### The Mizar article:

### The Ordering of Points on a Curve. Part II

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

- Received September 10, 1997
Copyright (c) 1997 Association of Mizar Users

- MML identifier: JORDAN5C
- [ MML identifier index ]

environ vocabulary ARYTM_1, EUCLID, PRE_TOPC, BORSUK_1, RELAT_1, TOPREAL1, FUNCT_1, TOPS_2, SUBSET_1, ORDINAL2, BOOLE, COMPTS_1, JORDAN3, RCOMP_1, FINSEQ_1, TREAL_1, SEQ_1, TOPMETR, JORDAN5C, FINSEQ_4; notation TARSKI, XBOOLE_0, SUBSET_1, XREAL_0, REAL_1, NAT_1, RCOMP_1, RELAT_1, FINSEQ_1, FUNCT_1, FUNCT_2, TOPMETR, FINSEQ_4, JORDAN3, STRUCT_0, TOPREAL1, PRE_TOPC, TOPS_2, COMPTS_1, EUCLID, TREAL_1; constructors REAL_1, NAT_1, TOPS_2, COMPTS_1, RCOMP_1, TREAL_1, JORDAN3, TOPREAL1, FINSEQ_4, MEMBERED; clusters BORSUK_1, EUCLID, FUNCT_1, PRE_TOPC, RELSET_1, STRUCT_0, XREAL_0, ARYTM_3, MEMBERED; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; definitions TARSKI, JORDAN3, XBOOLE_0; theorems AXIOMS, BORSUK_1, COMPTS_1, EUCLID, FUNCT_1, FUNCT_2, RELAT_1, JORDAN3, GOBOARD1, NAT_1, GOBOARD9, SPPOL_1, TOPS_1, GOBOARD2, PRE_TOPC, RCOMP_1, REAL_1, HEINE, PCOMPS_1, SPPOL_2, TARSKI, TOPMETR, TOPREAL1, TOPREAL3, RFUNCT_2, TSEP_1, JORDAN5B, TOPS_2, TREAL_1, JORDAN5A, ZFMISC_1, PARTFUN2, RELSET_1, XBOOLE_0, XBOOLE_1, FINSEQ_3, XCMPLX_1; schemes NAT_1; begin :: First and last point of a curve Lm1: for r be Real st 0 <= r & r <= 1 holds 0 <= 1-r & 1-r <= 1 proof let r be Real; assume 0 <= r & r <= 1; then 1-1 <= 1-r & 1-r <= 1-0 by REAL_1:92; hence thesis; end; theorem Th1: :: Characterization of First Point Intersecting Q for P, Q being Subset of TOP-REAL 2, p1, p2, q1 being Point of TOP-REAL 2, f being map of I[01], (TOP-REAL 2)|P, s1 be Real st P is_an_arc_of p1,p2 & q1 in P & q1 in Q & f.s1 = q1 & f is_homeomorphism & f.0 = p1 & f.1 = p2 & 0 <= s1 & s1 <= 1 & (for t being Real st 0 <= t & t < s1 holds not f.t in Q) holds for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = q1 & 0 <= s2 & s2 <= 1 holds (for t being Real st 0 <= t & t < s2 holds not g.t in Q) proof let P, Q be Subset of TOP-REAL 2, p1, p2, q1 be Point of TOP-REAL 2, f be map of I[01], (TOP-REAL 2)|P, s1 be Real; assume A1: P is_an_arc_of p1,p2 & q1 in P & q1 in Q & f.s1=q1 & f is_homeomorphism & f.0=p1 & f.1=p2 & 0 <= s1 & s1 <= 1 & (for t being Real st 0<=t & t<s1 holds not f.t in Q); then reconsider P'=P as non empty Subset of TOP-REAL 2; let g be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A2: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=q1 & 0<=s2 & s2<=1; let t be Real; assume A3: 0<=t & t<s2; then A4: t <= 1 by A2,AXIOMS:22; reconsider f,g as map of I[01], (TOP-REAL 2)|P'; set fg = f"*g; A5: f" is_homeomorphism by A1,TOPS_2:70; then A6: fg is_homeomorphism by A2,TOPS_2:71; A7: dom f = [#]I[01] & rng f = [#]((TOP-REAL 2)|P) by A1,TOPS_2:def 5; then A8: dom f = the carrier of I[01] & rng f = the carrier of ((TOP-REAL 2)|P) by PRE_TOPC:12; then A9: 0 in dom f & 1 in dom f by JORDAN5A:2; A10: f is one-to-one by A1,TOPS_2:def 5; A11: fg is continuous & fg is one-to-one by A6,TOPS_2:def 5; A12: f".p1 = (f qua Function)".p1 by A7,A10,TOPS_2:def 4 .= 0 by A1,A9,A10,FUNCT_1:54; A13: dom (f") = [#]((TOP-REAL 2)|P) by A5,TOPS_2:def 5; A14: f".p2 = (f qua Function)".p2 by A7,A10,TOPS_2:def 4 .= 1 by A1,A9,A10,FUNCT_1:54; A15: dom g = [#]I[01] & rng g = [#]((TOP-REAL 2)|P) by A2,TOPS_2:def 5; then A16: dom g = the carrier of I[01] by PRE_TOPC:12; then 0 in dom g by JORDAN5A:2; then A17: (f"*g).0 = 0 by A2,A12,FUNCT_1:23; 1 in dom g by A16,JORDAN5A:2; then A18: (f"*g).1 = 1 by A2,A14,FUNCT_1:23; A19: s1 in dom f by A1,A8,JORDAN5A:2; s2 in dom g by A2,A16,JORDAN5A:2; then A20: (f"*g).s2 = f".q1 by A2,FUNCT_1:23 .= (f qua Function)".q1 by A7,A10,TOPS_2:def 4 .= s1 by A1,A10,A19,FUNCT_1:54; reconsider t1 = t, s2' = s2 as Point of I[01] by A2,A3,A4,JORDAN5A:2; A21: t in the carrier of I[01] by A3,A4,JORDAN5A:2; then fg.t in the carrier of I[01] by FUNCT_2:7; then reconsider Ft = fg.t1 as Real by BORSUK_1:83; A22: 0 <= Ft proof per cases by A3; suppose 0<t; hence thesis by A11,A17,A18,BORSUK_1:def 17,JORDAN5A:16,TOPMETR:27; suppose 0 = t; hence thesis by A2,A12,A16,FUNCT_1:23; end; fg.s2' = s1 by A20; then A23: Ft < s1 by A3,A11,A17,A18,JORDAN5A:16,TOPMETR:27; A24: f*(f"*g) = (g qua Relation) * (f * f") by RELAT_1:55 .= (g qua Relation) * id rng f by A7,A10,TOPS_2:65 .= id rng g * g by A2,A7,TOPS_2:def 5 .= g by FUNCT_1:42; dom (f"*g) = dom g by A13,A15,RELAT_1:46; then f.((f"*g).t) = g.t by A16,A21,A24,FUNCT_1:23; hence thesis by A1,A22,A23; end; definition let P, Q be Subset of TOP-REAL 2, p1, p2 be Point of TOP-REAL 2; assume A1: P meets Q & P /\ Q is closed & P is_an_arc_of p1,p2; func First_Point(P,p1,p2,Q) -> Point of TOP-REAL 2 means :Def1: it in P /\ Q & for g being map of I[01], (TOP-REAL 2)|P, s2 being Real st g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = it & 0 <= s2 & s2 <= 1 holds (for t being Real st 0 <= t & t < s2 holds not g.t in Q); existence proof consider EX be Point of TOP-REAL 2 such that A2: EX in P /\ Q & ex g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=EX & 0<=s2 & s2<=1 & (for t being Real st 0<=t & t<s2 holds not g.t in Q) by A1,JORDAN5A:22; EX in P & EX in Q by A2,XBOOLE_0:def 3; then for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=EX & 0<=s2 & s2<=1 holds (for t being Real st 0<=t & t<s2 holds not g.t in Q) by A1,A2,Th1; hence thesis by A2; end; uniqueness proof let IT1, IT2 be Point of TOP-REAL 2; A3: P /\ Q c= Q & P /\ Q c= P by XBOOLE_1:17; assume A4: IT1 in P /\ Q & for g1 being map of I[01], (TOP-REAL 2)|P, s1 be Real st g1 is_homeomorphism & g1.0=p1 & g1.1=p2 & g1.s1=IT1 & 0<=s1 & s1<=1 holds (for t being Real st 0<=t & t<s1 holds not g1.t in Q); assume A5: IT2 in P /\ Q & for g2 being map of I[01], (TOP-REAL 2)|P, s2 be Real st g2 is_homeomorphism & g2.0=p1 & g2.1=p2 & g2.s2=IT2 & 0<=s2 & s2<=1 holds (for t being Real st 0<=t & t<s2 holds not g2.t in Q); consider g be map of I[01], (TOP-REAL 2)|P such that A6: g is_homeomorphism & g.0=p1 & g.1=p2 by A1,TOPREAL1:def 2; A7: rng g = [#]((TOP-REAL 2)|P) by A6,TOPS_2:def 5 .= P by PRE_TOPC:def 10; A8: dom g = [#]I[01] by A6,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; consider ss1 be set such that A9: ss1 in dom g & g.ss1 = IT1 by A3,A4,A7,FUNCT_1:def 5; reconsider ss1 as Real by A8,A9,BORSUK_1:83; consider ss2 be set such that A10: ss2 in dom g & g.ss2 = IT2 by A3,A5,A7,FUNCT_1:def 5; reconsider ss2 as Real by A8,A10,BORSUK_1:83; A11:0 <= ss1 & ss1 <= 1 & 0 <= ss2 & ss2 <= 1 by A8,A9,A10,JORDAN5A:2; per cases by REAL_1:def 5; suppose ss1 < ss2; hence thesis by A3,A4,A5,A6,A9,A10,A11; suppose ss1 = ss2; hence thesis by A9,A10; suppose ss1 > ss2; hence thesis by A3,A4,A5,A6,A9,A10,A11; end; end; theorem for P, Q being Subset of TOP-REAL 2, p, p1, p2 being Point of TOP-REAL 2 st p in P & P is_an_arc_of p1, p2 & Q = {p} holds First_Point (P,p1,p2,Q) = p proof let P, Q be Subset of TOP-REAL 2, p, p1, p2 be Point of TOP-REAL 2; A1: TOP-REAL 2 is_T2 by SPPOL_1:26; assume A2: p in P & P is_an_arc_of p1, p2 & Q = {p}; then A3: P /\ {p} = {p} by ZFMISC_1:52; then A4: p in P /\ {p} by TARSKI:def 1; then A5: P meets {p} & P /\ Q is closed & P is_an_arc_of p1,p2 by A1,A2,A3,PCOMPS_1:10,XBOOLE_0:3; for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p & 0<=s2 & s2<=1 holds (for t being Real st 0<=t & t<s2 holds not g.t in {p}) proof let g be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A6: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p & 0<=s2 & s2<=1; then A7: g is one-to-one by TOPS_2:def 5; [#]((TOP-REAL 2)|P) <> {} by A2,PRE_TOPC:def 10; then the carrier of I[01] <> {} & the carrier of ((TOP-REAL 2)|P) <> {} by PRE_TOPC:12; then A8: dom g = the carrier of I[01] by FUNCT_2:def 1; let t be Real; assume A9: 0<=t & t<s2; then t <= 1 by A6,AXIOMS:22; then s2 in dom g & t in dom g & s2 <> t by A6,A8,A9,JORDAN5A:2; then g.t <> g.s2 by A7,FUNCT_1:def 8; hence thesis by A6,TARSKI:def 1; end; hence thesis by A2,A4,A5,Def1; end; theorem Th3: for P being Subset of TOP-REAL 2, Q being Subset of TOP-REAL 2, p1, p2 being Point of TOP-REAL 2 st p1 in Q & P /\ Q is closed & P is_an_arc_of p1, p2 holds First_Point (P, p1, p2, Q) = p1 proof let P be Subset of TOP-REAL 2, Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2; assume A1: p1 in Q & P /\ Q is closed & P is_an_arc_of p1, p2; then A2:p1 in P by TOPREAL1:4; then A3: p1 in P /\ Q by A1,XBOOLE_0:def 3; A4: P meets Q by A1,A2,XBOOLE_0:3; for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p1 & 0<=s2 & s2<=1 holds (for t being Real st 0<=t & t<s2 holds not g.t in Q) proof let g be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A5: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p1 & 0<=s2 & s2<=1; then dom g = [#] I[01] by TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; then A6: s2 in dom g & 0 in dom g by A5,JORDAN5A:2; A7: g is one-to-one by A5,TOPS_2:def 5; let t be Real; assume 0<=t & t<s2; hence thesis by A5,A6,A7,FUNCT_1:def 8; end; hence thesis by A1,A3,A4,Def1; end; theorem ::Characterization of Last Point Intersecting Q Th4: for P, Q being Subset of TOP-REAL 2, p1, p2, q1 being Point of TOP-REAL 2, f being map of I[01], (TOP-REAL 2)|P, s1 be Real st P is_an_arc_of p1,p2 & q1 in P & q1 in Q & f.s1 = q1 & f is_homeomorphism & f.0 = p1 & f.1 = p2 & 0 <= s1 & s1 <= 1 & (for t being Real st 1 >= t & t > s1 holds not f.t in Q) holds for g being map of I[01], (TOP-REAL 2)|P, s2 being Real st g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = q1 & 0 <= s2 & s2 <= 1 holds (for t being Real st 1 >= t & t > s2 holds not g.t in Q) proof let P, Q be Subset of TOP-REAL 2, p1, p2, q1 be Point of TOP-REAL 2, f be map of I[01], (TOP-REAL 2)|P, s1 be Real; assume A1: P is_an_arc_of p1,p2 & q1 in P & q1 in Q & f.s1=q1 & f is_homeomorphism & f.0=p1 & f.1=p2 & 0 <= s1 & s1 <= 1 & (for t being Real st 1>=t & t>s1 holds not f.t in Q); let g be map of I[01], (TOP-REAL 2)|P, s2 be Real; reconsider P'=P as non empty Subset of TOP-REAL 2 by A1; assume A2: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=q1 & 0<=s2 & s2<=1; reconsider f, g as map of I[01], (TOP-REAL 2)|P'; let t be Real; assume A3: 1>=t & t>s2; then A4: 0 <= t by A2,AXIOMS:22; set fg = f"*g; A5: f" is_homeomorphism by A1,TOPS_2:70; then A6: fg is_homeomorphism by A2,TOPS_2:71; A7: dom f = [#]I[01] & rng f = [#]((TOP-REAL 2)|P) by A1,TOPS_2:def 5; then A8: dom f = the carrier of I[01] & rng f = the carrier of ((TOP-REAL 2)|P) by PRE_TOPC:12; then A9: 0 in dom f & 1 in dom f by JORDAN5A:2; A10: f is one-to-one by A1,TOPS_2:def 5; A11: fg is continuous & fg is one-to-one by A6,TOPS_2:def 5; A12: f".p1 = (f qua Function)".p1 by A7,A10,TOPS_2:def 4 .= 0 by A1,A9,A10,FUNCT_1:54; A13: dom (f") = [#]((TOP-REAL 2)|P) by A5,TOPS_2:def 5; A14: f".p2 = (f qua Function)".p2 by A7,A10,TOPS_2:def 4 .= 1 by A1,A9,A10,FUNCT_1:54; A15: dom g = [#]I[01] & rng g = [#]((TOP-REAL 2)|P) by A2,TOPS_2:def 5; then A16: dom g = the carrier of I[01] by PRE_TOPC:12; then 0 in dom g by JORDAN5A:2; then A17: (f"*g).0 = 0 by A2,A12,FUNCT_1:23; 1 in dom g by A16,JORDAN5A:2; then A18: (f"*g).1 = 1 by A2,A14,FUNCT_1:23; A19: s1 in dom f by A1,A8,JORDAN5A:2; s2 in dom g by A2,A16,JORDAN5A:2; then A20: (f"*g).s2 = f".q1 by A2,FUNCT_1:23 .= (f qua Function)".q1 by A7,A10,TOPS_2:def 4 .= s1 by A1,A10,A19,FUNCT_1:54; reconsider t1 = t, s2' = s2 as Point of I[01] by A2,A3,A4,JORDAN5A:2; A21: t in the carrier of I[01] by A3,A4,JORDAN5A:2; then fg.t in the carrier of I[01] by FUNCT_2:7; then reconsider Ft = fg.t1 as Real by BORSUK_1:83; A22: Ft <= 1 proof per cases by A3,REAL_1:def 5; suppose t<1; hence thesis by A11,A17,A18,BORSUK_1:def 18,JORDAN5A:16,TOPMETR:27 ; suppose t=1; hence thesis by A2,A14,A16,FUNCT_1:23; end; fg.s2' = s1 by A20; then A23: s1 < Ft by A3,A11,A17,A18,JORDAN5A:16,TOPMETR:27; A24: f*(f"*g) = (g qua Relation) * (f * f") by RELAT_1:55 .= (g qua Relation) * id rng f by A7,A10,TOPS_2:65 .= id rng g * g by A2,A7,TOPS_2:def 5 .= g by FUNCT_1:42; t in dom (f"*g) by A13,A15,A16,A21,RELAT_1:46; then f.((f"*g).t) = g.t by A24,FUNCT_1:23; hence thesis by A1,A22,A23; end; definition let P, Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2; assume A1: P meets Q & P /\ Q is closed & P is_an_arc_of p1,p2; func Last_Point (P,p1,p2,Q) -> Point of TOP-REAL 2 means :Def2: it in P /\ Q & for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s2 = it & 0 <= s2 & s2 <= 1 holds for t being Real st 1 >= t & t > s2 holds not g.t in Q; existence proof consider EX be Point of TOP-REAL 2 such that A2: EX in P /\ Q & ex g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=EX & 0<=s2 & s2<=1 & (for t being Real st 1>=t & t>s2 holds not g.t in Q) by A1,JORDAN5A:23; EX in P & EX in Q by A2,XBOOLE_0:def 3; then for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=EX & 0<=s2 & s2<=1 holds (for t being Real st 1>=t & t>s2 holds not g.t in Q) by A1,A2,Th4; hence thesis by A2; end; uniqueness proof let IT1, IT2 be Point of TOP-REAL 2; A3: P /\ Q c= Q & P /\ Q c= P by XBOOLE_1:17; assume A4: IT1 in P /\ Q & for g1 being map of I[01], (TOP-REAL 2)|P, s1 be Real st g1 is_homeomorphism & g1.0=p1 & g1.1=p2 & g1.s1=IT1 & 0<=s1 & s1<=1 holds (for t being Real st 1>=t & t>s1 holds not g1.t in Q); assume A5: IT2 in P /\ Q & for g2 being map of I[01], (TOP-REAL 2)|P, s2 be Real st g2 is_homeomorphism & g2.0=p1 & g2.1=p2 & g2.s2=IT2 & 0<=s2 & s2<=1 holds (for t being Real st 1>=t & t>s2 holds not g2.t in Q); consider g be map of I[01], (TOP-REAL 2)|P such that A6: g is_homeomorphism & g.0=p1 & g.1=p2 by A1,TOPREAL1:def 2; A7: rng g = [#]((TOP-REAL 2)|P) by A6,TOPS_2:def 5 .= P by PRE_TOPC:def 10; A8: dom g = [#]I[01] by A6,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; consider ss1 be set such that A9: ss1 in dom g & g.ss1 = IT1 by A3,A4,A7,FUNCT_1:def 5; reconsider ss1 as Real by A8,A9,BORSUK_1:83; consider ss2 be set such that A10: ss2 in dom g & g.ss2 = IT2 by A3,A5,A7,FUNCT_1:def 5; reconsider ss2 as Real by A8,A10,BORSUK_1:83; A11:0 <= ss1 & ss1 <= 1 & 0 <= ss2 & ss2 <= 1 by A8,A9,A10,JORDAN5A:2; per cases by REAL_1:def 5; suppose ss1 < ss2; hence thesis by A3,A4,A5,A6,A9,A10,A11; suppose ss1 = ss2; hence thesis by A9,A10; suppose ss1 > ss2; hence thesis by A3,A4,A5,A6,A9,A10,A11; end; end; theorem for P, Q being Subset of TOP-REAL 2, p, p1,p2 being Point of TOP-REAL 2 st p in P & P is_an_arc_of p1, p2 & Q = {p} holds Last_Point (P, p1, p2, Q) = p proof let P, Q be Subset of TOP-REAL 2, p, p1, p2 be Point of TOP-REAL 2; A1: TOP-REAL 2 is_T2 by SPPOL_1:26; assume A2: p in P & P is_an_arc_of p1, p2 & Q = {p}; then A3: P /\ {p} = {p} by ZFMISC_1:52; then A4: p in P /\ {p} by TARSKI:def 1; then A5: P meets {p} & P /\ Q is closed & P is_an_arc_of p1,p2 by A1,A2,A3,PCOMPS_1:10,XBOOLE_0:3; for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p & 0<=s2 & s2<=1 holds (for t being Real st 1>=t & t>s2 holds not g.t in {p}) proof let g be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A6: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p & 0<=s2 & s2<=1; then A7: g is one-to-one by TOPS_2:def 5; [#]((TOP-REAL 2)|P) <> {} by A2,PRE_TOPC:def 10; then the carrier of I[01] <> {} & the carrier of ((TOP-REAL 2)|P) <> {} by PRE_TOPC:12; then A8: dom g = the carrier of I[01] by FUNCT_2:def 1; let t be Real; assume A9: 1>=t & t>s2; then t >= 0 by A6,AXIOMS:22; then s2 in dom g & t in dom g & s2 <> t by A6,A8,A9,JORDAN5A:2; then g.t <> g.s2 by A7,FUNCT_1:def 8; hence thesis by A6,TARSKI:def 1; end; hence thesis by A2,A4,A5,Def2; end; theorem Th6: for P,Q being Subset of TOP-REAL 2, p1, p2 being Point of TOP-REAL 2 st p2 in Q & P /\ Q is closed & P is_an_arc_of p1, p2 holds Last_Point (P, p1, p2, Q) = p2 proof let P,Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2; assume A1: p2 in Q & P /\ Q is closed & P is_an_arc_of p1, p2; then A2:p2 in P by TOPREAL1:4; then A3: p2 in P /\ Q by A1,XBOOLE_0:def 3; A4: P meets Q by A1,A2,XBOOLE_0:3; for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p2 & 0<=s2 & s2<=1 holds (for t being Real st 1>=t & t>s2 holds not g.t in Q) proof let g be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A5: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=p2 & 0<=s2 & s2<=1; then dom g = [#] I[01] by TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; then A6: s2 in dom g & 1 in dom g by A5,JORDAN5A:2; A7: g is one-to-one by A5,TOPS_2:def 5; let t be Real; assume 1>=t & t>s2; hence thesis by A5,A6,A7,FUNCT_1:def 8; end; hence thesis by A1,A3,A4,Def2; end; theorem Th7: for P being Subset of TOP-REAL 2, Q being Subset of TOP-REAL 2, p1, p2 being Point of TOP-REAL 2 st P c= Q & P is closed & P is_an_arc_of p1, p2 holds First_Point (P, p1, p2, Q) = p1 & Last_Point (P, p1, p2, Q) = p2 proof let P be Subset of TOP-REAL 2, Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2; assume A1: P c= Q & P is closed & P is_an_arc_of p1, p2; then A2: P /\ Q = P by XBOOLE_1:28; p1 in P & p2 in P by A1,TOPREAL1:4; hence thesis by A1,A2,Th3,Th6; end; begin :: The ordering of points on a curve definition let P be Subset of TOP-REAL 2, p1, p2, q1, q2 be Point of TOP-REAL 2; pred LE q1, q2, P, p1, p2 means :Def3: q1 in P & q2 in P & for g being map of I[01], (TOP-REAL 2)|P, s1, s2 being Real st g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1 & 0 <= s1 & s1 <= 1 & g.s2 = q2 & 0 <= s2 & s2 <= 1 holds s1 <= s2; end; theorem Th8: for P being Subset of TOP-REAL 2, p1, p2, q1, q2 being Point of TOP-REAL 2, g being map of I[01], (TOP-REAL 2)|P, s1, s2 being Real st P is_an_arc_of p1,p2 & g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1 & 0 <= s1 & s1 <= 1 & g.s2 = q2 & 0 <= s2 & s2 <= 1 & s1 <= s2 holds LE q1,q2,P,p1,p2 proof let P be Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2, g be map of I[01], (TOP-REAL 2)|P, s1,s2 be Real; assume A1: P is_an_arc_of p1,p2 & g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q2 & 0<=s2 & s2<=1 & s1<=s2; then reconsider P' = P as non empty Subset of TOP-REAL 2 by TOPREAL1:4; reconsider g as map of I[01], (TOP-REAL 2)|P'; A2: s1 in the carrier of I[01] by A1,JORDAN5A:2; then g.s1 in the carrier of ((TOP-REAL 2)|P') by FUNCT_2:7; then A3: q1 in [#]((TOP-REAL 2)|P') by A1,PRE_TOPC:12; A4: s2 in the carrier of I[01] by A1,JORDAN5A:2; then g.s2 in the carrier of ((TOP-REAL 2)|P') by FUNCT_2:7; then q2 in [#]((TOP-REAL 2)|P') by A1,PRE_TOPC:12; then A5: q1 in P & q2 in P by A3,PRE_TOPC:def 10; now let h be map of I[01], (TOP-REAL 2)|P', t1,t2 be Real; assume A6: h is_homeomorphism & h.0=p1 & h.1=p2 & h.t1=q1 & 0<=t1 & t1<=1 & h.t2=q2 & 0<=t2 & t2<=1; set hg = h"*g; h" is_homeomorphism by A6,TOPS_2:70; then A7: hg is_homeomorphism by A1,TOPS_2:71; A8: dom h = [#]I[01] & rng h = [#]((TOP-REAL 2)|P) by A6,TOPS_2:def 5; then A9: dom h = the carrier of I[01] & rng h = the carrier of ((TOP-REAL 2)|P) by PRE_TOPC:12; then A10: 0 in dom h & 1 in dom h by JORDAN5A:2; A11: h is one-to-one by A6,TOPS_2:def 5; A12: hg is continuous & hg is one-to-one by A7,TOPS_2:def 5; A13: h".p1 = (h qua Function)".p1 by A8,A11,TOPS_2:def 4 .= 0 by A6,A10,A11,FUNCT_1:54; A14: h".p2 = (h qua Function)".p2 by A8,A11,TOPS_2:def 4 .= 1 by A6,A10,A11,FUNCT_1:54; dom g = [#]I[01] & rng g = [#]((TOP-REAL 2)|P) by A1,TOPS_2:def 5; then A15: dom g = the carrier of I[01] by PRE_TOPC:12; then 0 in dom g by JORDAN5A:2; then A16: (h"*g).0 = 0 by A1,A13,FUNCT_1:23; 1 in dom g by A15,JORDAN5A:2; then A17: (h"*g).1 = 1 by A1,A14,FUNCT_1:23; A18: t1 in dom h by A6,A9,JORDAN5A:2; s1 in dom g by A1,A15,JORDAN5A:2; then A19: (h"*g).s1 = h".q1 by A1,FUNCT_1:23 .= (h qua Function)".q1 by A8,A11,TOPS_2:def 4 .= t1 by A6,A11,A18,FUNCT_1:54; A20: t2 in dom h by A6,A9,JORDAN5A:2; s2 in dom g by A1,A15,JORDAN5A:2; then A21: (h"*g).s2 = h".q2 by A1,FUNCT_1:23 .= (h qua Function)".q2 by A8,A11,TOPS_2:def 4 .= t2 by A6,A11,A20,FUNCT_1:54; reconsider s1' = s1, s2' = s2 as Point of I[01] by A1,JORDAN5A:2; hg.s1 in the carrier of I[01] & hg.s2 in the carrier of I[01] by A2,A4,FUNCT_2:7; then reconsider hg1 = hg.s1', hg2 = hg.s2' as Real by BORSUK_1:83; per cases by A1,REAL_1:def 5; suppose s1 < s2; then hg1 < hg2 by A12,A16,A17,JORDAN5A:17; hence t1 <= t2 by A19,A21; suppose s1 = s2; hence t1 <= t2 by A19,A21; end; hence LE q1,q2,P,p1,p2 by A5,Def3; end; theorem Th9: for P being Subset of TOP-REAL 2, p1,p2,q1 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & q1 in P holds LE q1,q1,P,p1,p2 proof let P be Subset of TOP-REAL 2, p1,p2,q1 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2 & q1 in P; then reconsider P as non empty Subset of TOP-REAL 2; now let g be map of I[01], (TOP-REAL 2)|P, s1,s2 be Real; assume A2: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1; then A3: s1 in the carrier of I[01] & s2 in the carrier of I[01] by JORDAN5A:2; g is one-to-one by A2,TOPS_2:def 5; hence s1 <= s2 by A2,A3,FUNCT_2:25; end; hence thesis by A1,Def3; end; theorem Th10: for P being Subset of TOP-REAL 2, p1,p2,q1 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & q1 in P holds LE p1,q1,P,p1,p2 & LE q1,p2,P,p1,p2 proof let P be Subset of TOP-REAL 2, p1,p2,q1 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2 & q1 in P; then reconsider P as non empty Subset of TOP-REAL 2; A2:p1 in P & p2 in P by A1,TOPREAL1:4; A3:now let g be map of I[01], (TOP-REAL 2)|P,s1,s2 be Real; assume A4: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=p1 & 0<=s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1; then A5: s1 in the carrier of I[01] & s2 in the carrier of I[01] & 0 in the carrier of I[01] by JORDAN5A:2; g is one-to-one by A4,TOPS_2:def 5; hence s1 <= s2 by A4,A5,FUNCT_2:25; end; now let g be map of I[01], (TOP-REAL 2)|P,s1,s2 be Real; assume A6: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=p2 & 0<=s2 & s2<=1; then A7: s1 in the carrier of I[01] & s2 in the carrier of I[01] & 1 in the carrier of I[01] by JORDAN5A:2; g is one-to-one by A6,TOPS_2:def 5; hence s1 <= s2 by A6,A7,FUNCT_2:25; end; hence thesis by A1,A2,A3,Def3; end; theorem for P being Subset of TOP-REAL 2, p1,p2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 holds LE p1,p2,P,p1,p2 proof let P be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2; then p2 in P by TOPREAL1:4; hence thesis by A1,Th10; end; theorem Th12: for P being Subset of TOP-REAL 2, p1, p2, q1, q2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q1,P,p1,p2 holds q1=q2 proof let P be Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q1,P,p1,p2; then consider f being map of I[01], (TOP-REAL 2)|P such that A2: f is_homeomorphism & f.0 = p1 & f.1 = p2 by TOPREAL1:def 2; A3: dom f = [#]I[01] by A2,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; A4:rng f = [#]((TOP-REAL 2)|P) by A2,TOPS_2:def 5 .= P by PRE_TOPC:def 10; then q2 in rng f by A1,Def3; then consider x be set such that A5: x in dom f & q2 = f.x by FUNCT_1:def 5; [.0,1.] = { r1 where r1 is Real : 0 <= r1 & r1 <= 1 } by RCOMP_1:def 1; then consider s3 being Real such that A6: s3 = x & 0 <= s3 & s3 <= 1 by A3,A5,BORSUK_1:83; q1 in rng f by A1,A4,Def3; then consider y be set such that A7: y in dom f & q1 = f.y by FUNCT_1:def 5; [.0,1.] = { r1 where r1 is Real : 0 <= r1 & r1 <= 1 } by RCOMP_1:def 1; then consider s4 being Real such that A8: s4 = y & 0 <= s4 & s4 <= 1 by A3,A7,BORSUK_1:83; A9: s3 <= s4 by A1,A2,A5,A6,A7,A8,Def3; s4 <= s3 by A1,A2,A5,A6,A7,A8,Def3; hence q1=q2 by A5,A6,A7,A8,A9,AXIOMS:21; end; theorem Th13: for P being Subset of TOP-REAL 2, p1,p2,q1,q2,q3 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds LE q1,q3,P,p1,p2 proof let P be Subset of TOP-REAL 2, p1,p2,q1,q2,q3 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2 & LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2; then A2: q1 in P & q2 in P & for g being map of I[01], (TOP-REAL 2)|P,s1,s2 being Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q2 & 0<=s2 & s2<=1 holds s1<=s2 by Def3; A3: q3 in P & for g being map of I[01], (TOP-REAL 2)|P,s1,s2 being Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q2 & 0<=s1 & s1<=1 & g.s2=q3 & 0<=s2 & s2<=1 holds s1<=s2 by A1,Def3; now let g be map of I[01], (TOP-REAL 2)|P,s1,s2 be Real; assume A4: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q3 & 0<=s2 & s2<=1; then A5: dom g = [#]I[01] by TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; rng g = [#]((TOP-REAL 2)|P) by A4,TOPS_2:def 5 .= P by PRE_TOPC:def 10; then consider x be set such that A6: x in dom g & q2 = g.x by A2,FUNCT_1:def 5; [.0,1.] = { r1 where r1 is Real : 0 <= r1 & r1 <= 1 } by RCOMP_1:def 1; then consider s3 being Real such that A7: s3 = x & 0 <= s3 & s3 <= 1 by A5,A6,BORSUK_1:83; s1 <= s3 & s3 <= s2 by A1,A4,A6,A7,Def3; hence s1 <= s2 by AXIOMS:22; end; hence thesis by A2,A3,Def3; end; theorem for P being Subset of TOP-REAL 2, p1, p2, q1, q2 being Point of TOP-REAL 2 st P is_an_arc_of p1, p2 & q1 in P & q2 in P & q1 <> q2 holds LE q1,q2,P,p1,p2 & not LE q2,q1,P,p1,p2 or LE q2,q1,P,p1,p2 & not LE q1,q2,P,p1,p2 proof let P be Subset of TOP-REAL 2, p1,p2,q1,q2 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2 & q1 in P & q2 in P & q1 <> q2; then reconsider P as non empty Subset of TOP-REAL 2; not (LE q1,q2,P,p1,p2 iff LE q2,q1,P,p1,p2) proof assume A2: LE q1,q2,P,p1,p2 iff LE q2,q1,P,p1,p2; consider f be map of I[01], (TOP-REAL 2)|P such that A3: f is_homeomorphism & f.0 = p1 & f.1 = p2 by A1,TOPREAL1:def 2; A4: rng f = [#]((TOP-REAL 2)|P) by A3,TOPS_2:def 5 .= P by PRE_TOPC:def 10; then consider x be set such that A5: x in dom f & q1 = f.x by A1,FUNCT_1:def 5; consider y be set such that A6: y in dom f & q2 = f.y by A1,A4,FUNCT_1:def 5; A7: dom f = [#]I[01] by A3,TOPS_2:def 5 .= [.0,1.] by BORSUK_1:83,PRE_TOPC:12; then reconsider s1 = x, s2 = y as Real by A5,A6; A8: f.s1=q1 & 0<=s1 & s1<=1 & f.s2=q2 & 0<=s2 & s2<=1 by A5,A6,A7,BORSUK_1:83,JORDAN5A:2; per cases by REAL_1:def 5; suppose s1 < s2; hence thesis by A1,A2,A3,A8,Th8,Th12; suppose s1 = s2; hence thesis by A1,A5,A6; suppose s1 > s2; hence thesis by A1,A2,A3,A8,Th8,Th12; end; hence thesis; end; begin :: Some properties of the ordering of points on a curve theorem Th15: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2 st f is_S-Seq & L~f /\ Q is closed & q in L~f & q in Q holds LE First_Point(L~f,f/.1,f/.len f,Q), q, L~f, f/.1, f/.len f proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2; assume A1: f is_S-Seq & L~f /\ Q is closed & q in L~f & q in Q; then A2: L~f meets Q by XBOOLE_0:3; set P = L~f; A3: P is_an_arc_of f/.1,f/.len f by A1,TOPREAL1:31; A4: L~f /\ Q c= L~f by XBOOLE_1:17; set q1 = First_Point(P,f/.1,f/.len f,Q); A5: q1 in L~f /\ Q by A1,A2,A3,Def1; for g being map of I[01], (TOP-REAL 2)| P, s1, s2 being Real st g is_homeomorphism & g.0=f/.1 & g.1=f/.len f & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q & 0<=s2 & s2<=1 holds s1<=s2 by A1,A2,A3,Def1; hence thesis by A1,A4,A5,Def3; end; theorem Th16: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2 st f is_S-Seq & L~f /\ Q is closed & q in L~f & q in Q holds LE q, Last_Point(L~f,f/.1,f/.len f,Q), L~f, f/.1, f/.len f proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2; set P = L~f; assume A1: f is_S-Seq & L~f /\ Q is closed & q in L~f & q in Q; then A2: L~f meets Q by XBOOLE_0:3; A3: P is_an_arc_of f/.1,f/.len f by A1,TOPREAL1:31; A4: L~f /\ Q c= L~f & L~f /\ Q c= Q by XBOOLE_1:17; set q1 = Last_Point(P,f/.1,f/.len f,Q); A5: q1 in L~f /\ Q by A1,A2,A3,Def2; for g being map of I[01], (TOP-REAL 2)| P, s1, s2 being Real st g is_homeomorphism & g.0=f/.1 & g.1=f/.len f & g.s1=q & 0<=s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1 holds s1<=s2 by A1,A2,A3,Def2; hence thesis by A1,A4,A5,Def3; end; theorem Th17: for q1, q2, p1, p2 being Point of TOP-REAL 2 st p1 <> p2 holds LE q1, q2, LSeg(p1, p2), p1, p2 implies LE q1, q2, p1, p2 proof let q1, q2, p1, p2 be Point of TOP-REAL 2; set P = LSeg (p1, p2); assume A1: p1 <> p2; assume A2: LE q1, q2, P, p1, p2; hence q1 in P & q2 in P by Def3; let r1,r2 be Real; assume A3: 0<=r1 & r1<=1 & q1=(1-r1)*p1+r1*p2 & 0<=r2 & r2<=1 & q2=(1-r2)*p1+r2*p2; consider f be map of I[01], (TOP-REAL 2) | LSeg(p1,p2) such that A4: (for x being Real st x in [.0,1.] holds f.x = (1-x)*p1 + x*p2) & f is_homeomorphism & f.0 = p1 & f.1 = p2 by A1,JORDAN5A:4; r1 in [.0,1.] & r2 in [.0,1.] by A3,BORSUK_1:83,JORDAN5A:2; then q1 = f.r1 & q2 = f.r2 by A3,A4; hence thesis by A2,A3,A4,Def3; end; theorem for P, Q being Subset of TOP-REAL 2, p1, p2 being Point of TOP-REAL 2 st P is_an_arc_of p1,p2 & P meets Q & P /\ Q is closed holds First_Point(P,p1,p2,Q) = Last_Point(P,p2,p1,Q) & Last_Point(P,p1,p2,Q) = First_Point(P,p2,p1,Q) proof let P, Q be Subset of TOP-REAL 2, p1,p2 be Point of TOP-REAL 2; assume A1: P is_an_arc_of p1,p2 & P /\ Q <> {} & P /\ Q is closed; then consider x be set such that A2: x in P /\ Q by XBOOLE_0:def 1; reconsider P as non empty Subset of TOP-REAL 2 by A2; A3: P is_an_arc_of p2,p1 & P meets Q by A1,JORDAN5B:14,XBOOLE_0:def 7; then A4: Last_Point(P,p2,p1,Q) in P /\ Q by A1,Def2; A5: First_Point(P,p2,p1,Q) in P /\ Q & Last_Point(P,p1,p2,Q) in P /\ Q by A1,A3,Def1,Def2; A6: for g being map of I[01], (TOP-REAL 2)|P, s2 being Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=Last_Point(P,p2,p1,Q) & 0<=s2 & s2<=1 holds (for t being Real st 0<=t & t<s2 holds not g.t in Q) proof let f be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A7: f is_homeomorphism & f.0 = p1 & f.1 = p2 & f.s2 = Last_Point(P,p2,p1,Q) & 0<=s2 & s2<=1; set Ex = L[01]((0,1)(#),(#)(0,1)); A8: Ex is_homeomorphism by TREAL_1:21; set g = f * Ex; A9: Ex.(0,1)(#) = 0 by BORSUK_1:def 17,TREAL_1:8,12; A10: Ex.(#)(0,1) = 1 by BORSUK_1:def 18,TREAL_1:8,12; dom f = [#]I[01] by A7,TOPS_2:def 5; then A11: rng Ex = dom f by A8,TOPMETR:27,TOPS_2:def 5; then A12: dom g = dom Ex by RELAT_1:46; rng g = rng f by A11,RELAT_1:47; then A13: rng g = [#] ((TOP-REAL 2) | P) by A7,TOPS_2:def 5; dom g = [#]I[01] by A8,A12,TOPMETR:27,TOPS_2:def 5; then A14: dom g = the carrier of I[01] & rng g = the carrier of ((TOP-REAL 2) | P) by A13,PRE_TOPC:12; then g is Function of the carrier of I[01], the carrier of ((TOP-REAL 2) | P) by FUNCT_2:def 1,RELSET_1:11; then reconsider g as map of I[01], (TOP-REAL 2) | P; A15: g is_homeomorphism by A7,A8,TOPMETR:27,TOPS_2:71; A16: g.0 = p2 by A7,A10,A14,BORSUK_1:def 17,FUNCT_1:22,TREAL_1:8; A17: g.1 = p1 by A7,A9,A14,BORSUK_1:def 18,FUNCT_1:22,TREAL_1:8; let t be Real; assume A18: 0<=t & t<s2; then A19: t <= 1 by A7,AXIOMS:22; set s21 = 1 - s2; A20: 0 <= s21 & s21 <= 1 by A7,Lm1; A21: 0 <= 1-t & 1-t <= 1 by A18,A19,Lm1; then reconsider r2 = 1-s2, t' = 1-t as Point of Closed-Interval-TSpace(0,1) by A20,JORDAN5A:2,TOPMETR:27; A22: Ex.r2 = (1-(1-s2))*1 + (1-s2)*0 by BORSUK_1:def 17,def 18,TREAL_1:8,def 3 .= s2 by XCMPLX_1:18; A23: Ex.t' = (1-(1-t))*1 + (1-t)*0 by BORSUK_1:def 17,def 18,TREAL_1:8,def 3 .= t by XCMPLX_1:18; s21 in dom g by A14,A20,JORDAN5A:2; then A24: g.s21 = f.s2 by A22,FUNCT_1:22; 1-t in dom g by A14,A21,JORDAN5A:2; then A25: g.t' = f.t by A23,FUNCT_1:22; 1-s2 < 1-t by A18,REAL_1:92; hence thesis by A1,A3,A7,A15,A16,A17,A20,A21,A24,A25,Def2; end; for g being map of I[01], (TOP-REAL 2)|P, s2 being Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s2=First_Point(P,p2,p1,Q) & 0<=s2 & s2<=1 holds (for t being Real st 1>=t & t>s2 holds not g.t in Q) proof let f be map of I[01], (TOP-REAL 2)|P, s2 be Real; assume A26: f is_homeomorphism & f.0 = p1 & f.1 = p2 & f.s2 = First_Point(P,p2,p1,Q) & 0<=s2 & s2<=1; set Ex = L[01]((0,1)(#),(#)(0,1)); A27: Ex is_homeomorphism by TREAL_1:21; set g = f * Ex; A28: Ex.(0,1)(#) = 0 by BORSUK_1:def 17,TREAL_1:8,12; A29: Ex.(#)(0,1) = 1 by BORSUK_1:def 18,TREAL_1:8,12; dom f = [#]I[01] by A26,TOPS_2:def 5; then A30: rng Ex = dom f by A27,TOPMETR:27,TOPS_2:def 5; then A31: dom g = dom Ex by RELAT_1:46; rng g = rng f by A30,RELAT_1:47; then A32: rng g = [#] ((TOP-REAL 2) | P) by A26,TOPS_2:def 5; dom g = [#]I[01] by A27,A31,TOPMETR:27,TOPS_2:def 5; then A33: dom g = the carrier of I[01] & rng g = the carrier of ((TOP-REAL 2) | P) by A32,PRE_TOPC:12; then g is Function of the carrier of I[01], the carrier of ((TOP-REAL 2) | P) by FUNCT_2:def 1,RELSET_1:11; then reconsider g as map of I[01], (TOP-REAL 2) | P; A34: g is_homeomorphism by A26,A27,TOPMETR:27,TOPS_2:71; A35: g.0 = p2 by A26,A29,A33,BORSUK_1:def 17,FUNCT_1:22,TREAL_1:8; A36: g.1 = p1 by A26,A28,A33,BORSUK_1:def 18,FUNCT_1:22,TREAL_1:8; let t be Real; assume A37: 1>=t & t>s2; then A38: t >= 0 by A26,AXIOMS:22; set s21 = 1 - s2; A39: 0 <= s21 & s21 <= 1 by A26,Lm1; A40: 0 <= 1-t & 1-t <= 1 by A37,A38,Lm1; then reconsider r2 = 1-s2, t' = 1-t as Point of Closed-Interval-TSpace(0,1) by A39,JORDAN5A:2,TOPMETR:27; A41: Ex.r2 = (1-(1-s2))*1 + (1-s2)*0 by BORSUK_1:def 17,def 18,TREAL_1:8,def 3 .= s2 by XCMPLX_1:18; A42: Ex.t' = (1-(1-t))*1 + (1-t)*0 by BORSUK_1:def 17,def 18,TREAL_1:8,def 3 .= t by XCMPLX_1:18; s21 in dom g by A33,A39,JORDAN5A:2; then A43: g.s21 = f.s2 by A41,FUNCT_1:22; 1-t in dom g by A33,A40,JORDAN5A:2; then A44: g.t' = f.t by A42,FUNCT_1:22; 1-s2 > 1-t by A37,REAL_1:92; hence thesis by A1,A3,A26,A34,A35,A36,A39,A40,A43,A44,Def1; end; hence thesis by A1,A3,A4,A5,A6,Def1,Def2; end; theorem Th19: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, i being Nat st L~f meets Q & Q is closed & f is_S-Seq & 1 <= i & i+1 <= len f & First_Point (L~f, f/.1, f/.len f, Q) in LSeg (f, i) holds First_Point (L~f, f/.1, f/.len f, Q) = First_Point (LSeg (f, i), f/.i, f/.(i+1), Q) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, i be Nat; A1: L~f is closed by SPPOL_1:49; assume A2: L~f meets Q & Q is closed & f is_S-Seq & 1 <= i & i+1 <= len f & First_Point (L~f, f/.1, f/.len f, Q) in LSeg (f, i); then A3: f is one-to-one by TOPREAL1:def 10; len f >= 2 by A2,TOPREAL1:def 10; then reconsider P = L~f, R = LSeg (f, i) as non empty Subset of the carrier of TOP-REAL 2 by A2,TOPREAL1:29; A4: L~f meets Q & L~f /\ Q is closed & P is_an_arc_of f/.1, f/.len f by A1,A2,TOPREAL1:31,TOPS_1:35; then First_Point (P, f/.1, f/.len f, Q) in L~f /\ Q by Def1; then A5: First_Point (P, f/.1, f/.len f, Q) in L~f & First_Point (P, f/.1, f/.len f, Q) in Q by XBOOLE_0:def 3; set FPO = First_Point (R, f/.i, f/.(i+1), Q), FPG = First_Point (P, f/.1, f/.len f, Q); A6: i in dom f & i+1 in dom f by A2,GOBOARD2:3; A7: f/.i <> f/.(i+1) proof assume f/.i = f/.(i+1); then i = i+1 by A3,A6,PARTFUN2:17; hence thesis by REAL_1:69; end; FPG = FPO proof A8: LSeg (f,i) is closed by SPPOL_1:40; A9:LSeg (f,i) = LSeg (f/.i, f/.(i+1)) by A2,TOPREAL1:def 5; LSeg (f, i) /\ Q <> {} by A2,A5,XBOOLE_0:def 3; then A10: LSeg (f, i) meets Q & LSeg (f, i) /\ Q is closed & R is_an_arc_of f/.i, f/.(i+1) by A2,A7,A8,A9,TOPREAL1:15,TOPS_1:35,XBOOLE_0:def 7; FPG in L~f /\ Q & for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=f/.1 & g.1=f/.len f & g.s2= FPG & 0<=s2 & s2<=1 holds (for t be Real st 0<=t & t<s2 holds not g.t in Q) by A4,Def1; then A11:FPG in L~f by XBOOLE_0:def 3; consider F be map of I[01], (TOP-REAL 2)|P such that A12: F is_homeomorphism & F.0 = f/.1 & F.1 = f/.len f by A4,TOPREAL1:def 2; A13:dom F = [#]I[01] by A12,TOPS_2:def 5 .= [.0,1.] by BORSUK_1:83,PRE_TOPC:12; rng F = [#]((TOP-REAL 2)|P) by A12,TOPS_2:def 5 .= L~f by PRE_TOPC:def 10; then consider s21 be set such that A14: s21 in dom F & F.s21 = FPG by A11,FUNCT_1:def 5; reconsider s21 as Real by A13,A14; A15:0 <= s21 & s21 <= 1 by A13,A14,BORSUK_1:83,JORDAN5A:2; A16: FPG in LSeg (f,i) /\ Q by A2,A5,XBOOLE_0:def 3; for g being map of I[01], (TOP-REAL 2)|R, s2 be Real st g is_homeomorphism & g.0=f/.i & g.1=f/.(i+1) & g.s2 = FPG & 0<=s2 & s2<=1 holds (for t be Real st 0<=t & t<s2 holds not g.t in Q) proof let g be map of I[01], (TOP-REAL 2)|R, s2 be Real; assume A17: g is_homeomorphism & g.0=f/.i & g.1=f/.(i+1) & g.s2 = FPG & 0<=s2 & s2<=1; consider ppi, pi1 be Real such that A18: ppi < pi1 & 0 <= ppi & ppi <= 1 & 0 <= pi1 & pi1 <= 1 & LSeg (f, i) = F.:[.ppi, pi1.] & F.ppi = f/.i & F.pi1 = f/.(i+1) by A2,A12,JORDAN5B:7; A19: ppi in { dd where dd is Real : ppi <= dd & dd <= pi1 } & pi1 in { dd where dd is Real : ppi <= dd & dd <= pi1 } by A18; then A20: ppi in [.ppi,pi1.] & pi1 in [.ppi,pi1.] by RCOMP_1:def 1; A21: [.ppi,pi1.] c= [.0,1.] by A18,TREAL_1:1; then reconsider Poz = [.ppi, pi1.] as non empty Subset of I[01] by A19,BORSUK_1:83,RCOMP_1:def 1; the carrier of ((TOP-REAL 2)|P) = [#] ((TOP-REAL 2)|P) by PRE_TOPC:12 .= P by PRE_TOPC:def 10; then reconsider SEG = LSeg (f, i) as non empty Subset of the carrier of (TOP-REAL 2)|P by A2,TOPREAL3:26; A22: the carrier of (((TOP-REAL 2) | P) | SEG) = [#](((TOP-REAL 2) | P) | SEG) by PRE_TOPC:12 .= SEG by PRE_TOPC:def 10; reconsider SE = SEG as non empty Subset of TOP-REAL 2; reconsider SEG as non empty Subset of (TOP-REAL 2)|P; A23: ((TOP-REAL 2) | P) | SEG = (TOP-REAL 2) | SE by GOBOARD9:4; consider hh be map of I[01]|Poz, (TOP-REAL 2)|R such that A24: hh = F|Poz & hh is_homeomorphism by A2,A12,A18,JORDAN5B:8; A25: F is continuous & F is one-to-one by A12,TOPS_2:def 5; A26: hh is one-to-one by A24,TOPS_2:def 5; A27: hh is one-to-one by A24,TOPS_2:def 5; A28: dom hh = [#] (I[01] | Poz) by A24,TOPS_2:def 5; then A29:dom hh = Poz by PRE_TOPC:def 10; A30:rng hh = hh.:(the carrier of (I[01]|Poz)) by FUNCT_2:45 .= hh.:(dom hh) by A28,PRE_TOPC:12 .= SEG by A18,A24,A29,RFUNCT_2:5 .= [#](((TOP-REAL 2) | P) | SEG) by A22,PRE_TOPC:12; A31:the carrier of Closed-Interval-TSpace (ppi,pi1) = Poz by A18,TOPMETR:25; reconsider A = Closed-Interval-TSpace (ppi,pi1) as strict SubSpace of I[01] by A18,TOPMETR:27,TREAL_1:6; A32:Poz = [#] A by A31,PRE_TOPC:12; A33: rng g = [#]((TOP-REAL 2) | SE) by A17,TOPS_2:def 5; A34: rng (hh") = [#] (I[01] | Poz) by A23,A26,A30,TOPS_2:62 .= Poz by PRE_TOPC:def 10; set H = hh" * g; A35: id Poz is one-to-one by FUNCT_1:52; A36:hh = F * id Poz by A24,RELAT_1:94; A37: dom g = [#]I[01] by A17,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; let t be Real; assume A38: 0 <= t & t < s2; then A39: t < 1 by A17,AXIOMS:22; then A40: t in dom g by A37,A38,JORDAN5A:2; ppi in dom F /\ Poz & pi1 in dom F /\ Poz by A13,A20,A21,XBOOLE_0:def 3 ; then A41: ppi in dom hh & pi1 in dom hh by A24,RELAT_1:90; then A42: hh.ppi = f/.i & hh.pi1 = f/.(i+1) by A18,A24,FUNCT_1:70; 0 in dom g by A37,JORDAN5A:2; then A43:H.0 = hh".(f/.i) by A17,FUNCT_1:23 .= (hh qua Function)".(f/.i) by A23,A27,A30,TOPS_2:def 4 .= ppi by A27,A41,A42,FUNCT_1:54; 1 in dom g by A37,JORDAN5A:2; then A44: H.1 = hh".(f/.(i+1)) by A17,FUNCT_1:23 .= (hh qua Function)".(f/.(i+1)) by A23,A27,A30,TOPS_2:def 4 .= pi1 by A27,A41,A42,FUNCT_1:54; set ss = H.t; A45: dom (hh") = [#] (((TOP-REAL 2) | P) | SEG) by A23,A26,A30,TOPS_2:62; then A46: dom H = dom g & rng H = rng (hh") by A23,A33,RELAT_1:46,47; A47: rng H = Poz by A23,A33,A34,A45,RELAT_1:47; ss in Poz by A34,A40,A46,FUNCT_1:def 5; then ss in { l where l is Real : ppi <= l & l <= pi1 } by RCOMP_1:def 1; then consider ss' be Real such that A48: ss' = ss & ppi <= ss' & ss' <= pi1; A49: F is one-to-one by A12,TOPS_2:def 5; A50: rng F = [#]((TOP-REAL 2)|P) by A12,TOPS_2:def 5; g.t in [#] ((TOP-REAL 2) | SE) by A33,A40,FUNCT_1:def 5; then A51: g.t in SEG by PRE_TOPC:def 10; then g.t in the carrier of ((TOP-REAL 2)|P); then A52: g.t in rng F by A50,PRE_TOPC:12; A53: t in dom H by A23,A33,A40,A45,RELAT_1:46; A54: g.t in dom (F") by A49,A50,A52,TOPS_2:62; consider x be set such that A55: x in dom F & x in Poz & g.t = F.x by A18,A51,FUNCT_1:def 12; x = (F qua Function)".(g.t) by A25,A55,FUNCT_1:54; then A56: F".(g.t) in Poz by A49,A50,A55,TOPS_2:def 4; A57: (F qua Function)".(g.t) in Poz by A25,A55,FUNCT_1:54; A58: F".(g.t) in dom (id Poz) by A56,FUNCT_1:34; consider z be set such that A59: z in dom F & z in Poz & F.s21 = F.z by A2,A14,A18,FUNCT_1:def 12; A60:s21 in Poz by A14,A25,A59,FUNCT_1:def 8; A61:s2 in dom g by A17,A37,JORDAN5A:2; hh.s21 = g.s2 by A14,A17,A24,A60,FUNCT_1:72; then s21 = (hh qua Function)".(g.s2) by A26,A29,A60,FUNCT_1:54; then A62: s21 = hh".(g.s2) by A23,A26,A30,TOPS_2:def 4; A63: dom H = the carrier of Closed-Interval-TSpace(0,1) by A23,A33,A37,A45,RELAT_1:46,TOPMETR:27; rng H c= the carrier of Closed-Interval-TSpace(ppi,pi1) by A18,A47,TOPMETR:25; then H is Function of the carrier of Closed-Interval-TSpace(0,1), the carrier of Closed-Interval-TSpace(ppi,pi1) by A63,FUNCT_2:4; then reconsider H as map of Closed-Interval-TSpace(0,1), Closed-Interval-TSpace(ppi,pi1); reconsider w1 = s2, w2 = t as Point of Closed-Interval-TSpace(0,1) by A17,A38,A39,JORDAN5A:2,TOPMETR:27; A64: g is continuous one-to-one by A17,TOPS_2:def 5; hh" is_homeomorphism by A24,TOPS_2:70; then A65: hh" is continuous one-to-one by TOPS_2:def 5; I[01] | Poz = A by A32,PRE_TOPC:def 10; then A66: H is continuous one-to-one by A64,A65,FUNCT_1:46,TOPMETR:27,TOPS_2:58 ; s21 = H.w1 & ss' = H.w2 & t < s2 & ppi < pi1 by A18,A38,A48,A61,A62,FUNCT_1:23; then A67: ss' < s21 by A43,A44,A66,JORDAN5A:16; F.ss = (((hh"*g) qua Relation) * (F qua Relation)).t by A53,FUNCT_1:23 .= ( (g qua Relation) * ((hh" qua Relation) * (F qua Relation))).t by RELAT_1:55 .= (F*hh").(g.t) by A40,FUNCT_1:23 .= (F*(hh qua Function)").(g.t) by A23,A26,A30,TOPS_2:def 4 .= ( F * (((F qua Function)" qua Relation) * ((id Poz)" qua Relation))).(g.t) by A25,A35,A36,FUNCT_1:66 .= ( ((F" qua Relation) * ((id Poz)" qua Relation)) * (F qua Relation)).(g.t) by A49,A50,TOPS_2:def 4 .= ( (F" qua Relation) * (((id Poz)" qua Relation) * (F qua Relation))).(g.t) by RELAT_1:55 .= ( (F" qua Relation) * ((F*id Poz) qua Relation) ).(g.t) by FUNCT_1:67 .= (F*(id Poz)).(F".(g.t)) by A54,FUNCT_1:23 .= F.((id Poz).(F".(g.t))) by A58,FUNCT_1:23 .= F.((id Poz).((F qua Function)".(g.t))) by A49,A50,TOPS_2:def 4 .= F.((F qua Function)".(g.t)) by A57,FUNCT_1:34 .= g.t by A49,A52,FUNCT_1:57; hence thesis by A4,A12,A14,A15,A18,A48,A67,Def1; end; hence thesis by A10,A16,Def1; end; hence thesis; end; theorem Th20: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, i being Nat st L~f meets Q & Q is closed & f is_S-Seq & 1 <= i & i+1 <= len f & Last_Point (L~f, f/.1, f/.len f, Q) in LSeg (f, i) holds Last_Point (L~f, f/.1, f/.len f, Q) = Last_Point (LSeg (f, i), f/.i, f/.(i+1), Q) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, i be Nat; A1: L~f is closed by SPPOL_1:49; assume A2: L~f meets Q & Q is closed & f is_S-Seq & 1 <= i & i+1 <= len f & Last_Point (L~f, f/.1, f/.len f, Q) in LSeg (f, i); then A3: f is one-to-one by TOPREAL1:def 10; len f >= 2 by A2,TOPREAL1:def 10; then reconsider P = L~f, R = LSeg (f, i) as non empty Subset of TOP-REAL 2 by A2,TOPREAL1:29; A4: L~f meets Q & L~f /\ Q is closed & P is_an_arc_of f/.1, f/.len f by A1,A2,TOPREAL1:31,TOPS_1:35; then Last_Point (P, f/.1, f/.len f, Q) in L~f /\ Q by Def2; then A5: Last_Point (P, f/.1, f/.len f, Q) in L~f & Last_Point (P, f/.1, f/.len f, Q) in Q by XBOOLE_0:def 3; set FPO = Last_Point (R, f/.i, f/.(i+1), Q), FPG = Last_Point (P, f/.1, f/.len f, Q); A6: i in dom f & i+1 in dom f by A2,GOBOARD2:3; A7: f/.i <> f/.(i+1) proof assume f/.i = f/.(i+1); then i = i+1 by A3,A6,PARTFUN2:17; hence thesis by REAL_1:69; end; FPG = FPO proof A8: LSeg (f,i) is closed by SPPOL_1:40; A9:LSeg (f,i) = LSeg (f/.i, f/.(i+1)) by A2,TOPREAL1:def 5; LSeg (f, i) /\ Q <> {} by A2,A5,XBOOLE_0:def 3; then A10: LSeg (f, i) meets Q & LSeg (f, i) /\ Q is closed & R is_an_arc_of f/.i, f/.(i+1) by A2,A7,A8,A9,TOPREAL1:15,TOPS_1:35,XBOOLE_0:def 7; FPG in L~f /\ Q & for g being map of I[01], (TOP-REAL 2)|P, s2 be Real st g is_homeomorphism & g.0=f/.1 & g.1=f/.len f & g.s2= FPG & 0<=s2 & s2<=1 holds (for t be Real st 1>=t & t>s2 holds not g.t in Q) by A4,Def2; then A11:FPG in L~f by XBOOLE_0:def 3; consider F be map of I[01], (TOP-REAL 2)|P such that A12: F is_homeomorphism & F.0 = f/.1 & F.1 = f/.len f by A4,TOPREAL1:def 2; A13:dom F = [#]I[01] by A12,TOPS_2:def 5 .= [.0,1.] by BORSUK_1:83,PRE_TOPC:12; rng F = [#]((TOP-REAL 2)|P) by A12,TOPS_2:def 5 .= L~f by PRE_TOPC:def 10; then consider s21 be set such that A14: s21 in dom F & F.s21 = FPG by A11,FUNCT_1:def 5; reconsider s21 as Real by A13,A14; A15:0 <= s21 & s21 <= 1 by A13,A14,BORSUK_1:83,JORDAN5A:2; A16: FPG in LSeg (f,i) /\ Q by A2,A5,XBOOLE_0:def 3; for g being map of I[01], (TOP-REAL 2)|R, s2 be Real st g is_homeomorphism & g.0=f/.i & g.1=f/.(i+1) & g.s2 = FPG & 0<=s2 & s2<=1 holds (for t be Real st 1>=t & t>s2 holds not g.t in Q) proof let g be map of I[01], (TOP-REAL 2)|R, s2 be Real; assume A17: g is_homeomorphism & g.0=f/.i & g.1=f/.(i+1) & g.s2 = FPG & 0<=s2 & s2<=1; consider ppi, pi1 be Real such that A18: ppi < pi1 & 0 <= ppi & ppi <= 1 & 0 <= pi1 & pi1 <= 1 & LSeg (f, i) = F.:[.ppi, pi1.] & F.ppi = f/.i & F.pi1 = f/.(i+1) by A2,A12,JORDAN5B:7; A19: ppi in { dd where dd is Real : ppi <= dd & dd <= pi1 } & pi1 in { dd where dd is Real : ppi <= dd & dd <= pi1 } by A18; then A20: ppi in [.ppi,pi1.] & pi1 in [.ppi,pi1.] by RCOMP_1:def 1; A21: [.ppi,pi1.] c= [.0,1.] by A18,TREAL_1:1; then reconsider Poz = [.ppi, pi1.] as non empty Subset of I[01] by A19,BORSUK_1:83,RCOMP_1:def 1; A22: the carrier of (I[01]|Poz) = [#] (I[01]|Poz) by PRE_TOPC:12 .= Poz by PRE_TOPC:def 10; set gg = F | Poz; gg is Function of the carrier of (I[01]|Poz), the carrier of (TOP-REAL 2)| P by A22,FUNCT_2:38; then reconsider gg as map of I[01]|Poz, (TOP-REAL 2)| P; reconsider Lf = L~f as non empty Subset of TOP-REAL 2 by A4; the carrier of ((TOP-REAL 2)|Lf) = [#] ((TOP-REAL 2)|Lf) by PRE_TOPC:12 .= Lf by PRE_TOPC:def 10; then reconsider SEG = LSeg (f, i) as non empty Subset of (TOP-REAL 2)|Lf by A2,TOPREAL3:26; A23: the carrier of (((TOP-REAL 2) | Lf) | SEG) = [#](((TOP-REAL 2) | Lf) | SEG) by PRE_TOPC:12 .= SEG by PRE_TOPC:def 10; rng gg c= SEG proof let ii be set; assume ii in rng gg; then consider j be set such that A24: j in dom gg & ii = gg.j by FUNCT_1:def 5; j in dom F /\ Poz by A24,RELAT_1:90; then A25: j in dom F by XBOOLE_0:def 3; A26: dom gg = Poz by A22,FUNCT_2:def 1; then j in dom F & F.j in LSeg (f,i) by A18,A24,A25,FUNCT_1:def 12; hence thesis by A24,A26,FUNCT_1:72; end; then A27:gg is Function of the carrier of (I[01]|Poz), the carrier of (((TOP-REAL 2) | Lf) | SEG) by A23,FUNCT_2:8; reconsider SE = SEG as non empty Subset of TOP-REAL 2; reconsider SEG as non empty Subset of (TOP-REAL 2)|Lf; A28: ((TOP-REAL 2) | Lf) | SEG = (TOP-REAL 2) | SE by GOBOARD9:4; reconsider hh' = gg as map of I[01]|Poz, ((TOP-REAL 2) | Lf)| SEG by A27; A29: F is continuous & F is one-to-one by A12,TOPS_2:def 5; then gg is continuous by TOPMETR:10; then A30: hh' is continuous by TOPMETR:9; A31: hh' is one-to-one by A29,FUNCT_1:84; reconsider hh = hh' as map of I[01]|Poz, (TOP-REAL 2) | SE by A28; A32: hh is one-to-one by A29,FUNCT_1:84; A33: dom hh = the carrier of (I[01] | Poz) by FUNCT_2:def 1 .= [#] (I[01] | Poz) by PRE_TOPC:12; then A34:dom hh = Poz by PRE_TOPC:def 10; A35:rng hh = hh.:(the carrier of (I[01]|Poz)) by FUNCT_2:45 .= hh.:(dom hh) by A33,PRE_TOPC:12 .= SEG by A18,A34,RFUNCT_2:5 .= [#](((TOP-REAL 2) | Lf) | SEG) by A23,PRE_TOPC:12; A36:the carrier of Closed-Interval-TSpace (ppi,pi1) = Poz by A18,TOPMETR:25; reconsider A = Closed-Interval-TSpace (ppi,pi1) as strict SubSpace of I[01] by A18,TOPMETR:27,TREAL_1:6; A37:Poz = [#] A by A36,PRE_TOPC:12; Closed-Interval-TSpace (ppi,pi1) is compact by A18,HEINE:11; then [#] Closed-Interval-TSpace (ppi,pi1) is compact by COMPTS_1:10; then for P being Subset of A st P=Poz holds P is compact by A36,PRE_TOPC:12 ; then Poz is compact by A37,COMPTS_1:11; then A38: I[01]|Poz is compact by COMPTS_1:12; TOP-REAL 2 is_T2 by SPPOL_1:26; then (TOP-REAL 2)|SE is_T2 by TOPMETR:3; then A39: hh is_homeomorphism by A28,A30,A31,A33,A35,A38,COMPTS_1:26; A40: rng g = [#]((TOP-REAL 2) | SE) by A17,TOPS_2:def 5; A41: rng (hh") = [#] (I[01] | Poz) by A28,A31,A35,TOPS_2:62 .= Poz by PRE_TOPC:def 10; set H = hh" * g; A42: id Poz is one-to-one by FUNCT_1:52; A43:hh = F * id Poz by RELAT_1:94; A44: dom g = [#]I[01] by A17,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; let t be Real; assume A45: 1 >= t & t > s2; then A46: t > 0 by A17; then A47: t in dom g by A44,A45,JORDAN5A:2; ppi in dom F /\ Poz & pi1 in dom F /\ Poz by A13,A20,A21,XBOOLE_0:def 3 ; then A48: ppi in dom hh & pi1 in dom hh by RELAT_1:90; then A49: hh.ppi = f/.i & hh.pi1 = f/.(i+1) by A18,FUNCT_1:70; 0 in dom g by A44,JORDAN5A:2; then A50:H.0 = hh".(f/.i) by A17,FUNCT_1:23 .= (hh qua Function)".(f/.i) by A28,A32,A35,TOPS_2:def 4 .= ppi by A32,A48,A49,FUNCT_1:54; 1 in dom g by A44,JORDAN5A:2; then A51: H.1 = hh".(f/.(i+1)) by A17,FUNCT_1:23 .= (hh qua Function)".(f/.(i+1)) by A28,A32,A35,TOPS_2:def 4 .= pi1 by A32,A48,A49,FUNCT_1:54; set ss = H.t; A52: dom (hh") = [#] (((TOP-REAL 2) | Lf) | SEG) by A28,A31,A35,TOPS_2:62; then A53: dom H = dom g & rng H = rng (hh") by A28,A40,RELAT_1:46,47; A54: rng H = Poz by A28,A40,A41,A52,RELAT_1:47; ss in Poz by A41,A47,A53,FUNCT_1:def 5; then ss in { l where l is Real : ppi <= l & l <= pi1 } by RCOMP_1:def 1; then consider ss' be Real such that A55: ss' = ss & ppi <= ss' & ss' <= pi1; A56: F is one-to-one by A12,TOPS_2:def 5; A57: rng F = [#]((TOP-REAL 2)|P) by A12,TOPS_2:def 5; g.t in [#] ((TOP-REAL 2) | SE) by A40,A47,FUNCT_1:def 5; then A58: g.t in SEG by PRE_TOPC:def 10; then g.t in the carrier of ((TOP-REAL 2)|Lf); then A59: g.t in rng F by A57,PRE_TOPC:12; A60: t in dom H by A28,A40,A47,A52,RELAT_1:46; A61: g.t in dom (F") by A56,A57,A59,TOPS_2:62; consider x be set such that A62: x in dom F & x in Poz & g.t = F.x by A18,A58,FUNCT_1:def 12; x = (F qua Function)".(g.t) by A29,A62,FUNCT_1:54; then A63: F".(g.t) in Poz by A56,A57,A62,TOPS_2:def 4; A64: (F qua Function)".(g.t) in Poz by A29,A62,FUNCT_1:54; A65: F".(g.t) in dom (id Poz) by A63,FUNCT_1:34; consider z be set such that A66: z in dom F & z in Poz & F.s21 = F.z by A2,A14,A18,FUNCT_1:def 12; A67:s21 in Poz by A14,A29,A66,FUNCT_1:def 8; A68:s2 in dom g by A17,A44,JORDAN5A:2; hh.s21 = g.s2 by A14,A17,A67,FUNCT_1:72; then s21 = (hh qua Function)".(g.s2) by A31,A34,A67,FUNCT_1:54; then A69: s21 = hh".(g.s2) by A28,A31,A35,TOPS_2:def 4; A70: dom H = the carrier of Closed-Interval-TSpace(0,1) by A28,A40,A44,A52,RELAT_1:46,TOPMETR:27; rng H c= the carrier of Closed-Interval-TSpace(ppi,pi1) by A18,A54,TOPMETR:25; then H is Function of the carrier of Closed-Interval-TSpace(0,1), the carrier of Closed-Interval-TSpace(ppi,pi1) by A70,FUNCT_2:4; then reconsider H as map of Closed-Interval-TSpace(0,1), Closed-Interval-TSpace(ppi,pi1); reconsider w1 = s2, w2 = t as Point of Closed-Interval-TSpace(0,1) by A17,A45,A46,JORDAN5A:2,TOPMETR:27; A71: g is continuous one-to-one by A17,TOPS_2:def 5; hh" is_homeomorphism by A39,TOPS_2:70; then A72: hh" is continuous one-to-one by TOPS_2:def 5; I[01] | Poz = A by A37,PRE_TOPC:def 10; then A73: H is continuous one-to-one by A71,A72,FUNCT_1:46,TOPMETR:27,TOPS_2:58 ; s21 = H.w1 & ss' = H.w2 & t > s2 & ppi < pi1 by A18,A45,A55,A68,A69,FUNCT_1:23; then A74: ss' > s21 by A50,A51,A73,JORDAN5A:16; A75: 1 >= ss' by A18,A55,AXIOMS:22; F.ss = (((hh"*g) qua Relation) * (F qua Relation)).t by A60,FUNCT_1:23 .= ( (g qua Relation) * ((hh" qua Relation) * (F qua Relation))).t by RELAT_1:55 .= (F*hh").(g.t) by A47,FUNCT_1:23 .= (F*(hh qua Function)").(g.t) by A28,A31,A35,TOPS_2:def 4 .= ( F * (((F qua Function)" qua Relation) * ((id Poz)" qua Relation))).(g.t) by A29,A42,A43,FUNCT_1:66 .= ( ((F" qua Relation) * ((id Poz)" qua Relation)) * (F qua Relation)).(g.t) by A56,A57,TOPS_2:def 4 .= ( (F" qua Relation) * (((id Poz)" qua Relation) * (F qua Relation))).(g.t) by RELAT_1:55 .= ( (F" qua Relation) * ((F*id Poz) qua Relation) ).(g.t) by FUNCT_1:67 .= (F*(id Poz)).(F".(g.t)) by A61,FUNCT_1:23 .= F.((id Poz).(F".(g.t))) by A65,FUNCT_1:23 .= F.((id Poz).((F qua Function)".(g.t))) by A56,A57,TOPS_2:def 4 .= F.((F qua Function)".(g.t)) by A64,FUNCT_1:34 .= g.t by A56,A59,FUNCT_1:57; hence thesis by A4,A12,A14,A15,A55,A74,A75,Def2; end; hence thesis by A10,A16,Def2; end; hence thesis; end; theorem for f being FinSequence of TOP-REAL 2, i be Nat st 1 <= i & i+1 <= len f & f is_S-Seq & First_Point (L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i) holds First_Point (L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.i proof let f be FinSequence of TOP-REAL 2, i be Nat; assume A1: 1 <= i & i+1 <= len f & f is_S-Seq & First_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i); then reconsider Q = LSeg (f,i) as non empty Subset of TOP-REAL 2; Q = LSeg (f/.i, f/.(i+1)) by A1,TOPREAL1:def 5; then Q c= L~f by A1,SPPOL_2:16; then L~f meets Q & Q is closed by A1,SPPOL_1:40,XBOOLE_0:3; then A2: First_Point (L~f, f/.1, f/.len f, Q) = First_Point (Q, f/.i, f/.(i+1), Q) by A1,Th19; Q is closed & Q is_an_arc_of f/.i, f/.(i+1) by A1,JORDAN5B:15,SPPOL_1:40; hence thesis by A2,Th7; end; theorem for f being FinSequence of TOP-REAL 2, i be Nat st 1 <= i & i+1 <= len f & f is_S-Seq & Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i) holds Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) = f/.(i+1) proof let f be FinSequence of TOP-REAL 2, i be Nat; assume A1: 1 <= i & i+1 <= len f & f is_S-Seq & Last_Point ( L~f, f/.1, f/.len f, LSeg (f,i) ) in LSeg (f,i); then reconsider Q = LSeg (f,i) as non empty Subset of TOP-REAL 2; Q = LSeg (f/.i, f/.(i+1)) by A1,TOPREAL1:def 5; then Q c= L~f by A1,SPPOL_2:16; then L~f meets Q & Q is closed by A1,SPPOL_1:40,XBOOLE_0:3; then A2: Last_Point (L~f, f/.1, f/.len f, Q) = Last_Point (Q, f/.i, f/.(i+1), Q) by A1,Th20; Q is closed & Q is_an_arc_of f/.i, f/.(i+1) by A1,JORDAN5B:15,SPPOL_1:40; hence thesis by A2,Th7; end; theorem Th23: for f be FinSequence of TOP-REAL 2, i be Nat st f is_S-Seq & 1 <= i & i+1 <= len f holds LE f/.i, f/.(i+1), L~f, f/.1, f/.len f proof let f be FinSequence of TOP-REAL 2, i be Nat; assume A1: f is_S-Seq & 1 <= i & i+1 <= len f; set p1 = f/.1, p2 = f/.len f, q1 = f/.i, q2 = f/.(i+1); A2:len f >= 2 by A1,TOPREAL1:def 10; then reconsider P = L~f as non empty Subset of TOP-REAL 2 by TOPREAL1:29; A3:i in dom f & i+1 in dom f by A1,GOBOARD2:3; then A4: q1 in P by A2,GOBOARD1:16; A5: q2 in P by A2,A3,GOBOARD1:16; for g being map of I[01], (TOP-REAL 2)|P, s1,s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q2 & 0<=s2 & s2<=1 holds s1<=s2 proof let g be map of I[01], (TOP-REAL 2)|P, s1,s2 be Real; assume A6: g is_homeomorphism & g.0 = p1 & g.1 = p2 & g.s1 = q1 & 0 <= s1 & s1 <= 1 & g.s2 = q2 & 0 <= s2 & s2 <= 1; then consider r1, r2 be Real such that A7: r1 < r2 & 0 <= r1 & r1 <= 1 & 0 <= r2 & r2 <= 1 & LSeg (f, i) = g.:[.r1, r2.] & g.r1 = q1 & g.r2 = q2 by A1,JORDAN5B:7; dom g = [#]I[01] by A6,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; then A8: r1 in dom g & s1 in dom g & r2 in dom g & s2 in dom g by A6,A7, JORDAN5A:2; g is one-to-one by A6,TOPS_2:def 5; then s1 = r1 & s2 = r2 by A6,A7,A8,FUNCT_1:def 8; hence thesis by A7; end; hence thesis by A4,A5,Def3; end; theorem Th24: for f be FinSequence of TOP-REAL 2, i, j be Nat st f is_S-Seq & 1 <= i & i <= j & j <= len f holds LE f/.i, f/.j, L~f, f/.1, f/.len f proof let f be FinSequence of TOP-REAL 2, i, j be Nat such that A1: f is_S-Seq and A2: 1 <= i and A3: i <= j and A4: j <= len f; A5: len f >= 2 by A1,TOPREAL1:def 10; then reconsider P = L~f as non empty Subset of TOP-REAL 2 by TOPREAL1:29; consider k being Nat such that A6: i+k = j by A3,NAT_1:28; A7: P is_an_arc_of f/.1, f/.len f by A1,TOPREAL1:31; defpred ILE[Nat] means 1 <= i & i+$1 <= len f implies LE f/.i, f/.(i+$1), P, f/.1, f/.len f; A8: ILE[0] proof assume 1 <= i & i+0 <= len f; then i in dom f by FINSEQ_3:27; then f/.i in P by A5,GOBOARD1:16; hence thesis by A7,Th9; end; A9: for l be Nat st ILE[l] holds ILE[l + 1] proof let l be Nat; assume A10: ILE[l]; A11: i <= i+l by NAT_1:29; assume A12: 1 <= i & i+(l+1) <= len f; A13: f/.(i+l+1) = f/.(i+(l+1)) by XCMPLX_1:1; 1 <= i+l & i+l+1 <= len f by A11,A12,AXIOMS:22,XCMPLX_1:1; then A14: LE f/.(i+l), f/.(i+(l+1)), P, f/.1,f/.len f by A1,A13,Th23; A15: i+(l+1) = i+l+1 by XCMPLX_1:1; i+l <= i+l+1 by NAT_1:29; hence thesis by A7,A10,A12,A14,A15,Th13,AXIOMS:22; end; for l be Nat holds ILE[l] from Ind(A8, A9); hence thesis by A2,A4,A6; end; Lm2: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2, i being Nat st LSeg (f,i) meets Q & f is_S-Seq & Q is closed & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q holds LE First_Point(LSeg(f,i), f/.i,f/.(i+1),Q), q, f/.i, f/.(i+1) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2, i be Nat; assume A1: LSeg(f,i) meets Q & f is_S-Seq & Q is closed & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q; then A2: LSeg(f,i) = LSeg(f/.i, f/.(i+1)) by TOPREAL1:def 5; reconsider P = LSeg(f,i) as non empty Subset of TOP-REAL 2 by A1; P is closed by SPPOL_1:40; then A3: P /\ Q is closed by A1,TOPS_1:35; A4: P is_an_arc_of f/.i,f/.(i+1) by A1,JORDAN5B:15; A5: P /\ Q c= P by XBOOLE_1:17; set q1 = First_Point(P,f/.i,f/.(i+1),Q), p1 = f/.i, p2 = f/.(i+1); A6: f is one-to-one by A1,TOPREAL1:def 10; A7: i in dom f & i+1 in dom f by A1,GOBOARD2:3; A8: p1 <> p2 proof assume p1 = p2; then i = i+1 by A6,A7,PARTFUN2:17; hence thesis by REAL_1:69; end; A9: q1 in P /\ Q by A1,A3,A4,Def1; for g being map of I[01], (TOP-REAL 2)|P, s1, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q & 0<=s2 & s2<=1 holds s1<=s2 by A1,A3,A4,Def1; then LE q1, q, P, p1, p2 by A1,A5,A9,Def3; hence thesis by A2,A8,Th17; end; Lm3: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2, i being Nat st L~f meets Q & f is_S-Seq & Q is closed & First_Point(L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q holds LE First_Point(L~f,f/.1,f/.len f,Q), q, f/.i, f/.(i+1) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2, i be Nat; assume A1: L~f meets Q & f is_S-Seq & Q is closed & First_Point(L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q; then LSeg (f,i) /\ Q <> {} by XBOOLE_0:def 3; then A2: LSeg (f,i) meets Q by XBOOLE_0:def 7; len f >= 2 by A1,TOPREAL1:def 10; then reconsider P = L~f, R = LSeg (f,i) as non empty Subset of TOP-REAL 2 by A1,TOPREAL1:29; First_Point (P,f/.1,f/.len f,Q) = First_Point (R, f/.i, f/.(i+1), Q) by A1,Th19; hence thesis by A1,A2,Lm2; end; Lm4: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2, i being Nat st LSeg (f,i) meets Q & f is_S-Seq & Q is closed & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q holds LE q, Last_Point(LSeg(f,i),f/.i,f/.(i+1),Q), f/.i, f/.(i+1) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2, i be Nat; assume A1: LSeg(f,i) meets Q & f is_S-Seq & Q is closed & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q; then A2: LSeg(f,i) = LSeg(f/.i, f/.(i+1)) by TOPREAL1:def 5; reconsider P = LSeg(f,i) as non empty Subset of TOP-REAL 2 by A1; P is closed by SPPOL_1:40; then A3: P /\ Q is closed by A1,TOPS_1:35; A4: P is_an_arc_of f/.i,f/.(i+1) by A1,JORDAN5B:15; A5: P /\ Q c= P by XBOOLE_1:17; set q1 = Last_Point(P,f/.i,f/.(i+1),Q), p1 = f/.i, p2 = f/.(i+1); A6: f is one-to-one by A1,TOPREAL1:def 10; A7: i in dom f & i+1 in dom f by A1,GOBOARD2:3; A8: p1 <> p2 proof assume p1 = p2; then i = i+1 by A6,A7,PARTFUN2:17; hence thesis by REAL_1:69; end; A9: q1 in P /\ Q by A1,A3,A4,Def2; for g being map of I[01], (TOP-REAL 2)|P, s1, s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q & 0<=s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1 holds s1<=s2 by A1,A3,A4,Def2; then LE q, q1, P, p1, p2 by A1,A5,A9,Def3; hence thesis by A2,A8,Th17; end; Lm5: for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2, i being Nat st L~f meets Q & f is_S-Seq & Q is closed & Last_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1 <= i & i+1 <= len f & q in LSeg(f,i) & q in Q holds LE q, Last_Point(L~f,f/.1,f/.len f,Q), f/.i, f/.(i+1) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2, i be Nat; assume A1: L~f meets Q & f is_S-Seq & Q is closed & Last_Point(L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,i) & q in Q; then LSeg (f,i) /\ Q <> {} by XBOOLE_0:def 3; then A2: LSeg (f,i) meets Q by XBOOLE_0:def 7; len f >= 2 by A1,TOPREAL1:def 10; then reconsider P = L~f, R = LSeg (f,i) as non empty Subset of the carrier of TOP-REAL 2 by A1,TOPREAL1:29; Last_Point(P,f/.1,f/.len f,Q) = Last_Point (R, f/.i, f/.(i+1), Q) by A1,Th20; hence thesis by A1,A2,Lm4; end; theorem Th25: for f being FinSequence of TOP-REAL 2, q being Point of TOP-REAL 2, i being Nat st f is_S-Seq & 1 <= i & i+1 <= len f & q in LSeg(f,i) holds LE f/.i, q, L~f, f/.1, f/.len f proof let f be FinSequence of TOP-REAL 2, q be Point of TOP-REAL 2, i be Nat; assume A1: f is_S-Seq & 1<=i & i+1<=len f & q in LSeg(f,i); then A2: 2 <= len f by TOPREAL1:def 10; then reconsider P = L~f as non empty Subset of the carrier of TOP-REAL 2 by TOPREAL1:29; set p1 = f/.1, p2 = f/.len f, q1 = f/.i; i in dom f by A1,GOBOARD2:3; then A3: q1 in P by A2,GOBOARD1:16; A4: q in P by A1,SPPOL_2:17; for g being map of I[01], (TOP-REAL 2)|P, s1,s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q & 0<=s2 & s2<=1 holds s1<=s2 proof let g be map of I[01], (TOP-REAL 2)|P, s1,s2 be Real; assume A5: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q & 0<=s2 & s2<=1; then consider r1, r2 be Real such that A6: r1 < r2 & 0 <= r1 & r1 <= 1 & 0 <= r2 & r2 <= 1 & LSeg (f, i) = g.: [.r1, r2.] & g.r1 = q1 & g.r2 = f/.(i+1) by A1,JORDAN5B:7 ; consider r3' be set such that A7: r3' in dom g & r3' in [.r1, r2.] & g.r3' = q by A1,A6,FUNCT_1:def 12; r3' in { l where l is Real : r1 <= l & l <= r2 } by A7,RCOMP_1:def 1; then consider r3 be Real such that A8: r3 = r3' & r1 <= r3 & r3 <= r2; dom g = [#]I[01] by A5,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; then A9: r1 in dom g & s1 in dom g & r2 in dom g & s2 in dom g by A5,A6,JORDAN5A:2; A10: g is one-to-one by A5,TOPS_2:def 5; then s2 = r3 by A5,A7,A8,A9,FUNCT_1:def 8; hence thesis by A5,A6,A8,A9,A10,FUNCT_1:def 8; end; hence thesis by A3,A4,Def3; end; theorem Th26: for f being FinSequence of TOP-REAL 2, q being Point of TOP-REAL 2, i being Nat st f is_S-Seq & 1<=i & i+1<=len f & q in LSeg(f,i) holds LE q, f/.(i+1), L~f, f/.1, f/.len f proof let f be FinSequence of TOP-REAL 2, q be Point of TOP-REAL 2, i be Nat; assume A1: f is_S-Seq & 1<=i & i+1<=len f & q in LSeg(f,i); then len f >= 2 by TOPREAL1:def 10; then reconsider P = L~f as non empty Subset of the carrier of TOP-REAL 2 by TOPREAL1:29; set p1 = f/.1, p2 = f/.len f, q1 = f/.(i+1); A2: q1 in LSeg (f,i) by A1,TOPREAL1:27; A3: q in P by A1,SPPOL_2:17; A4: q1 in P by A2,SPPOL_2:17; for g being map of I[01], (TOP-REAL 2)|P, s1,s2 be Real st g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q & 0<=s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1 holds s1<=s2 proof let g be map of I[01], (TOP-REAL 2)|P, s1,s2 be Real; assume A5: g is_homeomorphism & g.0=p1 & g.1=p2 & g.s1=q & 0<=s1 & s1<=1 & g.s2=q1 & 0<=s2 & s2<=1; then consider r1, r2 be Real such that A6: r1 < r2 & 0 <= r1 & r1 <= 1 & 0 <= r2 & r2 <= 1 & LSeg (f, i) = g.:[.r1, r2.] & g.r1 = f/.i & g.r2 = q1 by A1,JORDAN5B:7; consider r3' be set such that A7: r3' in dom g & r3' in [.r1, r2.] & g.r3' = q by A1,A6,FUNCT_1:def 12; r3' in { l where l is Real : r1 <= l & l <= r2 } by A7,RCOMP_1:def 1; then consider r3 be Real such that A8: r3 = r3' & r1 <= r3 & r3 <= r2; dom g = [#]I[01] by A5,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; then A9: r1 in dom g & s1 in dom g & r2 in dom g & s2 in dom g by A5,A6, JORDAN5A:2; A10: g is one-to-one by A5,TOPS_2:def 5; then s1 = r3 by A5,A7,A8,A9,FUNCT_1:def 8; hence thesis by A5,A6,A8,A9,A10,FUNCT_1:def 8; end; hence thesis by A3,A4,Def3; end; theorem for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2, i, j being Nat st L~f meets Q & f is_S-Seq & Q is closed & First_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,j) & 1 <= j & j + 1 <= len f & q in Q & First_Point (L~f,f/.1,f/.len f,Q) <> q holds i <= j & (i=j implies LE First_Point(L~f,f/.1,f/.len f,Q), q, f/.i, f/.(i+1)) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2, i,j be Nat; assume A1: L~f meets Q & f is_S-Seq & Q is closed & First_Point(L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,j) & 1<=j & j+1<=len f & q in Q & First_Point (L~f,f/.1,f/.len f,Q) <> q; then A2: q in L~f by SPPOL_2:17; reconsider P = L~f as non empty Subset of TOP-REAL 2 by A1,SPPOL_2:17; set q1 = First_Point(P,f/.1,f/.len f,Q), p1 = f/.i; A3: P is_an_arc_of f/.1, f/.len f by A1,TOPREAL1:31; thus i <= j proof assume j < i; then A4: j+1 <= i by NAT_1:38; i <= i + 1 & j <= j + 1 by NAT_1:29; then i <= len f & 1 <= j+1 by A1,AXIOMS:22; then A5: LE f/.(j+1), p1, P, f/.1, f/.len f by A1,A4,Th24; LE q, f/.(j+1), P, f/.1, f/.len f by A1,Th26; then A6: LE q, p1, P, f/.1, f/.len f by A3,A5,Th13; LE p1, q1, P, f/.1, f/.len f by A1,Th25; then A7: LE q, q1, P, f/.1, f/.len f by A3,A6,Th13; P is closed by SPPOL_1:49; then A8: L~f meets Q & L~f /\ Q is closed & P is_an_arc_of f/.1, f/.len f by A1,TOPREAL1:31,TOPS_1:35; then LE q1, q, P, f/.1, f/.len f by A1,A2,Th15; hence contradiction by A1,A7,A8,Th12; end; assume i = j; hence thesis by A1,Lm3; end; theorem for f being FinSequence of TOP-REAL 2, Q being Subset of TOP-REAL 2, q being Point of TOP-REAL 2, i, j being Nat st L~f meets Q & f is_S-Seq & Q is closed & Last_Point (L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,j) & 1 <= j & j + 1 <= len f & q in Q & Last_Point (L~f,f/.1,f/.len f,Q) <> q holds i >= j & (i=j implies LE q, Last_Point(L~f,f/.1,f/.len f,Q), f/.i, f/.(i+1)) proof let f be FinSequence of TOP-REAL 2, Q be Subset of TOP-REAL 2, q be Point of TOP-REAL 2, i,j be Nat; assume A1: L~f meets Q & f is_S-Seq & Q is closed & Last_Point(L~f,f/.1,f/.len f,Q) in LSeg(f,i) & 1<=i & i+1<=len f & q in LSeg(f,j) & 1<=j & j+1<=len f & q in Q & Last_Point (L~f,f/.1,f/.len f,Q) <> q; then A2: q in L~f by SPPOL_2:17; reconsider P = L~f as non empty Subset of TOP-REAL 2 by A1, SPPOL_2:17; set q1 = Last_Point(P,f/.1,f/.len f,Q), p2 = f/.(i+1); A3: P is_an_arc_of f/.1, f/.len f by A1,TOPREAL1:31; thus i >= j proof assume j > i; then A4: i+1 <= j by NAT_1:38; i <= i + 1 & j <= j + 1 by NAT_1:29; then 1 <= i + 1 & j <= len f by A1,AXIOMS:22; then A5:LE p2, f/.j, P, f/.1, f/.len f by A1,A4,Th24; LE f/.j, q, P, f/.1, f/.len f by A1,Th25; then A6:LE p2, q, P, f/.1, f/.len f by A3,A5,Th13; LE q1, p2, P, f/.1, f/.len f by A1,Th26; then A7:LE q1, q, P, f/.1, f/.len f by A3,A6,Th13; L~f is closed by SPPOL_1:49; then A8: L~f meets Q & L~f /\ Q is closed & P is_an_arc_of f/.1, f/.len f by A1,TOPREAL1:31,TOPS_1:35; then LE q, q1, P, f/.1, f/.len f by A1,A2,Th16; hence contradiction by A1,A7,A8,Th12; end; assume i = j; hence thesis by A1,Lm5; end; theorem Th29: for f being FinSequence of TOP-REAL 2, q1, q2 being Point of TOP-REAL 2, i being Nat st q1 in LSeg(f,i) & q2 in LSeg(f,i) & f is_S-Seq & 1 <= i & i + 1 <= len f holds LE q1, q2, L~f, f/.1, f/.len f implies LE q1, q2, LSeg (f,i), f/.i, f/.(i+1) proof let f be FinSequence of TOP-REAL 2, q1, q2 be Point of TOP-REAL 2, i be Nat; assume A1: q1 in LSeg(f,i) & q2 in LSeg(f,i) & f is_S-Seq & 1 <= i & i + 1 <= len f; then A2: L~f is_an_arc_of f/.1, f/.len f by TOPREAL1:31; assume A3: LE q1, q2, L~f, f/.1, f/.len f; len f >= 2 by A1,TOPREAL1:def 10; then reconsider P = L~f, Q = LSeg(f,i) as non empty Subset of TOP-REAL 2 by A1,TOPREAL1:29; consider F being map of I[01], (TOP-REAL 2)|P such that A4: F is_homeomorphism & F.0 = f/.1 & F.1 = f/.len f by A2,TOPREAL1:def 2; consider ppi, pi1 be Real such that A5: ppi < pi1 & 0 <= ppi & ppi <= 1 & 0 <= pi1 & pi1 <= 1 & LSeg (f, i) = F.:[.ppi, pi1.] & F.ppi = f/.i & F.pi1 = f/.(i+1) by A1,A4,JORDAN5B:7; A6: ppi in { dd where dd is Real : ppi <= dd & dd <= pi1 } & pi1 in { dd where dd is Real : ppi <= dd & dd <= pi1 } by A5; then A7: ppi in [.ppi,pi1.] & pi1 in [.ppi,pi1.] by RCOMP_1:def 1; [.ppi,pi1.] c= [.0,1.] by A5,TREAL_1:1; then reconsider Poz = [.ppi,pi1.] as non empty Subset of I[01] by A6,BORSUK_1:83,RCOMP_1:def 1; consider G be map of I[01]|Poz, (TOP-REAL 2)|Q such that A8: G = F|Poz & G is_homeomorphism by A1,A4,A5,JORDAN5B:8; set Ex = L[01]((#)(ppi,pi1),(ppi,pi1)(#)); set H = G * Ex; A9: Ex is_homeomorphism by A5,TREAL_1:20; then A10: rng Ex = [#] Closed-Interval-TSpace(ppi,pi1) by TOPS_2:def 5; then A11: rng Ex = the carrier of Closed-Interval-TSpace(ppi,pi1) by PRE_TOPC:12; A12: dom G = [#](I[01]|Poz) by A8,TOPS_2:def 5 .= Poz by PRE_TOPC:def 10 .= the carrier of Closed-Interval-TSpace(ppi,pi1) by A5,TOPMETR:25 .= [#] Closed-Interval-TSpace(ppi,pi1) by PRE_TOPC:12; then A13: dom H = dom Ex by A10,RELAT_1:46 .= [#] I[01] by A9,TOPMETR:27,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; A14: rng H = rng G by A10,A12,RELAT_1:47 .= [#] ((TOP-REAL 2)|LSeg(f,i)) by A8,TOPS_2:def 5; then H is Function of the carrier of I[01], the carrier of (TOP-REAL 2)|Q by A13,FUNCT_2:4; then reconsider H as map of I[01], (TOP-REAL 2)|Q; reconsider K1 = Closed-Interval-TSpace(ppi,pi1), K2 = I[01]|Poz as SubSpace of I[01] by A5,TOPMETR:27,TREAL_1:6; the carrier of K1 = [.ppi,pi1.] by A5,TOPMETR:25 .= [#](I[01]|Poz) by PRE_TOPC:def 10 .= the carrier of K2 by PRE_TOPC:12; then I[01] = Closed-Interval-TSpace(0,1) & I[01]|Poz = Closed-Interval-TSpace(ppi,pi1) by TOPMETR:27,TSEP_1:5; then A15: H is_homeomorphism by A8,A9,TOPS_2:71; A16: Ex.0 = Ex.(#)(0,1) by TREAL_1:def 1 .= (#)(ppi,pi1) by A5,TREAL_1:12 .= ppi by A5,TREAL_1:def 1; 0 in dom H by A13,JORDAN5A:2; then A17: H.0 = G.ppi by A16,FUNCT_1:22 .= f/.i by A5,A7,A8,FUNCT_1:72; A18: Ex.1 = Ex.(0,1)(#) by TREAL_1:def 2 .= (ppi,pi1)(#) by A5,TREAL_1:12 .= pi1 by A5,TREAL_1:def 2; 1 in dom H by A13,JORDAN5A:2; then A19: H.1 = G.pi1 by A18,FUNCT_1:22 .= f/.(i+1) by A5,A7,A8,FUNCT_1:72; A20: Q is_an_arc_of f/.i, f/.(i+1) by A1,JORDAN5B:15; q1 in rng H by A1,A14,PRE_TOPC:def 10; then consider x1' be set such that A21: x1' in dom H & q1 = H.x1' by FUNCT_1:def 5; q2 in rng H by A1,A14,PRE_TOPC:def 10; then consider x2' be set such that A22: x2' in dom H & q2 = H.x2' by FUNCT_1:def 5; A23: x1' in { l where l is Real : 0 <= l & l <= 1 } & x2' in { l where l is Real : 0 <= l & l <= 1 } by A13,A21,A22,BORSUK_1:83,RCOMP_1:def 1; then consider x1 be Real such that A24: x1 = x1' & 0 <= x1 & x1 <= 1; consider x2 be Real such that A25: x2 = x2' & 0 <= x2 & x2 <= 1 by A23; A26: x1 in the carrier of I[01] & x2 in the carrier of I[01] by A24,A25,JORDAN5A:2; reconsider X1 = x1, X2 = x2 as Point of Closed-Interval-TSpace (0,1) by A24,A25,JORDAN5A:2,TOPMETR:27; x1 in dom Ex & x2 in dom Ex by A10,A12,A13,A26,RELAT_1:46; then Ex.x1 in the carrier of Closed-Interval-TSpace(ppi,pi1) & Ex.x2 in the carrier of Closed-Interval-TSpace(ppi,pi1) by A11,FUNCT_1:def 5; then A27: Ex.x1 in Poz & Ex.x2 in Poz by A5,TOPMETR:25; A28: q1 = G.(Ex.x1) by A21,A24,FUNCT_1:22 .= F.(Ex.x1) by A8,A27,FUNCT_1:72; A29: q2 = G.(Ex.x2) by A22,A25,FUNCT_1:22 .= F.(Ex.x2) by A8,A27,FUNCT_1:72; reconsider E1 = Ex.X1, E2 = Ex.X2 as Real by A27; 0 <= E1 & E1 <= 1 & 0 <= E2 & E2 <= 1 by A27,JORDAN5A:2; then A30: E1 <= E2 by A3,A4,A28,A29,Def3; Ex is one-to-one continuous by A9,TOPS_2:def 5; then x1 <= x2 by A5,A16,A18,A30,JORDAN5A:16; hence thesis by A15,A17,A19,A20,A21,A22,A24,A25,Th8; end; theorem for f being FinSequence of TOP-REAL 2, q1, q2 being Point of TOP-REAL 2 st q1 in L~f & q2 in L~f & f is_S-Seq & q1 <> q2 holds LE q1, q2, L~f, f/.1, f/.len f iff for i, j being Nat st q1 in LSeg(f,i) & q2 in LSeg(f,j) & 1 <= i & i+1 <= len f & 1 <= j & j+1 <= len f holds i <= j & (i = j implies LE q1,q2,f/.i,f/.(i+1)) proof let f be FinSequence of TOP-REAL 2, q1,q2 be Point of TOP-REAL 2; set p3 = f/.1, p4 = f/.len f; assume A1: q1 in L~f & q2 in L~f & f is_S-Seq & q1 <> q2; then A2: L~f is_an_arc_of p3, p4 by TOPREAL1:31; reconsider P = L~f as non empty Subset of TOP-REAL 2 by A1; hereby assume A3: LE q1,q2,L~f,f/.1,f/.len f; thus for i,j being Nat st q1 in LSeg(f,i) & q2 in LSeg(f,j) & 1<=i & i+1<=len f & 1<=j & j+1<=len f holds i<=j & (i=j implies LE q1,q2,f/.i,f/.(i+1)) proof let i,j be Nat; assume A4:q1 in LSeg(f,i) & q2 in LSeg(f,j) & 1<=i & i+1<=len f & 1<=j & j+1<=len f; thus i<=j proof assume j < i; then j+1 <= i by NAT_1:38; then consider m be Nat such that A5: j+1+m = i by NAT_1:28; A6: LE q2, f/.(j+1), P, p3, p4 by A1,A4,Th26; i <= i + 1 & j <= j + 1 by NAT_1:29; then 1 <= j+1 & j+1 <= j+1+m & j+1+m <= len f by A4,A5,AXIOMS:22,NAT_1:29; then LE f/.(j+1), f/.(j+1+m), P, p3, p4 by A1,Th24; then A7: LE q2, f/.(j+1+m), P, p3, p4 by A2,A6,Th13; LE f/.(j+1+m), q1, P, p3, p4 by A1,A4,A5,Th25; then LE q2, q1, P, p3, p4 by A2,A7,Th13; hence thesis by A1,A2,A3,Th12; end; assume A8: i = j; set p1 = f/.i, p2 = f/.(i+1); A9:i in dom f & i+1 in dom f by A4,GOBOARD2:3; A10:LSeg (f,i) = LSeg (p1, p2) by A4,TOPREAL1:def 5; reconsider Q = LSeg (f,i) as non empty Subset of TOP-REAL 2 by A4; A11:f is one-to-one by A1,TOPREAL1:def 10; A12:p1 <> p2 proof assume p1 = p2; then i = i+1 by A9,A11,PARTFUN2:17; hence thesis by REAL_1:69; end; LE q1, q2, Q, p1, p2 by A1,A3,A4,A8,Th29; hence LE q1,q2,p1,p2 by A10,A12,Th17; end; end; assume A13: for i,j being Nat st q1 in LSeg(f,i) & q2 in LSeg(f,j) & 1<=i & i+1<=len f & 1<=j & j+1<=len f holds i<=j & (i=j implies LE q1,q2,f/.i,f/.(i+1)); consider i be Nat such that A14: 1 <= i & i+1 <= len f & q1 in LSeg(f,i) by A1,SPPOL_2:13; consider j be Nat such that A15: 1 <= j & j+1 <= len f & q2 in LSeg(f,j) by A1,SPPOL_2:13; A16: i<=j & (i=j implies LE q1,q2,f/.i,f/.(i+1)) by A13,A14,A15; A17: P is_an_arc_of f/.1, f/.len f by A1,TOPREAL1:31; per cases by A16,REAL_1:def 5; suppose i < j; then i+1 <= j by NAT_1:38; then consider m be Nat such that A18: j = i+1 + m by NAT_1:28; A19: 1 <= i+1 & i+1+m+1 <= len f by A15,A18,NAT_1:37; A20: LE q1, f/.(i+1), P, f/.1, f/.len f by A1,A14,Th26; i <= i + 1 & j <= j + 1 by NAT_1:29; then i+1 <= i+1+m & i+1+m <= len f by A15,A18,AXIOMS:22,NAT_1:29; then LE f/.(i+1), f/.(i+1+m), P, f/.1, f/.len f by A1,A19,Th24; then A21: LE q1, f/.(i+1+m), P, f/.1, f/.len f by A17,A20,Th13; LE f/.(i+1+m), q2, P, f/.1, f/.len f by A1,A15,A18,Th25; hence LE q1, q2, L~f, f/.1, f/.len f by A17,A21,Th13; suppose A22: i = j; then A23: LE q1, q2, f/.i, f/.(i+1) by A13,A14,A15; set p1 = f/.i,p2 = f/.(i+1); A24:i in dom f & i+1 in dom f by A14,GOBOARD2:3; A25:f is one-to-one by A1,TOPREAL1:def 10; p1 <> p2 proof assume p1 = p2; then i = i+1 by A24,A25,PARTFUN2:17; hence thesis by REAL_1:69; end; then consider H be map of I[01], (TOP-REAL 2) | LSeg(p1, p2) such that A26: (for x being Real st x in [.0,1.] holds H.x = (1-x)*p1 + x*p2) & H is_homeomorphism & H.0 = p1 & H.1 = p2 by JORDAN5A:4; A27: LSeg(f,i) = LSeg(p1,p2) by A14,TOPREAL1:def 5; reconsider Q = LSeg (f,i) as non empty Subset of TOP-REAL 2 by A14; reconsider H as map of I[01], (TOP-REAL 2) | Q by A27; q1 in P & q2 in P & for g being map of I[01], (TOP-REAL 2)|P, s1,s2 be Real st g is_homeomorphism & g.0=f/.1 & g.1=f/.len f & g.s1=q1 & 0<=s1 & s1<=1 & g.s2=q2 & 0<=s2 & s2<=1 holds s1<=s2 proof thus q1 in P & q2 in P by A1; let F be map of I[01], (TOP-REAL 2)|P, s1, s2 be Real; assume A28: F is_homeomorphism & F.0 = f/.1 & F.1 = f/.len f & F.s1 = q1 & 0 <= s1 & s1 <= 1 & F.s2 = q2 & 0 <= s2 & s2 <= 1; then A29: s1 in the carrier of I[01] & s2 in the carrier of I[01] by JORDAN5A:2 ; A30: dom F = [#] I[01] by A28,TOPS_2:def 5 .= the carrier of I[01] by PRE_TOPC:12; A31: F is one-to-one by A28,TOPS_2:def 5; consider ppi, pi1 be Real such that A32: ppi < pi1 & 0 <= ppi & ppi <= 1 & 0 <= pi1 & pi1 <= 1 & LSeg (f, i) = F.:[.ppi, pi1.] & F.ppi = f/.i & F.pi1 = f/.(i+1) by A1,A14,A28,JORDAN5B:7; A33: ppi in { dd where dd is Real : ppi <= dd & dd <= pi1 } & pi1 in { dd where dd is Real : ppi <= dd & dd <= pi1 } by A32; then A34: ppi in [.ppi,pi1.] & pi1 in [.ppi,pi1.] by RCOMP_1:def 1; [.ppi,pi1.] c= [.0,1.] by A32,TREAL_1:1; then reconsider Poz = [.ppi,pi1.] as non empty Subset of I[01] by A33,BORSUK_1:83,RCOMP_1:def 1; consider G be map of I[01]|Poz, (TOP-REAL 2)|Q such that A35: G = F|Poz & G is_homeomorphism by A1,A14,A28,A32,JORDAN5B:8; A36:dom H = [#]I[01] by A26,TOPS_2:def 5 .= [.0,1.] by BORSUK_1:83,PRE_TOPC:12; A37:rng H = [#] ((TOP-REAL 2) | LSeg(f,i)) by A26,A27,TOPS_2:def 5 .= LSeg (f,i) by PRE_TOPC:def 10; then consider x1' be set such that A38: x1' in dom H & H.x1' = q1 by A14,FUNCT_1:def 5; x1' in { l where l is Real : 0 <= l & l <= 1 } by A36,A38,RCOMP_1:def 1; then consider x1 be Real such that A39: x1 = x1' & 0 <= x1 & x1 <= 1; consider x2' be set such that A40: x2' in dom H & H.x2' = q2 by A15,A22,A37,FUNCT_1:def 5; x2' in { l where l is Real : 0 <= l & l <= 1 } by A36,A40,RCOMP_1:def 1; then consider x2 be Real such that A41: x2 = x2' & 0 <= x2 & x2 <= 1; reconsider K1 = Closed-Interval-TSpace(ppi,pi1), K2 = I[01]|Poz as SubSpace of I[01] by A32,TOPMETR:27,TREAL_1:6; A42: the carrier of K1 = [.ppi,pi1.] by A32,TOPMETR:25 .= [#](I[01]|Poz) by PRE_TOPC:def 10 .= the carrier of K2 by PRE_TOPC:12; then A43:Closed-Interval-TSpace(ppi,pi1) = I[01]|Poz by TSEP_1:5; reconsider E=G" * H as map of Closed-Interval-TSpace(0,1), Closed-Interval-TSpace(ppi,pi1) by A42,TOPMETR:27; G" is_homeomorphism & H is_homeomorphism by A26,A27,A35,TOPS_2:70; then E is_homeomorphism by A43,TOPMETR:27,TOPS_2:71; then A44:E is continuous one-to-one by TOPS_2:def 5; A45:dom G = [#] (I[01]|Poz) & rng G = [#] ((TOP-REAL 2)|LSeg(f,i)) & G is one-to-one by A35,TOPS_2:def 5; then A46:dom G = Poz by PRE_TOPC:def 10; A47:G" = (G qua Function)" by A45,TOPS_2:def 4; 0 in { l where l is Real : 0 <= l & l <= 1 } & 1 in { l where l is Real : 0 <= l & l <= 1 }; then A48:0 in [.0,1.] & 1 in [.0,1.] by RCOMP_1:def 1; then A49:H.0 = (1-0)*p1 + 0 * p2 by A26 .= p1 + 0 * p2 by EUCLID:33 .= p1 + 0.REAL 2 by EUCLID:33 .= p1 by EUCLID:31; A50:H.1 = (1-1)*p1 + 1*p2 by A26,A48 .= 0.REAL 2 + 1*p2 by EUCLID:33 .= 0.REAL 2 + p2 by EUCLID:33 .= p2 by EUCLID:31; G.ppi = p1 by A32,A34,A35,FUNCT_1:72; then A51:ppi = G".(H.0) by A34,A45,A46,A47,A49,FUNCT_1:54 .= E.0 by A36,A48,FUNCT_1:23; G.pi1 = p2 by A32,A34,A35,FUNCT_1:72; then A52:pi1 = G".(H.1) by A34,A45,A46,A47,A50,FUNCT_1:54 .= E.1 by A36,A48,FUNCT_1:23; consider x be set such that A53: x in dom F & x in [.ppi,pi1.] & q1 = F.x by A14,A32,FUNCT_1:def 12; A54:x = s1 by A28,A29,A30,A31,A53,FUNCT_1:def 8; A55:s1 in dom G by A28,A29,A30,A31,A46,A53,FUNCT_1:def 8; G.s1 = q1 by A35,A53,A54,FUNCT_1:72; then A56:s1 = G".(H.x1) by A38,A39,A45,A47,A55,FUNCT_1:54 .= E.x1 by A38,A39,FUNCT_1:23; consider x' be set such that A57: x' in dom F & x' in [.ppi,pi1.] & q2 = F.x' by A15,A22,A32,FUNCT_1:def 12; A58:s2 in Poz by A28,A29,A30,A31,A57,FUNCT_1:def 8; A59:s2 in dom G by A28,A29,A30,A31,A46,A57,FUNCT_1:def 8; G.s2 = q2 by A28,A35,A58,FUNCT_1:72; then A60:s2 = G".(H.x2) by A40,A41,A45,A47,A59,FUNCT_1:54 .= E.x2 by A40,A41,FUNCT_1:23; reconsider X1 = x1, X2 = x2 as Point of Closed-Interval-TSpace(0,1) by A36,A38,A39,A40,A41,TOPMETR:25; A61:s1 = E.X1 & s2 = E.X2 by A56,A60; A62:q1 = (1-x1)*p1 + x1*p2 by A26,A36,A38,A39; q2 = (1-x2)*p1 + x2*p2 by A26,A36,A40,A41; then A63:x1 <= x2 by A23,A39,A41,A62,JORDAN3:def 6; per cases by A63,REAL_1:def 5; suppose x1 = x2; hence thesis by A56,A60; suppose x1 < x2; hence thesis by A32,A44,A51,A52,A61,JORDAN5A:16; end; hence LE q1, q2, L~f, f/.1, f/.len f by Def3; end;

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