORSUK_6:42; ::_thesis: verum end; suppose not p in LSeg (a1,b1) ; ::_thesis: a,b are_connected hence a,b are_connected by A4; ::_thesis: verum end; end; end; suppose a1 = b1 ; ::_thesis: a,b are_connected hence a,b are_connected ; ::_thesis: verum end; end; end; hence (TOP-REAL n) | S is pathwise_connected by BORSUK_2:def_3; ::_thesis: verum end; theorem Th10: :: TOPALG_6:10 for T being non empty TopSpace for t being Point of T for n being Nat for S being non empty Subset of T st n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic holds T | S is pathwise_connected proof let T be non empty TopSpace; ::_thesis: for t being Point of T for n being Nat for S being non empty Subset of T st n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic holds T | S is pathwise_connected let t be Point of T; ::_thesis: for n being Nat for S being non empty Subset of T st n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic holds T | S is pathwise_connected let n be Nat; ::_thesis: for S being non empty Subset of T st n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic holds T | S is pathwise_connected let S be non empty Subset of T; ::_thesis: ( n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic implies T | S is pathwise_connected ) assume A1: ( n >= 2 & S = ([#] T) \ {t} & TOP-REAL n,T are_homeomorphic ) ; ::_thesis: T | S is pathwise_connected then consider f being Function of T,(TOP-REAL n) such that A2: f is being_homeomorphism by T_0TOPSP:def_1; reconsider p = f . t as Point of (TOP-REAL n) ; reconsider SN = ([#] (TOP-REAL n)) \ {p} as non empty Subset of (TOP-REAL n) by A1, RAMSEY_1:4; A3: (TOP-REAL n) | SN is pathwise_connected by A1, Th9; A4: ( dom f = [#] T & rng f = [#] (TOP-REAL n) ) by A2, TOPS_2:58; then A5: f " ([#] (TOP-REAL n)) = [#] T by RELAT_1:134; consider x being set such that A6: f " {p} = {x} by A4, A2, FUNCT_1:74; A7: x in f " {p} by A6, TARSKI:def_1; then ( x in dom f & f . x in {p} ) by FUNCT_1:def_7; then p = f . x by TARSKI:def_1; then x = t by A2, A7, A4, FUNCT_1:def_4; then A8: f " SN = S by A1, A5, A6, FUNCT_1:69; A9: dom (SN |` f) = f " SN by MFOLD_2:1 .= [#] (T | (f " SN)) by PRE_TOPC:def_5 ; rng (SN |` f) c= SN ; then rng (SN |` f) c= [#] ((TOP-REAL n) | SN) by PRE_TOPC:def_5; then reconsider g = SN |` f as Function of (T | (f " SN)),((TOP-REAL n) | SN) by A9, FUNCT_2:2; g is being_homeomorphism by A2, MFOLD_2:4; then (TOP-REAL n) | SN,T | S are_homeomorphic by A8, T_0TOPSP:def_1; hence T | S is pathwise_connected by A3, TOPALG_3:9; ::_thesis: verum end; registration let n be Element of NAT ; let p, q be Point of (TOP-REAL n); cluster TPlane (p,q) -> convex ; correctness coherence TPlane (p,q) is convex ; proof set P = Plane (p,q); for w1, w2 being Point of (TOP-REAL n) st w1 in Plane (p,q) & w2 in Plane (p,q) holds LSeg (w1,w2) c= Plane (p,q) proof let w1, w2 be Point of (TOP-REAL n); ::_thesis: ( w1 in Plane (p,q) & w2 in Plane (p,q) implies LSeg (w1,w2) c= Plane (p,q) ) assume A1: ( w1 in Plane (p,q) & w2 in Plane (p,q) ) ; ::_thesis: LSeg (w1,w2) c= Plane (p,q) reconsider n0 = n as Nat ; reconsider p0 = p, q0 = q as Point of (TOP-REAL n0) ; A2: Plane (p,q) = { y where y is Point of (TOP-REAL n0) : |(p0,(y - q0))| = 0 } by MFOLD_2:def_2; consider v1 being Point of (TOP-REAL n0) such that A3: ( w1 = v1 & |(p0,(v1 - q0))| = 0 ) by A1, A2; consider v2 being Point of (TOP-REAL n0) such that A4: ( w2 = v2 & |(p0,(v2 - q0))| = 0 ) by A1, A2; for x being set st x in LSeg (w1,w2) holds x in Plane (p,q) proof let x be set ; ::_thesis: ( x in LSeg (w1,w2) implies x in Plane (p,q) ) assume A5: x in LSeg (w1,w2) ; ::_thesis: x in Plane (p,q) then reconsider w = x as Point of (TOP-REAL n0) ; reconsider r1 = 1 as real number ; consider r being Real such that A6: ( 0 <= r & r <= 1 & w = ((1 - r) * w1) + (r * w2) ) by A5, JGRAPH_1:35; A7: |(p0,((1 - r) * (v1 - q0)))| = (1 - r) * 0 by A3, EUCLID_2:20 .= 0 ; A8: |(p0,(r * (v2 - q0)))| = r * 0 by A4, EUCLID_2:20 .= 0 ; A9: ((1 - r) * (v1 - q0)) + (r * (v2 - q0)) = ((1 - r) * (w1 - q)) + ((r * w2) - (r * q)) by A3, A4, EUCLID:49 .= (((1 - r) * w1) - ((1 - r) * q)) + ((r * w2) - (r * q)) by EUCLID:49 .= (((1 - r) * w1) + (- ((1 - r) * q))) + ((r * w2) - (r * q)) by EUCLID:41 .= (((1 - r) * w1) + ((- (1 - r)) * q)) + ((r * w2) - (r * q)) by EUCLID:40 .= (((1 - r) * w1) + ((r - 1) * q)) + ((r * w2) - (r * q)) .= (((1 - r) * w1) + ((r * q) - (r1 * q))) + ((r * w2) - (r * q)) by EUCLID:50 .= ((1 - r) * w1) + (((r * q) - (r1 * q)) + ((r * w2) - (r * q))) by EUCLID:26 .= ((1 - r) * w1) + (((r * q) - (r1 * q)) + ((r * w2) + (- (r * q)))) by EUCLID:41 .= ((1 - r) * w1) + (((r * q) + (- (r1 * q))) + ((r * w2) + (- (r * q)))) by EUCLID:41 .= ((1 - r) * w1) + ((((- (r1 * q)) + (r * q)) + (- (r * q))) + (r * w2)) by EUCLID:26 .= ((1 - r) * w1) + (((- (r1 * q)) + ((r * q) + (- (r * q)))) + (r * w2)) by EUCLID:26 .= ((1 - r) * w1) + (((- (r1 * q)) + ((r * q) - (r * q))) + (r * w2)) by EUCLID:41 .= ((1 - r) * w1) + (((- (r1 * q)) + (0. (TOP-REAL n))) + (r * w2)) by EUCLID:42 .= ((1 - r) * w1) + ((- (r1 * q)) + (r * w2)) by EUCLID:27 .= (((1 - r) * w1) + (r * w2)) + (- (r1 * q)) by EUCLID:26 .= w + (- q0) by A6, EUCLID:29 .= w - q0 by EUCLID:41 ; 0 = |(p0,((1 - r) * (v1 - q0)))| + |(p0,(r * (v2 - q0)))| by A7, A8 .= |(p0,(w - q0))| by A9, EUCLID_2:26 ; hence x in Plane (p,q) by A2; ::_thesis: verum end; hence LSeg (w1,w2) c= Plane (p,q) by TARSKI:def_3; ::_thesis: verum end; then Plane (p,q) is convex Subset of (TOP-REAL n) by RLTOPSP1:22; then [#] ((TOP-REAL n) | (Plane (p,q))) is convex Subset of (TOP-REAL n) by PRE_TOPC:def_5; then [#] (TPlane (p,q)) is convex Subset of (TOP-REAL n) by MFOLD_2:def_3; hence TPlane (p,q) is convex by TOPALG_2:def_1; ::_thesis: verum end; end; begin definition let T be non empty TopSpace; attrT is having_trivial_Fundamental_Group means :Def1: :: TOPALG_6:def 1 for t being Point of T holds pi_1 (T,t) is trivial ; end; :: deftheorem Def1 defines having_trivial_Fundamental_Group TOPALG_6:def_1_:_ for T being non empty TopSpace holds ( T is having_trivial_Fundamental_Group iff for t being Point of T holds pi_1 (T,t) is trivial ); definition let T be non empty TopSpace; attrT is simply_connected means :Def2: :: TOPALG_6:def 2 ( T is having_trivial_Fundamental_Group & T is pathwise_connected ); end; :: deftheorem Def2 defines simply_connected TOPALG_6:def_2_:_ for T being non empty TopSpace holds ( T is simply_connected iff ( T is having_trivial_Fundamental_Group & T is pathwise_connected ) ); registration cluster non empty TopSpace-like simply_connected -> non empty pathwise_connected having_trivial_Fundamental_Group for TopStruct ; coherence for b1 being non empty TopSpace st b1 is simply_connected holds ( b1 is having_trivial_Fundamental_Group & b1 is pathwise_connected ) by Def2; cluster non empty TopSpace-like pathwise_connected having_trivial_Fundamental_Group -> non empty simply_connected for TopStruct ; coherence for b1 being non empty TopSpace st b1 is having_trivial_Fundamental_Group & b1 is pathwise_connected holds b1 is simply_connected by Def2; end; theorem Th11: :: TOPALG_6:11 for T being non empty TopSpace st T is having_trivial_Fundamental_Group holds for t being Point of T for P, Q being Loop of t holds P,Q are_homotopic proof let T be non empty TopSpace; ::_thesis: ( T is having_trivial_Fundamental_Group implies for t being Point of T for P, Q being Loop of t holds P,Q are_homotopic ) assume A1: T is having_trivial_Fundamental_Group ; ::_thesis: for t being Point of T for P, Q being Loop of t holds P,Q are_homotopic let t be Point of T; ::_thesis: for P, Q being Loop of t holds P,Q are_homotopic let P, Q be Loop of t; ::_thesis: P,Q are_homotopic set E = EqRel (T,t); A2: pi_1 (T,t) is trivial by A1, Def1; ( Class ((EqRel (T,t)),P) in the carrier of (pi_1 (T,t)) & Class ((EqRel (T,t)),Q) in the carrier of (pi_1 (T,t)) ) by TOPALG_1:47; then Class ((EqRel (T,t)),P) = Class ((EqRel (T,t)),Q) by A2, ZFMISC_1:def_10; hence P,Q are_homotopic by TOPALG_1:46; ::_thesis: verum end; registration let n be Nat; cluster TOP-REAL n -> having_trivial_Fundamental_Group ; coherence TOP-REAL n is having_trivial_Fundamental_Group proof let o be Point of (TOP-REAL n); :: according to TOPALG_6:def_1 ::_thesis: pi_1 ((TOP-REAL n),o) is trivial thus pi_1 ((TOP-REAL n),o) is trivial ; ::_thesis: verum end; end; registration cluster non empty trivial TopSpace-like -> non empty having_trivial_Fundamental_Group for TopStruct ; coherence for b1 being non empty TopSpace st b1 is trivial holds b1 is having_trivial_Fundamental_Group proof let T be non empty TopSpace; ::_thesis: ( T is trivial implies T is having_trivial_Fundamental_Group ) assume A1: T is trivial ; ::_thesis: T is having_trivial_Fundamental_Group let t be Point of T; :: according to TOPALG_6:def_1 ::_thesis: pi_1 (T,t) is trivial reconsider L = I[01] --> t as Loop of t by JORDAN:41; the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)),L))} by A1, Th8; hence pi_1 (T,t) is trivial ; ::_thesis: verum end; end; theorem Th12: :: TOPALG_6:12 for T being non empty TopSpace holds ( T is simply_connected iff for t1, t2 being Point of T holds ( t1,t2 are_connected & ( for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) ) ) ) proof let T be non empty TopSpace; ::_thesis: ( T is simply_connected iff for t1, t2 being Point of T holds ( t1,t2 are_connected & ( for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) ) ) ) hereby ::_thesis: ( ( for t1, t2 being Point of T holds ( t1,t2 are_connected & ( for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) ) ) ) implies T is simply_connected ) assume A1: T is simply_connected ; ::_thesis: for t1, t2 being Point of T holds ( t1,t2 are_connected & ( for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) ) ) let t1, t2 be Point of T; ::_thesis: ( t1,t2 are_connected & ( for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) ) ) thus A2: t1,t2 are_connected by A1, BORSUK_2:def_3; ::_thesis: for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) let P, Q be Path of t1,t2; ::_thesis: Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) set E = EqRel (T,t1,t2); A3: P,(P + (- Q)) + Q are_homotopic by A1, TOPALG_1:22; set C = the constant Loop of t1; P + (- Q), the constant Loop of t1 are_homotopic by A1, Th11; then A4: (P + (- Q)) + Q, the constant Loop of t1 + Q are_homotopic by A1, BORSUK_6:76; the constant Loop of t1 + Q,Q are_homotopic by A1, BORSUK_6:83; then (P + (- Q)) + Q,Q are_homotopic by A4, BORSUK_6:79; then P,Q are_homotopic by A3, BORSUK_6:79; hence Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) by A2, TOPALG_1:46; ::_thesis: verum end; assume A5: for t1, t2 being Point of T holds ( t1,t2 are_connected & ( for P, Q being Path of t1,t2 holds Class ((EqRel (T,t1,t2)),P) = Class ((EqRel (T,t1,t2)),Q) ) ) ; ::_thesis: T is simply_connected thus T is having_trivial_Fundamental_Group :: according to TOPALG_6:def_2 ::_thesis: T is pathwise_connected proof let t be Point of T; :: according to TOPALG_6:def_1 ::_thesis: pi_1 (T,t) is trivial let x, y be Element of (pi_1 (T,t)); :: according to STRUCT_0:def_10 ::_thesis: x = y ( ex P being Loop of t st x = Class ((EqRel (T,t)),P) & ex P being Loop of t st y = Class ((EqRel (T,t)),P) ) by TOPALG_1:47; hence x = y by A5; ::_thesis: verum end; thus T is pathwise_connected by A5, BORSUK_2:def_3; ::_thesis: verum end; registration let T be non empty having_trivial_Fundamental_Group TopSpace; let t be Point of T; cluster FundamentalGroup (T,t) -> trivial ; coherence pi_1 (T,t) is trivial by Def1; end; theorem Th13: :: TOPALG_6:13 for S, T being non empty TopSpace st S,T are_homeomorphic & S is having_trivial_Fundamental_Group holds T is having_trivial_Fundamental_Group proof let S, T be non empty TopSpace; ::_thesis: ( S,T are_homeomorphic & S is having_trivial_Fundamental_Group implies T is having_trivial_Fundamental_Group ) given f being Function of S,T such that A1: f is being_homeomorphism ; :: according to T_0TOPSP:def_1 ::_thesis: ( not S is having_trivial_Fundamental_Group or T is having_trivial_Fundamental_Group ) assume A2: for s being Point of S holds pi_1 (S,s) is trivial ; :: according to TOPALG_6:def_1 ::_thesis: T is having_trivial_Fundamental_Group let t be Point of T; :: according to TOPALG_6:def_1 ::_thesis: pi_1 (T,t) is trivial rng f = [#] T by A1, TOPS_2:def_5; then consider s being Point of S such that A3: f . s = t by FUNCT_2:113; A4: FundGrIso (f,s) is bijective by A1, TOPALG_3:31; pi_1 (S,s) is trivial by A2; hence pi_1 (T,t) is trivial by A3, A4, KNASTER:11, TOPREALC:1; ::_thesis: verum end; theorem Th14: :: TOPALG_6:14 for S, T being non empty TopSpace st S,T are_homeomorphic & S is simply_connected holds T is simply_connected proof let S, T be non empty TopSpace; ::_thesis: ( S,T are_homeomorphic & S is simply_connected implies T is simply_connected ) assume ( S,T are_homeomorphic & S is simply_connected ) ; ::_thesis: T is simply_connected hence ( T is having_trivial_Fundamental_Group & T is pathwise_connected ) by Th13, TOPALG_3:9; :: according to TOPALG_6:def_2 ::_thesis: verum end; theorem Th15: :: TOPALG_6:15 for T being non empty having_trivial_Fundamental_Group TopSpace for t being Point of T for P1, P2 being Loop of t holds P1,P2 are_homotopic proof let T be non empty having_trivial_Fundamental_Group TopSpace; ::_thesis: for t being Point of T for P1, P2 being Loop of t holds P1,P2 are_homotopic let t be Point of T; ::_thesis: for P1, P2 being Loop of t holds P1,P2 are_homotopic let P1, P2 be Loop of t; ::_thesis: P1,P2 are_homotopic ( Class ((EqRel (T,t)),P1) in the carrier of (pi_1 (T,t)) & Class ((EqRel (T,t)),P2) in the carrier of (pi_1 (T,t)) ) by TOPALG_1:47; then Class ((EqRel (T,t)),P1) = Class ((EqRel (T,t)),P2) by ZFMISC_1:def_10; hence P1,P2 are_homotopic by TOPALG_1:46; ::_thesis: verum end; definition let T be non empty TopSpace; let t be Point of T; let l be Loop of t; attrl is nullhomotopic means :Def3: :: TOPALG_6:def 3 ex c being constant Loop of t st l,c are_homotopic ; end; :: deftheorem Def3 defines nullhomotopic TOPALG_6:def_3_:_ for T being non empty TopSpace for t being Point of T for l being Loop of t holds ( l is nullhomotopic iff ex c being constant Loop of t st l,c are_homotopic ); registration let T be non empty TopSpace; let t be Point of T; cluster constant -> nullhomotopic for Path of t,t; coherence for b1 being Loop of t st b1 is constant holds b1 is nullhomotopic proof let l be Loop of t; ::_thesis: ( l is constant implies l is nullhomotopic ) assume l is constant ; ::_thesis: l is nullhomotopic then reconsider l = l as constant Loop of t ; take l ; :: according to TOPALG_6:def_3 ::_thesis: l,l are_homotopic thus l,l are_homotopic by BORSUK_6:88; ::_thesis: verum end; end; registration let T be non empty TopSpace; let t be Point of T; cluster non empty Relation-like the carrier of I[01] -defined the carrier of T -valued Function-like constant total quasi_total continuous for Path of t,t; existence ex b1 being Loop of t st b1 is constant proof reconsider l = I[01] --> t as constant Loop of t by JORDAN:41; take l ; ::_thesis: l is constant thus l is constant ; ::_thesis: verum end; end; theorem Th16: :: TOPALG_6:16 for T, U being non empty TopSpace for t being Point of T for f being Loop of t for g being continuous Function of T,U st f is nullhomotopic holds g * f is nullhomotopic proof let T, U be non empty TopSpace; ::_thesis: for t being Point of T for f being Loop of t for g being continuous Function of T,U st f is nullhomotopic holds g * f is nullhomotopic let t be Point of T; ::_thesis: for f being Loop of t for g being continuous Function of T,U st f is nullhomotopic holds g * f is nullhomotopic let f be Loop of t; ::_thesis: for g being continuous Function of T,U st f is nullhomotopic holds g * f is nullhomotopic let g be continuous Function of T,U; ::_thesis: ( f is nullhomotopic implies g * f is nullhomotopic ) given c being constant Loop of t such that A1: f,c are_homotopic ; :: according to TOPALG_6:def_3 ::_thesis: g * f is nullhomotopic consider F being Function of [:I[01],I[01]:],T such that A2: F is continuous and A3: for s being Point of I[01] holds ( F . (s,0) = f . s & F . (s,1) = c . s & F . (0,s) = t & F . (1,s) = t ) by A1, BORSUK_2:def_7; reconsider d = I[01] --> (g . t) as constant Loop of g . t by JORDAN:41; reconsider G = g * F as Function of [:I[01],I[01]:],U ; take d ; :: according to TOPALG_6:def_3 ::_thesis: g * f,d are_homotopic take G ; :: according to BORSUK_2:def_7 ::_thesis: ( G is continuous & ( for b1 being Element of the carrier of I[01] holds ( G . (b1,0) = (g * f) . b1 & G . (b1,1) = d . b1 & G . (0,b1) = g . t & G . (1,b1) = g . t ) ) ) thus G is continuous by A2; ::_thesis: for b1 being Element of the carrier of I[01] holds ( G . (b1,0) = (g * f) . b1 & G . (b1,1) = d . b1 & G . (0,b1) = g . t & G . (1,b1) = g . t ) let s be Point of I[01]; ::_thesis: ( G . (s,0) = (g * f) . s & G . (s,1) = d . s & G . (0,s) = g . t & G . (1,s) = g . t ) reconsider j0 = 0 , j1 = 1 as Point of I[01] by BORSUK_1:def_14, BORSUK_1:def_15; set I = the carrier of I[01]; A4: the carrier of [:I[01],I[01]:] = [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2; thus G . (s,0) = g . (F . (s,j0)) by A4, BINOP_1:18 .= g . (f . s) by A3 .= (g * f) . s by FUNCT_2:15 ; ::_thesis: ( G . (s,1) = d . s & G . (0,s) = g . t & G . (1,s) = g . t ) thus G . (s,1) = g . (F . (s,j1)) by A4, BINOP_1:18 .= g . (c . s) by A3 .= g . t by TOPALG_3:21 .= d . s by FUNCOP_1:7 ; ::_thesis: ( G . (0,s) = g . t & G . (1,s) = g . t ) thus G . (0,s) = g . (F . (j0,s)) by A4, BINOP_1:18 .= g . t by A3 ; ::_thesis: G . (1,s) = g . t thus G . (1,s) = g . (F . (j1,s)) by A4, BINOP_1:18 .= g . t by A3 ; ::_thesis: verum end; registration let T, U be non empty TopSpace; let t be Point of T; let f be nullhomotopic Loop of t; let g be continuous Function of T,U; clusterf * g -> nullhomotopic for Loop of g . t; coherence for b1 being Loop of g . t st b1 = g * f holds b1 is nullhomotopic by Th16; end; registration let T be non empty having_trivial_Fundamental_Group TopSpace; let t be Point of T; cluster -> nullhomotopic for Path of t,t; coherence for b1 being Loop of t holds b1 is nullhomotopic proof let l be Loop of t; ::_thesis: l is nullhomotopic reconsider c = I[01] --> t as constant Loop of t by JORDAN:41; take c ; :: according to TOPALG_6:def_3 ::_thesis: l,c are_homotopic thus l,c are_homotopic by Th15; ::_thesis: verum end; end; theorem Th17: :: TOPALG_6:17 for T being non empty TopSpace st ( for t being Point of T for f being Loop of t holds f is nullhomotopic ) holds T is having_trivial_Fundamental_Group proof let T be non empty TopSpace; ::_thesis: ( ( for t being Point of T for f being Loop of t holds f is nullhomotopic ) implies T is having_trivial_Fundamental_Group ) assume A1: for t being Point of T for f being Loop of t holds f is nullhomotopic ; ::_thesis: T is having_trivial_Fundamental_Group for t being Point of T holds pi_1 (T,t) is trivial proof let t be Point of T; ::_thesis: pi_1 (T,t) is trivial set C = the constant Loop of t; the carrier of (pi_1 (T,t)) = {(Class ((EqRel (T,t)), the constant Loop of t))} proof set E = EqRel (T,t); hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {(Class ((EqRel (T,t)), the constant Loop of t))} c= the carrier of (pi_1 (T,t)) let x be set ; ::_thesis: ( x in the carrier of (pi_1 (T,t)) implies x in {(Class ((EqRel (T,t)), the constant Loop of t))} ) assume x in the carrier of (pi_1 (T,t)) ; ::_thesis: x in {(Class ((EqRel (T,t)), the constant Loop of t))} then consider P being Loop of t such that A2: x = Class ((EqRel (T,t)),P) by TOPALG_1:47; P is nullhomotopic by A1; then consider C1 being constant Loop of t such that A3: P,C1 are_homotopic by Def3; C1, the constant Loop of t are_homotopic by Th5; then P, the constant Loop of t are_homotopic by A3, BORSUK_6:79; then x = Class ((EqRel (T,t)), the constant Loop of t) by A2, TOPALG_1:46; hence x in {(Class ((EqRel (T,t)), the constant Loop of t))} by TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(Class ((EqRel (T,t)), the constant Loop of t))} or x in the carrier of (pi_1 (T,t)) ) assume x in {(Class ((EqRel (T,t)), the constant Loop of t))} ; ::_thesis: x in the carrier of (pi_1 (T,t)) then A4: x = Class ((EqRel (T,t)), the constant Loop of t) by TARSKI:def_1; the constant Loop of t in Loops t by TOPALG_1:def_1; then x in Class (EqRel (T,t)) by A4, EQREL_1:def_3; hence x in the carrier of (pi_1 (T,t)) by TOPALG_1:def_5; ::_thesis: verum end; hence pi_1 (T,t) is trivial ; ::_thesis: verum end; hence T is having_trivial_Fundamental_Group by Def1; ::_thesis: verum end; registration let n be Element of NAT ; let p, q be Point of (TOP-REAL n); cluster TPlane (p,q) -> having_trivial_Fundamental_Group ; correctness coherence TPlane (p,q) is having_trivial_Fundamental_Group ; proof for t being Point of (TPlane (p,q)) holds pi_1 ((TPlane (p,q)),t) is trivial ; hence TPlane (p,q) is having_trivial_Fundamental_Group by Def1; ::_thesis: verum end; end; theorem Th18: :: TOPALG_6:18 for T being non empty TopSpace for t being Point of T for S being non empty SubSpace of T for s being Point of S for f being Loop of t for g being Loop of s st t = s & f = g & g is nullhomotopic holds f is nullhomotopic proof let T be non empty TopSpace; ::_thesis: for t being Point of T for S being non empty SubSpace of T for s being Point of S for f being Loop of t for g being Loop of s st t = s & f = g & g is nullhomotopic holds f is nullhomotopic let t be Point of T; ::_thesis: for S being non empty SubSpace of T for s being Point of S for f being Loop of t for g being Loop of s st t = s & f = g & g is nullhomotopic holds f is nullhomotopic let S be non empty SubSpace of T; ::_thesis: for s being Point of S for f being Loop of t for g being Loop of s st t = s & f = g & g is nullhomotopic holds f is nullhomotopic let s be Point of S; ::_thesis: for f being Loop of t for g being Loop of s st t = s & f = g & g is nullhomotopic holds f is nullhomotopic let f be Loop of t; ::_thesis: for g being Loop of s st t = s & f = g & g is nullhomotopic holds f is nullhomotopic let g be Loop of s; ::_thesis: ( t = s & f = g & g is nullhomotopic implies f is nullhomotopic ) assume that A1: ( t = s & f = g ) and A2: g is nullhomotopic ; ::_thesis: f is nullhomotopic consider c being constant Loop of s such that A3: g,c are_homotopic by A2, Def3; c = I[01] --> s by BORSUK_2:5 .= I[01] --> t by A1 ; then reconsider c = c as constant Loop of t by JORDAN:41; f,c are_homotopic by A1, A3, Th6; hence f is nullhomotopic by Def3; ::_thesis: verum end; begin definition let T be TopStruct ; let f be PartFunc of R^1,T; attrf is parametrized-curve means :Def4: :: TOPALG_6:def 4 ( dom f is interval Subset of REAL & ex S being SubSpace of R^1 ex g being Function of S,T st ( f = g & S = R^1 | (dom f) & g is continuous ) ); end; :: deftheorem Def4 defines parametrized-curve TOPALG_6:def_4_:_ for T being TopStruct for f being PartFunc of R^1,T holds ( f is parametrized-curve iff ( dom f is interval Subset of REAL & ex S being SubSpace of R^1 ex g being Function of S,T st ( f = g & S = R^1 | (dom f) & g is continuous ) ) ); Lm1: for T being TopStruct for f being PartFunc of R^1,T st f = {} holds f is parametrized-curve proof let T be TopStruct ; ::_thesis: for f being PartFunc of R^1,T st f = {} holds f is parametrized-curve let f be PartFunc of R^1,T; ::_thesis: ( f = {} implies f is parametrized-curve ) assume A1: f = {} ; ::_thesis: f is parametrized-curve reconsider f = f as PartFunc of R^1,T ; dom f = {} by A1; then A2: dom f c= REAL by XBOOLE_1:2; reconsider A = {} as Subset of R^1 by XBOOLE_1:2; reconsider S = R^1 | A as SubSpace of R^1 ; {} c= [:([#] S),([#] T):] ; then reconsider g = f as Relation of ([#] S),([#] T) by A1; A3: A = dom g ; reconsider g = g as Function of S,T ; for P1 being Subset of T st P1 is closed holds g " P1 is closed ; then g is continuous by PRE_TOPC:def_6; hence f is parametrized-curve by A3, A2, Def4; ::_thesis: verum end; registration let T be TopStruct ; cluster Relation-like the carrier of R^1 -defined the carrier of T -valued Function-like parametrized-curve for Element of bool [: the carrier of R^1, the carrier of T:]; correctness existence ex b1 being PartFunc of R^1,T st b1 is parametrized-curve ; proof reconsider c = {} as PartFunc of R^1,T by XBOOLE_1:2; take c ; ::_thesis: c is parametrized-curve thus c is parametrized-curve by Lm1; ::_thesis: verum end; end; theorem :: TOPALG_6:19 for T being TopStruct holds {} is parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2; definition let T be TopStruct ; func Curves T -> Subset of (PFuncs (REAL,([#] T))) equals :: TOPALG_6:def 5 { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ; coherence { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } is Subset of (PFuncs (REAL,([#] T))) proof set C = { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ; for x being set st x in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } holds x in PFuncs (REAL,([#] T)) proof let x be set ; ::_thesis: ( x in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } implies x in PFuncs (REAL,([#] T)) ) assume x in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ; ::_thesis: x in PFuncs (REAL,([#] T)) then ex f being Element of PFuncs (REAL,([#] T)) st ( x = f & f is parametrized-curve PartFunc of R^1,T ) ; hence x in PFuncs (REAL,([#] T)) ; ::_thesis: verum end; hence { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } is Subset of (PFuncs (REAL,([#] T))) by TARSKI:def_3; ::_thesis: verum end; end; :: deftheorem defines Curves TOPALG_6:def_5_:_ for T being TopStruct holds Curves T = { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ; registration let T be TopStruct ; cluster Curves T -> non empty ; coherence not Curves T is empty proof reconsider c = {} as PartFunc of R^1,T by XBOOLE_1:2; reconsider f1 = c as Element of PFuncs (REAL,([#] T)) by PARTFUN1:45, TOPMETR:17; f1 is parametrized-curve PartFunc of R^1,T by Lm1; then f1 in Curves T ; hence not Curves T is empty ; ::_thesis: verum end; end; definition let T be TopStruct ; mode Curve of T is Element of Curves T; correctness ; end; theorem Th20: :: TOPALG_6:20 for T being TopStruct for f being parametrized-curve PartFunc of R^1,T holds f is Curve of T proof let T be TopStruct ; ::_thesis: for f being parametrized-curve PartFunc of R^1,T holds f is Curve of T let f be parametrized-curve PartFunc of R^1,T; ::_thesis: f is Curve of T reconsider f1 = f as Element of PFuncs (REAL,([#] T)) by PARTFUN1:45, TOPMETR:17; f1 in Curves T ; hence f is Curve of T ; ::_thesis: verum end; theorem Th21: :: TOPALG_6:21 for T being TopStruct holds {} is Curve of T proof let T be TopStruct ; ::_thesis: {} is Curve of T reconsider f = {} as parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2; f is Curve of T by Th20; hence {} is Curve of T ; ::_thesis: verum end; theorem Th22: :: TOPALG_6:22 for T being TopStruct for t1, t2 being Point of T for p being Path of t1,t2 st t1,t2 are_connected holds p is Curve of T proof let T be TopStruct ; ::_thesis: for t1, t2 being Point of T for p being Path of t1,t2 st t1,t2 are_connected holds p is Curve of T let t1, t2 be Point of T; ::_thesis: for p being Path of t1,t2 st t1,t2 are_connected holds p is Curve of T let p be Path of t1,t2; ::_thesis: ( t1,t2 are_connected implies p is Curve of T ) assume t1,t2 are_connected ; ::_thesis: p is Curve of T then A1: ( p is continuous & p . 0 = t1 & p . 1 = t2 ) by BORSUK_2:def_2; percases ( not T is empty or T is empty ) ; suppose not T is empty ; ::_thesis: p is Curve of T then A2: [#] I[01] = dom p by FUNCT_2:def_1; then A3: dom p c= [#] R^1 by PRE_TOPC:def_4; then reconsider A = dom p as Subset of R^1 ; A4: I[01] = R^1 | A by A2, BORSUK_1:40, TOPMETR:19, TOPMETR:20; rng p c= [#] T ; then reconsider c = p as PartFunc of R^1,T by A3, RELSET_1:4; reconsider c = c as parametrized-curve PartFunc of R^1,T by A2, A4, Def4, A1, BORSUK_1:40; c is Element of Curves T by Th20; hence p is Curve of T ; ::_thesis: verum end; supposeA5: T is empty ; ::_thesis: p is Curve of T then reconsider c = p as PartFunc of R^1,T ; c = {} by A5; then reconsider c = c as parametrized-curve PartFunc of R^1,T by Lm1; c is Element of Curves T by Th20; hence p is Curve of T ; ::_thesis: verum end; end; end; theorem Th23: :: TOPALG_6:23 for T being TopStruct for c being Curve of T holds c is parametrized-curve PartFunc of R^1,T proof let T be TopStruct ; ::_thesis: for c being Curve of T holds c is parametrized-curve PartFunc of R^1,T let c be Curve of T; ::_thesis: c is parametrized-curve PartFunc of R^1,T c in { f where f is Element of PFuncs (REAL,([#] T)) : f is parametrized-curve PartFunc of R^1,T } ; then consider f being Element of PFuncs (REAL,([#] T)) such that A1: ( c = f & f is parametrized-curve PartFunc of R^1,T ) ; thus c is parametrized-curve PartFunc of R^1,T by A1; ::_thesis: verum end; theorem Th24: :: TOPALG_6:24 for T being TopStruct for c being Curve of T holds ( dom c c= REAL & rng c c= [#] T ) proof let T be TopStruct ; ::_thesis: for c being Curve of T holds ( dom c c= REAL & rng c c= [#] T ) let c be Curve of T; ::_thesis: ( dom c c= REAL & rng c c= [#] T ) reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23; ( dom f c= [#] R^1 & rng f c= [#] T ) ; hence ( dom c c= REAL & rng c c= [#] T ) by TOPMETR:17; ::_thesis: verum end; registration let T be TopStruct ; let c be Curve of T; cluster dom c -> real-membered ; correctness coherence dom c is real-membered ; proof for x being set st x in dom c holds x is real proof let x be set ; ::_thesis: ( x in dom c implies x is real ) assume A1: x in dom c ; ::_thesis: x is real dom c c= REAL by Th24; hence x is real by A1; ::_thesis: verum end; hence dom c is real-membered by MEMBERED:def_3; ::_thesis: verum end; end; definition let T be TopStruct ; let c be Curve of T; attrc is with_first_point means :Def6: :: TOPALG_6:def 6 dom c is left_end ; attrc is with_last_point means :Def7: :: TOPALG_6:def 7 dom c is right_end ; end; :: deftheorem Def6 defines with_first_point TOPALG_6:def_6_:_ for T being TopStruct for c being Curve of T holds ( c is with_first_point iff dom c is left_end ); :: deftheorem Def7 defines with_last_point TOPALG_6:def_7_:_ for T being TopStruct for c being Curve of T holds ( c is with_last_point iff dom c is right_end ); definition let T be TopStruct ; let c be Curve of T; attrc is with_endpoints means :Def8: :: TOPALG_6:def 8 ( c is with_first_point & c is with_last_point ); end; :: deftheorem Def8 defines with_endpoints TOPALG_6:def_8_:_ for T being TopStruct for c being Curve of T holds ( c is with_endpoints iff ( c is with_first_point & c is with_last_point ) ); registration let T be TopStruct ; cluster with_first_point with_last_point -> with_endpoints for Element of Curves T; correctness coherence for b1 being Curve of T st b1 is with_first_point & b1 is with_last_point holds b1 is with_endpoints ; by Def8; cluster with_endpoints -> with_first_point with_last_point for Element of Curves T; correctness coherence for b1 being Curve of T st b1 is with_endpoints holds ( b1 is with_first_point & b1 is with_last_point ); by Def8; end; registration let T be non empty TopStruct ; cluster Relation-like Function-like with_endpoints for Element of Curves T; correctness existence ex b1 being Curve of T st b1 is with_endpoints ; proof set t = the Point of T; set f = I[01] --> the Point of T; A1: ( (I[01] --> the Point of T) . 0 = the Point of T & (I[01] --> the Point of T) . 1 = the Point of T ) by BORSUK_1:def_14, BORSUK_1:def_15, FUNCOP_1:7; set p = the Path of the Point of T, the Point of T; the Point of T, the Point of T are_connected by A1, BORSUK_2:def_1; then reconsider c = the Path of the Point of T, the Point of T as Curve of T by Th22; take c ; ::_thesis: c is with_endpoints A2: dom c = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; set S = [.0,1.]; inf [.0,1.] = 0 by XXREAL_2:25; then inf [.0,1.] in [.0,1.] by XXREAL_1:1; hence dom c is left_end by A2, XXREAL_2:def_5; :: according to TOPALG_6:def_6,TOPALG_6:def_8 ::_thesis: c is with_last_point sup [.0,1.] = 1 by XXREAL_2:29; then sup [.0,1.] in [.0,1.] by XXREAL_1:1; hence dom c is right_end by A2, XXREAL_2:def_6; :: according to TOPALG_6:def_7 ::_thesis: verum end; end; registration let T be non empty TopStruct ; let c be with_first_point Curve of T; cluster dom c -> non empty ; correctness coherence not dom c is empty ; proof dom c is left_end by Def6; hence not dom c is empty ; ::_thesis: verum end; cluster inf (dom c) -> real ; correctness coherence inf (dom c) is real ; proof dom c is left_end by Def6; hence inf (dom c) is real ; ::_thesis: verum end; end; registration let T be non empty TopStruct ; let c be with_last_point Curve of T; cluster dom c -> non empty ; correctness coherence not dom c is empty ; proof dom c is right_end by Def7; hence not dom c is empty ; ::_thesis: verum end; cluster sup (dom c) -> real ; correctness coherence sup (dom c) is real ; proof dom c is right_end by Def7; hence sup (dom c) is real ; ::_thesis: verum end; end; registration let T be non empty TopStruct ; cluster with_first_point -> non empty for Element of Curves T; coherence for b1 being Curve of T st b1 is with_first_point holds not b1 is empty proof let C be Curve of T; ::_thesis: ( C is with_first_point implies not C is empty ) assume ( C is with_first_point & C is empty ) ; ::_thesis: contradiction then reconsider c = {} as with_first_point Curve of T ; dom c is left_end ; hence contradiction ; ::_thesis: verum end; cluster with_last_point -> non empty for Element of Curves T; coherence for b1 being Curve of T st b1 is with_last_point holds not b1 is empty proof let C be Curve of T; ::_thesis: ( C is with_last_point implies not C is empty ) assume ( C is with_last_point & C is empty ) ; ::_thesis: contradiction then reconsider c = {} as with_last_point Curve of T ; dom c is right_end ; hence contradiction ; ::_thesis: verum end; end; definition let T be non empty TopStruct ; let c be with_first_point Curve of T; func the_first_point_of c -> Point of T equals :: TOPALG_6:def 9 c . (inf (dom c)); correctness coherence c . (inf (dom c)) is Point of T; proof A1: rng c c= [#] T by Th24; dom c is left_end by Def6; then inf (dom c) in dom c by XXREAL_2:def_5; then c . (inf (dom c)) in rng c by FUNCT_1:3; hence c . (inf (dom c)) is Point of T by A1; ::_thesis: verum end; end; :: deftheorem defines the_first_point_of TOPALG_6:def_9_:_ for T being non empty TopStruct for c being with_first_point Curve of T holds the_first_point_of c = c . (inf (dom c)); definition let T be non empty TopStruct ; let c be with_last_point Curve of T; func the_last_point_of c -> Point of T equals :: TOPALG_6:def 10 c . (sup (dom c)); correctness coherence c . (sup (dom c)) is Point of T; proof A1: rng c c= [#] T by Th24; dom c is right_end by Def7; then sup (dom c) in dom c by XXREAL_2:def_6; then c . (sup (dom c)) in rng c by FUNCT_1:3; hence c . (sup (dom c)) is Point of T by A1; ::_thesis: verum end; end; :: deftheorem defines the_last_point_of TOPALG_6:def_10_:_ for T being non empty TopStruct for c being with_last_point Curve of T holds the_last_point_of c = c . (sup (dom c)); theorem Th25: :: TOPALG_6:25 for T being non empty TopStruct for t1, t2 being Point of T for p being Path of t1,t2 st t1,t2 are_connected holds p is with_endpoints Curve of T proof let T be non empty TopStruct ; ::_thesis: for t1, t2 being Point of T for p being Path of t1,t2 st t1,t2 are_connected holds p is with_endpoints Curve of T let t1, t2 be Point of T; ::_thesis: for p being Path of t1,t2 st t1,t2 are_connected holds p is with_endpoints Curve of T let p be Path of t1,t2; ::_thesis: ( t1,t2 are_connected implies p is with_endpoints Curve of T ) assume t1,t2 are_connected ; ::_thesis: p is with_endpoints Curve of T then reconsider c = p as Curve of T by Th22; A1: [.0,1.] = dom c by BORSUK_1:40, FUNCT_2:def_1; 0 in [.0,1.] by XXREAL_1:1; then inf (dom c) in dom c by A1, Th4; then dom c is left_end by XXREAL_2:def_5; then A2: c is with_first_point by Def6; 1 in [.0,1.] by XXREAL_1:1; then sup (dom c) in dom c by A1, Th4; then dom c is right_end by XXREAL_2:def_6; then c is with_last_point by Def7; hence p is with_endpoints Curve of T by A2; ::_thesis: verum end; theorem Th26: :: TOPALG_6:26 for T being non empty TopStruct for c being Curve of T for r1, r2 being real number holds c | [.r1,r2.] is Curve of T proof let T be non empty TopStruct ; ::_thesis: for c being Curve of T for r1, r2 being real number holds c | [.r1,r2.] is Curve of T let c be Curve of T; ::_thesis: for r1, r2 being real number holds c | [.r1,r2.] is Curve of T let r1, r2 be real number ; ::_thesis: c | [.r1,r2.] is Curve of T reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23; set f1 = f | [.r1,r2.]; reconsider A = dom f as interval Subset of REAL by Def4; reconsider B = [.r1,r2.] as interval Subset of REAL ; A1: A /\ B is interval Subset of REAL ; then A2: dom (f | [.r1,r2.]) is interval Subset of REAL by RELAT_1:61; consider S being SubSpace of R^1 , g being Function of S,T such that A3: ( f = g & S = R^1 | (dom f) & g is continuous ) by Def4; reconsider A0 = dom f as Subset of R^1 ; A4: [#] S = A0 by A3, PRE_TOPC:def_5; reconsider K0 = (dom f) /\ [.r1,r2.] as Subset of S by A4, XBOOLE_1:17; reconsider g1 = g | K0 as Function of (S | K0),T by PRE_TOPC:9; A5: g1 is continuous by A3, TOPMETR:7; A6: g1 = (f | (dom f)) | [.r1,r2.] by A3, RELAT_1:71 .= f | [.r1,r2.] ; A7: (dom f) /\ [.r1,r2.] = dom (f | [.r1,r2.]) by RELAT_1:61; reconsider K1 = K0 as Subset of (R^1 | A0) by A3; reconsider D1 = dom (f | [.r1,r2.]) as Subset of R^1 by A1, RELAT_1:61, TOPMETR:17; S | K0 = R^1 | D1 by A3, A7, PRE_TOPC:7, XBOOLE_1:17; then reconsider f1 = f | [.r1,r2.] as parametrized-curve PartFunc of R^1,T by A2, A5, A6, Def4; c | [.r1,r2.] = f1 ; hence c | [.r1,r2.] is Curve of T by Th20; ::_thesis: verum end; theorem Th27: :: TOPALG_6:27 for T being non empty TopStruct for c being with_endpoints Curve of T holds dom c = [.(inf (dom c)),(sup (dom c)).] proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T holds dom c = [.(inf (dom c)),(sup (dom c)).] let c be with_endpoints Curve of T; ::_thesis: dom c = [.(inf (dom c)),(sup (dom c)).] reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23; dom f is interval Subset of REAL by Def4; then reconsider A = dom c as ext-real-membered left_end right_end interval set by Def6, Def7; A = [.(min A),(max A).] by XXREAL_2:75; hence dom c = [.(inf (dom c)),(sup (dom c)).] ; ::_thesis: verum end; theorem Th28: :: TOPALG_6:28 for T being non empty TopStruct for c being with_endpoints Curve of T st dom c = [.0,1.] holds c is Path of the_first_point_of c, the_last_point_of c proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T st dom c = [.0,1.] holds c is Path of the_first_point_of c, the_last_point_of c let c be with_endpoints Curve of T; ::_thesis: ( dom c = [.0,1.] implies c is Path of the_first_point_of c, the_last_point_of c ) assume A1: dom c = [.0,1.] ; ::_thesis: c is Path of the_first_point_of c, the_last_point_of c set t1 = the_first_point_of c; set t2 = the_last_point_of c; reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23; consider S being SubSpace of R^1 , p being Function of S,T such that A2: ( f = p & S = R^1 | (dom f) & p is continuous ) by Def4; reconsider p = p as Function of I[01],T by A2, A1, BORSUK_1:40, FUNCT_2:def_1; A3: S = I[01] by A2, A1, TOPMETR:19, TOPMETR:20; A4: p . 0 = the_first_point_of c by A1, A2, XXREAL_2:25; A5: p . 1 = the_last_point_of c by A2, A1, XXREAL_2:29; then the_first_point_of c, the_last_point_of c are_connected by A2, A3, A4, BORSUK_2:def_1; hence c is Path of the_first_point_of c, the_last_point_of c by A3, A4, A5, A2, BORSUK_2:def_2; ::_thesis: verum end; theorem Th29: :: TOPALG_6:29 for T being non empty TopStruct for c being with_endpoints Curve of T holds c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of the_first_point_of c, the_last_point_of c proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T holds c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of the_first_point_of c, the_last_point_of c let c be with_endpoints Curve of T; ::_thesis: c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of the_first_point_of c, the_last_point_of c set t1 = the_first_point_of c; set t2 = the_last_point_of c; reconsider c0 = c as parametrized-curve PartFunc of R^1,T by Th23; consider S being SubSpace of R^1 , g being Function of S,T such that A1: ( c0 = g & S = R^1 | (dom c0) & g is continuous ) by Def4; reconsider S = S as non empty TopStruct by A1; A2: inf (dom c) <= sup (dom c) by XXREAL_2:40; then A3: L[01] (0,1,(inf (dom c)),(sup (dom c))) is continuous Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((inf (dom c)),(sup (dom c)))) by BORSUK_6:34; A4: dom c0 = [.(inf (dom c)),(sup (dom c)).] by Th27; then A5: Closed-Interval-TSpace ((inf (dom c)),(sup (dom c))) = S by A2, A1, TOPMETR:19; reconsider f = L[01] (0,1,(inf (dom c)),(sup (dom c))) as Function of I[01],S by A4, A2, A1, TOPMETR:19, TOPMETR:20; reconsider p = g * f as Function of I[01],T ; A6: ( 0 in [.0,1.] & 1 in [.0,1.] ) by XXREAL_1:1; A7: dom (L[01] (0,1,(inf (dom c)),(sup (dom c)))) = the carrier of (Closed-Interval-TSpace (0,1)) by FUNCT_2:def_1 .= [.0,1.] by TOPMETR:18 ; A8: (L[01] (0,1,(inf (dom c)),(sup (dom c)))) . 0 = ((((sup (dom c)) - (inf (dom c))) / (1 - 0)) * (0 - 0)) + (inf (dom c)) by A2, BORSUK_6:35 .= inf (dom c) ; A9: (L[01] (0,1,(inf (dom c)),(sup (dom c)))) . 1 = ((((sup (dom c)) - (inf (dom c))) / (1 - 0)) * (1 - 0)) + (inf (dom c)) by A2, BORSUK_6:35 .= sup (dom c) ; A10: p is continuous by A1, A3, A5, TOPMETR:20, TOPS_2:46; A11: p . 0 = the_first_point_of c by A8, A1, A6, A7, FUNCT_1:13; A12: p . 1 = the_last_point_of c by A9, A1, A6, A7, FUNCT_1:13; then the_first_point_of c, the_last_point_of c are_connected by A10, A11, BORSUK_2:def_1; hence c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of the_first_point_of c, the_last_point_of c by A1, A10, A11, A12, BORSUK_2:def_2; ::_thesis: verum end; theorem :: TOPALG_6:30 for T being non empty TopStruct for c being with_endpoints Curve of T for t1, t2 being Point of T st c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected holds ( t1 = the_first_point_of c & t2 = the_last_point_of c ) proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T for t1, t2 being Point of T st c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected holds ( t1 = the_first_point_of c & t2 = the_last_point_of c ) let c be with_endpoints Curve of T; ::_thesis: for t1, t2 being Point of T st c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected holds ( t1 = the_first_point_of c & t2 = the_last_point_of c ) let t1, t2 be Point of T; ::_thesis: ( c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected implies ( t1 = the_first_point_of c & t2 = the_last_point_of c ) ) assume A1: ( c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) is Path of t1,t2 & t1,t2 are_connected ) ; ::_thesis: ( t1 = the_first_point_of c & t2 = the_last_point_of c ) A2: inf (dom c) <= sup (dom c) by XXREAL_2:40; A3: ( 0 in [.0,1.] & 1 in [.0,1.] ) by XXREAL_1:1; A4: dom (L[01] (0,1,(inf (dom c)),(sup (dom c)))) = the carrier of (Closed-Interval-TSpace (0,1)) by FUNCT_2:def_1 .= [.0,1.] by TOPMETR:18 ; A5: (L[01] (0,1,(inf (dom c)),(sup (dom c)))) . 0 = ((((sup (dom c)) - (inf (dom c))) / (1 - 0)) * (0 - 0)) + (inf (dom c)) by A2, BORSUK_6:35 .= inf (dom c) ; A6: (L[01] (0,1,(inf (dom c)),(sup (dom c)))) . 1 = ((((sup (dom c)) - (inf (dom c))) / (1 - 0)) * (1 - 0)) + (inf (dom c)) by A2, BORSUK_6:35 .= sup (dom c) ; reconsider p = c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) as Path of t1,t2 by A1; A7: p . 0 = the_first_point_of c by A5, A3, A4, FUNCT_1:13; p . 1 = the_last_point_of c by A6, A3, A4, FUNCT_1:13; hence ( t1 = the_first_point_of c & t2 = the_last_point_of c ) by A7, A1, BORSUK_2:def_2; ::_thesis: verum end; theorem Th31: :: TOPALG_6:31 for T being non empty TopStruct for c being with_endpoints Curve of T holds ( the_first_point_of c in rng c & the_last_point_of c in rng c ) proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T holds ( the_first_point_of c in rng c & the_last_point_of c in rng c ) let c be with_endpoints Curve of T; ::_thesis: ( the_first_point_of c in rng c & the_last_point_of c in rng c ) A1: inf (dom c) <= sup (dom c) by XXREAL_2:40; dom c = [.(inf (dom c)),(sup (dom c)).] by Th27; then ( inf (dom c) in dom c & sup (dom c) in dom c ) by A1, XXREAL_1:1; hence ( the_first_point_of c in rng c & the_last_point_of c in rng c ) by FUNCT_1:3; ::_thesis: verum end; theorem Th32: :: TOPALG_6:32 for T being non empty TopStruct for r1, r2 being real number for t1, t2 being Point of T for p1 being Path of t1,t2 st t1,t2 are_connected & r1 < r2 holds p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T proof let T be non empty TopStruct ; ::_thesis: for r1, r2 being real number for t1, t2 being Point of T for p1 being Path of t1,t2 st t1,t2 are_connected & r1 < r2 holds p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T let r1, r2 be real number ; ::_thesis: for t1, t2 being Point of T for p1 being Path of t1,t2 st t1,t2 are_connected & r1 < r2 holds p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T let t1, t2 be Point of T; ::_thesis: for p1 being Path of t1,t2 st t1,t2 are_connected & r1 < r2 holds p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T let p1 be Path of t1,t2; ::_thesis: ( t1,t2 are_connected & r1 < r2 implies p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T ) assume A1: t1,t2 are_connected ; ::_thesis: ( not r1 < r2 or p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T ) assume A2: r1 < r2 ; ::_thesis: p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T then A3: L[01] (r1,r2,0,1) is continuous Function of (Closed-Interval-TSpace (r1,r2)),(Closed-Interval-TSpace (0,1)) by BORSUK_6:34; A4: ( p1 is continuous & p1 . 0 = t1 & p1 . 1 = t2 ) by A1, BORSUK_2:def_2; set c = p1 * (L[01] (r1,r2,0,1)); rng (L[01] (r1,r2,0,1)) c= [#] (Closed-Interval-TSpace (0,1)) by RELAT_1:def_19; then rng (L[01] (r1,r2,0,1)) c= dom p1 by FUNCT_2:def_1, TOPMETR:20; then dom (p1 * (L[01] (r1,r2,0,1))) = dom (L[01] (r1,r2,0,1)) by RELAT_1:27; then dom (p1 * (L[01] (r1,r2,0,1))) = [#] (Closed-Interval-TSpace (r1,r2)) by FUNCT_2:def_1; then A5: dom (p1 * (L[01] (r1,r2,0,1))) = [.r1,r2.] by A2, TOPMETR:18; A6: rng (p1 * (L[01] (r1,r2,0,1))) c= [#] T ; then reconsider c = p1 * (L[01] (r1,r2,0,1)) as PartFunc of R^1,T by A5, RELSET_1:4, TOPMETR:17; set S = R^1 | (dom c); dom c = [#] (R^1 | (dom c)) by PRE_TOPC:def_5; then reconsider g = c as Function of (R^1 | (dom c)),T by A6, FUNCT_2:2; A7: R^1 | (dom c) = Closed-Interval-TSpace (r1,r2) by A2, A5, TOPMETR:19; reconsider p2 = p1 as Function of (Closed-Interval-TSpace (0,1)),T by TOPMETR:20; g is continuous by A4, A7, A3, TOPMETR:20, TOPS_2:46; then c is parametrized-curve by A5, Def4; then reconsider c = c as Curve of T by Th20; ( dom c is left_end & dom c is right_end ) by A2, A5, XXREAL_2:33; then ( c is with_first_point & c is with_last_point ) by Def6, Def7; hence p1 * (L[01] (r1,r2,0,1)) is with_endpoints Curve of T ; ::_thesis: verum end; theorem Th33: :: TOPALG_6:33 for T being non empty TopStruct for c being with_endpoints Curve of T holds the_first_point_of c, the_last_point_of c are_connected proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T holds the_first_point_of c, the_last_point_of c are_connected let c be with_endpoints Curve of T; ::_thesis: the_first_point_of c, the_last_point_of c are_connected set t1 = the_first_point_of c; set t2 = the_last_point_of c; reconsider f = c as parametrized-curve PartFunc of R^1,T by Th23; consider S being SubSpace of R^1 , g being Function of S,T such that A1: ( f = g & S = R^1 | (dom f) & g is continuous ) by Def4; set r1 = inf (dom c); set r2 = sup (dom c); set p = g * (L[01] (0,1,(inf (dom c)),(sup (dom c)))); A2: inf (dom c) <= sup (dom c) by XXREAL_2:40; then A3: L[01] (0,1,(inf (dom c)),(sup (dom c))) is continuous Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((inf (dom c)),(sup (dom c)))) by BORSUK_6:34; rng (L[01] (0,1,(inf (dom c)),(sup (dom c)))) c= [#] (Closed-Interval-TSpace ((inf (dom c)),(sup (dom c)))) by RELAT_1:def_19; then rng (L[01] (0,1,(inf (dom c)),(sup (dom c)))) c= [.(inf (dom c)),(sup (dom c)).] by A2, TOPMETR:18; then rng (L[01] (0,1,(inf (dom c)),(sup (dom c)))) c= dom c by Th27; then dom (g * (L[01] (0,1,(inf (dom c)),(sup (dom c))))) = dom (L[01] (0,1,(inf (dom c)),(sup (dom c)))) by A1, RELAT_1:27; then A4: dom (g * (L[01] (0,1,(inf (dom c)),(sup (dom c))))) = [#] (Closed-Interval-TSpace (0,1)) by FUNCT_2:def_1; rng (g * (L[01] (0,1,(inf (dom c)),(sup (dom c))))) c= [#] T ; then reconsider p = g * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) as Function of I[01],T by A4, FUNCT_2:2, TOPMETR:20; dom f = [.(inf (dom c)),(sup (dom c)).] by Th27; then S = Closed-Interval-TSpace ((inf (dom c)),(sup (dom c))) by A1, A2, TOPMETR:19; then A5: p is continuous by A1, A3, TOPMETR:20, TOPS_2:46; dom p = [.0,1.] by A4, TOPMETR:18; then A6: ( 0 in dom p & 1 in dom p ) by XXREAL_1:1; A7: (L[01] (0,1,(inf (dom c)),(sup (dom c)))) . 0 = ((((sup (dom c)) - (inf (dom c))) / (1 - 0)) * (0 - 0)) + (inf (dom c)) by A2, BORSUK_6:35 .= inf (dom c) ; A8: (L[01] (0,1,(inf (dom c)),(sup (dom c)))) . 1 = ((((sup (dom c)) - (inf (dom c))) / (1 - 0)) * (1 - 0)) + (inf (dom c)) by A2, BORSUK_6:35 .= sup (dom c) ; A9: p . 0 = the_first_point_of c by A1, A7, A6, FUNCT_1:12; p . 1 = the_last_point_of c by A1, A8, A6, FUNCT_1:12; hence the_first_point_of c, the_last_point_of c are_connected by A5, A9, BORSUK_2:def_1; ::_thesis: verum end; definition let T be non empty TopStruct ; let c1, c2 be with_endpoints Curve of T; predc1,c2 are_homotopic means :Def11: :: TOPALG_6:def 11 ex a, b being Point of T ex p1, p2 being Path of a,b st ( p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p1,p2 are_homotopic ); symmetry for c1, c2 being with_endpoints Curve of T st ex a, b being Point of T ex p1, p2 being Path of a,b st ( p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p1,p2 are_homotopic ) holds ex a, b being Point of T ex p1, p2 being Path of a,b st ( p1 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p2 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p1,p2 are_homotopic ) ; end; :: deftheorem Def11 defines are_homotopic TOPALG_6:def_11_:_ for T being non empty TopStruct for c1, c2 being with_endpoints Curve of T holds ( c1,c2 are_homotopic iff ex a, b being Point of T ex p1, p2 being Path of a,b st ( p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p1,p2 are_homotopic ) ); definition let T be non empty TopSpace; let c1, c2 be with_endpoints Curve of T; :: original: are_homotopic redefine predc1,c2 are_homotopic ; reflexivity for c1 being with_endpoints Curve of T holds (T,b1,b1) proof let c be with_endpoints Curve of T; ::_thesis: (T,c,c) reconsider p = c * (L[01] (0,1,(inf (dom c)),(sup (dom c)))) as Path of the_first_point_of c, the_last_point_of c by Th29; p,p are_homotopic by Th33, BORSUK_2:12; hence (T,c,c) by Def11; ::_thesis: verum end; symmetry for c1, c2 being with_endpoints Curve of T st (T,b1,b2) holds (T,b2,b1) ; end; theorem Th34: :: TOPALG_6:34 for T being non empty TopStruct for c1, c2 being with_endpoints Curve of T for a, b being Point of T for p1, p2 being Path of a,b st c1 = p1 & c2 = p2 & a,b are_connected holds ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) proof let T be non empty TopStruct ; ::_thesis: for c1, c2 being with_endpoints Curve of T for a, b being Point of T for p1, p2 being Path of a,b st c1 = p1 & c2 = p2 & a,b are_connected holds ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) let c1, c2 be with_endpoints Curve of T; ::_thesis: for a, b being Point of T for p1, p2 being Path of a,b st c1 = p1 & c2 = p2 & a,b are_connected holds ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) let a, b be Point of T; ::_thesis: for p1, p2 being Path of a,b st c1 = p1 & c2 = p2 & a,b are_connected holds ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) let p1, p2 be Path of a,b; ::_thesis: ( c1 = p1 & c2 = p2 & a,b are_connected implies ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) ) assume A1: ( c1 = p1 & c2 = p2 ) ; ::_thesis: ( not a,b are_connected or ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) ) assume A2: a,b are_connected ; ::_thesis: ( c1,c2 are_homotopic iff p1,p2 are_homotopic ) A3: ( 0 is Point of I[01] & 1 is Point of I[01] ) by BORSUK_1:40, XXREAL_1:1; A4: ( inf (dom c1) = 0 & sup (dom c1) = 1 & inf (dom c2) = 0 & sup (dom c2) = 1 ) by A1, Th4; A5: ( dom p1 = the carrier of I[01] & dom p2 = the carrier of I[01] ) by FUNCT_2:def_1; thus ( c1,c2 are_homotopic implies p1,p2 are_homotopic ) ::_thesis: ( p1,p2 are_homotopic implies c1,c2 are_homotopic ) proof assume c1,c2 are_homotopic ; ::_thesis: p1,p2 are_homotopic then consider aa, bb being Point of T, pp1, pp2 being Path of aa,bb such that A6: ( pp1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & pp2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & pp1,pp2 are_homotopic ) by Def11; consider f being Function of [:I[01],I[01]:],T such that A7: ( f is continuous & ( for t being Point of I[01] holds ( f . (t,0) = pp1 . t & f . (t,1) = pp2 . t & f . (0,t) = aa & f . (1,t) = bb ) ) ) by A6, BORSUK_2:def_7; A8: ( pp1 = p1 & pp2 = p2 ) by A1, A6, A4, A5, Th1, RELAT_1:52, TOPMETR:20; A9: ( f . (0,0) = pp1 . 0 & f . (0,1) = pp2 . 0 & f . (0,0) = aa & f . (0,1) = aa ) by A7, A3; A10: ( f . (1,0) = pp1 . 1 & f . (1,1) = pp2 . 1 & f . (1,0) = bb & f . (1,1) = bb ) by A7, A3; ( aa = a & bb = b ) by A8, A9, A10, A2, BORSUK_2:def_2; hence p1,p2 are_homotopic by A7, A8, BORSUK_2:def_7; ::_thesis: verum end; assume A11: p1,p2 are_homotopic ; ::_thesis: c1,c2 are_homotopic ( p1 = p1 * (L[01] (0,1,0,1)) & p2 = p2 * (L[01] (0,1,0,1)) ) by A5, Th1, RELAT_1:52, TOPMETR:20; hence c1,c2 are_homotopic by A4, A1, A11, Def11; ::_thesis: verum end; theorem Th35: :: TOPALG_6:35 for T being non empty TopStruct for c1, c2 being with_endpoints Curve of T st c1,c2 are_homotopic holds ( the_first_point_of c1 = the_first_point_of c2 & the_last_point_of c1 = the_last_point_of c2 ) proof let T be non empty TopStruct ; ::_thesis: for c1, c2 being with_endpoints Curve of T st c1,c2 are_homotopic holds ( the_first_point_of c1 = the_first_point_of c2 & the_last_point_of c1 = the_last_point_of c2 ) let c1, c2 be with_endpoints Curve of T; ::_thesis: ( c1,c2 are_homotopic implies ( the_first_point_of c1 = the_first_point_of c2 & the_last_point_of c1 = the_last_point_of c2 ) ) assume c1,c2 are_homotopic ; ::_thesis: ( the_first_point_of c1 = the_first_point_of c2 & the_last_point_of c1 = the_last_point_of c2 ) then consider a, b being Point of T, p1, p2 being Path of a,b such that A1: ( p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p1,p2 are_homotopic ) by Def11; A2: ( 0 is Point of I[01] & 1 is Point of I[01] ) by BORSUK_1:40, XXREAL_1:1; consider f being Function of [:I[01],I[01]:],T such that A3: ( f is continuous & ( for t being Point of I[01] holds ( f . (t,0) = p1 . t & f . (t,1) = p2 . t & f . (0,t) = a & f . (1,t) = b ) ) ) by A1, BORSUK_2:def_7; A4: ( f . (0,0) = p1 . 0 & f . (0,1) = p2 . 0 & f . (0,0) = a & f . (0,1) = a ) by A3, A2; A5: ( f . (1,0) = p1 . 1 & f . (1,1) = p2 . 1 & f . (1,0) = b & f . (1,1) = b ) by A3, A2; A6: ( 0 in [.0,1.] & 1 in [.0,1.] ) by XXREAL_1:1; A7: dom (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) = the carrier of (Closed-Interval-TSpace (0,1)) by FUNCT_2:def_1 .= [.0,1.] by TOPMETR:18 ; A8: dom (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) = the carrier of (Closed-Interval-TSpace (0,1)) by FUNCT_2:def_1 .= [.0,1.] by TOPMETR:18 ; A9: inf (dom c1) <= sup (dom c1) by XXREAL_2:40; A10: inf (dom c2) <= sup (dom c2) by XXREAL_2:40; A11: (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) . 0 = ((((sup (dom c2)) - (inf (dom c2))) / (1 - 0)) * (0 - 0)) + (inf (dom c2)) by A10, BORSUK_6:35 .= inf (dom c2) ; (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) . 0 = ((((sup (dom c1)) - (inf (dom c1))) / (1 - 0)) * (0 - 0)) + (inf (dom c1)) by A9, BORSUK_6:35 .= inf (dom c1) ; then p1 . 0 = c1 . (inf (dom c1)) by A1, A6, A7, FUNCT_1:13; hence the_first_point_of c1 = the_first_point_of c2 by A4, A1, A11, A6, A8, FUNCT_1:13; ::_thesis: the_last_point_of c1 = the_last_point_of c2 A12: (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) . 1 = ((((sup (dom c2)) - (inf (dom c2))) / (1 - 0)) * (1 - 0)) + (inf (dom c2)) by A10, BORSUK_6:35 .= sup (dom c2) ; (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) . 1 = ((((sup (dom c1)) - (inf (dom c1))) / (1 - 0)) * (1 - 0)) + (inf (dom c1)) by A9, BORSUK_6:35 .= sup (dom c1) ; then p1 . 1 = c1 . (sup (dom c1)) by A1, A6, A7, FUNCT_1:13; hence the_last_point_of c1 = the_last_point_of c2 by A5, A1, A12, A6, A8, FUNCT_1:13; ::_thesis: verum end; theorem Th36: :: TOPALG_6:36 for T being non empty TopSpace for c1, c2 being with_endpoints Curve of T for S being Subset of R^1 st dom c1 = dom c2 & S = dom c1 holds ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ) proof let T be non empty TopSpace; ::_thesis: for c1, c2 being with_endpoints Curve of T for S being Subset of R^1 st dom c1 = dom c2 & S = dom c1 holds ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ) let c1, c2 be with_endpoints Curve of T; ::_thesis: for S being Subset of R^1 st dom c1 = dom c2 & S = dom c1 holds ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ) let S be Subset of R^1; ::_thesis: ( dom c1 = dom c2 & S = dom c1 implies ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ) ) assume A1: ( dom c1 = dom c2 & S = dom c1 ) ; ::_thesis: ( c1,c2 are_homotopic iff ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ) A2: inf (dom c1) <= sup (dom c1) by XXREAL_2:40; A3: S = [.(inf (dom c1)),(sup (dom c1)).] by A1, Th27; A4: 0 in [#] I[01] by BORSUK_1:40, XXREAL_1:1; A5: 1 in [#] I[01] by BORSUK_1:40, XXREAL_1:1; A6: inf S in S by A3, A2, A1, XXREAL_1:1; A7: sup S in S by A3, A2, A1, XXREAL_1:1; thus ( c1,c2 are_homotopic implies ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ) ::_thesis: ( ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) implies c1,c2 are_homotopic ) proof assume A8: c1,c2 are_homotopic ; ::_thesis: ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) percases ( inf (dom c1) < sup (dom c1) or not inf (dom c1) < sup (dom c1) ) ; supposeA9: inf (dom c1) < sup (dom c1) ; ::_thesis: ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) consider a, b being Point of T, p1, p2 being Path of a,b such that A10: ( p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) & p2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) & p1,p2 are_homotopic ) by A8, Def11; consider f being Function of [:I[01],I[01]:],T such that A11: ( f is continuous & ( for t being Point of I[01] holds ( f . (t,0) = p1 . t & f . (t,1) = p2 . t & f . (0,t) = a & f . (1,t) = b ) ) ) by A10, BORSUK_2:def_7; set f1 = L[01] ((inf (dom c1)),(sup (dom c1)),0,1); set f2 = L[01] (0,1,0,1); reconsider S1 = R^1 | S as non empty TopSpace by A1; A12: Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1))) = S1 by A3, A9, TOPMETR:19; reconsider f1 = L[01] ((inf (dom c1)),(sup (dom c1)),0,1) as continuous Function of S1,I[01] by A9, A12, BORSUK_6:34, TOPMETR:20; reconsider f2 = L[01] (0,1,0,1) as continuous Function of I[01],I[01] by BORSUK_6:34, TOPMETR:20; set h = f * [:f1,f2:]; reconsider h = f * [:f1,f2:] as Function of [:(R^1 | S),I[01]:],T ; take h ; ::_thesis: ex a, b being Point of T st ( h is continuous & ( for t being Point of (R^1 | S) holds ( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) ) take a ; ::_thesis: ex b being Point of T st ( h is continuous & ( for t being Point of (R^1 | S) holds ( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) ) take b ; ::_thesis: ( h is continuous & ( for t being Point of (R^1 | S) holds ( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) ) thus h is continuous by A11; ::_thesis: ( ( for t being Point of (R^1 | S) holds ( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( h . ((inf S),t) = a & h . ((sup S),t) = b ) ) ) A13: dom f1 = [#] (R^1 | S) by FUNCT_2:def_1; A14: dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; A15: f2 . 0 = (((1 - 0) / (1 - 0)) * (0 - 0)) + 0 by BORSUK_6:35 .= 0 ; A16: f2 . 1 = (((1 - 0) / (1 - 0)) * (1 - 0)) + 0 by BORSUK_6:35 .= 1 ; A17: rng f1 c= [#] I[01] by RELAT_1:def_19; A18: 0 in dom f2 by A14, XXREAL_1:1; A19: 1 in dom f2 by A14, XXREAL_1:1; A20: (sup (dom c1)) - (inf (dom c1)) <> 0 by A9; thus for t being Point of (R^1 | S) holds ( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) ::_thesis: for t being Point of I[01] holds ( h . ((inf S),t) = a & h . ((sup S),t) = b ) proof let t be Point of (R^1 | S); ::_thesis: ( h . (t,0) = c1 . t & h . (t,1) = c2 . t ) A21: t in dom f1 by A13; [t,0] in [:(dom f1),(dom f2):] by A13, A18, ZFMISC_1:def_2; then A22: [t,0] in dom [:f1,f2:] by FUNCT_3:def_8; [t,1] in [:(dom f1),(dom f2):] by A13, A19, ZFMISC_1:def_2; then A23: [t,1] in dom [:f1,f2:] by FUNCT_3:def_8; A24: f1 . t in rng f1 by A13, FUNCT_1:3; A25: t in S by A21, A13, PRE_TOPC:def_5; t in [#] (Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1)))) by A12; then A26: t in dom (L[01] ((inf (dom c1)),(sup (dom c1)),(inf (dom c1)),(sup (dom c1)))) by FUNCT_2:def_1; A27: ( inf (dom c1) <= t & t <= sup (dom c1) ) by A25, A3, XXREAL_1:1; A28: (L[01] ((inf (dom c1)),(sup (dom c1)),(inf (dom c1)),(sup (dom c1)))) . t = ((((sup (dom c1)) - (inf (dom c1))) / ((sup (dom c1)) - (inf (dom c1)))) * (t - (inf (dom c1)))) + (inf (dom c1)) by A27, A9, BORSUK_6:35 .= (1 * (t - (inf (dom c1)))) + (inf (dom c1)) by A20, XCMPLX_1:60 .= t ; thus h . (t,0) = h . [t,0] by BINOP_1:def_1 .= f . ([:f1,f2:] . [t,0]) by A22, FUNCT_1:13 .= f . ([:f1,f2:] . (t,0)) by BINOP_1:def_1 .= f . [(f1 . t),(f2 . 0)] by A13, A18, FUNCT_3:def_8 .= f . ((f1 . t),0) by A15, BINOP_1:def_1 .= p1 . (f1 . t) by A24, A11, A17 .= ((c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1))))) * f1) . t by A10, A13, FUNCT_1:13 .= (c1 * ((L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) * f1)) . t by RELAT_1:36 .= (c1 * (L[01] ((inf (dom c1)),(sup (dom c1)),(inf (dom c1)),(sup (dom c1))))) . t by Th2, A9 .= c1 . t by A28, A26, FUNCT_1:13 ; ::_thesis: h . (t,1) = c2 . t thus h . (t,1) = h . [t,1] by BINOP_1:def_1 .= f . ([:f1,f2:] . [t,1]) by A23, FUNCT_1:13 .= f . ([:f1,f2:] . (t,1)) by BINOP_1:def_1 .= f . [(f1 . t),(f2 . 1)] by A13, A19, FUNCT_3:def_8 .= f . ((f1 . t),1) by A16, BINOP_1:def_1 .= p2 . (f1 . t) by A24, A11, A17 .= ((c2 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1))))) * f1) . t by A10, A1, A13, FUNCT_1:13 .= (c2 * ((L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) * f1)) . t by RELAT_1:36 .= (c2 * (L[01] ((inf (dom c1)),(sup (dom c1)),(inf (dom c1)),(sup (dom c1))))) . t by Th2, A9 .= c2 . t by A28, A26, FUNCT_1:13 ; ::_thesis: verum end; thus for t being Point of I[01] holds ( h . ((inf S),t) = a & h . ((sup S),t) = b ) ::_thesis: verum proof let t be Point of I[01]; ::_thesis: ( h . ((inf S),t) = a & h . ((sup S),t) = b ) A29: inf S in dom f1 by A6, A13, PRE_TOPC:def_5; A30: sup S in dom f1 by A7, A13, PRE_TOPC:def_5; t in [#] I[01] ; then A31: t in dom f2 by FUNCT_2:def_1; [(inf S),t] in [:(dom f1),(dom f2):] by A31, A29, ZFMISC_1:def_2; then A32: [(inf S),t] in dom [:f1,f2:] by FUNCT_3:def_8; [(sup S),t] in [:(dom f1),(dom f2):] by A31, A30, ZFMISC_1:def_2; then A33: [(sup S),t] in dom [:f1,f2:] by FUNCT_3:def_8; ( 0 <= t & t <= 1 ) by BORSUK_1:40, XXREAL_1:1; then A34: f2 . t = (((1 - 0) / (1 - 0)) * (t - 0)) + 0 by BORSUK_6:35 .= t ; A35: f1 . (inf S) = (((1 - 0) / ((sup (dom c1)) - (inf (dom c1)))) * ((inf (dom c1)) - (inf (dom c1)))) + 0 by A1, A9, BORSUK_6:35 .= 0 ; A36: f1 . (sup S) = (((1 - 0) / ((sup (dom c1)) - (inf (dom c1)))) * ((sup (dom c1)) - (inf (dom c1)))) + 0 by A1, A9, BORSUK_6:35 .= (((sup (dom c1)) - (inf (dom c1))) / ((sup (dom c1)) - (inf (dom c1)))) * 1 by XCMPLX_1:75 .= 1 by A20, XCMPLX_1:60 ; thus h . ((inf S),t) = h . [(inf S),t] by BINOP_1:def_1 .= f . ([:f1,f2:] . [(inf S),t]) by A32, FUNCT_1:13 .= f . ([:f1,f2:] . ((inf S),t)) by BINOP_1:def_1 .= f . [(f1 . (inf S)),(f2 . t)] by A31, A29, FUNCT_3:def_8 .= f . ((f1 . (inf S)),t) by A34, BINOP_1:def_1 .= a by A11, A35 ; ::_thesis: h . ((sup S),t) = b thus h . ((sup S),t) = h . [(sup S),t] by BINOP_1:def_1 .= f . ([:f1,f2:] . [(sup S),t]) by A33, FUNCT_1:13 .= f . ([:f1,f2:] . ((sup S),t)) by BINOP_1:def_1 .= f . [(f1 . (sup S)),(f2 . t)] by A31, A30, FUNCT_3:def_8 .= f . ((f1 . (sup S)),t) by A34, BINOP_1:def_1 .= b by A11, A36 ; ::_thesis: verum end; end; suppose not inf (dom c1) < sup (dom c1) ; ::_thesis: ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) then inf (dom c1) = sup (dom c1) by A2, XXREAL_0:1; then dom c1 = [.(inf (dom c1)),(inf (dom c1)).] by Th27; then A37: dom c1 = {(inf (dom c1))} by XXREAL_1:17; set a = the_first_point_of c1; set f = [:(R^1 | S),I[01]:] --> (the_first_point_of c1); A38: for t being Point of (R^1 | S) holds ( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,0) = c1 . t & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = c2 . t ) proof let t be Point of (R^1 | S); ::_thesis: ( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,0) = c1 . t & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = c2 . t ) A39: t in [#] (R^1 | S) by A1, SUBSET_1:def_1; A40: [t,0] in [:([#] (R^1 | S)),([#] I[01]):] by A4, A1, ZFMISC_1:def_2; A41: [t,1] in [:([#] (R^1 | S)),([#] I[01]):] by A5, A1, ZFMISC_1:def_2; A42: t in S by A39, PRE_TOPC:def_5; then A43: t = inf (dom c1) by A1, A37, TARSKI:def_1; thus ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,0) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [t,0] by BINOP_1:def_1 .= c1 . t by A43, A40, FUNCOP_1:7 ; ::_thesis: ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = c2 . t thus ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . (t,1) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [t,1] by BINOP_1:def_1 .= the_first_point_of c1 by A41, FUNCOP_1:7 .= the_first_point_of c2 by A8, Th35 .= c2 . t by A1, A42, A37, TARSKI:def_1 ; ::_thesis: verum end; for t being Point of I[01] holds ( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((inf S),t) = the_first_point_of c1 & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = the_first_point_of c1 ) proof let t be Point of I[01]; ::_thesis: ( ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((inf S),t) = the_first_point_of c1 & ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = the_first_point_of c1 ) A44: inf S in [#] (R^1 | S) by A6, PRE_TOPC:def_5; A45: sup S in [#] (R^1 | S) by A7, PRE_TOPC:def_5; A46: [(inf S),t] in [:([#] (R^1 | S)),([#] I[01]):] by A44, ZFMISC_1:def_2; A47: [(sup S),t] in [:([#] (R^1 | S)),([#] I[01]):] by A45, ZFMISC_1:def_2; thus ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((inf S),t) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [(inf S),t] by BINOP_1:def_1 .= the_first_point_of c1 by A46, FUNCOP_1:7 ; ::_thesis: ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = the_first_point_of c1 thus ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . ((sup S),t) = ([:(R^1 | S),I[01]:] --> (the_first_point_of c1)) . [(sup S),t] by BINOP_1:def_1 .= the_first_point_of c1 by A47, FUNCOP_1:7 ; ::_thesis: verum end; hence ex f being Function of [:(R^1 | S),I[01]:],T ex a, b being Point of T st ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) by A38; ::_thesis: verum end; end; end; given f being Function of [:(R^1 | S),I[01]:],T, a, b being Point of T such that A48: ( f is continuous & ( for t being Point of (R^1 | S) holds ( f . (t,0) = c1 . t & f . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( f . ((inf S),t) = a & f . ((sup S),t) = b ) ) ) ; ::_thesis: c1,c2 are_homotopic A49: inf S in [#] (R^1 | S) by A6, PRE_TOPC:def_5; A50: sup S in [#] (R^1 | S) by A7, PRE_TOPC:def_5; A51: a = f . ((inf S),0) by A4, A48 .= the_first_point_of c1 by A1, A49, A48 ; b = f . ((sup S),0) by A4, A48 .= the_last_point_of c1 by A1, A50, A48 ; then reconsider p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) as Path of a,b by A51, Th29; A52: a = f . ((inf S),1) by A5, A48 .= the_first_point_of c2 by A1, A49, A48 ; b = f . ((sup S),1) by A5, A48 .= the_last_point_of c2 by A1, A50, A48 ; then reconsider p2 = c2 * (L[01] (0,1,(inf (dom c2)),(sup (dom c2)))) as Path of a,b by A52, Th29; set f1 = L[01] (0,1,(inf (dom c1)),(sup (dom c1))); set f2 = L[01] (0,1,0,1); reconsider S1 = R^1 | S as non empty TopSpace by A1; A53: Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1))) = S1 by A3, A2, TOPMETR:19; reconsider f1 = L[01] (0,1,(inf (dom c1)),(sup (dom c1))) as continuous Function of I[01],S1 by A2, A53, BORSUK_6:34, TOPMETR:20; reconsider f2 = L[01] (0,1,0,1) as continuous Function of I[01],I[01] by BORSUK_6:34, TOPMETR:20; set h = f * [:f1,f2:]; reconsider h = f * [:f1,f2:] as Function of [:I[01],I[01]:],T ; A54: dom f1 = [#] I[01] by FUNCT_2:def_1; A55: dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1; A56: f2 . 0 = (((1 - 0) / (1 - 0)) * (0 - 0)) + 0 by BORSUK_6:35 .= 0 ; A57: f2 . 1 = (((1 - 0) / (1 - 0)) * (1 - 0)) + 0 by BORSUK_6:35 .= 1 ; A58: 0 in dom f2 by A55, XXREAL_1:1; A59: 1 in dom f2 by A55, XXREAL_1:1; for t being Point of I[01] holds ( h . (t,0) = p1 . t & h . (t,1) = p2 . t & h . (0,t) = a & h . (1,t) = b ) proof let t be Point of I[01]; ::_thesis: ( h . (t,0) = p1 . t & h . (t,1) = p2 . t & h . (0,t) = a & h . (1,t) = b ) [t,0] in [:(dom f1),(dom f2):] by A54, A58, ZFMISC_1:def_2; then A60: [t,0] in dom [:f1,f2:] by FUNCT_3:def_8; [t,1] in [:(dom f1),(dom f2):] by A54, A59, ZFMISC_1:def_2; then A61: [t,1] in dom [:f1,f2:] by FUNCT_3:def_8; A62: 0 in dom f1 by A54, BORSUK_1:40, XXREAL_1:1; A63: 1 in dom f1 by A54, BORSUK_1:40, XXREAL_1:1; [0,t] in [:(dom f1),(dom f2):] by A62, A55, BORSUK_1:40, ZFMISC_1:def_2; then A64: [0,t] in dom [:f1,f2:] by FUNCT_3:def_8; [1,t] in [:(dom f1),(dom f2):] by A63, A55, BORSUK_1:40, ZFMISC_1:def_2; then A65: [1,t] in dom [:f1,f2:] by FUNCT_3:def_8; A66: f1 . 0 = ((((sup (dom c1)) - (inf (dom c1))) / (1 - 0)) * (0 - 0)) + (inf (dom c1)) by A2, BORSUK_6:35 .= inf S by A1 ; A67: f1 . 1 = ((((sup (dom c1)) - (inf (dom c1))) / (1 - 0)) * (1 - 0)) + (inf (dom c1)) by A2, BORSUK_6:35 .= sup S by A1 ; ( 0 <= t & t <= 1 ) by BORSUK_1:40, XXREAL_1:1; then A68: f2 . t = (((1 - 0) / (1 - 0)) * (t - 0)) + 0 by BORSUK_6:35 .= t ; thus h . (t,0) = h . [t,0] by BINOP_1:def_1 .= f . ([:f1,f2:] . [t,0]) by A60, FUNCT_1:13 .= f . ([:f1,f2:] . (t,0)) by BINOP_1:def_1 .= f . [(f1 . t),(f2 . 0)] by A54, A58, FUNCT_3:def_8 .= f . ((f1 . t),0) by A56, BINOP_1:def_1 .= c1 . (f1 . t) by A48 .= p1 . t by A54, FUNCT_1:13 ; ::_thesis: ( h . (t,1) = p2 . t & h . (0,t) = a & h . (1,t) = b ) thus h . (t,1) = h . [t,1] by BINOP_1:def_1 .= f . ([:f1,f2:] . [t,1]) by A61, FUNCT_1:13 .= f . ([:f1,f2:] . (t,1)) by BINOP_1:def_1 .= f . [(f1 . t),(f2 . 1)] by A54, A59, FUNCT_3:def_8 .= f . ((f1 . t),1) by A57, BINOP_1:def_1 .= c2 . (f1 . t) by A48 .= p2 . t by A1, A54, FUNCT_1:13 ; ::_thesis: ( h . (0,t) = a & h . (1,t) = b ) thus h . (0,t) = h . [0,t] by BINOP_1:def_1 .= f . ([:f1,f2:] . [0,t]) by A64, FUNCT_1:13 .= f . ([:f1,f2:] . (0,t)) by BINOP_1:def_1 .= f . [(f1 . 0),(f2 . t)] by A62, A55, BORSUK_1:40, FUNCT_3:def_8 .= f . ((inf S),t) by A66, A68, BINOP_1:def_1 .= a by A48 ; ::_thesis: h . (1,t) = b thus h . (1,t) = h . [1,t] by BINOP_1:def_1 .= f . ([:f1,f2:] . [1,t]) by A65, FUNCT_1:13 .= f . ([:f1,f2:] . (1,t)) by BINOP_1:def_1 .= f . [(f1 . 1),(f2 . t)] by A63, A55, BORSUK_1:40, FUNCT_3:def_8 .= f . ((sup S),t) by A67, A68, BINOP_1:def_1 .= b by A48 ; ::_thesis: verum end; then p1,p2 are_homotopic by A48, BORSUK_2:def_7; hence c1,c2 are_homotopic by Def11; ::_thesis: verum end; definition let T be TopStruct ; let c1, c2 be Curve of T; funcc1 + c2 -> Curve of T equals :Def12: :: TOPALG_6:def 12 c1 \/ c2 if c1 \/ c2 is Curve of T otherwise {} ; correctness coherence ( ( c1 \/ c2 is Curve of T implies c1 \/ c2 is Curve of T ) & ( c1 \/ c2 is not Curve of T implies {} is Curve of T ) ); consistency for b1 being Curve of T holds verum; proof now__::_thesis:_(_c1_\/_c2_is_not_Curve_of_T_implies_{}_is_Curve_of_T_) assume c1 \/ c2 is not Curve of T ; ::_thesis: {} is Curve of T {} is parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2; hence {} is Curve of T by Th20; ::_thesis: verum end; hence ( ( c1 \/ c2 is Curve of T implies c1 \/ c2 is Curve of T ) & ( c1 \/ c2 is not Curve of T implies {} is Curve of T ) & ( for b1 being Curve of T holds verum ) ) ; ::_thesis: verum end; end; :: deftheorem Def12 defines + TOPALG_6:def_12_:_ for T being TopStruct for c1, c2 being Curve of T holds ( ( c1 \/ c2 is Curve of T implies c1 + c2 = c1 \/ c2 ) & ( c1 \/ c2 is not Curve of T implies c1 + c2 = {} ) ); theorem Th37: :: TOPALG_6:37 for T being non empty TopStruct for c being with_endpoints Curve of T for r being real number ex c1, c2 being Element of Curves T st ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T for r being real number ex c1, c2 being Element of Curves T st ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) let c be with_endpoints Curve of T; ::_thesis: for r being real number ex c1, c2 being Element of Curves T st ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) let r be real number ; ::_thesis: ex c1, c2 being Element of Curves T st ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) set c1 = c | [.(inf (dom c)),r.]; set c2 = c | [.r,(sup (dom c)).]; reconsider c1 = c | [.(inf (dom c)),r.] as Curve of T by Th26; reconsider c2 = c | [.r,(sup (dom c)).] as Curve of T by Th26; take c1 ; ::_thesis: ex c2 being Element of Curves T st ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) take c2 ; ::_thesis: ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) c1 \/ c2 = c proof percases ( r < inf (dom c) or r >= inf (dom c) ) ; supposeA1: r < inf (dom c) ; ::_thesis: c1 \/ c2 = c then [.(inf (dom c)),r.] = {} by XXREAL_1:29; then A2: c1 = {} ; [.(inf (dom c)),(sup (dom c)).] c= [.r,(sup (dom c)).] by A1, XXREAL_1:34; then dom c c= [.r,(sup (dom c)).] by Th27; hence c1 \/ c2 = c by A2, RELAT_1:68; ::_thesis: verum end; supposeA3: r >= inf (dom c) ; ::_thesis: c1 \/ c2 = c percases ( r > sup (dom c) or r <= sup (dom c) ) ; supposeA4: r > sup (dom c) ; ::_thesis: c1 \/ c2 = c then [.r,(sup (dom c)).] = {} by XXREAL_1:29; then A5: c2 = {} ; [.(inf (dom c)),(sup (dom c)).] c= [.(inf (dom c)),r.] by A4, XXREAL_1:34; then dom c c= [.(inf (dom c)),r.] by Th27; hence c1 \/ c2 = c by A5, RELAT_1:68; ::_thesis: verum end; supposeA6: r <= sup (dom c) ; ::_thesis: c1 \/ c2 = c dom c = [.(inf (dom c)),(sup (dom c)).] by Th27 .= [.(inf (dom c)),r.] \/ [.r,(sup (dom c)).] by A6, A3, XXREAL_1:165 ; then c | (dom c) = c1 \/ c2 by RELAT_1:78; hence c1 \/ c2 = c ; ::_thesis: verum end; end; end; end; end; hence ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r.] & c2 = c | [.r,(sup (dom c)).] ) by Def12; ::_thesis: verum end; theorem Th38: :: TOPALG_6:38 for T being non empty TopSpace for c1, c2 being with_endpoints Curve of T st sup (dom c1) = inf (dom c2) & the_last_point_of c1 = the_first_point_of c2 holds ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] & (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) proof let T be non empty TopSpace; ::_thesis: for c1, c2 being with_endpoints Curve of T st sup (dom c1) = inf (dom c2) & the_last_point_of c1 = the_first_point_of c2 holds ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] & (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) let c1, c2 be with_endpoints Curve of T; ::_thesis: ( sup (dom c1) = inf (dom c2) & the_last_point_of c1 = the_first_point_of c2 implies ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] & (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) ) assume A1: sup (dom c1) = inf (dom c2) ; ::_thesis: ( not the_last_point_of c1 = the_first_point_of c2 or ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] & (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) ) assume A2: the_last_point_of c1 = the_first_point_of c2 ; ::_thesis: ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] & (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) set f = c1 \/ c2; A3: dom (c1 \/ c2) = (dom c1) \/ (dom c2) by RELAT_1:1; reconsider f1 = c1 as parametrized-curve PartFunc of R^1,T by Th23; A4: dom f1 is interval Subset of REAL by Def4; reconsider f2 = c2 as parametrized-curve PartFunc of R^1,T by Th23; A5: dom f2 is interval Subset of REAL by Def4; A6: (dom c1) \/ (dom c2) c= REAL by A4, A5, XBOOLE_1:8; ( rng f1 c= [#] T & rng f2 c= [#] T ) ; then (rng c1) \/ (rng c2) c= [#] T by XBOOLE_1:8; then A7: rng (c1 \/ c2) c= [#] T by RELAT_1:12; A8: dom (c1 \/ c2) c= [#] R^1 by A6, RELAT_1:1, TOPMETR:17; reconsider S0 = dom (c1 \/ c2) as Subset of R^1 by A6, RELAT_1:1, TOPMETR:17; A9: inf (dom c2) <= sup (dom c2) by XXREAL_2:40; A10: inf (dom c1) <= sup (dom c1) by XXREAL_2:40; A11: dom f1 = [.(inf (dom c1)),(sup (dom c1)).] by Th27; A12: dom f2 = [.(inf (dom c2)),(sup (dom c2)).] by Th27; A13: (dom f1) /\ (dom f2) = {(sup (dom c1))} by A11, A12, A1, A9, A10, XXREAL_1:418; A14: (dom f1) /\ (dom f2) c= dom (c1 \/ c2) by A3, XBOOLE_1:29; set S = R^1 | S0; consider S1 being SubSpace of R^1 , g1 being Function of S1,T such that A15: ( f1 = g1 & S1 = R^1 | (dom f1) & g1 is continuous ) by Def4; consider S2 being SubSpace of R^1 , g2 being Function of S2,T such that A16: ( f2 = g2 & S2 = R^1 | (dom f2) & g2 is continuous ) by Def4; sup (dom c1) in dom (c1 \/ c2) by A13, A14, ZFMISC_1:31; then sup (dom c1) in [#] (R^1 | S0) by PRE_TOPC:def_5; then reconsider p = sup (dom c1) as Point of (R^1 | S0) ; reconsider S1 = S1, S2 = S2 as SubSpace of R^1 | S0 by A15, A16, A3, TOPMETR:22, XBOOLE_1:7; A17: ([#] S1) \/ ([#] S2) = (dom f1) \/ ([#] S2) by A15, PRE_TOPC:def_5 .= (dom f1) \/ (dom f2) by A16, PRE_TOPC:def_5 .= [#] (R^1 | S0) by A3, PRE_TOPC:def_5 ; A18: ([#] S1) /\ ([#] S2) = (dom f1) /\ ([#] S2) by A15, PRE_TOPC:def_5 .= {p} by A13, A16, PRE_TOPC:def_5 ; S1 = Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1))) by A11, A10, A15, TOPMETR:19; then A19: S1 is compact by A10, HEINE:4; S2 = Closed-Interval-TSpace ((inf (dom c2)),(sup (dom c2))) by A12, A9, A16, TOPMETR:19; then A20: S2 is compact by A9, HEINE:4; A21: R^1 | S0 is T_2 by TOPMETR:2; A22: g1 +* g2 is continuous Function of (R^1 | S0),T by A17, A18, A19, A20, A21, A15, A16, A1, A2, COMPTS_1:20; for x, y1, y2 being set st [x,y1] in c1 \/ c2 & [x,y2] in c1 \/ c2 holds y1 = y2 proof let x, y1, y2 be set ; ::_thesis: ( [x,y1] in c1 \/ c2 & [x,y2] in c1 \/ c2 implies y1 = y2 ) assume A23: ( [x,y1] in c1 \/ c2 & [x,y2] in c1 \/ c2 ) ; ::_thesis: y1 = y2 percases ( ( [x,y1] in c1 & [x,y2] in c1 ) or ( [x,y1] in c2 & [x,y2] in c2 ) or ( [x,y1] in c1 & [x,y2] in c2 ) or ( [x,y1] in c2 & [x,y2] in c1 ) ) by A23, XBOOLE_0:def_3; suppose ( [x,y1] in c1 & [x,y2] in c1 ) ; ::_thesis: y1 = y2 hence y1 = y2 by FUNCT_1:def_1; ::_thesis: verum end; suppose ( [x,y1] in c2 & [x,y2] in c2 ) ; ::_thesis: y1 = y2 hence y1 = y2 by FUNCT_1:def_1; ::_thesis: verum end; supposeA24: ( [x,y1] in c1 & [x,y2] in c2 ) ; ::_thesis: y1 = y2 then ( x in dom c1 & x in dom c2 ) by XTUPLE_0:def_12; then x in (dom c1) /\ (dom c2) by XBOOLE_0:def_4; then x = p by A13, TARSKI:def_1; then ( c1 . p = y1 & c2 . p = y2 ) by A24, FUNCT_1:1; hence y1 = y2 by A1, A2; ::_thesis: verum end; supposeA25: ( [x,y1] in c2 & [x,y2] in c1 ) ; ::_thesis: y1 = y2 then ( x in dom c2 & x in dom c1 ) by XTUPLE_0:def_12; then x in (dom c2) /\ (dom c1) by XBOOLE_0:def_4; then x = p by A13, TARSKI:def_1; then ( c2 . p = y1 & c1 . p = y2 ) by A25, FUNCT_1:1; hence y1 = y2 by A1, A2; ::_thesis: verum end; end; end; then reconsider f = c1 \/ c2 as Function by FUNCT_1:def_1; A26: dom f = (dom g1) \/ (dom g2) by A15, A16, RELAT_1:1; for x being set st x in (dom g1) \/ (dom g2) holds ( ( x in dom g2 implies f . x = g2 . x ) & ( not x in dom g2 implies f . x = g1 . x ) ) proof let x be set ; ::_thesis: ( x in (dom g1) \/ (dom g2) implies ( ( x in dom g2 implies f . x = g2 . x ) & ( not x in dom g2 implies f . x = g1 . x ) ) ) assume A27: x in (dom g1) \/ (dom g2) ; ::_thesis: ( ( x in dom g2 implies f . x = g2 . x ) & ( not x in dom g2 implies f . x = g1 . x ) ) thus ( x in dom g2 implies f . x = g2 . x ) ::_thesis: ( not x in dom g2 implies f . x = g1 . x ) proof assume x in dom g2 ; ::_thesis: f . x = g2 . x then [x,(g2 . x)] in g2 by FUNCT_1:1; then [x,(g2 . x)] in f by A16, XBOOLE_0:def_3; hence f . x = g2 . x by FUNCT_1:1; ::_thesis: verum end; thus ( not x in dom g2 implies f . x = g1 . x ) ::_thesis: verum proof assume not x in dom g2 ; ::_thesis: f . x = g1 . x then x in dom g1 by A27, XBOOLE_0:def_3; then [x,(g1 . x)] in g1 by FUNCT_1:1; then [x,(g1 . x)] in f by A15, XBOOLE_0:def_3; hence f . x = g1 . x by FUNCT_1:1; ::_thesis: verum end; end; then A28: f = g1 +* g2 by A26, FUNCT_4:def_1; reconsider f = f as PartFunc of R^1,T by A7, A8, RELSET_1:4; dom c1 meets dom c2 by A13, XBOOLE_0:def_7; then dom f is interval Subset of REAL by A4, A5, A3, XBOOLE_1:8, XXREAL_2:89; then f is parametrized-curve by A22, A28, Def4; then A29: c1 \/ c2 is Curve of T by Th20; then A30: c1 + c2 = c1 \/ c2 by Def12; A31: dom (c1 \/ c2) = [.(inf (dom c1)),(sup (dom c2)).] by A3, A11, A12, A1, A9, A10, XXREAL_1:165; A32: inf (dom c1) in dom f1 by A11, A10, XXREAL_1:1; then inf (dom c1) in dom f by A3, XBOOLE_0:def_3; then inf (dom f) in dom f by A31, A1, A9, A10, XXREAL_0:2, XXREAL_2:25; then dom (c1 + c2) is left_end by A30, XXREAL_2:def_5; then A33: c1 + c2 is with_first_point by Def6; A34: sup (dom c2) in dom f2 by A12, A9, XXREAL_1:1; then sup (dom c2) in dom f by A3, XBOOLE_0:def_3; then sup [.(inf (dom c1)),(sup (dom c2)).] in dom f by A1, A9, A10, XXREAL_0:2, XXREAL_2:29; then dom (c1 + c2) is right_end by A31, A30, XXREAL_2:def_6; then A35: c1 + c2 is with_last_point by Def7; thus ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] ) by A33, A35, A30, A3, A11, A12, A1, A9, A10, XXREAL_1:165; ::_thesis: ( (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) A36: (c1 + c2) . (inf (dom c1)) = c1 . (inf (dom c1)) proof A37: (c1 + c2) . (inf (dom c1)) = f . (inf (dom c1)) by A29, Def12; [(inf (dom c1)),(c1 . (inf (dom c1)))] in c1 by A32, FUNCT_1:1; then [(inf (dom c1)),(c1 . (inf (dom c1)))] in f by XBOOLE_0:def_3; hence (c1 + c2) . (inf (dom c1)) = c1 . (inf (dom c1)) by A37, FUNCT_1:1; ::_thesis: verum end; thus (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 by A36; ::_thesis: (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 A38: (c1 + c2) . (sup (dom c2)) = c2 . (sup (dom c2)) proof A39: (c1 + c2) . (sup (dom c2)) = f . (sup (dom c2)) by A29, Def12; [(sup (dom c2)),(c2 . (sup (dom c2)))] in c2 by A34, FUNCT_1:1; then [(sup (dom c2)),(c2 . (sup (dom c2)))] in f by XBOOLE_0:def_3; hence (c1 + c2) . (sup (dom c2)) = c2 . (sup (dom c2)) by A39, FUNCT_1:1; ::_thesis: verum end; thus (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 by A38; ::_thesis: verum end; theorem Th39: :: TOPALG_6:39 for T being non empty TopSpace for c1, c2, c3, c4, c5, c6 being with_endpoints Curve of T st c1,c2 are_homotopic & dom c1 = dom c2 & c3,c4 are_homotopic & dom c3 = dom c4 & c5 = c1 + c3 & c6 = c2 + c4 & the_last_point_of c1 = the_first_point_of c3 & sup (dom c1) = inf (dom c3) holds c5,c6 are_homotopic proof let T be non empty TopSpace; ::_thesis: for c1, c2, c3, c4, c5, c6 being with_endpoints Curve of T st c1,c2 are_homotopic & dom c1 = dom c2 & c3,c4 are_homotopic & dom c3 = dom c4 & c5 = c1 + c3 & c6 = c2 + c4 & the_last_point_of c1 = the_first_point_of c3 & sup (dom c1) = inf (dom c3) holds c5,c6 are_homotopic let c1, c2, c3, c4, c5, c6 be with_endpoints Curve of T; ::_thesis: ( c1,c2 are_homotopic & dom c1 = dom c2 & c3,c4 are_homotopic & dom c3 = dom c4 & c5 = c1 + c3 & c6 = c2 + c4 & the_last_point_of c1 = the_first_point_of c3 & sup (dom c1) = inf (dom c3) implies c5,c6 are_homotopic ) assume A1: ( c1,c2 are_homotopic & dom c1 = dom c2 ) ; ::_thesis: ( not c3,c4 are_homotopic or not dom c3 = dom c4 or not c5 = c1 + c3 or not c6 = c2 + c4 or not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic ) reconsider S1 = [.(inf (dom c1)),(sup (dom c1)).] as non empty Subset of R^1 by Th27, TOPMETR:17; A2: dom c1 = S1 by Th27; then consider H1 being Function of [:(R^1 | S1),I[01]:],T, a1, b1 being Point of T such that A3: ( H1 is continuous & ( for t being Point of (R^1 | S1) holds ( H1 . (t,0) = c1 . t & H1 . (t,1) = c2 . t ) ) & ( for t being Point of I[01] holds ( H1 . ((inf S1),t) = a1 & H1 . ((sup S1),t) = b1 ) ) ) by A1, Th36; assume A4: ( c3,c4 are_homotopic & dom c3 = dom c4 ) ; ::_thesis: ( not c5 = c1 + c3 or not c6 = c2 + c4 or not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic ) reconsider S2 = [.(inf (dom c3)),(sup (dom c3)).] as non empty Subset of R^1 by Th27, TOPMETR:17; A5: dom c3 = S2 by Th27; then consider H2 being Function of [:(R^1 | S2),I[01]:],T, a2, b2 being Point of T such that A6: ( H2 is continuous & ( for t being Point of (R^1 | S2) holds ( H2 . (t,0) = c3 . t & H2 . (t,1) = c4 . t ) ) & ( for t being Point of I[01] holds ( H2 . ((inf S2),t) = a2 & H2 . ((sup S2),t) = b2 ) ) ) by A4, Th36; assume A7: c5 = c1 + c3 ; ::_thesis: ( not c6 = c2 + c4 or not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic ) A8: c5 = c1 \/ c3 proof percases ( c1 \/ c3 is Curve of T or not c1 \/ c3 is Curve of T ) ; suppose c1 \/ c3 is Curve of T ; ::_thesis: c5 = c1 \/ c3 hence c5 = c1 \/ c3 by A7, Def12; ::_thesis: verum end; suppose c1 \/ c3 is not Curve of T ; ::_thesis: c5 = c1 \/ c3 hence c5 = c1 \/ c3 by A7, Def12; ::_thesis: verum end; end; end; assume A9: c6 = c2 + c4 ; ::_thesis: ( not the_last_point_of c1 = the_first_point_of c3 or not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic ) A10: c6 = c2 \/ c4 proof percases ( c2 \/ c4 is Curve of T or not c2 \/ c4 is Curve of T ) ; suppose c2 \/ c4 is Curve of T ; ::_thesis: c6 = c2 \/ c4 hence c6 = c2 \/ c4 by A9, Def12; ::_thesis: verum end; suppose c2 \/ c4 is not Curve of T ; ::_thesis: c6 = c2 \/ c4 hence c6 = c2 \/ c4 by A9, Def12; ::_thesis: verum end; end; end; assume A11: the_last_point_of c1 = the_first_point_of c3 ; ::_thesis: ( not sup (dom c1) = inf (dom c3) or c5,c6 are_homotopic ) assume A12: sup (dom c1) = inf (dom c3) ; ::_thesis: c5,c6 are_homotopic A13: dom c5 = (dom c1) \/ (dom c3) by A8, RELAT_1:1 .= dom c6 by A10, A1, A4, RELAT_1:1 ; reconsider S3 = S1 \/ S2 as Subset of R^1 ; A14: dom c5 = (dom c1) \/ (dom c3) by A8, RELAT_1:1 .= S3 by A5, Th27 ; set T1 = [:(R^1 | S1),I[01]:]; set T2 = [:(R^1 | S2),I[01]:]; set T3 = [:(R^1 | S3),I[01]:]; A15: I[01] is SubSpace of I[01] by TSEP_1:2; R^1 | S1 is SubSpace of R^1 | S3 by TOPMETR:4; then A16: [:(R^1 | S1),I[01]:] is SubSpace of [:(R^1 | S3),I[01]:] by A15, BORSUK_3:21; R^1 | S2 is SubSpace of R^1 | S3 by TOPMETR:4; then A17: [:(R^1 | S2),I[01]:] is SubSpace of [:(R^1 | S3),I[01]:] by A15, BORSUK_3:21; A18: ([#] (R^1 | S1)) \/ ([#] (R^1 | S2)) = S1 \/ ([#] (R^1 | S2)) by PRE_TOPC:def_5 .= S3 by PRE_TOPC:def_5 .= [#] (R^1 | S3) by PRE_TOPC:def_5 ; A19: ([#] [:(R^1 | S1),I[01]:]) \/ ([#] [:(R^1 | S2),I[01]:]) = [:([#] (R^1 | S1)),([#] I[01]):] \/ ([#] [:(R^1 | S2),I[01]:]) by BORSUK_1:def_2 .= [:([#] (R^1 | S1)),([#] I[01]):] \/ [:([#] (R^1 | S2)),([#] I[01]):] by BORSUK_1:def_2 .= [:([#] (R^1 | S3)),([#] I[01]):] by A18, ZFMISC_1:97 .= [#] [:(R^1 | S3),I[01]:] by BORSUK_1:def_2 ; A20: inf (dom c1) <= sup (dom c1) by XXREAL_2:40; R^1 | S1 = Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1))) by A20, TOPMETR:19; then A21: R^1 | S1 is compact by A20, HEINE:4; A22: inf (dom c3) <= sup (dom c3) by XXREAL_2:40; R^1 | S2 = Closed-Interval-TSpace ((inf (dom c3)),(sup (dom c3))) by A22, TOPMETR:19; then A23: R^1 | S2 is compact by A22, HEINE:4; [:(R^1 | S3),I[01]:] is SubSpace of [:R^1,I[01]:] by A15, BORSUK_3:21; then A24: [:(R^1 | S3),I[01]:] is T_2 by TOPMETR:2; for p being set st p in ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) holds H1 . p = H2 . p proof let p be set ; ::_thesis: ( p in ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) implies H1 . p = H2 . p ) assume A25: p in ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) ; ::_thesis: H1 . p = H2 . p A26: ([#] [:(R^1 | S1),I[01]:]) /\ ([#] [:(R^1 | S2),I[01]:]) = [:([#] (R^1 | S1)),([#] I[01]):] /\ ([#] [:(R^1 | S2),I[01]:]) by BORSUK_1:def_2 .= [:([#] (R^1 | S1)),([#] I[01]):] /\ [:([#] (R^1 | S2)),([#] I[01]):] by BORSUK_1:def_2 .= [:(([#] (R^1 | S1)) /\ ([#] (R^1 | S2))),([#] I[01]):] by ZFMISC_1:99 ; A27: ([#] (R^1 | S1)) /\ ([#] (R^1 | S2)) = S1 /\ ([#] (R^1 | S2)) by PRE_TOPC:def_5 .= S1 /\ S2 by PRE_TOPC:def_5 ; A28: S1 /\ S2 = {(sup (dom c1))} by A22, A20, A12, XXREAL_1:418; then consider x, y being set such that A29: ( x in {(sup (dom c1))} & y in [#] I[01] & p = [x,y] ) by A25, A27, A26, ZFMISC_1:def_2; reconsider y = y as Point of I[01] by A29; A30: x = sup S1 by A2, A29, TARSKI:def_1; A31: x = inf S2 by A5, A12, A29, TARSKI:def_1; A32: 0 in [#] I[01] by BORSUK_1:40, XXREAL_1:1; A33: sup S1 in [#] (R^1 | S1) by A30, A27, A28, A29, XBOOLE_0:def_4; thus H1 . p = H1 . ((sup S1),y) by A29, A30, BINOP_1:def_1 .= b1 by A3 .= H1 . ((sup S1),0) by A3, A32 .= the_last_point_of c1 by A2, A3, A33 .= H2 . ((inf S2),0) by A6, A31, A27, A28, A29, A11, A5 .= a2 by A32, A6 .= H2 . ((inf S2),y) by A6 .= H2 . p by A29, A30, A2, A5, A12, BINOP_1:def_1 ; ::_thesis: verum end; then consider H3 being Function of [:(R^1 | S3),I[01]:],T such that A34: ( H3 = H1 +* H2 & H3 is continuous ) by A16, A17, A19, A21, A23, A24, A3, A6, BORSUK_2:1; A35: for t being Point of (R^1 | S3) holds ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t ) proof let t be Point of (R^1 | S3); ::_thesis: ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t ) A36: 0 in [#] I[01] by BORSUK_1:40, XXREAL_1:1; then [t,0] in [:([#] (R^1 | S3)),([#] I[01]):] by ZFMISC_1:def_2; then [t,0] in [#] [:(R^1 | S3),I[01]:] ; then [t,0] in dom H3 by FUNCT_2:def_1; then A37: [t,0] in (dom H1) \/ (dom H2) by A34, FUNCT_4:def_1; A38: 1 in [#] I[01] by BORSUK_1:40, XXREAL_1:1; then [t,1] in [:([#] (R^1 | S3)),([#] I[01]):] by ZFMISC_1:def_2; then [t,1] in [#] [:(R^1 | S3),I[01]:] ; then [t,1] in dom H3 by FUNCT_2:def_1; then A39: [t,1] in (dom H1) \/ (dom H2) by A34, FUNCT_4:def_1; percases ( [t,0] in dom H2 or not [t,0] in dom H2 ) ; supposeA40: [t,0] in dom H2 ; ::_thesis: ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t ) then [t,0] in [#] [:(R^1 | S2),I[01]:] ; then [t,0] in [:([#] (R^1 | S2)),([#] I[01]):] by BORSUK_1:def_2; then A41: t in [#] (R^1 | S2) by ZFMISC_1:87; then A42: t in dom c3 by A5, PRE_TOPC:def_5; then t in (dom c1) \/ (dom c3) by XBOOLE_0:def_3; then A43: t in dom c5 by A8, RELAT_1:1; [t,(c3 . t)] in c3 by A42, FUNCT_1:1; then A44: [t,(c3 . t)] in c5 by A8, XBOOLE_0:def_3; thus H3 . (t,0) = H3 . [t,0] by BINOP_1:def_1 .= H2 . [t,0] by A37, A40, A34, FUNCT_4:def_1 .= H2 . (t,0) by BINOP_1:def_1 .= c3 . t by A6, A41 .= c5 . t by A44, A43, FUNCT_1:def_2 ; ::_thesis: H3 . (t,1) = c6 . t [t,1] in [:([#] (R^1 | S2)),([#] I[01]):] by A41, A38, ZFMISC_1:def_2; then [t,1] in [#] [:(R^1 | S2),I[01]:] ; then A45: [t,1] in dom H2 by FUNCT_2:def_1; t in (dom c2) \/ (dom c4) by A42, A4, XBOOLE_0:def_3; then A46: t in dom c6 by A10, RELAT_1:1; [t,(c4 . t)] in c4 by A42, A4, FUNCT_1:1; then A47: [t,(c4 . t)] in c6 by A10, XBOOLE_0:def_3; thus H3 . (t,1) = H3 . [t,1] by BINOP_1:def_1 .= H2 . [t,1] by A39, A45, A34, FUNCT_4:def_1 .= H2 . (t,1) by BINOP_1:def_1 .= c4 . t by A6, A41 .= c6 . t by A47, A46, FUNCT_1:def_2 ; ::_thesis: verum end; supposeA48: not [t,0] in dom H2 ; ::_thesis: ( H3 . (t,0) = c5 . t & H3 . (t,1) = c6 . t ) ( [t,0] in dom H1 or [t,0] in dom H2 ) by A37, XBOOLE_0:def_3; then [t,0] in [#] [:(R^1 | S1),I[01]:] by A48; then [t,0] in [:([#] (R^1 | S1)),([#] I[01]):] by BORSUK_1:def_2; then A49: t in [#] (R^1 | S1) by ZFMISC_1:87; then A50: t in dom c1 by A2, PRE_TOPC:def_5; then t in (dom c1) \/ (dom c3) by XBOOLE_0:def_3; then A51: t in dom c5 by A8, RELAT_1:1; [t,(c1 . t)] in c1 by A50, FUNCT_1:1; then A52: [t,(c1 . t)] in c5 by A8, XBOOLE_0:def_3; thus H3 . (t,0) = H3 . [t,0] by BINOP_1:def_1 .= H1 . [t,0] by A48, A37, A34, FUNCT_4:def_1 .= H1 . (t,0) by BINOP_1:def_1 .= c1 . t by A3, A49 .= c5 . t by A52, A51, FUNCT_1:def_2 ; ::_thesis: H3 . (t,1) = c6 . t t in (dom c2) \/ (dom c4) by A50, A1, XBOOLE_0:def_3; then A53: t in dom c6 by A10, RELAT_1:1; [t,(c2 . t)] in c2 by A50, A1, FUNCT_1:1; then A54: [t,(c2 . t)] in c6 by A10, XBOOLE_0:def_3; A55: not [t,1] in dom H2 proof assume [t,1] in dom H2 ; ::_thesis: contradiction then [t,1] in [#] [:(R^1 | S2),I[01]:] ; then [t,1] in [:([#] (R^1 | S2)),([#] I[01]):] by BORSUK_1:def_2; then t in [#] (R^1 | S2) by ZFMISC_1:87; then [t,0] in [:([#] (R^1 | S2)),([#] I[01]):] by A36, ZFMISC_1:def_2; then [t,0] in [#] [:(R^1 | S2),I[01]:] ; hence contradiction by A48, FUNCT_2:def_1; ::_thesis: verum end; thus H3 . (t,1) = H3 . [t,1] by BINOP_1:def_1 .= H1 . [t,1] by A55, A39, A34, FUNCT_4:def_1 .= H1 . (t,1) by BINOP_1:def_1 .= c2 . t by A3, A49 .= c6 . t by A54, A53, FUNCT_1:def_2 ; ::_thesis: verum end; end; end; for t being Point of I[01] holds ( H3 . ((inf S3),t) = a1 & H3 . ((sup S3),t) = b2 ) proof let t be Point of I[01]; ::_thesis: ( H3 . ((inf S3),t) = a1 & H3 . ((sup S3),t) = b2 ) A56: inf S1 = inf (dom c1) by Th27 .= inf [.(inf (dom c1)),(sup (dom c3)).] by A22, A20, A12, XXREAL_0:2, XXREAL_2:25 .= inf S3 by A22, A20, A12, XXREAL_1:165 ; inf S1 in S1 by A2, A20, XXREAL_1:1; then inf S1 in [#] (R^1 | S1) by PRE_TOPC:def_5; then [(inf S1),t] in [:([#] (R^1 | S1)),([#] I[01]):] by ZFMISC_1:def_2; then [(inf S1),t] in [#] [:(R^1 | S1),I[01]:] ; then [(inf S1),t] in dom H1 by FUNCT_2:def_1; then A57: [(inf S3),t] in (dom H1) \/ (dom H2) by A56, XBOOLE_0:def_3; thus H3 . ((inf S3),t) = a1 ::_thesis: H3 . ((sup S3),t) = b2 proof percases ( [(inf S3),t] in dom H2 or not [(inf S3),t] in dom H2 ) ; supposeA58: [(inf S3),t] in dom H2 ; ::_thesis: H3 . ((inf S3),t) = a1 then [(inf S3),t] in [#] [:(R^1 | S2),I[01]:] ; then [(inf S3),t] in [:([#] (R^1 | S2)),([#] I[01]):] by BORSUK_1:def_2; then inf S3 in [#] (R^1 | S2) by ZFMISC_1:87; then inf S3 in S2 by PRE_TOPC:def_5; then ( inf (dom c3) <= inf S1 & inf S1 <= sup (dom c3) ) by A56, XXREAL_1:1; then A59: inf (dom c3) <= inf (dom c1) by Th27; A60: inf (dom c3) = inf (dom c1) by A59, A12, A20, XXREAL_0:1; A61: inf S2 = inf (dom c3) by Th27 .= inf S1 by A60, Th27 ; A62: inf S1 = inf (dom c1) by Th27 .= inf (dom c3) by A59, A12, A20, XXREAL_0:1 .= sup S1 by A12, Th27 ; A63: 0 in [#] I[01] by BORSUK_1:40, XXREAL_1:1; sup (dom c1) = sup S1 by Th27; then sup S1 in S1 by A20, XXREAL_1:1; then A64: sup S1 in [#] (R^1 | S1) by PRE_TOPC:def_5; inf (dom c3) = inf S2 by Th27; then inf S2 in S2 by A22, XXREAL_1:1; then A65: inf S2 in [#] (R^1 | S2) by PRE_TOPC:def_5; A66: sup S1 = sup (dom c1) by Th27; A67: inf S2 = inf (dom c3) by Th27; A68: a1 = H1 . ((inf S1),0) by A3, A63 .= the_last_point_of c1 by A66, A62, A64, A3 .= H2 . ((inf S2),0) by A65, A6, A67, A11 .= a2 by A6, A63 ; H3 . ((inf S3),t) = H3 . [(inf S3),t] by BINOP_1:def_1 .= H2 . [(inf S3),t] by A57, A58, A34, FUNCT_4:def_1 .= H2 . ((inf S2),t) by A56, A61, BINOP_1:def_1 .= a1 by A6, A68 ; hence H3 . ((inf S3),t) = a1 ; ::_thesis: verum end; supposeA69: not [(inf S3),t] in dom H2 ; ::_thesis: H3 . ((inf S3),t) = a1 H3 . ((inf S3),t) = H3 . [(inf S3),t] by BINOP_1:def_1 .= H1 . [(inf S3),t] by A57, A69, A34, FUNCT_4:def_1 .= H1 . ((inf S3),t) by BINOP_1:def_1 .= a1 by A56, A3 ; hence H3 . ((inf S3),t) = a1 ; ::_thesis: verum end; end; end; A70: sup S2 = sup (dom c3) by Th27 .= sup [.(inf (dom c1)),(sup (dom c3)).] by A22, A20, A12, XXREAL_0:2, XXREAL_2:29 .= sup S3 by A22, A20, A12, XXREAL_1:165 ; sup S2 in S2 by A5, A22, XXREAL_1:1; then sup S2 in [#] (R^1 | S2) by PRE_TOPC:def_5; then [(sup S2),t] in [:([#] (R^1 | S2)),([#] I[01]):] by ZFMISC_1:def_2; then A71: [(sup S2),t] in [#] [:(R^1 | S2),I[01]:] ; then [(sup S2),t] in dom H2 by FUNCT_2:def_1; then A72: [(sup S3),t] in (dom H1) \/ (dom H2) by A70, XBOOLE_0:def_3; A73: [(sup S3),t] in dom H2 by A71, A70, FUNCT_2:def_1; H3 . ((sup S3),t) = H3 . [(sup S3),t] by BINOP_1:def_1 .= H2 . [(sup S3),t] by A72, A73, A34, FUNCT_4:def_1 .= H2 . ((sup S2),t) by A70, BINOP_1:def_1 .= b2 by A6 ; hence H3 . ((sup S3),t) = b2 ; ::_thesis: verum end; hence c5,c6 are_homotopic by A13, A14, A35, A34, Th36; ::_thesis: verum end; definition let T be TopStruct ; let f be FinSequence of Curves T; func Partial_Sums f -> FinSequence of Curves T means :Def13: :: TOPALG_6:def 13 ( len f = len it & f . 1 = it . 1 & ( for i being Nat st 1 <= i & i < len f holds it . (i + 1) = (it /. i) + (f /. (i + 1)) ) ); existence ex b1 being FinSequence of Curves T st ( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) ) proof percases ( len f > 0 or len f <= 0 ) ; supposeA1: len f > 0 ; ::_thesis: ex b1 being FinSequence of Curves T st ( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) ) reconsider q = <*(f /. 1)*> as FinSequence of Curves T ; A2: 0 + 1 <= len f by A1, NAT_1:13; then f /. 1 = f . 1 by FINSEQ_4:15; then A3: q . 1 = f . 1 by FINSEQ_1:40; defpred S1[ Nat] means ( $1 + 1 <= len f implies ex g being FinSequence of Curves T st ( $1 + 1 = len g & f . 1 = g . 1 & ( for i being Nat st 1 <= i & i < $1 + 1 holds g . (i + 1) = (g /. i) + (f /. (i + 1)) ) ) ); A4: for i being Nat st 1 <= i & i < 0 + 1 holds q . (i + 1) = (q /. i) + (f /. (i + 1)) ; A5: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: S1[k] ; ::_thesis: S1[k + 1] now__::_thesis:_(_(_(k_+_1)_+_1_<=_len_f_&_S1[k_+_1]_)_or_(_(k_+_1)_+_1_>_len_f_&_S1[k_+_1]_)_) percases ( (k + 1) + 1 <= len f or (k + 1) + 1 > len f ) ; caseA7: (k + 1) + 1 <= len f ; ::_thesis: S1[k + 1] k + 1 < (k + 1) + 1 by XREAL_1:29; then consider g being FinSequence of Curves T such that A8: k + 1 = len g and A9: f . 1 = g . 1 and A10: for i being Nat st 1 <= i & i < k + 1 holds g . (i + 1) = (g /. i) + (f /. (i + 1)) by A6, A7, XXREAL_0:2; reconsider g2 = g ^ <*((g /. (k + 1)) + (f /. ((k + 1) + 1)))*> as FinSequence of Curves T ; A11: Seg (len g) = dom g by FINSEQ_1:def_3; A12: len g2 = (len g) + (len <*((g /. (k + 1)) + (f /. ((k + 1) + 1)))*>) by FINSEQ_1:22 .= (k + 1) + 1 by A8, FINSEQ_1:40 ; A13: for i being Nat st 1 <= i & i < (k + 1) + 1 holds g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) proof let i be Nat; ::_thesis: ( 1 <= i & i < (k + 1) + 1 implies g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ) assume that A14: 1 <= i and A15: i < (k + 1) + 1 ; ::_thesis: g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) A16: i <= k + 1 by A15, NAT_1:13; percases ( i < k + 1 or i = k + 1 ) by A16, XXREAL_0:1; supposeA17: i < k + 1 ; ::_thesis: g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) A18: 1 <= i + 1 by NAT_1:12; i + 1 <= k + 1 by A17, NAT_1:13; then i + 1 in Seg (len g) by A8, A18, FINSEQ_1:1; then A19: g2 . (i + 1) = g . (i + 1) by A11, FINSEQ_1:def_7; i in Seg (len g) by A8, A14, A16, FINSEQ_1:1; then A20: g2 . i = g . i by A11, FINSEQ_1:def_7; A21: g /. i = g . i by A8, A14, A17, FINSEQ_4:15; g2 /. i = g2 . i by A12, A14, A15, FINSEQ_4:15; hence g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) by A10, A14, A17, A19, A20, A21; ::_thesis: verum end; supposeA22: i = k + 1 ; ::_thesis: g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) A23: g2 /. i = g2 . i by A12, A14, A15, FINSEQ_4:15; i in Seg (len g) by A8, A14, A16, FINSEQ_1:1; then A24: g . i = g2 . i by A11, FINSEQ_1:def_7; g /. i = g . i by A8, A14, A16, FINSEQ_4:15; hence g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) by A8, A22, A24, A23, FINSEQ_1:42; ::_thesis: verum end; end; end; 1 <= k + 1 by NAT_1:11; then 1 in Seg (len g) by A8, FINSEQ_1:1; then g2 . 1 = f . 1 by A9, A11, FINSEQ_1:def_7; hence S1[k + 1] by A12, A13; ::_thesis: verum end; case (k + 1) + 1 > len f ; ::_thesis: S1[k + 1] hence S1[k + 1] ; ::_thesis: verum end; end; end; hence S1[k + 1] ; ::_thesis: verum end; (len f) -' 1 = (len f) - 1 by A2, XREAL_1:233; then A25: ((len f) -' 1) + 1 = len f ; len q = 1 by FINSEQ_1:40; then A26: S1[ 0 ] by A3, A4; for k being Nat holds S1[k] from NAT_1:sch_2(A26, A5); hence ex b1 being FinSequence of Curves T st ( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) ) by A25; ::_thesis: verum end; supposeA27: len f <= 0 ; ::_thesis: ex b1 being FinSequence of Curves T st ( len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) ) take f ; ::_thesis: ( len f = len f & f . 1 = f . 1 & ( for i being Nat st 1 <= i & i < len f holds f . (i + 1) = (f /. i) + (f /. (i + 1)) ) ) thus ( len f = len f & f . 1 = f . 1 ) ; ::_thesis: for i being Nat st 1 <= i & i < len f holds f . (i + 1) = (f /. i) + (f /. (i + 1)) let i be Nat; ::_thesis: ( 1 <= i & i < len f implies f . (i + 1) = (f /. i) + (f /. (i + 1)) ) thus ( 1 <= i & i < len f implies f . (i + 1) = (f /. i) + (f /. (i + 1)) ) by A27; ::_thesis: verum end; end; end; uniqueness for b1, b2 being FinSequence of Curves T st len f = len b1 & f . 1 = b1 . 1 & ( for i being Nat st 1 <= i & i < len f holds b1 . (i + 1) = (b1 /. i) + (f /. (i + 1)) ) & len f = len b2 & f . 1 = b2 . 1 & ( for i being Nat st 1 <= i & i < len f holds b2 . (i + 1) = (b2 /. i) + (f /. (i + 1)) ) holds b1 = b2 proof let g1, g2 be FinSequence of Curves T; ::_thesis: ( len f = len g1 & f . 1 = g1 . 1 & ( for i being Nat st 1 <= i & i < len f holds g1 . (i + 1) = (g1 /. i) + (f /. (i + 1)) ) & len f = len g2 & f . 1 = g2 . 1 & ( for i being Nat st 1 <= i & i < len f holds g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ) implies g1 = g2 ) assume that A28: len f = len g1 and A29: f . 1 = g1 . 1 and A30: for i being Nat st 1 <= i & i < len f holds g1 . (i + 1) = (g1 /. i) + (f /. (i + 1)) ; ::_thesis: ( not len f = len g2 or not f . 1 = g2 . 1 or ex i being Nat st ( 1 <= i & i < len f & not g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ) or g1 = g2 ) defpred S1[ Nat] means ( 1 <= $1 & $1 <= len f implies g1 . $1 = g2 . $1 ); assume that A31: len f = len g2 and A32: f . 1 = g2 . 1 and A33: for i being Nat st 1 <= i & i < len f holds g2 . (i + 1) = (g2 /. i) + (f /. (i + 1)) ; ::_thesis: g1 = g2 A34: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A35: S1[k] ; ::_thesis: S1[k + 1] ( 1 <= k + 1 & k + 1 <= len f implies g1 . (k + 1) = g2 . (k + 1) ) proof assume that 1 <= k + 1 and A36: k + 1 <= len f ; ::_thesis: g1 . (k + 1) = g2 . (k + 1) A37: k < k + 1 by XREAL_1:29; then A38: k < len f by A36, XXREAL_0:2; percases ( 1 <= k or 1 > k ) ; supposeA39: 1 <= k ; ::_thesis: g1 . (k + 1) = g2 . (k + 1) then A40: g2 . (k + 1) = (g2 /. k) + (f /. (k + 1)) by A33, A38; A41: k <= len g2 by A31, A36, A37, XXREAL_0:2; A42: g1 /. k = g1 . k by A28, A38, A39, FINSEQ_4:15; g1 . (k + 1) = (g1 /. k) + (f /. (k + 1)) by A30, A38, A39; hence g1 . (k + 1) = g2 . (k + 1) by A35, A36, A37, A39, A40, A42, A41, FINSEQ_4:15, XXREAL_0:2; ::_thesis: verum end; suppose 1 > k ; ::_thesis: g1 . (k + 1) = g2 . (k + 1) then 0 + 1 > k ; then k = 0 by NAT_1:13; hence g1 . (k + 1) = g2 . (k + 1) by A29, A32; ::_thesis: verum end; end; end; hence S1[k + 1] ; ::_thesis: verum end; A43: S1[ 0 ] ; for k being Nat holds S1[k] from NAT_1:sch_2(A43, A34); hence g1 = g2 by A28, A31, FINSEQ_1:14; ::_thesis: verum end; end; :: deftheorem Def13 defines Partial_Sums TOPALG_6:def_13_:_ for T being TopStruct for f, b3 being FinSequence of Curves T holds ( b3 = Partial_Sums f iff ( len f = len b3 & f . 1 = b3 . 1 & ( for i being Nat st 1 <= i & i < len f holds b3 . (i + 1) = (b3 /. i) + (f /. (i + 1)) ) ) ); definition let T be TopStruct ; let f be FinSequence of Curves T; func Sum f -> Curve of T equals :Def14: :: TOPALG_6:def 14 (Partial_Sums f) . (len f) if len f > 0 otherwise {} ; coherence ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) ) proof A1: len f = len (Partial_Sums f) by Def13; now__::_thesis:_(_(_len_f_>_0_&_(Partial_Sums_f)_._(len_f)_is_Element_of_Curves_T_)_or_(_len_f_<=_0_&_(_len_f_>_0_implies_(Partial_Sums_f)_._(len_f)_is_Curve_of_T_)_&_(_not_len_f_>_0_implies_{}_is_Curve_of_T_)_)_) percases ( len f > 0 or len f <= 0 ) ; case len f > 0 ; ::_thesis: (Partial_Sums f) . (len f) is Element of Curves T then 0 + 1 <= len f by NAT_1:13; then len f in dom (Partial_Sums f) by A1, FINSEQ_3:25; then (Partial_Sums f) . (len f) in rng (Partial_Sums f) by FUNCT_1:def_3; hence (Partial_Sums f) . (len f) is Element of Curves T ; ::_thesis: verum end; caseA2: len f <= 0 ; ::_thesis: ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) ) {} is parametrized-curve PartFunc of R^1,T by Lm1, XBOOLE_1:2; hence ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) ) by A2, Th20; ::_thesis: verum end; end; end; hence ( ( len f > 0 implies (Partial_Sums f) . (len f) is Curve of T ) & ( not len f > 0 implies {} is Curve of T ) ) ; ::_thesis: verum end; consistency for b1 being Curve of T holds verum ; end; :: deftheorem Def14 defines Sum TOPALG_6:def_14_:_ for T being TopStruct for f being FinSequence of Curves T holds ( ( len f > 0 implies Sum f = (Partial_Sums f) . (len f) ) & ( not len f > 0 implies Sum f = {} ) ); theorem Th40: :: TOPALG_6:40 for T being non empty TopStruct for c being Curve of T holds Sum <*c*> = c proof let T be non empty TopStruct ; ::_thesis: for c being Curve of T holds Sum <*c*> = c let c be Curve of T; ::_thesis: Sum <*c*> = c set f = <*c*>; len <*c*> = 1 by FINSEQ_1:40; hence Sum <*c*> = (Partial_Sums <*c*>) . 1 by Def14 .= <*c*> . 1 by Def13 .= c by FINSEQ_1:40 ; ::_thesis: verum end; Lm2: for T being non empty TopStruct for f1, f2 being FinSequence of Curves T holds (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) proof let T be non empty TopStruct ; ::_thesis: for f1, f2 being FinSequence of Curves T holds (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) defpred S1[ Nat] means for f1, f2 being FinSequence of Curves T st len f1 = $1 holds (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1); A1: S1[ 0 ] proof let f1, f2 be FinSequence of Curves T; ::_thesis: ( len f1 = 0 implies (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) ) assume A2: len f1 = 0 ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) then not len f1 in Seg (len (Partial_Sums (f1 ^ f2))) by FINSEQ_1:1; then A3: not len f1 in dom (Partial_Sums (f1 ^ f2)) by FINSEQ_1:def_3; not len f1 in Seg (len (Partial_Sums f1)) by A2, FINSEQ_1:1; then A4: not len f1 in dom (Partial_Sums f1) by FINSEQ_1:def_3; thus (Partial_Sums (f1 ^ f2)) . (len f1) = {} by A3, FUNCT_1:def_2 .= (Partial_Sums f1) . (len f1) by A4, FUNCT_1:def_2 ; ::_thesis: verum end; A5: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A6: S1[k] ; ::_thesis: S1[k + 1] let f1, f2 be FinSequence of Curves T; ::_thesis: ( len f1 = k + 1 implies (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) ) assume A7: len f1 = k + 1 ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) then consider f3 being FinSequence of Curves T, c being Element of Curves T such that A8: f1 = f3 ^ <*c*> by FINSEQ_2:19; set f4 = <*c*> ^ f2; A9: f1 ^ f2 = f3 ^ (<*c*> ^ f2) by A8, FINSEQ_1:32; A10: len f1 = (len f3) + 1 by A8, FINSEQ_2:16; percases ( 1 > k or 1 <= k ) ; supposeA11: 1 > k ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) then A12: len f3 = 0 by A10, A7, NAT_1:14; f3 = {} by A11, A10, A7, FINSEQ_1:20; then A13: f1 = <*c*> by A8, FINSEQ_1:34; thus (Partial_Sums (f1 ^ f2)) . (len f1) = (f1 ^ f2) . 1 by A12, A10, Def13 .= c by A13, FINSEQ_1:41 .= f1 . 1 by A13, FINSEQ_1:40 .= (Partial_Sums f1) . (len f1) by A12, A10, Def13 ; ::_thesis: verum end; supposeA14: 1 <= k ; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) A15: k < len f1 by A7, NAT_1:16; A16: len (<*c*> ^ f2) = (len <*c*>) + (len f2) by FINSEQ_1:22 .= 1 + (len f2) by FINSEQ_1:39 ; len (f3 ^ (<*c*> ^ f2)) = k + (len (<*c*> ^ f2)) by A10, A7, FINSEQ_1:22; then A17: k < len (f3 ^ (<*c*> ^ f2)) by A16, NAT_1:16; then k in Seg (len (f3 ^ (<*c*> ^ f2))) by A14, FINSEQ_1:1; then k in Seg (len (Partial_Sums (f3 ^ (<*c*> ^ f2)))) by Def13; then A18: k in dom (Partial_Sums (f3 ^ (<*c*> ^ f2))) by FINSEQ_1:def_3; k in Seg (len f3) by A14, A10, A7, FINSEQ_1:1; then k in Seg (len (Partial_Sums f3)) by Def13; then A19: k in dom (Partial_Sums f3) by FINSEQ_1:def_3; k in Seg (len f1) by A14, A15, FINSEQ_1:1; then k in Seg (len (Partial_Sums f1)) by Def13; then A20: k in dom (Partial_Sums f1) by FINSEQ_1:def_3; A21: (Partial_Sums (f3 ^ (<*c*> ^ f2))) /. k = (Partial_Sums (f3 ^ (<*c*> ^ f2))) . k by A18, PARTFUN1:def_6 .= (Partial_Sums f3) . k by A10, A7, A6 .= (Partial_Sums f3) /. k by A19, PARTFUN1:def_6 ; A22: (Partial_Sums f1) /. k = (Partial_Sums f1) . k by A20, PARTFUN1:def_6 .= (Partial_Sums f3) . k by A8, A10, A7, A6 .= (Partial_Sums f3) /. k by A19, PARTFUN1:def_6 ; 1 + 1 <= k + 1 by A14, XREAL_1:6; then A23: 1 <= k + 1 by XXREAL_0:2; 0 + (len f1) <= (len f1) + (len f2) by XREAL_1:6; then k + 1 <= len (f1 ^ f2) by A7, FINSEQ_1:22; then k + 1 in Seg (len (f1 ^ f2)) by A23, FINSEQ_1:1; then A24: k + 1 in dom (f1 ^ f2) by FINSEQ_1:def_3; k + 1 in Seg (len f1) by A7, A23, FINSEQ_1:1; then A25: k + 1 in dom f1 by FINSEQ_1:def_3; A26: (f1 ^ f2) /. (k + 1) = (f1 ^ f2) . (k + 1) by A24, PARTFUN1:def_6 .= f1 . (k + 1) by A25, FINSEQ_1:def_7 .= f1 /. (k + 1) by A25, PARTFUN1:def_6 ; thus (Partial_Sums (f1 ^ f2)) . (len f1) = ((Partial_Sums f1) /. k) + (f1 /. (k + 1)) by A7, A9, A21, A22, A26, A14, A17, Def13 .= (Partial_Sums f1) . (len f1) by A14, A7, A15, Def13 ; ::_thesis: verum end; end; end; A27: for k being Nat holds S1[k] from NAT_1:sch_2(A1, A5); let f1, f2 be FinSequence of Curves T; ::_thesis: (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) thus (Partial_Sums (f1 ^ f2)) . (len f1) = (Partial_Sums f1) . (len f1) by A27; ::_thesis: verum end; theorem Th41: :: TOPALG_6:41 for T being non empty TopStruct for c being Curve of T for f being FinSequence of Curves T holds Sum (f ^ <*c*>) = (Sum f) + c proof let T be non empty TopStruct ; ::_thesis: for c being Curve of T for f being FinSequence of Curves T holds Sum (f ^ <*c*>) = (Sum f) + c let c be Curve of T; ::_thesis: for f being FinSequence of Curves T holds Sum (f ^ <*c*>) = (Sum f) + c let f be FinSequence of Curves T; ::_thesis: Sum (f ^ <*c*>) = (Sum f) + c percases ( len f <= 0 or len f > 0 ) ; supposeA1: len f <= 0 ; ::_thesis: Sum (f ^ <*c*>) = (Sum f) + c A2: f = {} by A1, FINSEQ_1:20; reconsider c0 = {} as Curve of T by Th21; thus Sum (f ^ <*c*>) = Sum <*c*> by A2, FINSEQ_1:34 .= c0 \/ c by Th40 .= c0 + c by Def12 .= (Sum f) + c by Def14, A1 ; ::_thesis: verum end; supposeA3: len f > 0 ; ::_thesis: Sum (f ^ <*c*>) = (Sum f) + c set f1 = f ^ <*c*>; A4: len (f ^ <*c*>) = (len f) + (len <*c*>) by FINSEQ_1:22 .= (len f) + 1 by FINSEQ_1:39 ; A5: Sum (f ^ <*c*>) = (Partial_Sums (f ^ <*c*>)) . (len (f ^ <*c*>)) by A4, Def14; 0 < 0 + (len f) by A3; then A6: 1 <= len f by NAT_1:19; A7: len f < len (f ^ <*c*>) by A4, NAT_1:13; len f in Seg (len (f ^ <*c*>)) by A6, A7, FINSEQ_1:1; then len f in Seg (len (Partial_Sums (f ^ <*c*>))) by Def13; then len f in dom (Partial_Sums (f ^ <*c*>)) by FINSEQ_1:def_3; then A8: (Partial_Sums (f ^ <*c*>)) /. (len f) = (Partial_Sums (f ^ <*c*>)) . (len f) by PARTFUN1:def_6 .= (Partial_Sums f) . (len f) by Lm2 .= Sum f by A3, Def14 ; len (f ^ <*c*>) in Seg (len (f ^ <*c*>)) by A4, FINSEQ_1:3; then len (f ^ <*c*>) in dom (f ^ <*c*>) by FINSEQ_1:def_3; then A9: (f ^ <*c*>) /. ((len f) + 1) = (f ^ <*c*>) . ((len f) + 1) by A4, PARTFUN1:def_6 .= c by FINSEQ_1:42 ; thus Sum (f ^ <*c*>) = (Sum f) + c by A8, A9, A5, A7, A4, A6, Def13; ::_thesis: verum end; end; end; theorem Th42: :: TOPALG_6:42 for T being non empty TopStruct for X being set for f being FinSequence of Curves T st ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) holds rng (Sum f) c= X proof let T be non empty TopStruct ; ::_thesis: for X being set for f being FinSequence of Curves T st ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) holds rng (Sum f) c= X let X be set ; ::_thesis: for f being FinSequence of Curves T st ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) holds rng (Sum f) c= X defpred S1[ Nat] means for f being FinSequence of Curves T st len f = $1 & ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) holds rng (Sum f) c= X; A1: S1[ 0 ] proof let f be FinSequence of Curves T; ::_thesis: ( len f = 0 & ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) implies rng (Sum f) c= X ) assume len f = 0 ; ::_thesis: ( ex i being Nat st ( 1 <= i & i <= len f & not rng (f /. i) c= X ) or rng (Sum f) c= X ) then Sum f = {} by Def14; then rng (Sum f) = {} ; hence ( ex i being Nat st ( 1 <= i & i <= len f & not rng (f /. i) c= X ) or rng (Sum f) c= X ) by XBOOLE_1:2; ::_thesis: verum end; A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] let f be FinSequence of Curves T; ::_thesis: ( len f = k + 1 & ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) implies rng (Sum f) c= X ) assume A4: len f = k + 1 ; ::_thesis: ( ex i being Nat st ( 1 <= i & i <= len f & not rng (f /. i) c= X ) or rng (Sum f) c= X ) then consider f1 being FinSequence of Curves T, c being Element of Curves T such that A5: f = f1 ^ <*c*> by FINSEQ_2:19; assume A6: for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ; ::_thesis: rng (Sum f) c= X A7: len f = (len f1) + (len <*c*>) by A5, FINSEQ_1:22 .= (len f1) + 1 by FINSEQ_1:39 ; A8: Sum f = (Sum f1) + c by A5, Th41; percases ( not (Sum f1) \/ c is Curve of T or (Sum f1) \/ c is Curve of T ) ; suppose (Sum f1) \/ c is not Curve of T ; ::_thesis: rng (Sum f) c= X then Sum f = {} by A8, Def12; then rng (Sum f) = {} ; hence rng (Sum f) c= X by XBOOLE_1:2; ::_thesis: verum end; suppose (Sum f1) \/ c is Curve of T ; ::_thesis: rng (Sum f) c= X then A9: Sum f = (Sum f1) \/ c by A8, Def12; A10: for i being Nat st 1 <= i & i <= len f1 holds rng (f1 /. i) c= X proof let i be Nat; ::_thesis: ( 1 <= i & i <= len f1 implies rng (f1 /. i) c= X ) assume A11: ( 1 <= i & i <= len f1 ) ; ::_thesis: rng (f1 /. i) c= X then A12: i + 1 <= (len f1) + 1 by XREAL_1:6; i <= i + 1 by NAT_1:12; then A13: i <= len f by A12, A7, XXREAL_0:2; then A14: rng (f /. i) c= X by A6, A11; i in Seg (len f) by A11, A13, FINSEQ_1:1; then i in dom f by FINSEQ_1:def_3; then A15: rng (f . i) c= X by A14, PARTFUN1:def_6; i in Seg (len f1) by A11, FINSEQ_1:1; then A16: i in dom f1 by FINSEQ_1:def_3; then f . i = f1 . i by A5, FINSEQ_1:def_7; hence rng (f1 /. i) c= X by A15, A16, PARTFUN1:def_6; ::_thesis: verum end; A17: rng (Sum f1) c= X by A3, A7, A4, A10; len f in Seg (len f) by A7, FINSEQ_1:3; then A18: len f in dom f by FINSEQ_1:def_3; f . (len f) = c by A7, A5, FINSEQ_1:42; then A19: f /. (len f) = c by A18, PARTFUN1:def_6; 0 + 1 <= (len f1) + 1 by XREAL_1:6; then A20: rng c c= X by A7, A19, A6; rng (Sum f) = (rng (Sum f1)) \/ (rng c) by A9, RELAT_1:12; hence rng (Sum f) c= X by A17, A20, XBOOLE_1:8; ::_thesis: verum end; end; end; A21: for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2); let f be FinSequence of Curves T; ::_thesis: ( ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) implies rng (Sum f) c= X ) thus ( ( for i being Nat st 1 <= i & i <= len f holds rng (f /. i) c= X ) implies rng (Sum f) c= X ) by A21; ::_thesis: verum end; theorem Th43: :: TOPALG_6:43 for T being non empty TopSpace for f being FinSequence of Curves T st len f > 0 & ( for i being Nat st 1 <= i & i < len f holds ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f holds f /. i is with_endpoints ) holds ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) proof let T be non empty TopSpace; ::_thesis: for f being FinSequence of Curves T st len f > 0 & ( for i being Nat st 1 <= i & i < len f holds ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f holds f /. i is with_endpoints ) holds ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) defpred S1[ Nat] means for f being FinSequence of Curves T st len f = $1 & len f > 0 & ( for i being Nat st 1 <= i & i < len f holds ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f holds f /. i is with_endpoints ) holds ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ); A1: S1[ 0 ] ; A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] let f be FinSequence of Curves T; ::_thesis: ( len f = k + 1 & len f > 0 & ( for i being Nat st 1 <= i & i < len f holds ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f holds f /. i is with_endpoints ) implies ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) ) assume A4: ( len f = k + 1 & len f > 0 ) ; ::_thesis: ( ex i being Nat st ( 1 <= i & i < len f & not ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i <= len f & not f /. i is with_endpoints ) or ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) ) consider f1 being FinSequence of Curves T, c2 being Element of Curves T such that A5: f = f1 ^ <*c2*> by A4, FINSEQ_2:19; A6: len f = (len f1) + (len <*c2*>) by A5, FINSEQ_1:22 .= (len f1) + 1 by FINSEQ_1:39 ; assume A7: for i being Nat st 1 <= i & i < len f holds ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ; ::_thesis: ( ex i being Nat st ( 1 <= i & i <= len f & not f /. i is with_endpoints ) or ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) ) assume A8: for i being Nat st 1 <= i & i <= len f holds f /. i is with_endpoints ; ::_thesis: ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) A9: 1 <= len f by A4, NAT_1:12; len f in Seg (len f) by A4, FINSEQ_1:3; then A10: len f in dom f by FINSEQ_1:def_3; c2 = f . (len f) by A5, A6, FINSEQ_1:42 .= f /. (len f) by A10, PARTFUN1:def_6 ; then reconsider c2 = c2 as with_endpoints Curve of T by A9, A8; A11: for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) proof let i be Nat; ::_thesis: ( 1 <= i & i < len f1 implies ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) assume A12: ( 1 <= i & i < len f1 ) ; ::_thesis: ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) A13: i < len f by A6, A12, NAT_1:13; i in Seg (len f1) by A12, FINSEQ_1:1; then A14: i in dom f1 by FINSEQ_1:def_3; A15: dom f1 c= dom f by A5, FINSEQ_1:26; A16: f /. i = f . i by A15, A14, PARTFUN1:def_6 .= f1 . i by A5, A14, FINSEQ_1:def_7 .= f1 /. i by A14, PARTFUN1:def_6 ; 1 + 1 <= i + 1 by A12, XREAL_1:6; then A17: 1 <= i + 1 by XXREAL_0:2; i + 1 <= len f1 by A12, NAT_1:13; then i + 1 in Seg (len f1) by A17, FINSEQ_1:1; then A18: i + 1 in dom f1 by FINSEQ_1:def_3; A19: f /. (i + 1) = f . (i + 1) by A18, A15, PARTFUN1:def_6 .= f1 . (i + 1) by A5, A18, FINSEQ_1:def_7 .= f1 /. (i + 1) by A18, PARTFUN1:def_6 ; thus ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) by A16, A19, A13, A12, A7; ::_thesis: verum end; A20: for i being Nat st 1 <= i & i <= len f1 holds f1 /. i is with_endpoints proof let i be Nat; ::_thesis: ( 1 <= i & i <= len f1 implies f1 /. i is with_endpoints ) assume A21: ( 1 <= i & i <= len f1 ) ; ::_thesis: f1 /. i is with_endpoints A22: i <= len f by A6, A21, NAT_1:13; i in Seg (len f1) by A21, FINSEQ_1:1; then A23: i in dom f1 by FINSEQ_1:def_3; A24: dom f1 c= dom f by A5, FINSEQ_1:26; f /. i = f . i by A24, A23, PARTFUN1:def_6 .= f1 . i by A5, A23, FINSEQ_1:def_7 .= f1 /. i by A23, PARTFUN1:def_6 ; hence f1 /. i is with_endpoints by A22, A21, A8; ::_thesis: verum end; percases ( len f1 = 0 or not len f1 = 0 ) ; suppose len f1 = 0 ; ::_thesis: ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) then f1 = {} by FINSEQ_1:20; then A25: f = <*c2*> by A5, FINSEQ_1:34; take c2 ; ::_thesis: ( Sum f = c2 & dom c2 = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c2 = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c2 = (f /. (len f)) . (sup (dom (f /. (len f)))) ) 1 in Seg 1 by FINSEQ_1:3; then A26: 1 in dom f by A25, FINSEQ_1:38; A27: f /. 1 = f . 1 by A26, PARTFUN1:def_6 .= c2 by A25, FINSEQ_1:40 ; A28: f /. (len f) = c2 by A27, A25, FINSEQ_1:40; thus ( Sum f = c2 & dom c2 = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c2 = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c2 = (f /. (len f)) . (sup (dom (f /. (len f)))) ) by A25, Th40, A27, A28, Th27; ::_thesis: verum end; supposeA29: not len f1 = 0 ; ::_thesis: ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) consider c1 being with_endpoints Curve of T such that A30: ( Sum f1 = c1 & dom c1 = [.(inf (dom (f1 /. 1))),(sup (dom (f1 /. (len f1)))).] & the_first_point_of c1 = (f1 /. 1) . (inf (dom (f1 /. 1))) & the_last_point_of c1 = (f1 /. (len f1)) . (sup (dom (f1 /. (len f1)))) ) by A4, A6, A11, A20, A3, A29; set c = c1 + c2; A31: 0 + 1 < (len f1) + 1 by A29, XREAL_1:6; then A32: 1 <= len f1 by NAT_1:13; A33: len f1 < len f by A6, NAT_1:13; then A34: ( (f /. (len f1)) . (sup (dom (f /. (len f1)))) = (f /. ((len f1) + 1)) . (inf (dom (f /. ((len f1) + 1)))) & sup (dom (f /. (len f1))) = inf (dom (f /. ((len f1) + 1))) ) by A32, A7; (len f1) + 1 in Seg (len f) by A6, A31, FINSEQ_1:1; then A35: (len f1) + 1 in dom f by FINSEQ_1:def_3; A36: f /. ((len f1) + 1) = f . ((len f1) + 1) by A35, PARTFUN1:def_6 .= c2 by A5, FINSEQ_1:42 ; A37: inf (dom (f1 /. 1)) <= sup (dom (f1 /. (len f1))) by A30, XXREAL_1:29; A38: dom f1 c= dom f by A5, FINSEQ_1:26; len f1 in Seg (len f1) by A29, FINSEQ_1:3; then A39: len f1 in dom f1 by FINSEQ_1:def_3; A40: f1 /. (len f1) = f1 . (len f1) by A39, PARTFUN1:def_6 .= f . (len f1) by A5, A39, FINSEQ_1:def_7 .= f /. (len f1) by A39, A38, PARTFUN1:def_6 ; A41: sup (dom c1) = inf (dom c2) by A36, A34, A40, A30, XXREAL_1:29, XXREAL_2:29; A42: the_last_point_of c1 = the_first_point_of c2 by A36, A30, A40, A33, A32, A7; A43: ( c1 + c2 is with_endpoints & dom (c1 + c2) = [.(inf (dom c1)),(sup (dom c2)).] & (c1 + c2) . (inf (dom c1)) = the_first_point_of c1 & (c1 + c2) . (sup (dom c2)) = the_last_point_of c2 ) by A41, A42, Th38; 1 in Seg (len f1) by A32, FINSEQ_1:1; then A44: 1 in dom f1 by FINSEQ_1:def_3; A45: f1 /. 1 = f1 . 1 by A44, PARTFUN1:def_6 .= f . 1 by A44, A5, FINSEQ_1:def_7 .= f /. 1 by A44, A38, PARTFUN1:def_6 ; reconsider c = c1 + c2 as with_endpoints Curve of T by A41, A42, Th38; take c ; ::_thesis: ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) inf (dom c1) <= sup (dom c2) by A43, XXREAL_1:29; hence ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) by A43, A45, A37, A30, A5, Th41, A36, A6, XXREAL_2:25, XXREAL_2:29; ::_thesis: verum end; end; end; for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2); hence for f being FinSequence of Curves T st len f > 0 & ( for i being Nat st 1 <= i & i < len f holds ( (f /. i) . (sup (dom (f /. i))) = (f /. (i + 1)) . (inf (dom (f /. (i + 1)))) & sup (dom (f /. i)) = inf (dom (f /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f holds f /. i is with_endpoints ) holds ex c being with_endpoints Curve of T st ( Sum f = c & dom c = [.(inf (dom (f /. 1))),(sup (dom (f /. (len f)))).] & the_first_point_of c = (f /. 1) . (inf (dom (f /. 1))) & the_last_point_of c = (f /. (len f)) . (sup (dom (f /. (len f)))) ) ; ::_thesis: verum end; theorem Th44: :: TOPALG_6:44 for T being non empty TopSpace for f1, f2 being FinSequence of Curves T for c1, c2 being with_endpoints Curve of T st len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 & ( for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) holds c1,c2 are_homotopic proof let T be non empty TopSpace; ::_thesis: for f1, f2 being FinSequence of Curves T for c1, c2 being with_endpoints Curve of T st len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 & ( for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) holds c1,c2 are_homotopic defpred S1[ Nat] means for f1, f2 being FinSequence of Curves T for c1, c2 being with_endpoints Curve of T st len f1 = $1 & len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 & ( for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) holds c1,c2 are_homotopic ; A1: S1[ 0 ] ; A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] let f1, f2 be FinSequence of Curves T; ::_thesis: for c1, c2 being with_endpoints Curve of T st len f1 = k + 1 & len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 & ( for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) holds c1,c2 are_homotopic let c1, c2 be with_endpoints Curve of T; ::_thesis: ( len f1 = k + 1 & len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 & ( for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) implies c1,c2 are_homotopic ) assume A4: ( len f1 = k + 1 & len f1 > 0 ) ; ::_thesis: ( not len f1 = len f2 or not Sum f1 = c1 or not Sum f2 = c2 or ex i being Nat st ( 1 <= i & i < len f1 & not ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds ( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic ) consider f1a being FinSequence of Curves T, c1b being Element of Curves T such that A5: f1 = f1a ^ <*c1b*> by A4, FINSEQ_2:19; A6: len f1 = (len f1a) + (len <*c1b*>) by A5, FINSEQ_1:22 .= (len f1a) + 1 by FINSEQ_1:39 ; assume A7: len f1 = len f2 ; ::_thesis: ( not Sum f1 = c1 or not Sum f2 = c2 or ex i being Nat st ( 1 <= i & i < len f1 & not ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds ( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic ) consider f2a being FinSequence of Curves T, c2b being Element of Curves T such that A8: f2 = f2a ^ <*c2b*> by A7, A4, FINSEQ_2:19; A9: len f2 = (len f2a) + (len <*c2b*>) by A8, FINSEQ_1:22 .= (len f2a) + 1 by FINSEQ_1:39 ; assume A10: ( Sum f1 = c1 & Sum f2 = c2 ) ; ::_thesis: ( ex i being Nat st ( 1 <= i & i < len f1 & not ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds ( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic ) assume A11: for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ; ::_thesis: ( ex i being Nat st ( 1 <= i & i < len f2 & not ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) or ex i being Nat st ( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds ( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic ) assume A12: for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ; ::_thesis: ( ex i being Nat st ( 1 <= i & i <= len f1 & ( for c3, c4 being with_endpoints Curve of T holds ( not c3 = f1 /. i or not c4 = f2 /. i or not c3,c4 are_homotopic or not dom c3 = dom c4 ) ) ) or c1,c2 are_homotopic ) assume A13: for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ; ::_thesis: c1,c2 are_homotopic A14: dom f1 = Seg (len f1) by FINSEQ_1:def_3 .= dom f2 by A7, FINSEQ_1:def_3 ; A15: 1 <= len f1 by A4, NAT_1:11; then len f1 in Seg (len f1) by FINSEQ_1:1; then A16: len f1 in dom f1 by FINSEQ_1:def_3; then A17: f1 /. (len f1) = f1 . (len f1) by PARTFUN1:def_6; consider c1bb, c2bb being with_endpoints Curve of T such that A18: ( c1bb = f1 /. (len f1) & c2bb = f2 /. (len f1) & c1bb,c2bb are_homotopic & dom c1bb = dom c2bb ) by A13, A15; A19: f1 . (len f1) = c1b by A5, A6, FINSEQ_1:42; A20: f2 . (len f2) = c2b by A8, A9, FINSEQ_1:42; A21: ( c1bb = c1b & c2bb = c2b ) by A7, A16, A14, A18, A19, A20, PARTFUN1:def_6; reconsider c1b = c1b, c2b = c2b as with_endpoints Curve of T by A7, A20, A18, A14, A16, A17, A5, A6, FINSEQ_1:42, PARTFUN1:def_6; percases ( len f1a > 0 or not len f1a > 0 ) ; supposeA22: len f1a > 0 ; ::_thesis: c1,c2 are_homotopic A23: for i being Nat st 1 <= i & i < len f1a holds ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) ) proof let i be Nat; ::_thesis: ( 1 <= i & i < len f1a implies ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) ) ) assume A24: ( 1 <= i & i < len f1a ) ; ::_thesis: ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) ) then A25: i + 1 < (len f1a) + 1 by XREAL_1:6; i <= i + 1 by NAT_1:11; then A26: i < len f1 by A6, A25, XXREAL_0:2; i in Seg (len f1) by A24, A26, FINSEQ_1:1; then A27: i in dom f1 by FINSEQ_1:def_3; i in Seg (len f1a) by A24, FINSEQ_1:1; then A28: i in dom f1a by FINSEQ_1:def_3; A29: f1 /. i = f1 . i by A27, PARTFUN1:def_6 .= f1a . i by A28, A5, FINSEQ_1:def_7 .= f1a /. i by A28, PARTFUN1:def_6 ; A30: 1 <= i + 1 by NAT_1:11; i + 1 in Seg (len f1) by A30, A25, A6, FINSEQ_1:1; then A31: i + 1 in dom f1 by FINSEQ_1:def_3; i + 1 <= len f1a by A24, NAT_1:13; then i + 1 in Seg (len f1a) by A30, FINSEQ_1:1; then A32: i + 1 in dom f1a by FINSEQ_1:def_3; f1 /. (i + 1) = f1 . (i + 1) by A31, PARTFUN1:def_6 .= f1a . (i + 1) by A32, A5, FINSEQ_1:def_7 .= f1a /. (i + 1) by A32, PARTFUN1:def_6 ; hence ( (f1a /. i) . (sup (dom (f1a /. i))) = (f1a /. (i + 1)) . (inf (dom (f1a /. (i + 1)))) & sup (dom (f1a /. i)) = inf (dom (f1a /. (i + 1))) ) by A26, A29, A24, A11; ::_thesis: verum end; for i being Nat st 1 <= i & i <= len f1a holds f1a /. i is with_endpoints proof let i be Nat; ::_thesis: ( 1 <= i & i <= len f1a implies f1a /. i is with_endpoints ) assume A33: ( 1 <= i & i <= len f1a ) ; ::_thesis: f1a /. i is with_endpoints then A34: i + 1 <= (len f1a) + 1 by XREAL_1:6; i <= i + 1 by NAT_1:11; then A35: i <= len f1 by A6, A34, XXREAL_0:2; i in Seg (len f1) by A33, A35, FINSEQ_1:1; then A36: i in dom f1 by FINSEQ_1:def_3; i in Seg (len f1a) by A33, FINSEQ_1:1; then A37: i in dom f1a by FINSEQ_1:def_3; A38: ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) by A33, A35, A13; f1 /. i = f1 . i by A36, PARTFUN1:def_6 .= f1a . i by A37, A5, FINSEQ_1:def_7 .= f1a /. i by A37, PARTFUN1:def_6 ; hence f1a /. i is with_endpoints by A38; ::_thesis: verum end; then consider c1a being with_endpoints Curve of T such that A39: ( Sum f1a = c1a & dom c1a = [.(inf (dom (f1a /. 1))),(sup (dom (f1a /. (len f1a)))).] & the_first_point_of c1a = (f1a /. 1) . (inf (dom (f1a /. 1))) & the_last_point_of c1a = (f1a /. (len f1a)) . (sup (dom (f1a /. (len f1a)))) ) by A22, A23, Th43; A40: for i being Nat st 1 <= i & i < len f2a holds ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) ) proof let i be Nat; ::_thesis: ( 1 <= i & i < len f2a implies ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) ) ) assume A41: ( 1 <= i & i < len f2a ) ; ::_thesis: ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) ) then A42: i + 1 < (len f2a) + 1 by XREAL_1:6; i <= i + 1 by NAT_1:11; then A43: i < len f2 by A9, A42, XXREAL_0:2; i in Seg (len f2) by A41, A43, FINSEQ_1:1; then A44: i in dom f2 by FINSEQ_1:def_3; i in Seg (len f2a) by A41, FINSEQ_1:1; then A45: i in dom f2a by FINSEQ_1:def_3; A46: f2 /. i = f2 . i by A44, PARTFUN1:def_6 .= f2a . i by A45, A8, FINSEQ_1:def_7 .= f2a /. i by A45, PARTFUN1:def_6 ; A47: 1 <= i + 1 by NAT_1:11; i + 1 in Seg (len f2) by A47, A42, A9, FINSEQ_1:1; then A48: i + 1 in dom f2 by FINSEQ_1:def_3; i + 1 <= len f2a by A41, NAT_1:13; then i + 1 in Seg (len f2a) by A47, FINSEQ_1:1; then A49: i + 1 in dom f2a by FINSEQ_1:def_3; f2 /. (i + 1) = f2 . (i + 1) by A48, PARTFUN1:def_6 .= f2a . (i + 1) by A49, A8, FINSEQ_1:def_7 .= f2a /. (i + 1) by A49, PARTFUN1:def_6 ; hence ( (f2a /. i) . (sup (dom (f2a /. i))) = (f2a /. (i + 1)) . (inf (dom (f2a /. (i + 1)))) & sup (dom (f2a /. i)) = inf (dom (f2a /. (i + 1))) ) by A43, A46, A41, A12; ::_thesis: verum end; for i being Nat st 1 <= i & i <= len f2a holds f2a /. i is with_endpoints proof let i be Nat; ::_thesis: ( 1 <= i & i <= len f2a implies f2a /. i is with_endpoints ) assume A50: ( 1 <= i & i <= len f2a ) ; ::_thesis: f2a /. i is with_endpoints then A51: i + 1 <= (len f2a) + 1 by XREAL_1:6; i <= i + 1 by NAT_1:11; then A52: i <= len f2 by A9, A51, XXREAL_0:2; i in Seg (len f2) by A50, A52, FINSEQ_1:1; then A53: i in dom f2 by FINSEQ_1:def_3; i in Seg (len f2a) by A50, FINSEQ_1:1; then A54: i in dom f2a by FINSEQ_1:def_3; A55: ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) by A7, A50, A52, A13; f2 /. i = f2 . i by A53, PARTFUN1:def_6 .= f2a . i by A54, A8, FINSEQ_1:def_7 .= f2a /. i by A54, PARTFUN1:def_6 ; hence f2a /. i is with_endpoints by A55; ::_thesis: verum end; then consider c2a being with_endpoints Curve of T such that A56: ( Sum f2a = c2a & dom c2a = [.(inf (dom (f2a /. 1))),(sup (dom (f2a /. (len f2a)))).] & the_first_point_of c2a = (f2a /. 1) . (inf (dom (f2a /. 1))) & the_last_point_of c2a = (f2a /. (len f2a)) . (sup (dom (f2a /. (len f2a)))) ) by A6, A7, A9, A22, A40, Th43; for i being Nat st 1 <= i & i <= len f1a holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) proof let i be Nat; ::_thesis: ( 1 <= i & i <= len f1a implies ex c3, c4 being with_endpoints Curve of T st ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) assume A57: ( 1 <= i & i <= len f1a ) ; ::_thesis: ex c3, c4 being with_endpoints Curve of T st ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) then A58: i + 1 <= (len f1a) + 1 by XREAL_1:6; i <= i + 1 by NAT_1:11; then A59: i <= len f1 by A6, A58, XXREAL_0:2; i in Seg (len f1) by A57, A59, FINSEQ_1:1; then A60: i in dom f1 by FINSEQ_1:def_3; i in Seg (len f1a) by A57, FINSEQ_1:1; then A61: i in dom f1a by FINSEQ_1:def_3; i in Seg (len f2) by A57, A59, A7, FINSEQ_1:1; then A62: i in dom f2 by FINSEQ_1:def_3; i in Seg (len f2a) by A57, A6, A7, A9, FINSEQ_1:1; then A63: i in dom f2a by FINSEQ_1:def_3; consider c3, c4 being with_endpoints Curve of T such that A64: ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) by A57, A59, A13; take c3 ; ::_thesis: ex c4 being with_endpoints Curve of T st ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) take c4 ; ::_thesis: ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) A65: f1 /. i = f1 . i by A60, PARTFUN1:def_6 .= f1a . i by A61, A5, FINSEQ_1:def_7 .= f1a /. i by A61, PARTFUN1:def_6 ; f2 /. i = f2 . i by A62, PARTFUN1:def_6 .= f2a . i by A63, A8, FINSEQ_1:def_7 .= f2a /. i by A63, PARTFUN1:def_6 ; hence ( c3 = f1a /. i & c4 = f2a /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) by A64, A65; ::_thesis: verum end; then A66: c1a,c2a are_homotopic by A3, A4, A23, A6, A22, A40, A7, A9, A39, A56; A67: c1 = c1a + c1b by A10, A5, A39, Th41; A68: c2 = c2a + c2b by A10, A8, A56, Th41; A69: f1 /. (len f1) = c1b by A5, A6, A17, FINSEQ_1:42; A70: 0 + 1 < (len f1a) + 1 by A22, XREAL_1:6; then A71: ( 1 <= len f1a & len f1a < len f1 ) by A6, NAT_1:13; then len f1a in Seg (len f1) by FINSEQ_1:1; then A72: len f1a in dom f1 by FINSEQ_1:def_3; len f1a in Seg (len f1a) by A71, FINSEQ_1:1; then A73: len f1a in dom f1a by FINSEQ_1:def_3; then A74: f1a /. (len f1a) = f1a . (len f1a) by PARTFUN1:def_6 .= f1 . (len f1a) by A5, A73, FINSEQ_1:def_7 .= f1 /. (len f1a) by A72, PARTFUN1:def_6 ; len f2a in Seg (len f2) by A71, A6, A9, A7, FINSEQ_1:1; then A75: len f2a in dom f2 by FINSEQ_1:def_3; len f2a in Seg (len f2a) by A71, A6, A7, A9, FINSEQ_1:1; then A76: len f2a in dom f2a by FINSEQ_1:def_3; then A77: f2a /. (len f2a) = f2a . (len f2a) by PARTFUN1:def_6 .= f2 . (len f2a) by A8, A76, FINSEQ_1:def_7 .= f2 /. (len f2a) by A75, PARTFUN1:def_6 ; 1 in Seg (len f1) by A70, A6, FINSEQ_1:1; then A78: 1 in dom f1 by FINSEQ_1:def_3; 1 in Seg (len f1a) by A71, FINSEQ_1:1; then A79: 1 in dom f1a by FINSEQ_1:def_3; then A80: f1a /. 1 = f1a . 1 by PARTFUN1:def_6 .= f1 . 1 by A5, A79, FINSEQ_1:def_7 .= f1 /. 1 by A78, PARTFUN1:def_6 ; 1 in Seg (len f2) by A70, A7, A6, FINSEQ_1:1; then A81: 1 in dom f2 by FINSEQ_1:def_3; 1 in Seg (len f2a) by A71, A6, A7, A9, FINSEQ_1:1; then A82: 1 in dom f2a by FINSEQ_1:def_3; then A83: f2a /. 1 = f2a . 1 by PARTFUN1:def_6 .= f2 . 1 by A8, A82, FINSEQ_1:def_7 .= f2 /. 1 by A81, PARTFUN1:def_6 ; A84: ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. 1 & c4 = f2 /. 1 & c3,c4 are_homotopic & dom c3 = dom c4 ) by A13, A15; A85: ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. (len f1a) & c4 = f2 /. (len f1a) & c3,c4 are_homotopic & dom c3 = dom c4 ) by A71, A13; A86: the_last_point_of c1a = the_first_point_of c1b by A69, A6, A74, A11, A71, A39; sup (dom c1a) = sup (dom (f1 /. (len f1a))) by A74, A39, XXREAL_1:29, XXREAL_2:29 .= inf (dom (f1 /. ((len f1a) + 1))) by A11, A71 .= inf (dom c1b) by A5, A6, A17, FINSEQ_1:42 ; hence c1,c2 are_homotopic by A66, A67, A68, A18, A21, A86, Th39, A56, A84, A85, A80, A83, A6, A7, A9, A74, A77, A39; ::_thesis: verum end; supposeA87: not len f1a > 0 ; ::_thesis: c1,c2 are_homotopic then f1a = {} by FINSEQ_1:20; then f1 = <*c1b*> by A5, FINSEQ_1:34; then A88: Sum f1 = c1b by Th40; f2a = {} by A87, A6, A7, A9, FINSEQ_1:20; then f2 = <*c2b*> by A8, FINSEQ_1:34; hence c1,c2 are_homotopic by A88, A18, A21, A10, Th40; ::_thesis: verum end; end; end; for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2); hence for f1, f2 being FinSequence of Curves T for c1, c2 being with_endpoints Curve of T st len f1 > 0 & len f1 = len f2 & Sum f1 = c1 & Sum f2 = c2 & ( for i being Nat st 1 <= i & i < len f1 holds ( (f1 /. i) . (sup (dom (f1 /. i))) = (f1 /. (i + 1)) . (inf (dom (f1 /. (i + 1)))) & sup (dom (f1 /. i)) = inf (dom (f1 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i < len f2 holds ( (f2 /. i) . (sup (dom (f2 /. i))) = (f2 /. (i + 1)) . (inf (dom (f2 /. (i + 1)))) & sup (dom (f2 /. i)) = inf (dom (f2 /. (i + 1))) ) ) & ( for i being Nat st 1 <= i & i <= len f1 holds ex c3, c4 being with_endpoints Curve of T st ( c3 = f1 /. i & c4 = f2 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) holds c1,c2 are_homotopic ; ::_thesis: verum end; theorem Th45: :: TOPALG_6:45 for T being non empty TopStruct for c being with_endpoints Curve of T for h being FinSequence of REAL st len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing holds ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) proof let T be non empty TopStruct ; ::_thesis: for c being with_endpoints Curve of T for h being FinSequence of REAL st len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing holds ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) defpred S1[ Nat] means for c being with_endpoints Curve of T for h being FinSequence of REAL st len h = $1 & len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing holds ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ); A1: S1[ 0 ] ; A2: for k being Nat st S1[k] holds S1[k + 1] proof let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] let c be with_endpoints Curve of T; ::_thesis: for h being FinSequence of REAL st len h = k + 1 & len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing holds ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) let h be FinSequence of REAL ; ::_thesis: ( len h = k + 1 & len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing implies ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) ) assume A4: ( len h = k + 1 & len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing ) ; ::_thesis: ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) consider h1 being FinSequence of REAL , r being Element of REAL such that A5: h = h1 ^ <*r*> by A4, FINSEQ_2:19; A6: len h = (len h1) + 1 by A5, FINSEQ_2:16; reconsider r1 = h . k as real number ; consider c1, c2 being Element of Curves T such that A7: ( c = c1 + c2 & c1 = c | [.(inf (dom c)),r1.] & c2 = c | [.r1,(sup (dom c)).] ) by Th37; A8: k < k + 1 by NAT_1:13; 1 <= 1 + k by NAT_1:12; then 1 in Seg (len h) by A4, FINSEQ_1:1; then A9: 1 in dom h by FINSEQ_1:def_3; percases ( len h1 < 2 or len h1 >= 2 ) ; suppose len h1 < 2 ; ::_thesis: ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) then len h1 < 1 + 1 ; then A10: len h1 <= 1 by NAT_1:13; percases ( h1 = {} or h1 <> {} ) ; suppose h1 = {} ; ::_thesis: ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) then h = <*r*> by A5, FINSEQ_1:34; then len h = 1 by FINSEQ_1:40; hence ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) by A4; ::_thesis: verum end; suppose h1 <> {} ; ::_thesis: ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) then len h1 >= 1 by FINSEQ_1:20; then A11: len h1 = 1 by A10, XXREAL_0:1; set f = <*c*>; take <*c*> ; ::_thesis: ( len <*c*> = (len h) - 1 & c = Sum <*c*> & ( for i being Nat st 1 <= i & i <= len <*c*> holds <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) A12: len <*c*> = 1 by FINSEQ_1:40; thus len <*c*> = (len h) - 1 by A11, A6, FINSEQ_1:40; ::_thesis: ( c = Sum <*c*> & ( for i being Nat st 1 <= i & i <= len <*c*> holds <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) thus c = Sum <*c*> by Th40; ::_thesis: for i being Nat st 1 <= i & i <= len <*c*> holds <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] thus for i being Nat st 1 <= i & i <= len <*c*> holds <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ::_thesis: verum proof let i be Nat; ::_thesis: ( 1 <= i & i <= len <*c*> implies <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] ) assume A13: ( 1 <= i & i <= len <*c*> ) ; ::_thesis: <*c*> /. i = c | [.(h /. i),(h /. (i + 1)).] then A14: i = 1 by A12, XXREAL_0:1; i in Seg (len <*c*>) by A13, FINSEQ_1:1; then A15: i in dom <*c*> by FINSEQ_1:def_3; A16: h /. i = inf (dom c) by A4, A14, A9, PARTFUN1:def_6; len h in Seg (len h) by A6, FINSEQ_1:3; then len h in dom h by FINSEQ_1:def_3; then A17: h /. (i + 1) = sup (dom c) by A4, A6, A11, A14, PARTFUN1:def_6; thus <*c*> /. i = <*c*> . i by A15, PARTFUN1:def_6 .= c by A14, FINSEQ_1:40 .= c | (dom c) .= c | [.(h /. i),(h /. (i + 1)).] by A16, A17, Th27 ; ::_thesis: verum end; end; end; end; supposeA18: len h1 >= 2 ; ::_thesis: ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) then A19: 1 < k by A4, A6, XXREAL_0:2; then k in Seg (len h) by A4, A8, FINSEQ_1:1; then A20: k in dom h by FINSEQ_1:def_3; k + 1 in Seg (len h) by A4, FINSEQ_1:4; then A21: k + 1 in dom h by FINSEQ_1:def_3; h . k <= h . (k + 1) by A8, A20, A21, A4, VALUED_0:def_13; then [.(inf (dom c)),r1.] c= [.(inf (dom c)),(sup (dom c)).] by A4, XXREAL_1:34; then A22: [.(inf (dom c)),r1.] c= dom c by Th27; A23: dom c1 = (dom c) /\ [.(inf (dom c)),r1.] by A7, RELAT_1:61 .= [.(inf (dom c)),r1.] by A22, XBOOLE_1:28 ; A24: inf (dom c) <= r1 by A4, A19, A9, A20, VALUED_0:def_13; then A25: r1 = sup (dom c1) by A23, XXREAL_2:29; A26: inf (dom c1) = inf (dom c) by A24, A23, XXREAL_2:25; then inf (dom c1) in [.(inf (dom c)),r1.] by A24, XXREAL_1:1; then dom c1 is left_end by A23, XXREAL_2:def_5; then A27: c1 is with_first_point by Def6; r1 in [.(inf (dom c)),r1.] by A24, XXREAL_1:1; then dom c1 is right_end by A25, A23, XXREAL_2:def_6; then A28: c1 is with_last_point by Def7; reconsider c1 = c1 as with_endpoints Curve of T by A27, A28; A29: h1 = h | (dom h1) by A5, FINSEQ_1:21; 1 in Seg k by A19, FINSEQ_1:1; then A30: 1 in dom h1 by A4, A6, FINSEQ_1:def_3; k in Seg k by A19, FINSEQ_1:1; then A31: len h1 in dom h1 by A4, A6, FINSEQ_1:def_3; A32: h1 . 1 = inf (dom c1) by A4, A26, A29, A30, FUNCT_1:49; A33: h1 . (len h1) = h . k by A6, A4, A29, A31, FUNCT_1:49 .= sup (dom c1) by A24, A23, XXREAL_2:29 ; A34: dom h c= REAL by XBOOLE_1:1; rng h c= REAL ; then reconsider h0 = h as PartFunc of REAL,REAL by A34, RELSET_1:4; A35: h0 | (dom h0) is increasing by A4; len h1 <= len h by A6, NAT_1:19; then Seg (len h1) c= Seg (len h) by FINSEQ_1:5; then Seg (len h1) c= dom h by FINSEQ_1:def_3; then A36: dom h1 c= dom h by FINSEQ_1:def_3; then A37: h1 is increasing by A29, A35, RFUNCT_2:28; consider f1 being FinSequence of Curves T such that A38: ( len f1 = (len h1) - 1 & c1 = Sum f1 & ( for i being Nat st 1 <= i & i <= len f1 holds f1 /. i = c1 | [.(h1 /. i),(h1 /. (i + 1)).] ) ) by A3, A4, A6, A18, A32, A33, A37; set f = f1 ^ <*c2*>; take f1 ^ <*c2*> ; ::_thesis: ( len (f1 ^ <*c2*>) = (len h) - 1 & c = Sum (f1 ^ <*c2*>) & ( for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) A39: len (f1 ^ <*c2*>) = (len f1) + (len <*c2*>) by FINSEQ_1:22 .= (len f1) + 1 by FINSEQ_1:40 ; thus len (f1 ^ <*c2*>) = (len h) - 1 by A6, A38, A39; ::_thesis: ( c = Sum (f1 ^ <*c2*>) & ( for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) thus c = Sum (f1 ^ <*c2*>) by Th41, A7, A38; ::_thesis: for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] thus for i being Nat st 1 <= i & i <= len (f1 ^ <*c2*>) holds (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ::_thesis: verum proof let i be Nat; ::_thesis: ( 1 <= i & i <= len (f1 ^ <*c2*>) implies (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] ) assume A40: ( 1 <= i & i <= len (f1 ^ <*c2*>) ) ; ::_thesis: (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] then i in Seg (len (f1 ^ <*c2*>)) by FINSEQ_1:1; then A41: i in dom (f1 ^ <*c2*>) by FINSEQ_1:def_3; percases ( i = len (f1 ^ <*c2*>) or i <> len (f1 ^ <*c2*>) ) ; supposeA42: i = len (f1 ^ <*c2*>) ; ::_thesis: (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] A43: h /. i = r1 by A42, A39, A38, A4, A6, A20, PARTFUN1:def_6; 1 + 1 <= i + 1 by A40, XREAL_1:6; then 1 < len h by A42, A39, A38, A6, XXREAL_0:2; then len h in Seg (len h) by FINSEQ_1:1; then len h in dom h by FINSEQ_1:def_3; then A44: h /. (i + 1) = sup (dom c) by A4, A42, A39, A38, A6, PARTFUN1:def_6; thus (f1 ^ <*c2*>) /. i = (f1 ^ <*c2*>) . i by A41, PARTFUN1:def_6 .= c | [.(h /. i),(h /. (i + 1)).] by A43, A44, A7, A42, A39, FINSEQ_1:42 ; ::_thesis: verum end; suppose i <> len (f1 ^ <*c2*>) ; ::_thesis: (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] then A45: i < (len f1) + 1 by A39, A40, XXREAL_0:1; then A46: i <= len f1 by NAT_1:13; then i in Seg (len f1) by A40, FINSEQ_1:1; then A47: i in dom f1 by FINSEQ_1:def_3; i in Seg (len h1) by A38, A40, A39, FINSEQ_1:1; then A48: i in dom h1 by FINSEQ_1:def_3; A49: h1 /. i = h1 . i by A48, PARTFUN1:def_6 .= (h | (dom h1)) . i by A5, FINSEQ_1:21 .= h . i by A48, FUNCT_1:49 .= h /. i by A48, A36, PARTFUN1:def_6 ; A50: i + 1 <= len h1 by A38, A45, NAT_1:13; 1 <= i + 1 by NAT_1:12; then i + 1 in Seg (len h1) by A50, FINSEQ_1:1; then A51: i + 1 in dom h1 by FINSEQ_1:def_3; A52: h1 /. (i + 1) = h1 . (i + 1) by A51, PARTFUN1:def_6 .= (h | (dom h1)) . (i + 1) by A5, FINSEQ_1:21 .= h . (i + 1) by A51, FUNCT_1:49 .= h /. (i + 1) by A51, A36, PARTFUN1:def_6 ; A53: i + 1 <= (len f1) + 1 by A45, NAT_1:13; h . (i + 1) <= h . k proof percases ( i + 1 = k or i + 1 <> k ) ; suppose i + 1 = k ; ::_thesis: h . (i + 1) <= h . k hence h . (i + 1) <= h . k ; ::_thesis: verum end; suppose i + 1 <> k ; ::_thesis: h . (i + 1) <= h . k then i + 1 < k by A38, A6, A4, A53, XXREAL_0:1; hence h . (i + 1) <= h . k by A51, A36, A20, A4, VALUED_0:def_13; ::_thesis: verum end; end; end; then A54: h /. (i + 1) <= r1 by A51, A36, PARTFUN1:def_6; h . 1 <= h . i proof percases ( i = 1 or i <> 1 ) ; suppose i = 1 ; ::_thesis: h . 1 <= h . i hence h . 1 <= h . i ; ::_thesis: verum end; suppose i <> 1 ; ::_thesis: h . 1 <= h . i then 1 < i by A40, XXREAL_0:1; hence h . 1 <= h . i by A36, A48, A9, A4, VALUED_0:def_13; ::_thesis: verum end; end; end; then A55: inf (dom c) <= h /. i by A4, A36, A48, PARTFUN1:def_6; (f1 ^ <*c2*>) . i = f1 . i by A47, FINSEQ_1:def_7 .= f1 /. i by A47, PARTFUN1:def_6 .= (c | [.(inf (dom c)),r1.]) | [.(h1 /. i),(h1 /. (i + 1)).] by A7, A38, A40, A46 .= c | [.(h1 /. i),(h1 /. (i + 1)).] by A55, A52, A49, A54, RELAT_1:74, XXREAL_1:34 ; hence (f1 ^ <*c2*>) /. i = c | [.(h /. i),(h /. (i + 1)).] by A41, A52, A49, PARTFUN1:def_6; ::_thesis: verum end; end; end; end; end; end; for k being Nat holds S1[k] from NAT_1:sch_2(A1, A2); hence for c being with_endpoints Curve of T for h being FinSequence of REAL st len h >= 2 & h . 1 = inf (dom c) & h . (len h) = sup (dom c) & h is increasing holds ex f being FinSequence of Curves T st ( len f = (len h) - 1 & c = Sum f & ( for i being Nat st 1 <= i & i <= len f holds f /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) ; ::_thesis: verum end; Lm3: for n being Nat for t being Point of (TUnitSphere n) for f being Loop of t st rng f <> the carrier of (TUnitSphere n) holds f is nullhomotopic proof let n be Nat; ::_thesis: for t being Point of (TUnitSphere n) for f being Loop of t st rng f <> the carrier of (TUnitSphere n) holds f is nullhomotopic let t be Point of (TUnitSphere n); ::_thesis: for f being Loop of t st rng f <> the carrier of (TUnitSphere n) holds f is nullhomotopic let f be Loop of t; ::_thesis: ( rng f <> the carrier of (TUnitSphere n) implies f is nullhomotopic ) assume A1: rng f <> the carrier of (TUnitSphere n) ; ::_thesis: f is nullhomotopic for x being set st x in rng f holds x in the carrier of (TUnitSphere n) ; then consider x being set such that A2: x in the carrier of (TUnitSphere n) and A3: not x in rng f by A1, TARSKI:1; reconsider n1 = n + 1 as Nat ; A4: [#] (Tunit_circle n1) c= [#] (TOP-REAL n1) by PRE_TOPC:def_4; A5: x in the carrier of (Tunit_circle n1) by A2, MFOLD_2:def_4; then reconsider p = x as Point of (TOP-REAL n1) by A4; p in the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by A5, TOPREALB:def_7; then A6: p in Sphere ((0. (TOP-REAL n1)),1) by TOPREALB:9; then - p in (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by Th3; then reconsider S = (TOP-REAL n1) | ((Sphere ((0. (TOP-REAL n1)),1)) \ {p}) as non empty SubSpace of TOP-REAL n1 ; A7: [#] S = (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by PRE_TOPC:def_5; then stereographic_projection (S,p) is being_homeomorphism by A6, MFOLD_2:31; then A8: TPlane (p,(0. (TOP-REAL n1))),S are_homeomorphic by T_0TOPSP:def_1; A9: S is having_trivial_Fundamental_Group by A8, Th13; Tunit_circle n1 is SubSpace of TOP-REAL n1 ; then A10: TUnitSphere n is SubSpace of TOP-REAL n1 by MFOLD_2:def_4; (Sphere ((0. (TOP-REAL n1)),1)) \ {p} c= Sphere ((0. (TOP-REAL n1)),1) by XBOOLE_1:36; then (Sphere ((0. (TOP-REAL n1)),1)) \ {p} c= the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by TOPREALB:9; then (Sphere ((0. (TOP-REAL n1)),1)) \ {p} c= the carrier of (Tunit_circle n1) by TOPREALB:def_7; then (Sphere ((0. (TOP-REAL n1)),1)) \ {p} c= the carrier of (TUnitSphere n) by MFOLD_2:def_4; then reconsider S0 = S as non empty SubSpace of TUnitSphere n by A7, A10, TOPMETR:3; 0 in the carrier of I[01] by BORSUK_1:43; then A11: 0 in dom f by FUNCT_2:def_1; t,t are_connected ; then A12: ( f is continuous & f . 0 = t & f . 1 = t ) by BORSUK_2:def_2; then t in rng f by A11, FUNCT_1:3; then A13: not t in {p} by A3, TARSKI:def_1; A14: the carrier of (TUnitSphere n) = the carrier of (Tunit_circle n1) by MFOLD_2:def_4 .= the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by TOPREALB:def_7 .= Sphere ((0. (TOP-REAL n1)),1) by TOPREALB:9 ; reconsider t0 = t as Point of S0 by A7, A14, A13, XBOOLE_0:def_5; dom f = the carrier of I[01] by FUNCT_2:def_1; then reconsider f0 = f as Function of I[01],S0 by A7, A3, A14, FUNCT_2:2, ZFMISC_1:34; A15: t0,t0 are_connected ; f0 is continuous by JORDAN16:8; then reconsider f0 = f as Loop of t0 by A12, A15, BORSUK_2:def_2; f0 is nullhomotopic by A9; hence f is nullhomotopic by Th18; ::_thesis: verum end; Lm4: for n being Nat for t being Point of (TUnitSphere n) for f being Loop of t st n >= 2 & rng f = the carrier of (TUnitSphere n) holds ex g being Loop of t st ( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) proof let n be Nat; ::_thesis: for t being Point of (TUnitSphere n) for f being Loop of t st n >= 2 & rng f = the carrier of (TUnitSphere n) holds ex g being Loop of t st ( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) let t be Point of (TUnitSphere n); ::_thesis: for f being Loop of t st n >= 2 & rng f = the carrier of (TUnitSphere n) holds ex g being Loop of t st ( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) let f be Loop of t; ::_thesis: ( n >= 2 & rng f = the carrier of (TUnitSphere n) implies ex g being Loop of t st ( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) ) assume that A1: n >= 2 and A2: rng f = the carrier of (TUnitSphere n) ; ::_thesis: ex g being Loop of t st ( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) reconsider n1 = n + 1 as Element of NAT ; Tunit_circle n1 is SubSpace of TOP-REAL n1 ; then A3: TUnitSphere n is SubSpace of TOP-REAL n1 by MFOLD_2:def_4; [#] (Tunit_circle n1) c= [#] (TOP-REAL n1) by PRE_TOPC:def_4; then A4: rng f c= the carrier of (TOP-REAL n1) by A2, MFOLD_2:def_4; dom f = the carrier of I[01] by FUNCT_2:def_1; then reconsider f1 = f as Function of I[01],(TOP-REAL n1) by A4, FUNCT_2:2; f1 is continuous by A3, PRE_TOPC:26; then consider h being FinSequence of REAL such that A5: ( h . 1 = 0 & h . (len h) = 1 & 5 <= len h & rng h c= the carrier of I[01] & h is increasing & ( for i being Element of NAT for Q being Subset of I[01] for W being Subset of (Euclid n1) st 1 <= i & i < len h & Q = [.(h /. i),(h /. (i + 1)).] & W = f1 .: Q holds diameter W < 1 ) ) by JGRAPH_8:1; set f2 = f * h; for x being set st x in rng (f * h) holds x in the carrier of (TUnitSphere n) ; then consider x being set such that A6: ( x in the carrier of (TUnitSphere n) & not x in rng (f * h) ) by A1, TARSKI:1; A7: [#] (Tunit_circle n1) c= [#] (TOP-REAL n1) by PRE_TOPC:def_4; A8: x in the carrier of (Tunit_circle n1) by A6, MFOLD_2:def_4; then reconsider p = x as Point of (TOP-REAL n1) by A7; p in the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by A8, TOPREALB:def_7; then A9: p in Sphere ((0. (TOP-REAL n1)),1) by TOPREALB:9; then A10: - p in (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by Th3; then reconsider U = (TOP-REAL n1) | ((Sphere ((0. (TOP-REAL n1)),1)) \ {p}) as non empty SubSpace of TOP-REAL n1 ; A11: [#] U = (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by PRE_TOPC:def_5; A12: - p in Sphere ((0. (TOP-REAL n1)),1) by A10, XBOOLE_0:def_5; then A13: - (- p) in (Sphere ((0. (TOP-REAL n1)),1)) \ {(- p)} by Th3; then reconsider V = (TOP-REAL n1) | ((Sphere ((0. (TOP-REAL n1)),1)) \ {(- p)}) as non empty SubSpace of TOP-REAL n1 ; A14: [#] V = (Sphere ((0. (TOP-REAL n1)),1)) \ {(- p)} by PRE_TOPC:def_5; A15: for i being Element of NAT st 1 <= i & i < len h & not f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U holds f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V proof let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len h & not f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U implies f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V ) assume A16: ( 1 <= i & i < len h ) ; ::_thesis: ( f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U or f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V ) i in Seg (len h) by A16, FINSEQ_1:1; then A17: i in dom h by FINSEQ_1:def_3; reconsider h1 = h as real-valued FinSequence ; reconsider i1 = i + 1 as Nat ; A18: ( i1 <= len h & 1 <= i1 ) by A16, NAT_1:13; then i1 in Seg (len h) by FINSEQ_1:1; then A19: i1 in dom h by FINSEQ_1:def_3; h1 . (i + 1) <= 1 by A5, A18, EUCLID_7:7; then A20: h /. (i + 1) <= 1 by A19, PARTFUN1:def_6; h1 . 1 <= h1 . i by A5, A16, EUCLID_7:7; then 0 <= h /. i by A5, A17, PARTFUN1:def_6; then reconsider Q = [.(h /. i),(h /. (i + 1)).] as Subset of I[01] by A20, BORSUK_1:40, XXREAL_1:34; f .: Q c= the carrier of (TUnitSphere n) ; then f .: Q c= [#] (Tunit_circle n1) by MFOLD_2:def_4; then f .: Q c= the carrier of (Tcircle ((0. (TOP-REAL n1)),1)) by TOPREALB:def_7; then A21: f .: Q c= Sphere ((0. (TOP-REAL n1)),1) by TOPREALB:9; reconsider W = f1 .: Q as Subset of (Euclid n1) by EUCLID:67; A22: diameter W < 1 by A16, A5; Sphere ((0. (TOP-REAL n1)),1) is bounded Subset of (Euclid n1) by JORDAN2C:11; then A23: W is bounded by A21, TBSP_1:14; ( not p in f .: Q or not - p in f .: Q ) proof assume A24: ( p in f .: Q & - p in f .: Q ) ; ::_thesis: contradiction reconsider p1 = p, p2 = - p as Point of (Euclid n1) by EUCLID:67; A25: dist (p1,p2) <= diameter W by A24, A23, TBSP_1:def_8; A26: Euclid n1 = MetrStruct(# (REAL n1),(Pitag_dist n1) #) by EUCLID:def_7; reconsider p3 = p1, p4 = p2 as Element of REAL n1 by A26; reconsider r1 = 1 as real number ; dist (p1,p2) = the distance of (Euclid n1) . (p1,p2) by METRIC_1:def_1 .= |.(p3 - p4).| by A26, EUCLID:def_6 ; then |.(p - (- p)).| < 1 by A25, A22, XXREAL_0:2; then |.(p + (- (- p))).| < 1 by EUCLID:41; then |.((r1 * p) + p).| < 1 by EUCLID:29; then |.((r1 * p) + (r1 * p)).| < 1 by EUCLID:29; then |.((1 + 1) * p).| < 1 by EUCLID:33; then A27: (abs 2) * |.p.| < 1 by EUCLID:11; |.(p - (0. (TOP-REAL n1))).| = 1 by A9, TOPREAL9:9; then |.(p + (- (0. (TOP-REAL n1)))).| = 1 by EUCLID:41; then |.(p + ((- 1) * (0. (TOP-REAL n1)))).| = 1 by EUCLID:39; then |.(p + (0. (TOP-REAL n1))).| = 1 by EUCLID:28; then A28: |.p.| = 1 by EUCLID:27; |.2.| = 2 by COMPLEX1:43; hence contradiction by A28, A27; ::_thesis: verum end; hence ( f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of U or f .: [.(h /. i),(h /. (i + 1)).] c= the carrier of V ) by A14, A11, A21, ZFMISC_1:34; ::_thesis: verum end; ( f is Path of t,t & t,t are_connected ) ; then reconsider c = f as with_endpoints Curve of (TUnitSphere n) by Th25; A29: 2 <= len h by A5, XXREAL_0:2; A30: ( inf (dom f) = 0 & sup (dom f) = 1 ) by Th4; then consider fc1 being FinSequence of Curves (TUnitSphere n) such that A31: ( len fc1 = (len h) - 1 & c = Sum fc1 & ( for i being Nat st 1 <= i & i <= len fc1 holds fc1 /. i = c | [.(h /. i),(h /. (i + 1)).] ) ) by A5, A29, Th45; A32: for i being Nat st 1 <= i & i <= len fc1 & not rng (fc1 /. i) c= the carrier of U holds rng (fc1 /. i) c= the carrier of V proof let i be Nat; ::_thesis: ( 1 <= i & i <= len fc1 & not rng (fc1 /. i) c= the carrier of U implies rng (fc1 /. i) c= the carrier of V ) assume A33: ( 1 <= i & i <= len fc1 ) ; ::_thesis: ( rng (fc1 /. i) c= the carrier of U or rng (fc1 /. i) c= the carrier of V ) then A34: i < ((len h) - 1) + 1 by A31, NAT_1:13; reconsider i0 = i as Element of NAT by ORDINAL1:def_12; f .: [.(h /. i0),(h /. (i0 + 1)).] = rng (f | [.(h /. i0),(h /. (i0 + 1)).]) by RELAT_1:115 .= rng (fc1 /. i) by A33, A31 ; hence ( rng (fc1 /. i) c= the carrier of U or rng (fc1 /. i) c= the carrier of V ) by A33, A34, A15; ::_thesis: verum end; A35: for c1 being with_endpoints Curve of (TUnitSphere n) st rng c1 c= the carrier of V & the_first_point_of c1 <> p & the_last_point_of c1 <> p & not inf (dom c1) = sup (dom c1) holds ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) proof let c1 be with_endpoints Curve of (TUnitSphere n); ::_thesis: ( rng c1 c= the carrier of V & the_first_point_of c1 <> p & the_last_point_of c1 <> p & not inf (dom c1) = sup (dom c1) implies ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) ) assume A36: rng c1 c= the carrier of V ; ::_thesis: ( not the_first_point_of c1 <> p or not the_last_point_of c1 <> p or inf (dom c1) = sup (dom c1) or ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) ) assume A37: ( the_first_point_of c1 <> p & the_last_point_of c1 <> p ) ; ::_thesis: ( inf (dom c1) = sup (dom c1) or ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) ) assume A38: not inf (dom c1) = sup (dom c1) ; ::_thesis: ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) set t1 = the_first_point_of c1; set t2 = the_last_point_of c1; reconsider p1 = c1 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) as Path of the_first_point_of c1, the_last_point_of c1 by Th29; stereographic_projection (V,(- p)) is being_homeomorphism by A12, A14, MFOLD_2:31; then A39: TPlane ((- p),(0. (TOP-REAL n1))),V are_homeomorphic by T_0TOPSP:def_1; - p <> 0. (TOP-REAL n1) proof assume - p = 0. (TOP-REAL n1) ; ::_thesis: contradiction then (- p) - (0. (TOP-REAL n1)) = 0. (TOP-REAL n1) by EUCLID:42; then |.(0. (TOP-REAL n1)).| = 1 by A12, TOPREAL9:9; hence contradiction by EUCLID_2:39; ::_thesis: verum end; then A40: TOP-REAL n, TPlane ((- p),(0. (TOP-REAL n1))) are_homeomorphic by MFOLD_2:29; then TOP-REAL n,V are_homeomorphic by A39, BORSUK_3:3; then consider fh being Function of (TOP-REAL n),V such that A41: fh is being_homeomorphism by T_0TOPSP:def_1; A42: ( dom fh = [#] (TOP-REAL n) & rng fh = [#] V ) by A41, TOPS_2:58; A43: [#] V is infinite by A1, A41, A42, CARD_1:59; reconsider v = p as Point of V by A13, A14; reconsider S = ([#] V) \ {v} as non empty Subset of V by A43, RAMSEY_1:4; A44: V | S is pathwise_connected by A1, A40, Th10, A39, BORSUK_3:3; A45: the_first_point_of c1 in rng c1 by Th31; A46: not the_first_point_of c1 in {v} by A37, TARSKI:def_1; A47: the_last_point_of c1 in rng c1 by Th31; A48: not the_last_point_of c1 in {v} by A37, TARSKI:def_1; ( the_first_point_of c1 in S & the_last_point_of c1 in S ) by A45, A46, A47, A36, A48, XBOOLE_0:def_5; then ( the_first_point_of c1 in [#] (V | S) & the_last_point_of c1 in [#] (V | S) ) by PRE_TOPC:def_5; then reconsider v1 = the_first_point_of c1, v2 = the_last_point_of c1 as Point of (V | S) ; A49: v1,v2 are_connected by A44, BORSUK_2:def_3; then consider p3 being Function of I[01],(V | S) such that A50: ( p3 is continuous & p3 . 0 = v1 & p3 . 1 = v2 ) by BORSUK_2:def_1; reconsider p3 = p3 as Path of v1,v2 by A50, A49, BORSUK_2:def_2; A51: Tcircle ((0. (TOP-REAL n1)),1) = Tunit_circle n1 by TOPREALB:def_7 .= TUnitSphere n by MFOLD_2:def_4 ; A52: V is SubSpace of (TOP-REAL n1) | (Sphere ((0. (TOP-REAL n1)),1)) by TOPMETR:22, XBOOLE_1:36; then A53: V is SubSpace of Tcircle ((0. (TOP-REAL n1)),1) by TOPREALB:def_6; reconsider S0 = V as non empty SubSpace of TUnitSphere n by A51, A52, TOPREALB:def_6; reconsider s1 = the_first_point_of c1, s2 = the_last_point_of c1 as Point of S0 by A45, A47, A36; A54: dom p3 = [#] I[01] by FUNCT_2:def_1; A55: [#] S0 c= [#] (TUnitSphere n) by PRE_TOPC:def_4; rng p3 c= [#] (V | S) ; then A56: rng p3 c= S by PRE_TOPC:def_5; then rng p3 c= [#] S0 by XBOOLE_1:1; then reconsider p3 = p3 as Function of I[01],(TUnitSphere n) by A54, A55, FUNCT_2:2, XBOOLE_1:1; V | S is SubSpace of TUnitSphere n by A53, A51, TSEP_1:7; then A57: p3 is continuous by A50, PRE_TOPC:26; then A58: the_first_point_of c1, the_last_point_of c1 are_connected by A50, BORSUK_2:def_1; then reconsider p2 = p3 as Path of the_first_point_of c1, the_last_point_of c1 by A50, A57, BORSUK_2:def_2; rng p1 c= rng c1 by RELAT_1:26; then A59: rng p1 c= [#] V by A36, XBOOLE_1:1; A60: rng p2 c= [#] V by A56, XBOOLE_1:1; A61: s1,s2 are_connected by A58, A60, JORDAN:29; reconsider p5 = p1, p6 = p2 as Path of s1,s2 by A58, A60, A59, JORDAN:29; reconsider n0 = n as Element of NAT by ORDINAL1:def_12; S0 is simply_connected by Th14, A39; then Class ((EqRel (S0,s1,s2)),p5) = Class ((EqRel (S0,s1,s2)),p6) by Th12; then p5,p6 are_homotopic by A61, TOPALG_1:46; then A62: p1,p2 are_homotopic by A58, A61, Th6; set r1 = inf (dom c1); set r2 = sup (dom c1); A63: inf (dom c1) <= sup (dom c1) by XXREAL_2:40; then A64: inf (dom c1) < sup (dom c1) by A38, XXREAL_0:1; then reconsider c2 = p2 * (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) as with_endpoints Curve of (TUnitSphere n) by A58, Th32; take c2 ; ::_thesis: ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) rng (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) c= [#] (Closed-Interval-TSpace (0,1)) by RELAT_1:def_19; then rng (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) c= dom p2 by FUNCT_2:def_1, TOPMETR:20; then dom c2 = dom (L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) by RELAT_1:27; then dom c2 = [#] (Closed-Interval-TSpace ((inf (dom c1)),(sup (dom c1)))) by FUNCT_2:def_1; then A65: dom c2 = [.(inf (dom c1)),(sup (dom c1)).] by A63, TOPMETR:18; A66: c2 * (L[01] (0,1,(inf (dom c1)),(sup (dom c1)))) = p2 * ((L[01] ((inf (dom c1)),(sup (dom c1)),0,1)) * (L[01] (0,1,(inf (dom c1)),(sup (dom c1))))) by RELAT_1:36 .= p2 * (id (Closed-Interval-TSpace (0,1))) by Th1, A64, Th2 .= p2 by FUNCT_2:17, TOPMETR:20 ; ( inf (dom c1) = inf (dom c2) & sup (dom c1) = sup (dom c2) ) by A65, Th27; hence c1,c2 are_homotopic by A62, A66, Def11; ::_thesis: ( rng c2 c= the carrier of U & dom c1 = dom c2 ) A67: rng c2 c= rng p2 by RELAT_1:26; A68: ((Sphere ((0. (TOP-REAL n1)),1)) \ {p}) \ {(- p)} c= (Sphere ((0. (TOP-REAL n1)),1)) \ {p} by XBOOLE_1:36; rng c2 c= ([#] V) \ {p} by A56, A67, XBOOLE_1:1; then rng c2 c= (Sphere ((0. (TOP-REAL n1)),1)) \ ({(- p)} \/ {p}) by A14, XBOOLE_1:41; then rng c2 c= ((Sphere ((0. (TOP-REAL n1)),1)) \ {p}) \ {(- p)} by XBOOLE_1:41; hence rng c2 c= the carrier of U by A11, A68, XBOOLE_1:1; ::_thesis: dom c1 = dom c2 thus dom c1 = dom c2 by A65, Th27; ::_thesis: verum end; A69: for i being Nat st 1 <= i & i <= len fc1 holds ( i + 1 in dom h & i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) ) proof let i be Nat; ::_thesis: ( 1 <= i & i <= len fc1 implies ( i + 1 in dom h & i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) ) ) assume A70: ( 1 <= i & i <= len fc1 ) ; ::_thesis: ( i + 1 in dom h & i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) ) A71: 1 <= 1 + i by NAT_1:11; A72: i + 1 <= ((len h) - 1) + 1 by A70, A31, XREAL_1:6; then i + 1 in Seg (len h) by A71, FINSEQ_1:1; hence A73: i + 1 in dom h by FINSEQ_1:def_3; ::_thesis: ( i in dom h & dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) ) A74: i < i + 1 by NAT_1:13; i <= len h by A72, NAT_1:13; then i in Seg (len h) by A70, FINSEQ_1:1; hence A75: i in dom h by FINSEQ_1:def_3; ::_thesis: ( dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] & h /. i < h /. (i + 1) ) A76: h /. i = h . i by A75, PARTFUN1:def_6; A77: h /. (i + 1) = h . (i + 1) by A73, PARTFUN1:def_6; A78: 0 <= h . i proof percases ( i = 1 or not i = 1 ) ; suppose i = 1 ; ::_thesis: 0 <= h . i hence 0 <= h . i by A5; ::_thesis: verum end; suppose not i = 1 ; ::_thesis: 0 <= h . i then A79: 1 < i by A70, XXREAL_0:1; 1 <= len h by A72, A71, XXREAL_0:2; then 1 in Seg (len h) by FINSEQ_1:1; then 1 in dom h by FINSEQ_1:def_3; hence 0 <= h . i by A5, A75, A79, VALUED_0:def_13; ::_thesis: verum end; end; end; A80: h . (i + 1) <= 1 proof percases ( i + 1 = len h or not i + 1 = len h ) ; suppose i + 1 = len h ; ::_thesis: h . (i + 1) <= 1 hence h . (i + 1) <= 1 by A5; ::_thesis: verum end; suppose not i + 1 = len h ; ::_thesis: h . (i + 1) <= 1 then A81: i + 1 < len h by A72, XXREAL_0:1; len h in Seg (len h) by A5, FINSEQ_1:3; then A82: len h in dom h by FINSEQ_1:def_3; i + 1 in Seg (len h) by A72, A71, FINSEQ_1:1; then i + 1 in dom h by FINSEQ_1:def_3; hence h . (i + 1) <= 1 by A5, A81, A82, VALUED_0:def_13; ::_thesis: verum end; end; end; [.(h . i),(h . (i + 1)).] c= [.0,1.] by A78, A80, XXREAL_1:34; then A83: [.(h . i),(h . (i + 1)).] c= dom c by A30, Th27; A84: fc1 /. i = c | [.(h /. i),(h /. (i + 1)).] by A31, A70; thus dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] by A84, A83, A76, A77, RELAT_1:62; ::_thesis: h /. i < h /. (i + 1) thus h /. i < h /. (i + 1) by A77, A76, A75, A73, A74, A5, VALUED_0:def_13; ::_thesis: verum end; A85: for i being Nat st 1 <= i & i <= len fc1 holds ( fc1 /. i is with_endpoints & ( for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i holds ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) ) proof let i be Nat; ::_thesis: ( 1 <= i & i <= len fc1 implies ( fc1 /. i is with_endpoints & ( for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i holds ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) ) ) assume A86: ( 1 <= i & i <= len fc1 ) ; ::_thesis: ( fc1 /. i is with_endpoints & ( for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i holds ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) ) A87: fc1 /. i = c | [.(h /. i),(h /. (i + 1)).] by A31, A86; A88: i + 1 in dom h by A69, A86; A89: i in dom h by A69, A86; A90: i < i + 1 by NAT_1:13; A91: h . i < h . (i + 1) by A89, A88, A90, A5, VALUED_0:def_13; A92: dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] by A69, A86; h /. i < h /. (i + 1) by A69, A86; then ( dom (fc1 /. i) is left_end & dom (fc1 /. i) is right_end ) by A92, XXREAL_2:33; then ( fc1 /. i is with_first_point & fc1 /. i is with_last_point ) by Def7, Def6; hence fc1 /. i is with_endpoints ; ::_thesis: for c1 being with_endpoints Curve of (TUnitSphere n) st c1 = fc1 /. i holds ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) let c1 be with_endpoints Curve of (TUnitSphere n); ::_thesis: ( c1 = fc1 /. i implies ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) ) assume A93: c1 = fc1 /. i ; ::_thesis: ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) A94: dom c1 = [.(inf (dom c1)),(sup (dom c1)).] by Th27; A95: inf (dom c1) <= sup (dom c1) by XXREAL_2:40; A96: inf (dom c1) = h /. i by A93, A94, A95, A92, XXREAL_1:66; A97: h /. i = h . i by A89, PARTFUN1:def_6; A98: sup (dom c1) = h /. (i + 1) by A93, A94, A95, A92, XXREAL_1:66; A99: h /. (i + 1) = h . (i + 1) by A88, PARTFUN1:def_6; percases ( rng c1 c= the carrier of U or rng c1 c= the carrier of V ) by A32, A86, A93; supposeA100: rng c1 c= the carrier of U ; ::_thesis: ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) take c1 ; ::_thesis: ( c1,c1 are_homotopic & rng c1 c= the carrier of U & dom c1 = dom c1 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) thus ( c1,c1 are_homotopic & rng c1 c= the carrier of U & dom c1 = dom c1 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) by A100, A93, A69, A86; ::_thesis: verum end; supposeA101: rng c1 c= the carrier of V ; ::_thesis: ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) A102: rng h c= dom f by A5, FUNCT_2:def_1; then A103: dom (f * h) = dom h by RELAT_1:27; A104: i + 1 in dom (f * h) by A102, A88, RELAT_1:27; A105: the_first_point_of c1 <> p proof assume A106: the_first_point_of c1 = p ; ::_thesis: contradiction inf (dom c1) in dom c1 by A94, A95, XXREAL_1:1; then c1 . (inf (dom c1)) = f . (h . i) by A96, A97, A93, A87, FUNCT_1:47 .= (f * h) . i by A103, A89, FUNCT_1:12 ; hence contradiction by A6, A106, A103, A89, FUNCT_1:3; ::_thesis: verum end; A107: the_last_point_of c1 <> p proof assume A108: the_last_point_of c1 = p ; ::_thesis: contradiction sup (dom c1) in dom c1 by A94, A95, XXREAL_1:1; then c1 . (sup (dom c1)) = f . (h . (i + 1)) by A98, A99, A93, A87, FUNCT_1:47 .= (f * h) . (i + 1) by A104, FUNCT_1:12 ; hence contradiction by A6, A108, A104, FUNCT_1:3; ::_thesis: verum end; consider c2 being with_endpoints Curve of (TUnitSphere n) such that A109: ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 ) by A35, A101, A105, A107, A96, A98, A91, A97, A99; take c2 ; ::_thesis: ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) thus ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) by A109, A93, A69, A86; ::_thesis: verum end; end; end; defpred S1[ set , set ] means for i being Nat for c1 being with_endpoints Curve of (TUnitSphere n) st i = $1 & c1 = fc1 /. i holds ex c2 being with_endpoints Curve of (TUnitSphere n) st ( c2 = $2 & c2,c1 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ); A110: for k being Nat st k in Seg (len fc1) holds ex x being Element of Curves (TUnitSphere n) st S1[k,x] proof let k be Nat; ::_thesis: ( k in Seg (len fc1) implies ex x being Element of Curves (TUnitSphere n) st S1[k,x] ) assume k in Seg (len fc1) ; ::_thesis: ex x being Element of Curves (TUnitSphere n) st S1[k,x] then A111: ( 1 <= k & k <= len fc1 ) by FINSEQ_1:1; set c1 = fc1 /. k; reconsider c1 = fc1 /. k as with_endpoints Curve of (TUnitSphere n) by A111, A85; consider c2 being with_endpoints Curve of (TUnitSphere n) such that A112: ( c1,c2 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. k),(h /. (k + 1)).] ) by A85, A111; reconsider x = c2 as Element of Curves (TUnitSphere n) ; take x ; ::_thesis: S1[k,x] thus S1[k,x] by A112; ::_thesis: verum end; ex p being FinSequence of Curves (TUnitSphere n) st ( dom p = Seg (len fc1) & ( for k being Nat st k in Seg (len fc1) holds S1[k,p . k] ) ) from FINSEQ_1:sch_5(A110); then consider fc2 being FinSequence of Curves (TUnitSphere n) such that A113: ( dom fc2 = Seg (len fc1) & ( for k being Nat st k in Seg (len fc1) holds S1[k,fc2 . k] ) ) ; A114: len fc2 = len fc1 by A113, FINSEQ_1:def_3; A115: 2 - 1 <= (len h) - 1 by A29, XREAL_1:9; then A116: len fc2 > 0 by A31, A113, FINSEQ_1:def_3; A117: for i being Nat st 1 <= i & i < len fc2 holds ( (fc2 /. i) . (sup (dom (fc2 /. i))) = (fc2 /. (i + 1)) . (inf (dom (fc2 /. (i + 1)))) & sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1))) ) proof let i be Nat; ::_thesis: ( 1 <= i & i < len fc2 implies ( (fc2 /. i) . (sup (dom (fc2 /. i))) = (fc2 /. (i + 1)) . (inf (dom (fc2 /. (i + 1)))) & sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1))) ) ) assume A118: ( 1 <= i & i < len fc2 ) ; ::_thesis: ( (fc2 /. i) . (sup (dom (fc2 /. i))) = (fc2 /. (i + 1)) . (inf (dom (fc2 /. (i + 1)))) & sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1))) ) then ( 1 <= i & i <= len fc1 ) by A113, FINSEQ_1:def_3; then A119: i in Seg (len fc1) by FINSEQ_1:1; set ci = fc1 /. i; reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A118, A114, A85; consider di being with_endpoints Curve of (TUnitSphere n) such that A120: ( di = fc2 . i & di,ci are_homotopic & rng di c= the carrier of U & dom ci = dom di & dom ci = [.(h /. i),(h /. (i + 1)).] ) by A119, A113; 1 + 1 <= i + 1 by A118, XREAL_1:6; then A121: 1 <= i + 1 by XXREAL_0:2; A122: i + 1 <= len fc2 by A118, NAT_1:13; then A123: i + 1 in Seg (len fc1) by A114, A121, FINSEQ_1:1; set ci1 = fc1 /. (i + 1); reconsider ci1 = fc1 /. (i + 1) as with_endpoints Curve of (TUnitSphere n) by A121, A122, A114, A85; consider di1 being with_endpoints Curve of (TUnitSphere n) such that A124: ( di1 = fc2 . (i + 1) & di1,ci1 are_homotopic & rng di1 c= the carrier of U & dom ci1 = dom di1 & dom ci1 = [.(h /. (i + 1)),(h /. ((i + 1) + 1)).] ) by A123, A113; A125: i + 1 in dom fc2 by A122, A113, A114, A121, FINSEQ_1:1; A126: h /. i < h /. (i + 1) by A69, A118, A114; A127: h /. (i + 1) < h /. ((i + 1) + 1) by A69, A121, A122, A114; A128: dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] by A69, A118, A114; A129: dom (fc1 /. (i + 1)) = [.(h /. (i + 1)),(h /. ((i + 1) + 1)).] by A69, A121, A122, A114; A130: h /. (i + 1) in [.(h /. i),(h /. (i + 1)).] by A126, XXREAL_1:1; A131: h /. (i + 1) in [.(h /. (i + 1)),(h /. ((i + 1) + 1)).] by A127, XXREAL_1:1; A132: fc2 /. i = fc2 . i by A119, A113, PARTFUN1:def_6; A133: fc2 /. (i + 1) = fc2 . (i + 1) by A125, PARTFUN1:def_6; thus (fc2 /. i) . (sup (dom (fc2 /. i))) = the_last_point_of di by A120, A132 .= the_last_point_of ci by A120, Th35 .= (fc1 /. i) . (h /. (i + 1)) by A126, A128, XXREAL_2:29 .= (c | [.(h /. i),(h /. (i + 1)).]) . (h /. (i + 1)) by A31, A118, A114 .= c . (h /. (i + 1)) by A130, FUNCT_1:49 .= (c | [.(h /. (i + 1)),(h /. ((i + 1) + 1)).]) . (h /. (i + 1)) by A131, FUNCT_1:49 .= (fc1 /. (i + 1)) . (h /. (i + 1)) by A31, A121, A122, A114 .= the_first_point_of ci1 by A127, A129, XXREAL_2:25 .= the_first_point_of di1 by A124, Th35 .= (fc2 /. (i + 1)) . (inf (dom (fc2 /. (i + 1)))) by A124, A133 ; ::_thesis: sup (dom (fc2 /. i)) = inf (dom (fc2 /. (i + 1))) A134: dom (fc2 /. i) = [.(h /. i),(h /. (i + 1)).] by A120, A119, A113, PARTFUN1:def_6; A135: dom (fc2 /. (i + 1)) = [.(h /. (i + 1)),(h /. (i + 2)).] by A124, A125, PARTFUN1:def_6; thus sup (dom (fc2 /. i)) = h /. (i + 1) by A134, A126, XXREAL_2:29 .= inf (dom (fc2 /. (i + 1))) by A135, A127, XXREAL_2:25 ; ::_thesis: verum end; A136: for i being Nat st 1 <= i & i <= len fc2 holds fc2 /. i is with_endpoints proof let i be Nat; ::_thesis: ( 1 <= i & i <= len fc2 implies fc2 /. i is with_endpoints ) assume A137: ( 1 <= i & i <= len fc2 ) ; ::_thesis: fc2 /. i is with_endpoints then A138: i in Seg (len fc1) by A114, FINSEQ_1:1; set ci = fc1 /. i; reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A137, A114, A85; consider di being with_endpoints Curve of (TUnitSphere n) such that A139: ( di = fc2 . i & di,ci are_homotopic & rng di c= the carrier of U & dom ci = dom di & dom ci = [.(h /. i),(h /. (i + 1)).] ) by A138, A113; thus fc2 /. i is with_endpoints by A139, A138, A113, PARTFUN1:def_6; ::_thesis: verum end; consider c0 being with_endpoints Curve of (TUnitSphere n) such that A140: ( Sum fc2 = c0 & dom c0 = [.(inf (dom (fc2 /. 1))),(sup (dom (fc2 /. (len fc2)))).] & the_first_point_of c0 = (fc2 /. 1) . (inf (dom (fc2 /. 1))) & the_last_point_of c0 = (fc2 /. (len fc2)) . (sup (dom (fc2 /. (len fc2)))) ) by A117, A136, A116, Th43; A141: for i being Nat st 1 <= i & i < len fc1 holds ( (fc1 /. i) . (sup (dom (fc1 /. i))) = (fc1 /. (i + 1)) . (inf (dom (fc1 /. (i + 1)))) & sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1))) ) proof let i be Nat; ::_thesis: ( 1 <= i & i < len fc1 implies ( (fc1 /. i) . (sup (dom (fc1 /. i))) = (fc1 /. (i + 1)) . (inf (dom (fc1 /. (i + 1)))) & sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1))) ) ) assume A142: ( 1 <= i & i < len fc1 ) ; ::_thesis: ( (fc1 /. i) . (sup (dom (fc1 /. i))) = (fc1 /. (i + 1)) . (inf (dom (fc1 /. (i + 1)))) & sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1))) ) A143: i in Seg (len fc1) by A142, FINSEQ_1:1; set ci = fc1 /. i; reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A142, A85; consider di being with_endpoints Curve of (TUnitSphere n) such that A144: ( di = fc2 . i & di,ci are_homotopic & rng di c= the carrier of U & dom ci = dom di & dom ci = [.(h /. i),(h /. (i + 1)).] ) by A143, A113; 1 + 1 <= i + 1 by A142, XREAL_1:6; then A145: 1 <= i + 1 by XXREAL_0:2; A146: i + 1 <= len fc2 by A114, A142, NAT_1:13; then A147: i + 1 in Seg (len fc1) by A114, A145, FINSEQ_1:1; set ci1 = fc1 /. (i + 1); reconsider ci1 = fc1 /. (i + 1) as with_endpoints Curve of (TUnitSphere n) by A145, A146, A114, A85; consider di1 being with_endpoints Curve of (TUnitSphere n) such that A148: ( di1 = fc2 . (i + 1) & di1,ci1 are_homotopic & rng di1 c= the carrier of U & dom ci1 = dom di1 & dom ci1 = [.(h /. (i + 1)),(h /. ((i + 1) + 1)).] ) by A147, A113; A149: h /. i < h /. (i + 1) by A69, A142; A150: h /. (i + 1) < h /. ((i + 1) + 1) by A69, A145, A146, A114; A151: dom (fc1 /. i) = [.(h /. i),(h /. (i + 1)).] by A69, A142; A152: dom (fc1 /. (i + 1)) = [.(h /. (i + 1)),(h /. ((i + 1) + 1)).] by A69, A145, A146, A114; A153: h /. (i + 1) in [.(h /. i),(h /. (i + 1)).] by A149, XXREAL_1:1; A154: h /. (i + 1) in [.(h /. (i + 1)),(h /. ((i + 1) + 1)).] by A150, XXREAL_1:1; thus (fc1 /. i) . (sup (dom (fc1 /. i))) = (fc1 /. i) . (h /. (i + 1)) by A149, A151, XXREAL_2:29 .= (c | [.(h /. i),(h /. (i + 1)).]) . (h /. (i + 1)) by A31, A142 .= c . (h /. (i + 1)) by A153, FUNCT_1:49 .= (c | [.(h /. (i + 1)),(h /. ((i + 1) + 1)).]) . (h /. (i + 1)) by A154, FUNCT_1:49 .= (fc1 /. (i + 1)) . (h /. (i + 1)) by A31, A145, A146, A114 .= (fc1 /. (i + 1)) . (inf (dom (fc1 /. (i + 1)))) by A150, A152, XXREAL_2:25 ; ::_thesis: sup (dom (fc1 /. i)) = inf (dom (fc1 /. (i + 1))) thus sup (dom (fc1 /. i)) = h /. (i + 1) by A144, A149, XXREAL_2:29 .= inf (dom (fc1 /. (i + 1))) by A148, A150, XXREAL_2:25 ; ::_thesis: verum end; for i being Nat st 1 <= i & i <= len fc2 holds ex c3, c4 being with_endpoints Curve of (TUnitSphere n) st ( c3 = fc2 /. i & c4 = fc1 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) proof let i be Nat; ::_thesis: ( 1 <= i & i <= len fc2 implies ex c3, c4 being with_endpoints Curve of (TUnitSphere n) st ( c3 = fc2 /. i & c4 = fc1 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) ) assume A155: ( 1 <= i & i <= len fc2 ) ; ::_thesis: ex c3, c4 being with_endpoints Curve of (TUnitSphere n) st ( c3 = fc2 /. i & c4 = fc1 /. i & c3,c4 are_homotopic & dom c3 = dom c4 ) then A156: i in Seg (len fc1) by A114, FINSEQ_1:1; set ci = fc1 /. i; reconsider ci = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A155, A114, A85; consider di being with_endpoints Curve of (TUnitSphere n) such that A157: ( di = fc2 . i & di,ci are_homotopic & rng di c= the carrier of U & dom ci = dom di & dom ci = [.(h /. i),(h /. (i + 1)).] ) by A156, A113; A158: i in dom fc2 by A155, A114, A113, FINSEQ_1:1; take di ; ::_thesis: ex c4 being with_endpoints Curve of (TUnitSphere n) st ( di = fc2 /. i & c4 = fc1 /. i & di,c4 are_homotopic & dom di = dom c4 ) take ci ; ::_thesis: ( di = fc2 /. i & ci = fc1 /. i & di,ci are_homotopic & dom di = dom ci ) thus ( di = fc2 /. i & ci = fc1 /. i & di,ci are_homotopic & dom di = dom ci ) by A157, A158, PARTFUN1:def_6; ::_thesis: verum end; then A159: c0,c are_homotopic by A117, A141, A140, Th44, A114, A115, A31; A160: dom c0 = [.0,1.] proof A161: 0 + 1 < (len fc1) + 1 by A115, A31, XREAL_1:6; A162: 1 in Seg (len fc1) by A115, A31, FINSEQ_1:1; set ci = fc1 /. 1; reconsider ci = fc1 /. 1 as with_endpoints Curve of (TUnitSphere n) by A115, A31, A85; consider di being with_endpoints Curve of (TUnitSphere n) such that A163: ( di = fc2 . 1 & di,ci are_homotopic & rng di c= the carrier of U & dom ci = dom di & dom ci = [.(h /. 1),(h /. (1 + 1)).] ) by A162, A113; 1 in Seg (len fc2) by A115, A31, A114, FINSEQ_1:1; then 1 in dom fc2 by FINSEQ_1:def_3; then A164: dom (fc2 /. 1) = [.(h /. 1),(h /. (1 + 1)).] by A163, PARTFUN1:def_6; A165: h /. 1 < h /. (1 + 1) by A69, A115, A31; 1 in Seg (len h) by A161, A31, FINSEQ_1:1; then A166: 1 in dom h by FINSEQ_1:def_3; A167: inf (dom (fc2 /. 1)) = h /. 1 by A165, A164, XXREAL_2:25 .= 0 by A5, A166, PARTFUN1:def_6 ; A168: len fc1 in Seg (len fc1) by A115, A31, FINSEQ_1:1; set ci1 = fc1 /. (len fc1); reconsider ci1 = fc1 /. (len fc1) as with_endpoints Curve of (TUnitSphere n) by A115, A31, A85; consider di1 being with_endpoints Curve of (TUnitSphere n) such that A169: ( di1 = fc2 . (len fc1) & di1,ci1 are_homotopic & rng di1 c= the carrier of U & dom ci1 = dom di1 & dom ci1 = [.(h /. (len fc1)),(h /. ((len fc1) + 1)).] ) by A168, A113; len fc1 in Seg (len fc2) by A114, A115, A31, FINSEQ_1:1; then len fc1 in dom fc2 by FINSEQ_1:def_3; then A170: dom (fc2 /. (len fc2)) = [.(h /. (len fc1)),(h /. ((len fc1) + 1)).] by A169, A114, PARTFUN1:def_6; A171: h /. (len fc1) < h /. ((len fc1) + 1) by A69, A115, A31; len h in Seg (len h) by A161, A31, FINSEQ_1:1; then A172: len h in dom h by FINSEQ_1:def_3; A173: sup (dom (fc2 /. (len fc2))) = h /. ((len fc1) + 1) by A171, A170, XXREAL_2:29 .= 1 by A5, A31, A172, PARTFUN1:def_6 ; thus dom c0 = [.0,1.] by A140, A167, A173; ::_thesis: verum end; for i being Nat st 1 <= i & i <= len fc2 holds rng (fc2 /. i) c= the carrier of U proof let i be Nat; ::_thesis: ( 1 <= i & i <= len fc2 implies rng (fc2 /. i) c= the carrier of U ) assume A174: ( 1 <= i & i <= len fc2 ) ; ::_thesis: rng (fc2 /. i) c= the carrier of U then i in Seg (len fc2) by FINSEQ_1:1; then A175: i in dom fc2 by FINSEQ_1:def_3; A176: i in Seg (len fc1) by A114, A174, FINSEQ_1:1; reconsider c1 = fc1 /. i as with_endpoints Curve of (TUnitSphere n) by A85, A174, A114; consider c2 being with_endpoints Curve of (TUnitSphere n) such that A177: ( c2 = fc2 . i & c2,c1 are_homotopic & rng c2 c= the carrier of U & dom c1 = dom c2 & dom c1 = [.(h /. i),(h /. (i + 1)).] ) by A176, A113; thus rng (fc2 /. i) c= the carrier of U by A175, A177, PARTFUN1:def_6; ::_thesis: verum end; then A178: rng c0 c= the carrier of U by A140, Th42; A179: t,t are_connected ; A180: t = the_first_point_of c by A30, A179, BORSUK_2:def_2 .= the_first_point_of c0 by Th35, A159 ; A181: t = the_last_point_of c by A30, A179, BORSUK_2:def_2 .= the_last_point_of c0 by Th35, A159 ; reconsider f0 = c0 as Loop of t by A160, A180, A181, Th28; A182: f0,f are_homotopic by A159, A179, Th34; not p in rng f0 proof assume p in rng f0 ; ::_thesis: contradiction then not p in {p} by A11, A178, XBOOLE_0:def_5; hence contradiction by TARSKI:def_1; ::_thesis: verum end; hence ex g being Loop of t st ( g,f are_homotopic & rng g <> the carrier of (TUnitSphere n) ) by A6, A182; ::_thesis: verum end; theorem Th46: :: TOPALG_6:46 for n being Nat st n >= 2 holds TUnitSphere n is having_trivial_Fundamental_Group proof let n be Nat; ::_thesis: ( n >= 2 implies TUnitSphere n is having_trivial_Fundamental_Group ) assume A1: n >= 2 ; ::_thesis: TUnitSphere n is having_trivial_Fundamental_Group set T = TUnitSphere n; for t being Point of (TUnitSphere n) for f being Loop of t holds f is nullhomotopic proof let t be Point of (TUnitSphere n); ::_thesis: for f being Loop of t holds f is nullhomotopic let f be Loop of t; ::_thesis: f is nullhomotopic percases ( rng f <> the carrier of (TUnitSphere n) or rng f = the carrier of (TUnitSphere n) ) ; suppose rng f <> the carrier of (TUnitSphere n) ; ::_thesis: f is nullhomotopic hence f is nullhomotopic by Lm3; ::_thesis: verum end; suppose rng f = the carrier of (TUnitSphere n) ; ::_thesis: f is nullhomotopic then consider g being Loop of t such that A2: g,f are_homotopic and A3: rng g <> the carrier of (TUnitSphere n) by A1, Lm4; g is nullhomotopic by A3, Lm3; then consider C being constant Loop of t such that A4: g,C are_homotopic by Def3; f,C are_homotopic by A2, A4, BORSUK_6:79; hence f is nullhomotopic by Def3; ::_thesis: verum end; end; end; hence TUnitSphere n is having_trivial_Fundamental_Group by Th17; ::_thesis: verum end; theorem :: TOPALG_6:47 for n being non empty Nat for r being real positive number for x being Point of (TOP-REAL n) st n >= 3 holds Tcircle (x,r) is having_trivial_Fundamental_Group proof let n be non empty Nat; ::_thesis: for r being real positive number for x being Point of (TOP-REAL n) st n >= 3 holds Tcircle (x,r) is having_trivial_Fundamental_Group let r be real positive number ; ::_thesis: for x being Point of (TOP-REAL n) st n >= 3 holds Tcircle (x,r) is having_trivial_Fundamental_Group let x be Point of (TOP-REAL n); ::_thesis: ( n >= 3 implies Tcircle (x,r) is having_trivial_Fundamental_Group ) assume A1: n >= 3 ; ::_thesis: Tcircle (x,r) is having_trivial_Fundamental_Group then n - 1 >= 3 - 1 by XREAL_1:9; then 0 <= n - 1 by XXREAL_0:2; then A2: (n -' 1) + 1 = (n - 1) + 1 by XREAL_0:def_2; 2 + 1 = 3 ; then 2 <= n -' 1 by A1, NAT_D:49; then A3: TUnitSphere (n -' 1) is having_trivial_Fundamental_Group by Th46; A4: TUnitSphere (n -' 1) = Tunit_circle ((n -' 1) + 1) by MFOLD_2:def_4; A5: Tunit_circle n = Tcircle ((0. (TOP-REAL n)),1) by TOPREALB:def_7; n in NAT by ORDINAL1:def_12; then Tcircle (x,r), Tcircle ((0. (TOP-REAL n)),1) are_homeomorphic by TOPREALB:20; hence Tcircle (x,r) is having_trivial_Fundamental_Group by A2, A3, A4, A5, Th13; ::_thesis: verum end;