:: The Topological Space ${\calE}^2_{\rm T}$. Simple Closed Curves :: by Agata Darmochwa{\l} and Yatsuka Nakamura :: :: Received December 30, 1991 :: Copyright (c) 1991-2019 Association of Mizar Users :: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland). :: This code can be distributed under the GNU General Public Licence :: version 3.0 or later, or the Creative Commons Attribution-ShareAlike :: License version 3.0 or later, subject to the binding interpretation :: detailed in file COPYING.interpretation. :: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies NUMBERS, PRE_TOPC, EUCLID, FINSEQ_1, XBOOLE_0, RLTOPSP1, CARD_1, MCART_1, XXREAL_0, TOPREAL1, SUBSET_1, TARSKI, ARYTM_3, RCOMP_1, BORSUK_1, FUNCT_1, RELAT_1, TOPS_2, ORDINAL2, STRUCT_0, PARTFUN1, FUNCT_4, TOPREAL2, FUNCT_2; notations TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, FUNCT_4, ORDINAL1, NUMBERS, XXREAL_0, FINSEQ_1, STRUCT_0, PRE_TOPC, TOPS_2, COMPTS_1, RLTOPSP1, EUCLID, TOPMETR, TOPREAL1; constructors FUNCT_4, RCOMP_1, TOPS_2, COMPTS_1, TOPMETR, TOPREAL1, CONVEX1; registrations XBOOLE_0, FUNCT_1, RELSET_1, FUNCT_2, XXREAL_0, STRUCT_0, PRE_TOPC, BORSUK_1, EUCLID, TOPREAL1, RLTOPSP1, XREAL_0, ORDINAL1; requirements NUMERALS, REAL, BOOLE, SUBSET; definitions TARSKI, PRE_TOPC, TOPS_2; equalities STRUCT_0, RELAT_1; expansions TARSKI, PRE_TOPC, TOPS_2; theorems BORSUK_1, COMPTS_1, ENUMSET1, EUCLID, FUNCT_1, FUNCT_2, HEINE, PRE_TOPC, TARSKI, TOPMETR, TOPMETR2, TOPS_2, ZFMISC_1, TOPREAL1, RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1, XXREAL_0, XXREAL_1, RLTOPSP1; begin reserve a for set; reserve p,p1,p2,q,q1,q2 for Point of TOP-REAL 2; reserve h1,h2 for FinSequence of TOP-REAL 2; Lm1: for x,X being set st not x in X holds {x} /\ X = {} by XBOOLE_0:def 7,ZFMISC_1:50; Lm2: LSeg(|[0,0]|,|[1,0]|) /\ LSeg(|[0,1]|,|[1,1]|) = {} by TOPREAL1:19 ,XBOOLE_0:def 7; Lm3: LSeg(|[0,0]|,|[0,1]|) /\ LSeg(|[1,0]|,|[1,1]|) = {} by TOPREAL1:20 ,XBOOLE_0:def 7; set p00 = |[ 0,0 ]|, p01 = |[ 0,1 ]|, p10 = |[ 1,0 ]|, p11 = |[ 1,1 ]|, L1 = LSeg(p00,p01),L2 = LSeg(p01,p11),L3 = LSeg(p00,p10),L4 = LSeg(p10,p11); Lm4: p00`1 = 0 by EUCLID:52; Lm5: p00`2 = 0 by EUCLID:52; Lm6: p01`1 = 0 by EUCLID:52; Lm7: p01`2 = 1 by EUCLID:52; Lm8: p10`1 = 1 by EUCLID:52; Lm9: p10`2 = 0 by EUCLID:52; Lm10: p11`1 = 1 by EUCLID:52; Lm11: p11`2 = 1 by EUCLID:52; Lm12: not p00 in L4 by Lm4,Lm8,Lm10,TOPREAL1:3; Lm13: not p00 in L2 by Lm5,Lm7,Lm11,TOPREAL1:4; Lm14: not p01 in L3 by Lm5,Lm7,Lm9,TOPREAL1:4; Lm15: not p01 in L4 by Lm6,Lm8,Lm10,TOPREAL1:3; Lm16: not p10 in L1 by Lm4,Lm6,Lm8,TOPREAL1:3; Lm17: not p10 in L2 by Lm7,Lm9,Lm11,TOPREAL1:4; Lm18: not p11 in L1 by Lm4,Lm6,Lm10,TOPREAL1:3; Lm19: not p11 in L3 by Lm5,Lm9,Lm11,TOPREAL1:4; Lm20: p00 in L1 by RLTOPSP1:68; Lm21: p00 in L3 by RLTOPSP1:68; Lm22: p01 in L1 by RLTOPSP1:68; Lm23: p01 in L2 by RLTOPSP1:68; Lm24: p10 in L3 by RLTOPSP1:68; Lm25: p10 in L4 by RLTOPSP1:68; Lm26: p11 in L2 by RLTOPSP1:68; Lm27: p11 in L4 by RLTOPSP1:68; set L = { p : p`1 = 0 & p`2 <= 1 & p`2 >= 0 or p`1 <= 1 & p`1 >= 0 & p`2 = 1 or p`1 <= 1 & p`1 >= 0 & p`2 = 0 or p`1 = 1 & p`2 <= 1 & p`2 >= 0 }; Lm28: p00 in L by Lm4,Lm5; Lm29: p11 in L by Lm10,Lm11; Lm30: p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg(p00, p01) implies ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof assume that A1: p1 <> p2 and A2: p2 in R^2-unit_square and A3: p1 in LSeg(p00, p01); A4: LSeg(p00,p1) c= L1 by A3,Lm20,TOPREAL1:6; p00 in LSeg(p1,p00) by RLTOPSP1:68; then p00 in LSeg(p1,p00) /\ L3 by Lm21,XBOOLE_0:def 4; then A5: {p00} c= LSeg(p1,p00) /\ L3 by ZFMISC_1:31; A6: LSeg(p1,p00) /\ L3 c= L1 /\ L3 by A3,Lm20,TOPREAL1:6,XBOOLE_1:26; then A7: LSeg(p1,p00) /\ L3 = {p00} by A5,TOPREAL1:17,XBOOLE_0:def 10; A8: L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A9: LSeg(p1,p00) /\ L4 = {} by A4,XBOOLE_1:3,26; p01 in LSeg(p01,p1) by RLTOPSP1:68; then p01 in LSeg(p01,p1) /\ L2 by Lm23,XBOOLE_0:def 4; then A10: {p01} c= LSeg(p01,p1) /\ L2 by ZFMISC_1:31; A11: p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:def 2,XBOOLE_0:def 3; A12: LSeg(p01,p1) /\ L2 c= {p01} by A3,Lm22,TOPREAL1:6,15,XBOOLE_1:26; A13: LSeg(p1,p01) c= L1 by A3,Lm22,TOPREAL1:6; then A14: LSeg(p01,p1) /\ L4 = {} by A8,XBOOLE_1:3,26; consider p such that A15: p = p1 and A16: p`1 = 0 and A17: p`2 <= 1 and A18: p`2 >= 0 by A3,TOPREAL1:13; per cases by A11,XBOOLE_0:def 3; suppose A19: p2 in L1; then A20: LSeg(p2,p1) c= L1 by A3,TOPREAL1:6; A21: p = |[p`1,p`2]| by EUCLID:53; consider q such that A22: q = p2 and A23: q`1 = 0 and A24: q`2 <= 1 and A25: q`2 >= 0 by A19,TOPREAL1:13; A26: q = |[q`1,q`2]| by EUCLID:53; now per cases by A1,A15,A16,A22,A23,A21,A26,XXREAL_0:1; case A27: p`2 < q`2; A28: LSeg(p1,p2) /\ LSeg(p1,p00) c= {p1} proof let a be object; assume A29: a in LSeg(p1,p2) /\ LSeg(p1,p00); then reconsider p = a as Point of TOP-REAL 2; A30: p in LSeg(p00,p1) by A29,XBOOLE_0:def 4; p00`2 <= p1`2 by A15,A18,EUCLID:52; then A31: p`2 <= p1`2 by A30,TOPREAL1:4; A32: p in LSeg(p1,p2) by A29,XBOOLE_0:def 4; then p1`2 <= p`2 by A15,A22,A27,TOPREAL1:4; then A33: p1`2 = p`2 by A31,XXREAL_0:1; p1`1 <= p`1 by A15,A16,A22,A23,A32,TOPREAL1:3; then p`1 = 0 by A15,A16,A22,A23,A32,TOPREAL1:3; then p = |[ 0, p1`2]| by A33,EUCLID:53 .= p1 by A15,A16,EUCLID:53; hence thesis by TARSKI:def 1; end; A34: LSeg(p01,p2) /\ L2 c= L1 /\ L2 by A19,Lm22,TOPREAL1:6,XBOOLE_1:26; A35: now set a = the Element of LSeg(p1,p00) /\ LSeg(p01,p2); assume A36: LSeg(p1,p00) /\ LSeg(p01,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A37: p in LSeg(p00,p1) by A36,XBOOLE_0:def 4; A38: p in LSeg(p2,p01) by A36,XBOOLE_0:def 4; p2`2 <= p01`2 by A22,A24,EUCLID:52; then A39: p2`2 <= p`2 by A38,TOPREAL1:4; p00`2 <= p1`2 by A15,A18,EUCLID:52; then p`2 <= p1`2 by A37,TOPREAL1:4; hence contradiction by A15,A22,A27,A39,XXREAL_0:2; end; p01 in LSeg(p01,p2) by RLTOPSP1:68; then p01 in LSeg(p01,p2) /\ L2 by Lm23,XBOOLE_0:def 4; then A40: {p01} c= LSeg(p01,p2) /\ L2 by ZFMISC_1:31; now assume p00 in LSeg(p01,p2) /\ L3; then A41: p00 in LSeg(p2,p01) by XBOOLE_0:def 4; p2`2 <= p01`2 by A22,A24,EUCLID:52; hence contradiction by A18,A22,A27,A41,Lm5,TOPREAL1:4; end; then A42: {p00} <> LSeg(p01,p2) /\ L3 by ZFMISC_1:31; LSeg(p01,p2) /\ L3 c= {p00} by A19,Lm22,TOPREAL1:6,17,XBOOLE_1:26; then A43: LSeg(p01,p2) /\ L3 = {} by A42,ZFMISC_1:33; A44: LSeg(p1,p2) /\ L3 c= {p00} by A3,A19,TOPREAL1:6,17,XBOOLE_1:26; A45: LSeg(p1,p2) /\ LSeg(p01,p2) c= {p2} proof let a be object; assume A46: a in LSeg(p1,p2) /\ LSeg(p01,p2); then reconsider p = a as Point of TOP-REAL 2; A47: p in LSeg(p2,p01) by A46,XBOOLE_0:def 4; p2`2 <= p01`2 by A22,A24,EUCLID:52; then A48: p2`2 <= p`2 by A47,TOPREAL1:4; A49: p in LSeg(p1,p2) by A46,XBOOLE_0:def 4; then p`2 <= p2`2 by A15,A22,A27,TOPREAL1:4; then A50: p2`2 = p`2 by A48,XXREAL_0:1; p1`1 <= p`1 by A15,A16,A22,A23,A49,TOPREAL1:3; then p`1 = 0 by A15,A16,A22,A23,A49,TOPREAL1:3; then p = |[0,p2`2]| by A50,EUCLID:53 .= p2 by A22,A23,EUCLID:53; hence thesis by TARSKI:def 1; end; A51: LSeg(p1,p00) /\ L2 c= {p01} by A3,Lm20,TOPREAL1:6,15,XBOOLE_1:26; now assume p01 in LSeg(p1,p00) /\ L2; then A52: p01 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`2 <= p1`2 by A15,A18,EUCLID:52; then p01`2 <= p1`2 by A52,TOPREAL1:4; hence contradiction by A15,A17,A24,A27,Lm7,XXREAL_0:1; end; then A53: {p01} <> LSeg(p1,p00) /\ L2 by ZFMISC_1:31; set P1 = LSeg(p1,p2), P2 = LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2)); A54: p1 in LSeg(p1,p00) by RLTOPSP1:68; A55: LSeg(p01,p2) c= L1 by A19,Lm22,TOPREAL1:6; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p00) by A54,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p00) by ZFMISC_1:31; then A56: LSeg(p1,p2) /\ LSeg(p1,p00) = {p1} by A28,XBOOLE_0:def 10; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A57: (L3 \/ L4) /\ LSeg(p11,p01) = {} \/ {p11} by Lm2,TOPREAL1:18 ,XBOOLE_1:23 .= {p11}; L3 \/ L4 is_an_arc_of p00,p11 by Lm4,Lm8,TOPREAL1:12,16; then A58: L3 \/ L4 \/ LSeg(p11,p01) is_an_arc_of p00,p01 by A57,TOPREAL1:10; (L3 \/ L4 \/ LSeg(p11,p01)) /\ LSeg(p01,p2) = LSeg(p01,p2) /\ ( L3 \/ L4) \/ (LSeg(p01,p2) /\ LSeg(p11,p01)) by XBOOLE_1:23 .= {} \/ (LSeg(p01,p2) /\ L4) \/ (LSeg(p01,p2) /\ LSeg(p11,p01)) by A43,XBOOLE_1:23 .= {} \/ (LSeg(p01,p2) /\ LSeg(p11,p01)) by A55,Lm3,XBOOLE_1:3,26 .= {p01} by A40,A34,TOPREAL1:15,XBOOLE_0:def 10; then A59: L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2) is_an_arc_of p00,p2 by A58, TOPREAL1:10; LSeg(p1,p00) /\ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2)) = LSeg(p1,p00) /\ (L3 \/ L4 \/ LSeg(p11,p01)) \/ LSeg(p1,p00) /\ LSeg(p01,p2) by XBOOLE_1:23 .= LSeg(p1,p00) /\ (L3 \/ L4) \/ LSeg(p1,p00) /\ LSeg(p11,p01) \/ LSeg(p1,p00) /\ LSeg(p01,p2) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ L3) \/ (LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p11,p01)) \/ (LSeg(p1,p00) /\ LSeg(p01,p2)) by XBOOLE_1:23 .= {p00} by A9,A7,A35,A51,A53,ZFMISC_1:33; hence LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2)) is_an_arc_of p1,p2 by A59,TOPREAL1:11; LSeg(p01,p2) \/ LSeg(p2,p1) \/ LSeg(p1,p00) = L1 by A3,A19,TOPREAL1:7; hence R^2-unit_square = LSeg(p1,p2) \/ LSeg(p01,p2) \/ LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01)) by TOPREAL1:def 2,XBOOLE_1:4 .= LSeg(p1,p2) \/ (LSeg(p1,p00) \/ LSeg(p01,p2)) \/ (L3 \/ L4 \/ LSeg(p11,p01)) by XBOOLE_1:4 .= LSeg(p1,p2) \/ ((LSeg(p1,p00) \/ LSeg(p01,p2)) \/ (L3 \/ L4 \/ LSeg(p11,p01))) by XBOOLE_1:4 .= P1 \/ P2 by XBOOLE_1:4; A60: p2 in LSeg(p01,p2) by RLTOPSP1:68; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p01,p2) by A60,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p01,p2) by ZFMISC_1:31; then A61: LSeg(p1,p2) /\ LSeg(p01,p2) = {p2} by A45,XBOOLE_0:def 10; A62: LSeg(p1,p2) c= L1 by A3,A19,TOPREAL1:6; A63: P1 /\ P2 = (LSeg(p1,p2) /\ LSeg(p1,p00)) \/ LSeg(p1,p2) /\ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p2)) by XBOOLE_1:23 .= (LSeg(p1,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p1,p2) /\ (L3 \/ L4 \/ LSeg(p11,p01))) \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by XBOOLE_1:23 .= (LSeg(p1,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p1,p2) /\ (L3 \/ L4)) \/ (LSeg(p1,p2) /\ LSeg(p11,p01)) \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ (LSeg(p1,p2) /\ L4) \/ (LSeg(p1 ,p2) /\ LSeg(p11,p01)) \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by A56,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ {} \/ (LSeg(p1,p2) /\ LSeg(p11, p01)) \/ (LSeg(p1,p2) /\ LSeg(p01,p2))) by A62,Lm3,XBOOLE_1:3,26 .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2})) by A61,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by XBOOLE_1:4; A64: now per cases; suppose A65: p1 = p00; then p00 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L3 <> {} by Lm21,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L3 = {p1} by A44,A65,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A63; end; suppose A66: p1 <> p00; now assume p00 in LSeg(p1,p2) /\ L3; then p00 in LSeg(p1,p2) by XBOOLE_0:def 4; then p1`2 <= p00`2 by A15,A22,A27,TOPREAL1:4; then p00`2 = p1`2 by A3,Lm5,Lm7,TOPREAL1:4; hence contradiction by A15,A16,A66,Lm5,EUCLID:53; end; then LSeg(p1,p2) /\ L3 <> {p00} by ZFMISC_1:31; then LSeg(p1,p2) /\ L3 = {} by A44,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A63; end; end; A67: LSeg(p1,p2) /\ L2 c= {p01} by A3,A19,TOPREAL1:6,15,XBOOLE_1:26; now per cases; suppose A68: p2 <> p01; now assume p01 in LSeg(p1,p2) /\ L2; then p01 in LSeg(p1,p2) by XBOOLE_0:def 4; then A69: p01`2 <= p2`2 by A15,A22,A27,TOPREAL1:4; p2`2 <= p01`2 by A19,Lm5,Lm7,TOPREAL1:4; then A70: p01`2 = p2`2 by A69,XXREAL_0:1; p2 = |[p2`1,p2`2]| by EUCLID:53 .= |[0,1]| by A22,A23,A70,EUCLID:52; hence contradiction by A68; end; then LSeg(p1,p2) /\ L2 <> {p01} by ZFMISC_1:31; then LSeg(p1,p2) /\ L2 = {} by A67,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A64,ENUMSET1:1; end; suppose A71: p2 = p01; then p01 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L2 <> {} by Lm23,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L2 = {p2} by A67,A71,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A64,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; case A72: p`2 > q`2; A73: LSeg(p2,p1) /\ LSeg(p01,p1) c= {p1} proof let a be object; assume A74: a in LSeg(p2,p1) /\ LSeg(p01,p1); then reconsider p = a as Point of TOP-REAL 2; A75: p in LSeg(p1,p01) by A74,XBOOLE_0:def 4; p1`2 <= p01`2 by A15,A17,EUCLID:52; then A76: p1`2 <= p`2 by A75,TOPREAL1:4; A77: p in LSeg(p2,p1) by A74,XBOOLE_0:def 4; then p`2 <= p1`2 by A15,A22,A72,TOPREAL1:4; then A78: p1`2 = p`2 by A76,XXREAL_0:1; p2`1 <= p`1 by A15,A16,A22,A23,A77,TOPREAL1:3; then p`1 = 0 by A15,A16,A22,A23,A77,TOPREAL1:3; then p = |[0,p1`2]| by A78,EUCLID:53 .= p1 by A15,A16,EUCLID:53; hence thesis by TARSKI:def 1; end; A79: LSeg(p2,p00) c= L1 by A19,Lm20,TOPREAL1:6; A80: now set a = the Element of LSeg(p2,p00) /\ LSeg(p01,p1); assume A81: LSeg(p2,p00) /\ LSeg(p01,p1) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A82: p in LSeg(p00,p2) by A81,XBOOLE_0:def 4; A83: p in LSeg(p1,p01) by A81,XBOOLE_0:def 4; p1`2 <= p01`2 by A15,A17,EUCLID:52; then A84: p1`2 <= p`2 by A83,TOPREAL1:4; p00`2 <= p2`2 by A22,A25,EUCLID:52; then p`2 <= p2`2 by A82,TOPREAL1:4; hence contradiction by A15,A22,A72,A84,XXREAL_0:2; end; A85: LSeg(p2,p1) /\ L3 c= {p00} by A3,A19,TOPREAL1:6,17,XBOOLE_1:26; now assume p01 in LSeg(p2,p00) /\ L2; then A86: p01 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`2 <= p2`2 by A22,A25,EUCLID:52; then p01`2 <= p2`2 by A86,TOPREAL1:4; hence contradiction by A17,A22,A24,A72,Lm7,XXREAL_0:1; end; then A87: {p01} <> LSeg(p2,p00) /\ L2 by ZFMISC_1:31; A88: LSeg(p2,p00) /\ L3 c= {p00} by A19,Lm20,TOPREAL1:6,17,XBOOLE_1:26; now assume p00 in LSeg(p01,p1) /\ L3; then A89: p00 in LSeg(p1,p01) by XBOOLE_0:def 4; p1`2 <= p01`2 by A15,A17,EUCLID:52; hence contradiction by A15,A25,A72,A89,Lm5,TOPREAL1:4; end; then A90: {p00} <> LSeg(p01,p1) /\ L3 by ZFMISC_1:31; A91: LSeg(p2,p00) /\ L2 c= {p01} by A19,Lm20,TOPREAL1:6,15,XBOOLE_1:26; A92: LSeg(p2,p1) /\ LSeg(p2,p00) c= {p2} proof let a be object; assume A93: a in LSeg(p2,p1) /\ LSeg(p2,p00); then reconsider p = a as Point of TOP-REAL 2; A94: p in LSeg(p00,p2) by A93,XBOOLE_0:def 4; p00`2 <= p2`2 by A22,A25,EUCLID:52; then A95: p`2 <= p2`2 by A94,TOPREAL1:4; A96: p in LSeg(p2,p1) by A93,XBOOLE_0:def 4; then p2`2 <= p`2 by A15,A22,A72,TOPREAL1:4; then A97: p2`2 = p`2 by A95,XXREAL_0:1; p2`1 <= p`1 by A15,A16,A22,A23,A96,TOPREAL1:3; then p`1 = 0 by A15,A16,A22,A23,A96,TOPREAL1:3; then p = |[ 0, p2`2]| by A97,EUCLID:53 .= p2 by A22,A23,EUCLID:53; hence thesis by TARSKI:def 1; end; A98: LSeg(p01,p1) /\ L3 c= {p00} by A3,Lm22,TOPREAL1:6,17,XBOOLE_1:26; take P1 = LSeg(p2,p1),P2 = LSeg(p2,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p1)); A99: p2 in LSeg(p2,p00) by RLTOPSP1:68; p2 in LSeg(p2,p1) by RLTOPSP1:68; then p2 in LSeg(p2,p1) /\ LSeg(p2,p00) by A99,XBOOLE_0:def 4; then A100: {p2} c= LSeg(p2,p1) /\ LSeg(p2,p00) by ZFMISC_1:31; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A101: L2 /\ (L3 \/ L4) = {} \/ {p11} by Lm2,TOPREAL1:18,XBOOLE_1:23 .= {p11}; L3 \/ L4 is_an_arc_of p11,p00 by Lm4,Lm8,TOPREAL1:12,16; then A102: L3 \/ L4 \/ LSeg(p11,p01) is_an_arc_of p01,p00 by A101,TOPREAL1:11; p00 in LSeg(p2,p00) by RLTOPSP1:68; then p00 in LSeg(p2,p00) /\ L3 by Lm21,XBOOLE_0:def 4; then A103: {p00} c= LSeg(p2,p00) /\ L3 by ZFMISC_1:31; LSeg(p1,p01) /\ (L3 \/ L4 \/ LSeg(p11,p01)) = LSeg(p01,p1) /\ ( L3 \/ L4) \/ (LSeg(p01,p1) /\ LSeg(p11,p01)) by XBOOLE_1:23 .= (LSeg(p01,p1) /\ L3) \/ (LSeg(p01,p1) /\ L4) \/ (LSeg(p01,p1) /\ LSeg(p11,p01)) by XBOOLE_1:23 .= {} \/ (LSeg(p01,p1) /\ L4) \/ (LSeg(p01,p1) /\ LSeg(p11,p01)) by A98,A90,ZFMISC_1:33 .= {p01} by A14,A10,A12,XBOOLE_0:def 10; then A104: L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p1) is_an_arc_of p1,p00 by A102, TOPREAL1:11; (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p1)) /\ LSeg(p00,p2) = LSeg(p2,p00) /\ (L3 \/ L4 \/ LSeg(p11,p01)) \/ LSeg(p2,p00) /\ LSeg(p01,p1) by XBOOLE_1:23 .= LSeg(p2,p00) /\ (L3 \/ L4) \/ LSeg(p2,p00) /\ LSeg(p11,p01) \/ LSeg(p2,p00) /\ LSeg(p01,p1) by XBOOLE_1:23 .= (LSeg(p2,p00) /\ L3) \/ (LSeg(p2,p00) /\ L4) \/ (LSeg(p2,p00) /\ LSeg(p11,p01)) \/ (LSeg(p2,p00) /\ LSeg(p01,p1)) by XBOOLE_1:23 .= (LSeg(p2,p00) /\ L3) \/ {} \/ (LSeg(p2,p00) /\ LSeg(p11,p01)) \/ (LSeg(p2,p00) /\ LSeg(p01,p1)) by A79,Lm3,XBOOLE_1:3,26 .= LSeg(p2,p00) /\ L3 \/ {} by A80,A91,A87,ZFMISC_1:33 .= {p00} by A103,A88,XBOOLE_0:def 10; hence P2 is_an_arc_of p1,p2 by A104,TOPREAL1:10; LSeg(p01,p1) \/ LSeg(p1,p2) \/ LSeg(p2,p00) = L1 by A3,A19,TOPREAL1:7; hence R^2-unit_square = LSeg(p2,p1) \/ LSeg(p01,p1) \/ LSeg(p2,p00) \/ (L3 \/ L4 \/ LSeg(p11,p01)) by TOPREAL1:def 2,XBOOLE_1:4 .= LSeg(p2,p1) \/ (LSeg(p2,p00) \/ LSeg(p01,p1)) \/ (L3 \/ L4 \/ LSeg(p11,p01)) by XBOOLE_1:4 .= LSeg(p2,p1) \/ ((LSeg(p2,p00) \/ LSeg(p01,p1)) \/ (L3 \/ L4 \/ LSeg(p11,p01))) by XBOOLE_1:4 .= P1 \/ P2 by XBOOLE_1:4; A105: p1 in LSeg(p01,p1) by RLTOPSP1:68; p1 in LSeg(p2,p1) by RLTOPSP1:68; then p1 in LSeg(p2,p1) /\ LSeg(p01,p1) by A105,XBOOLE_0:def 4; then A106: {p1} c= LSeg(p2,p1) /\ LSeg(p01,p1) by ZFMISC_1:31; A107: P1 /\ P2 = (LSeg(p2,p1) /\ LSeg(p2,p00)) \/ LSeg(p2,p1) /\ (L3 \/ L4 \/ LSeg(p11,p01) \/ LSeg(p01,p1)) by XBOOLE_1:23 .= (LSeg(p2,p1) /\ LSeg(p2,p00)) \/ ((LSeg(p2,p1) /\ (L3 \/ L4 \/ LSeg(p11,p01))) \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by XBOOLE_1:23 .= (LSeg(p2,p1) /\ LSeg(p2,p00)) \/ ((LSeg(p2,p1) /\ (L3 \/ L4)) \/ (LSeg(p2,p1) /\ LSeg(p11,p01)) \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by XBOOLE_1:23 .= (LSeg(p2,p1) /\ LSeg(p2,p00)) \/ ((LSeg(p2,p1) /\ L3) \/ (LSeg( p2,p1) /\ L4) \/ (LSeg(p2,p1) /\ LSeg(p11,p01)) \/ (LSeg(p2,p1) /\ LSeg(p01,p1) )) by XBOOLE_1:23 .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ (LSeg(p2,p1) /\ L4) \/ (LSeg(p2 ,p1) /\ LSeg(p11,p01)) \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by A100,A92, XBOOLE_0:def 10 .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ {} \/ (LSeg(p2,p1) /\ LSeg(p11, p01)) \/ (LSeg(p2,p1) /\ LSeg(p01,p1))) by A20,Lm3,XBOOLE_1:3,26 .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ LSeg(p2,p1) /\ L2 \/ {p1}) by A106 ,A73,XBOOLE_0:def 10 .= {p2} \/ ((LSeg(p2,p1) /\ L3) \/ ((LSeg(p2,p1) /\ L2) \/ {p1})) by XBOOLE_1:4 .= {p2} \/ (LSeg(p2,p1) /\ L3) \/ ((LSeg(p2,p1) /\ L2) \/ {p1}) by XBOOLE_1:4; A108: now per cases; suppose A109: p2 = p00; p2 in LSeg(p2,p1) by RLTOPSP1:68; then LSeg(p2,p1) /\ L3 <> {} by A109,Lm21,XBOOLE_0:def 4; then LSeg(p2,p1) /\ L3 = {p2} by A85,A109,ZFMISC_1:33; hence P1 /\ P2 = {p2} \/ ((LSeg(p2,p1) /\ L2) \/ {p1}) by A107; end; suppose A110: p2 <> p00; now assume p00 in LSeg(p2,p1) /\ L3; then p00 in LSeg(p2,p1) by XBOOLE_0:def 4; then p2`2 <= p00`2 by A15,A22,A72,TOPREAL1:4; then p00`2 = p2`2 by A19,Lm5,Lm7,TOPREAL1:4; hence contradiction by A22,A23,A110,Lm5,EUCLID:53; end; then LSeg(p2,p1) /\ L3 <> {p00} by ZFMISC_1:31; then LSeg(p2,p1) /\ L3 = {} by A85,ZFMISC_1:33; hence P1 /\ P2 = {p2} \/ ((LSeg(p2,p1) /\ L2) \/ {p1}) by A107; end; end; A111: LSeg(p2,p1) /\ L2 c= {p01} by A3,A19,TOPREAL1:6,15,XBOOLE_1:26; now per cases; suppose A112: p1 <> p01; now assume p01 in LSeg(p2,p1) /\ L2; then p01 in LSeg(p2,p1) by XBOOLE_0:def 4; then A113: p01`2 <= p1`2 by A15,A22,A72,TOPREAL1:4; p1`2 <= p01`2 by A3,Lm5,Lm7,TOPREAL1:4; then A114: p01`2 = p1`2 by A113,XXREAL_0:1; p1 = |[p1`1,p1`2]| by EUCLID:53 .= |[0,1]| by A15,A16,A114,EUCLID:52; hence contradiction by A112; end; then LSeg(p2,p1) /\ L2 <> {p01} by ZFMISC_1:31; then LSeg(p2,p1) /\ L2 = {} by A111,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A108,ENUMSET1:1; end; suppose A115: p1 = p01; then p01 in LSeg(p2,p1) by RLTOPSP1:68; then LSeg(p2,p1) /\ L2 <> {} by Lm23,XBOOLE_0:def 4; then LSeg(p2,p1) /\ L2 = {p1} by A111,A115,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A108,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; end; hence thesis; end; suppose A116: p2 in L2; then A117: LSeg(p01,p2) c= L2 by Lm23,TOPREAL1:6; LSeg(p1,p01) c= L1 by A3,Lm22,TOPREAL1:6; then A118: LSeg(p1,p01) /\ LSeg(p01,p2) c= L1 /\ L2 by A117,XBOOLE_1:27; take P1 = LSeg(p1,p01) \/ LSeg(p01,p2),P2 = LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2)); A119: p01 in LSeg(p01,p2) by RLTOPSP1:68; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A120: p11 in L4 /\ LSeg(p11,p2) by Lm27,XBOOLE_0:def 4; p01 in LSeg(p1,p01) by RLTOPSP1:68; then LSeg(p1,p01) /\ LSeg(p01,p2) <> {} by A119,XBOOLE_0:def 4; then A121: LSeg(p1,p01) /\ LSeg(p01,p2) = {p01} by A118,TOPREAL1:15,ZFMISC_1:33; p1 <> p01 or p2 <> p01 by A1; hence P1 is_an_arc_of p1,p2 by A121,TOPREAL1:12; A122: L1 = LSeg(p1,p01) \/ LSeg(p1,p00) by A3,TOPREAL1:5; A123: L4 is_an_arc_of p10,p11 by Lm9,Lm11,TOPREAL1:9; L3 is_an_arc_of p00,p10 by Lm4,Lm8,TOPREAL1:9; then A124: L3 \/ L4 is_an_arc_of p00,p11 by A123,TOPREAL1:2,16; A125: LSeg(p11,p2) c= L2 by A116,Lm26,TOPREAL1:6; then A126: L4 /\ LSeg(p11,p2) c= L4 /\ L2 by XBOOLE_1:27; A127: L3 /\ LSeg(p11,p2) = {} by A125,Lm2,XBOOLE_1:3,26; (L3 \/ L4) /\ LSeg(p11,p2) = (L3 /\ LSeg(p11,p2)) \/ (L4 /\ LSeg(p11 ,p2)) by XBOOLE_1:23 .= {p11} by A127,A126,A120,TOPREAL1:18,ZFMISC_1:33; then A128: L3 \/ L4 \/ LSeg(p11,p2) is_an_arc_of p00,p2 by A124,TOPREAL1:10; A129: LSeg(p01,p2) /\ LSeg(p11,p2) = {p2} by A116,TOPREAL1:8; A130: L2 = LSeg(p11,p2) \/ LSeg(p01,p2) by A116,TOPREAL1:5; LSeg(p1,p00) /\ LSeg(p11,p2) c= {p01} by A4,A125,TOPREAL1:15,XBOOLE_1:27; then A131: LSeg(p1,p00) /\ LSeg(p11,p2) = {p01} or LSeg(p1,p00) /\ LSeg(p11,p2) = {} by ZFMISC_1:33; A132: LSeg(p01,p2) c= L2 by A116,Lm23,TOPREAL1:6; then A133: LSeg(p01,p2) /\ L3 = {} by Lm2,XBOOLE_1:3,27; A134: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 1 by A116,TOPREAL1:13; A135: now A136: p2`1 <= p11`1 by A134,EUCLID:52; assume A137: p01 in LSeg(p1,p00) /\ LSeg(p11,p2); then A138: p01 in LSeg(p00,p1) by XBOOLE_0:def 4; p01 in LSeg(p2,p11) by A137,XBOOLE_0:def 4; then A139: p01`1 = p2`1 by A134,A136,Lm6,TOPREAL1:3; p00`2 <= p1`2 by A15,A18,EUCLID:52; then p01`2 <= p1`2 by A138,TOPREAL1:4; then p01`2 = p1`2 by A15,A17,Lm7,XXREAL_0:1; then p1 = |[p01`1,p01`2]| by A15,A16,Lm6,EUCLID:53 .= p2 by A134,A139,Lm7,EUCLID:53; hence contradiction by A1; end; LSeg(p1,p00) /\ (L3 \/ L4 \/ LSeg(p11,p2)) = (LSeg(p1,p00) /\ (L3 \/ L4)) \/ (LSeg(p1,p00) /\ LSeg(p11,p2)) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ L3) \/ (LSeg(p1,p00) /\ L4) by A131,A135,XBOOLE_1:23 ,ZFMISC_1:31 .= {p00} by A9,A5,A6,TOPREAL1:17,XBOOLE_0:def 10; hence P2 is_an_arc_of p1,p2 by A128,TOPREAL1:11; thus P1 \/ P2 = LSeg(p01,p2) \/ (LSeg(p1,p01) \/ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2)))) by XBOOLE_1:4 .= L1 \/ (L3 \/ L4 \/ LSeg(p11,p2)) \/ LSeg(p01,p2) by A122,XBOOLE_1:4 .= L1 \/ ((L3 \/ L4 \/ LSeg(p11,p2)) \/ LSeg(p01,p2)) by XBOOLE_1:4 .= L1 \/ (L2 \/ (L3 \/ L4)) by A130,XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A140: {p1} = LSeg(p1,p01) /\ LSeg(p1,p00) by A3,TOPREAL1:8; A141: P1 /\ P2 = (LSeg(p1,p01) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11, p2)))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p01) /\ LSeg(p1,p00)) \/ (LSeg(p1,p01) /\ (L3 \/ L4 \/ LSeg(p11,p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ (L3 \/ L4)) \/ (LSeg(p1,p01) /\ LSeg(p11, p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg(p11,p2)))) by A140 ,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4) \/ (LSeg(p1, p01) /\ LSeg(p11,p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p00) \/ (L3 \/ L4 \/ LSeg( p11,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4) \/ (LSeg(p1, p01) /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/ (LSeg(p01,p2) /\ ( L3 \/ L4 \/ LSeg(p11,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4) \/ (LSeg(p1, p01) /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p01,p2) /\ (L3 \/ L4)) \/ {p2})) by A129,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4) \/ (LSeg(p1, p01) /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p01,p2) /\ L3) \/ (LSeg(p01,p2) /\ L4) \/ {p2})) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/ (LSeg(p01,p2) /\ L4 \/ {p2})) by A14,A133; A142: now per cases; suppose A143: p2 = p11; then A144: not p2 in LSeg(p1,p01) by A13,Lm4,Lm6,Lm10,TOPREAL1:3; LSeg(p1,p01) /\ LSeg(p11,p2) = LSeg(p1,p01) /\ {p2} by A143,RLTOPSP1:70 .= {} by A144,Lm1; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p01) /\ L3) \/ ((LSeg(p01,p2) /\ LSeg (p1,p00)) \/ {p2}) by A141,A143,TOPREAL1:18; end; suppose A145: p2 <> p11 & p2 <> p01; now assume p01 in LSeg(p1,p01) /\ LSeg(p11,p2); then A146: p01 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`1 <= p11`1 by A134,EUCLID:52; then p2`1 = 0 by A134,A146,Lm6,TOPREAL1:3; hence contradiction by A134,A145,EUCLID:53; end; then A147: {p01} <> LSeg(p1,p01) /\ LSeg(p11,p2) by ZFMISC_1:31; LSeg(p1,p01) /\ LSeg(p11,p2) c= {p01} by A13,A125,TOPREAL1:15 ,XBOOLE_1:27; then A148: LSeg(p1,p01) /\ LSeg(p11,p2) = {} by A147,ZFMISC_1:33; now assume p11 in LSeg(p01,p2) /\ L4; then A149: p11 in LSeg(p01,p2) by XBOOLE_0:def 4; p01`1 <= p2`1 by A134,EUCLID:52; then p11`1 <= p2`1 by A149,TOPREAL1:3; then p2`1 = p11`1 by A134,Lm10,XXREAL_0:1; hence contradiction by A134,A145,Lm10,EUCLID:53; end; then A150: {p11} <> LSeg(p01,p2) /\ L4 by ZFMISC_1:31; LSeg(p01,p2) /\ L4 c= {p11} by A132,TOPREAL1:18,XBOOLE_1:27; then LSeg(p01,p2) /\ L4 = {} by A150,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p01) /\ L3) \/ ((LSeg(p01,p2) /\ LSeg (p1,p00)) \/ {p2}) by A141,A148; end; suppose A151: p2 = p01; then p2 in LSeg(p1,p01) by RLTOPSP1:68; then A152: LSeg(p1,p01) /\ LSeg(p11,p2) <> {} by A151,Lm23,XBOOLE_0:def 4; LSeg(p1,p01) /\ LSeg(p11,p2) c= {p2} by A13,A151,TOPREAL1:15 ,XBOOLE_1:27; then A153: LSeg(p1,p01) /\ LSeg(p11,p2) = {p2} by A152,ZFMISC_1:33; LSeg(p01,p2) /\ L4 = {p01} /\ L4 by A151,RLTOPSP1:70 .= {} by Lm1,Lm15; hence P1 /\ P2 = ({p1} \/ (LSeg(p1,p01) /\ L3)) \/ {p2} \/ ((LSeg(p01, p2) /\ LSeg(p1,p00)) \/ {p2}) by A141,A153,XBOOLE_1:4 .= ({p1} \/ (LSeg(p1,p01) /\ L3)) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00 )) \/ {p2} \/ {p2}) by XBOOLE_1:4 .= ({p1} \/ (LSeg(p1,p01) /\ L3)) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00 )) \/ ({p2} \/ {p2})) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p01) /\ L3) \/ ((LSeg(p01,p2) /\ LSeg(p1,p00)) \/ {p2}); end; end; now per cases; suppose A154: p1 = p01; then p1 in LSeg(p01,p2 ) by RLTOPSP1:68; then A155: LSeg(p01,p2) /\ LSeg(p1,p00) <> {} by A154,Lm22,XBOOLE_0:def 4; LSeg(p01,p2) /\ LSeg(p1,p00) c= {p1} by A132,A154,TOPREAL1:15 ,XBOOLE_1:27; then A156: LSeg(p01,p2) /\ LSeg(p1,p00) = {p1} by A155,ZFMISC_1:33; LSeg(p1,p01) /\ L3 = {p1} /\ L3 by A154,RLTOPSP1:70 .= {} by A154,Lm1,Lm14; hence P1 /\ P2 = {p1} \/ {p1} \/ {p2} by A142,A156,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A157: p1 = p00; A158: not p00 in LSeg(p01,p2) by A132,Lm5,Lm7,Lm11,TOPREAL1:4; LSeg(p01,p2) /\ LSeg(p1,p00) = LSeg(p01,p2) /\ {p00} by A157, RLTOPSP1:70 .= {} by A158,Lm1; hence thesis by A142,A157,ENUMSET1:1,TOPREAL1:17; end; suppose A159: p1 <> p00 & p1 <> p01; now assume p01 in LSeg(p01,p2) /\ LSeg(p1,p00); then A160: p01 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`2 <= p1`2 by A15,A18,EUCLID:52; then p01`2 <= p1`2 by A160,TOPREAL1:4; then p1`2 = 1 by A15,A17,Lm7,XXREAL_0:1; hence contradiction by A15,A16,A159,EUCLID:53; end; then A161: {p01} <> LSeg(p01,p2) /\ LSeg(p1,p00) by ZFMISC_1:31; LSeg(p01,p2) /\ LSeg(p1,p00) c= {p01} by A4,A132,TOPREAL1:15 ,XBOOLE_1:27; then A162: LSeg(p01,p2) /\ LSeg(p1,p00) = {} by A161,ZFMISC_1:33; now assume p00 in LSeg(p1,p01) /\ L3; then A163: p00 in LSeg(p1,p01) by XBOOLE_0:def 4; p1`2 <= p01`2 by A15,A17,EUCLID:52; then p1`2 = 0 by A15,A18,A163,Lm5,TOPREAL1:4; hence contradiction by A15,A16,A159,EUCLID:53; end; then A164: {p00} <> LSeg(p1,p01) /\ L3 by ZFMISC_1:31; LSeg(p1,p01) /\ L3 c= {p00} by A13,TOPREAL1:17,XBOOLE_1:27; then LSeg(p1,p01) /\ L3 = {} by A164,ZFMISC_1:33; hence thesis by A142,A162,ENUMSET1:1; end; end; hence thesis; end; suppose A165: p2 in L3; then A166: LSeg(p00,p2) c= L3 by Lm21,TOPREAL1:6; LSeg(p1,p00) c= L1 by A3,Lm20,TOPREAL1:6; then A167: LSeg(p1,p00) /\ LSeg(p00,p2) c= L1 /\ L3 by A166,XBOOLE_1:27; take P1 = LSeg(p1,p00) \/ LSeg(p00,p2),P2 = LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2)); A168: p00 in LSeg(p00,p2) by RLTOPSP1:68; p10 in LSeg(p10,p2) by RLTOPSP1:68; then A169: p10 in L4 /\ LSeg(p10,p2) by Lm25,XBOOLE_0:def 4; p00 in LSeg(p1,p00) by RLTOPSP1:68; then LSeg(p1,p00) /\ LSeg(p00,p2) <> {} by A168,XBOOLE_0:def 4; then A170: LSeg(p1,p00) /\ LSeg(p00,p2) = {p00} by A167,TOPREAL1:17,ZFMISC_1:33; p1 <> p00 or p00 <> p2 by A1; hence P1 is_an_arc_of p1,p2 by A170,TOPREAL1:12; A171: L1 = LSeg(p1,p00) \/ LSeg(p1,p01) by A3,TOPREAL1:5; A172: L4 is_an_arc_of p11,p10 by Lm9,Lm11,TOPREAL1:9; L2 is_an_arc_of p01,p11 by Lm6,Lm10,TOPREAL1:9; then A173: L2 \/ L4 is_an_arc_of p01,p10 by A172,TOPREAL1:2,18; A174: LSeg(p10,p2) c= L3 by A165,Lm24,TOPREAL1:6; then A175: L4 /\ LSeg(p10,p2) c= L4 /\ L3 by XBOOLE_1:27; A176: L2 /\ LSeg(p10,p2) = {} by A174,Lm2,XBOOLE_1:3,26; (L2 \/ L4) /\ LSeg(p10,p2) = (L2 /\ LSeg(p10,p2)) \/ (L4 /\ LSeg(p10 ,p2)) by XBOOLE_1:23 .= {p10} by A176,A175,A169,TOPREAL1:16,ZFMISC_1:33; then A177: L2 \/ L4 \/ LSeg(p10,p2) is_an_arc_of p01,p2 by A173,TOPREAL1:10; A178: LSeg(p00,p2) /\ LSeg(p10,p2) = {p2} by A165,TOPREAL1:8; A179: L3 = LSeg(p10,p2) \/ LSeg(p00,p2) by A165,TOPREAL1:5; LSeg(p1,p01) /\ LSeg(p10,p2) c= {p00} by A13,A174,TOPREAL1:17,XBOOLE_1:27; then A180: LSeg(p1,p01) /\ LSeg(p10,p2) = {p00} or LSeg(p1,p01) /\ LSeg(p10,p2) = {} by ZFMISC_1:33; A181: LSeg(p00,p2) c= L3 by A165,Lm21,TOPREAL1:6; then A182: LSeg(p00,p2) /\ L2 = {} by Lm2,XBOOLE_1:3,27; A183: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 0 by A165,TOPREAL1:13; A184: now A185: p2`1 <= p10`1 by A183,EUCLID:52; assume A186: p00 in LSeg(p1,p01) /\ LSeg(p10,p2); then A187: p00 in LSeg(p1,p01) by XBOOLE_0:def 4; p00 in LSeg(p2,p10) by A186,XBOOLE_0:def 4; then A188: p00`1 = p2`1 by A183,A185,Lm4,TOPREAL1:3; p1`2 <= p01`2 by A15,A17,EUCLID:52; then p00`2 = p1`2 by A15,A18,A187,Lm5,TOPREAL1:4; then p1 = |[p00`1,p00`2]| by A15,A16,Lm4,EUCLID:53 .= p2 by A183,A188,Lm5,EUCLID:53; hence contradiction by A1; end; LSeg(p1,p01) /\ (L2 \/ L4 \/ LSeg(p10,p2)) = (LSeg(p1,p01) /\ (L2 \/ L4)) \/ (LSeg(p1,p01) /\ LSeg(p10,p2)) by XBOOLE_1:23 .= (LSeg(p1,p01) /\ L2) \/ (LSeg(p01,p1) /\ L4) by A180,A184,XBOOLE_1:23 ,ZFMISC_1:31 .= {p01} by A14,A10,A12,XBOOLE_0:def 10; hence P2 is_an_arc_of p1,p2 by A177,TOPREAL1:11; thus P1 \/ P2 = LSeg(p00,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2)))) by XBOOLE_1:4 .= L1 \/ (L2 \/ L4 \/ LSeg(p10,p2)) \/ LSeg(p00,p2) by A171,XBOOLE_1:4 .= L1 \/ ((L2 \/ L4 \/ LSeg(p10,p2)) \/ LSeg(p00,p2)) by XBOOLE_1:4 .= L1 \/ (L2 \/ L4 \/ (LSeg(p10,p2) \/ LSeg(p00,p2))) by XBOOLE_1:4 .= L1 \/ (L2 \/ (L3 \/ L4)) by A179,XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A189: {p1} = LSeg(p1,p00) /\ LSeg(p1,p01) by A3,TOPREAL1:8; A190: P1 /\ P2 = (LSeg(p1,p00) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10, p2)))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ LSeg(p1,p01)) \/ (LSeg(p1,p00) /\ (L2 \/ L4 \/ LSeg(p10,p2))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ (L2 \/ L4)) \/ (LSeg(p1,p00) /\ LSeg(p10, p2))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg(p10,p2)))) by A189 ,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4) \/ (LSeg(p1, p00) /\ LSeg(p10,p2))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p01) \/ (L2 \/ L4 \/ LSeg( p10,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4) \/ (LSeg(p1, p00) /\ LSeg(p10,p2))) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/ (LSeg(p00,p2) /\ ( L2 \/ L4 \/ LSeg(p10,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4) \/ (LSeg(p1, p00) /\ LSeg(p10,p2))) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/ ((LSeg(p00,p2) /\ (L2 \/ L4)) \/ {p2})) by A178,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ L4) \/ (LSeg(p1, p00) /\ LSeg(p10,p2))) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/ ((LSeg(p00,p2) /\ L2) \/ (LSeg(p00,p2) /\ L4) \/ {p2})) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L2) \/ (LSeg(p1,p00) /\ LSeg(p10,p2))) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/ (LSeg(p00,p2) /\ L4 \/ {p2})) by A9,A182; A191: now per cases; suppose A192: p2 = p10; then not p2 in LSeg(p1,p00) by A4,Lm4,Lm6,Lm8,TOPREAL1:3; then A193: LSeg(p1,p00) misses {p2} by ZFMISC_1:50; LSeg(p1,p00) /\ LSeg(p10,p2) = LSeg(p1,p00) /\ {p2} by A192,RLTOPSP1:70 .= {} by A193,XBOOLE_0:def 7; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((LSeg(p00,p2) /\ LSeg (p1,p01)) \/ {p2}) by A190,A192,TOPREAL1:16; end; suppose A194: p2 <> p10 & p2 <> p00; now assume p00 in LSeg(p1,p00) /\ LSeg(p10,p2); then A195: p00 in LSeg(p2,p10) by XBOOLE_0:def 4; p2`1 <= p10`1 by A183,EUCLID:52; then p2`1 = 0 by A183,A195,Lm4,TOPREAL1:3; hence contradiction by A183,A194,EUCLID:53; end; then A196: {p00} <> LSeg(p1,p00) /\ LSeg(p10,p2) by ZFMISC_1:31; LSeg(p1,p00) /\ LSeg(p10,p2) c= {p00} by A4,A174,TOPREAL1:17 ,XBOOLE_1:27; then A197: LSeg(p1,p00) /\ LSeg(p10,p2) = {} by A196,ZFMISC_1:33; now assume p10 in LSeg(p00,p2) /\ L4; then A198: p10 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`1 <= p2`1 by A183,EUCLID:52; then p10`1 <= p2`1 by A198,TOPREAL1:3; then p2`1 = p10`1 by A183,Lm8,XXREAL_0:1; hence contradiction by A183,A194,Lm8,EUCLID:53; end; then A199: {p10} <> LSeg(p00,p2) /\ L4 by ZFMISC_1:31; LSeg(p00,p2) /\ L4 c= {p10} by A181,TOPREAL1:16,XBOOLE_1:27; then LSeg(p00,p2) /\ L4 = {} by A199,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((LSeg(p00,p2) /\ LSeg (p1,p01)) \/ {p2}) by A190,A197; end; suppose A200: p2 = p00; then p2 in LSeg(p1,p00) by RLTOPSP1:68; then A201: LSeg(p1,p00) /\ LSeg(p10,p2) <> {} by A200,Lm21,XBOOLE_0:def 4; LSeg(p1,p00) /\ LSeg(p10,p2) c= {p2} by A4,A200,TOPREAL1:17,XBOOLE_1:27 ; then A202: LSeg(p1,p00) /\ LSeg(p10,p2) = {p2} by A201,ZFMISC_1:33; LSeg(p00,p2) /\ L4 = {p00} /\ L4 by A200,RLTOPSP1:70 .= {} by Lm1,Lm12; hence P1 /\ P2 = ({p1} \/ (LSeg(p1,p00) /\ L2)) \/ {p2} \/ ((LSeg(p00, p2) /\ LSeg(p1,p01)) \/ {p2}) by A190,A202,XBOOLE_1:4 .= ({p1} \/ (LSeg(p1,p00) /\ L2)) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01 )) \/ {p2} \/ {p2}) by XBOOLE_1:4 .= ({p1} \/ (LSeg(p1,p00) /\ L2)) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01 )) \/ ({p2} \/ {p2})) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((LSeg(p00,p2) /\ LSeg(p1,p01)) \/ {p2}); end; end; now per cases; suppose A203: p1 = p01; then A204: LSeg(p00,p2) /\ LSeg(p1,p01) = LSeg(p00,p2) /\ {p1} by RLTOPSP1:70; not p1 in LSeg(p00,p2) by A181,A203,Lm5,Lm7,Lm9,TOPREAL1:4; then LSeg(p00,p2) /\ LSeg(p1,p01) = {} by A204,Lm1; hence thesis by A191,A203,ENUMSET1:1,TOPREAL1:15; end; suppose A205: p1 = p00; p00 in LSeg(p00,p2) by RLTOPSP1:68; then A206: LSeg(p00,p2) /\ LSeg(p1,p01) <> {} by A205,Lm20,XBOOLE_0:def 4; LSeg(p1,p00) /\ L2 = {p1} /\ L2 by A205,RLTOPSP1:70; then A207: LSeg(p1,p00) /\ L2 = {} by A205,Lm1,Lm13; LSeg(p00,p2) /\ LSeg(p1,p01) c= L3 /\ L1 by A165,A205,Lm21,TOPREAL1:6 ,XBOOLE_1:26; then LSeg(p00,p2) /\ LSeg(p1,p01) = {p1} by A205,A206,TOPREAL1:17 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ {p1} \/ {p2} by A191,A207,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A208: p1 <> p00 & p1 <> p01; now assume p00 in LSeg(p00,p2) /\ LSeg(p1,p01); then A209: p00 in LSeg(p1,p01) by XBOOLE_0:def 4; p1`2 <= p01`2 by A15,A17,EUCLID:52; then p1`2 = 0 by A15,A18,A209,Lm5,TOPREAL1:4; hence contradiction by A15,A16,A208,EUCLID:53; end; then A210: {p00} <> LSeg(p00,p2) /\ LSeg(p1,p01) by ZFMISC_1:31; LSeg(p00,p2) /\ LSeg(p1,p01) c= L3 /\ L1 by A13,A181,XBOOLE_1:27; then A211: LSeg(p00,p2) /\ LSeg(p1,p01) = {} by A210,TOPREAL1:17,ZFMISC_1:33; now assume p01 in LSeg(p1,p00) /\ L2; then A212: p01 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`2 <= p1`2 by A15,A18,EUCLID:52; then p01`2 <= p1`2 by A212,TOPREAL1:4; then p1`2 = 1 by A15,A17,Lm7,XXREAL_0:1; hence contradiction by A15,A16,A208,EUCLID:53; end; then A213: {p01} <> LSeg(p1,p00) /\ L2 by ZFMISC_1:31; LSeg(p1,p00) /\ L2 c= {p01} by A4,TOPREAL1:15,XBOOLE_1:27; then LSeg(p1,p00) /\ L2 = {} by A213,ZFMISC_1:33; hence thesis by A191,A211,ENUMSET1:1; end; end; hence thesis; end; suppose A214: p2 in L4; now let a be object; assume A215: a in LSeg(p1,p00) /\ LSeg(p1,p01); then reconsider p = a as Point of TOP-REAL 2; a in LSeg(p1,p01) by A215,XBOOLE_0:def 4; then A216: p1`2 <= p`2 by A15,A17,Lm7,TOPREAL1:4; A217: a in LSeg(p00,p1) by A215,XBOOLE_0:def 4; then p`2 <= p1`2 by A15,A18,Lm5,TOPREAL1:4; then A218: p`2 = p1`2 by A216,XXREAL_0:1; p`1 <= p1`1 by A15,A16,A217,Lm4,TOPREAL1:3; then p`1 = p1`1 by A15,A16,A217,Lm4,TOPREAL1:3; then a = |[p1`1,p1`2]| by A218,EUCLID:53 .= p1 by EUCLID:53; hence a in {p1} by TARSKI:def 1; end; then A219: LSeg(p1,p00) /\ LSeg(p1,p01) c= {p1}; A220: p2 in LSeg(p11,p2) by RLTOPSP1:68; p2 in LSeg(p10,p2) by RLTOPSP1:68; then p2 in LSeg(p10,p2) /\ LSeg(p11,p2) by A220,XBOOLE_0:def 4; then A221: {p2} c= LSeg(p10,p2) /\ LSeg(p11,p2) by ZFMISC_1:31; A222: ex q st q = p2 & q`1 = 1 & q`2 <= 1 & q`2 >= 0 by A214,TOPREAL1:13; now let a be object; assume A223: a in LSeg(p10,p2) /\ LSeg(p11,p2); then reconsider p = a as Point of TOP-REAL 2; A224: a in LSeg(p10,p2) by A223,XBOOLE_0:def 4; then A225: p2`1 <= p`1 by A222,Lm8,TOPREAL1:3; a in LSeg(p2,p11) by A223,XBOOLE_0:def 4; then A226: p2`2 <= p`2 by A222,Lm11,TOPREAL1:4; p`1 <= p2`1 by A222,A224,Lm8,TOPREAL1:3; then A227: p`1 = p2`1 by A225,XXREAL_0:1; p`2 <= p2`2 by A222,A224,Lm9,TOPREAL1:4; then p`2 = p2`2 by A226,XXREAL_0:1; then a = |[p2`1,p2`2]| by A227,EUCLID:53 .= p2 by EUCLID:53; hence a in {p2} by TARSKI:def 1; end; then A228: LSeg(p10,p2) /\ LSeg(p11,p2) c= {p2}; LSeg(p10,p2) c= L4 by A214,Lm25,TOPREAL1:6; then A229: L3 /\ LSeg(p10,p2) c= {p10} by TOPREAL1:16,XBOOLE_1:27; take P1 = LSeg(p1,p00) \/ L3 \/ LSeg(p10,p2),P2 = LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2); A230: p10 in LSeg(p10,p2) by RLTOPSP1:68; p10 in L3 by RLTOPSP1:68; then L3 /\ LSeg(p10,p2) <> {} by A230,XBOOLE_0:def 4; then L3 /\ LSeg(p10,p2) = {p10} by A229,ZFMISC_1:33; then A231: L3 \/ LSeg(p10,p2) is_an_arc_of p00,p2 by Lm4,Lm8,TOPREAL1:12; LSeg(p10,p2) c= L4 by A214,Lm25,TOPREAL1:6; then A232: LSeg(p1,p00) /\ LSeg(p10,p2) = {} by A4,Lm3,XBOOLE_1:3,27; LSeg(p1,p00) /\ (L3 \/ LSeg(p10,p2)) = (LSeg(p1,p00) /\ L3) \/ (LSeg (p1,p00) /\ LSeg(p10,p2)) by XBOOLE_1:23 .= {p00} by A5,A6,A232,TOPREAL1:17,XBOOLE_0:def 10; then LSeg(p1,p00) \/ (L3 \/ LSeg(p10,p2)) is_an_arc_of p1,p2 by A231,TOPREAL1:11; hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A233: L2 /\ LSeg(p11,p2) <> {} by Lm26,XBOOLE_0:def 4; A234: LSeg(p11,p2) c= L4 by A214,Lm27,TOPREAL1:6; then A235: LSeg(p1,p01) /\ LSeg(p11,p2) = {} by A13,Lm3,XBOOLE_1:3,27; L2 /\ LSeg(p11,p2) c= {p11} by A234,TOPREAL1:18,XBOOLE_1:27; then L2 /\ LSeg(p11,p2) = {p11} by A233,ZFMISC_1:33; then A236: L2 \/ LSeg(p11,p2) is_an_arc_of p01,p2 by Lm6,Lm10,TOPREAL1:12; LSeg(p1,p01) /\ (L2 \/ LSeg(p11,p2)) = (LSeg(p1,p01) /\ L2) \/ (LSeg (p1,p01) /\ LSeg(p11,p2)) by XBOOLE_1:23 .= {p01} by A10,A12,A235,XBOOLE_0:def 10; then LSeg(p1,p01) \/ (L2 \/ LSeg(p11,p2)) is_an_arc_of p1,p2 by A236,TOPREAL1:11; hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4; thus R^2-unit_square = LSeg(p1,p00) \/ LSeg(p1,p01) \/ L2 \/ (L3 \/ L4) by A3,TOPREAL1:5,def 2 .= LSeg(p1,p00) \/ (LSeg(p1,p01) \/ L2) \/ (L3 \/ L4) by XBOOLE_1:4 .= LSeg(p1,p00) \/ ((LSeg(p1,p01) \/ L2) \/ (L3 \/ L4)) by XBOOLE_1:4 .= LSeg(p1,p00) \/ (L3 \/ (LSeg(p1,p01) \/ L2 \/ L4)) by XBOOLE_1:4 .= LSeg(p1,p00) \/ L3 \/ (LSeg(p1,p01) \/ L2 \/ L4) by XBOOLE_1:4 .= LSeg(p1,p00) \/ L3 \/ (LSeg(p1,p01) \/ L2 \/ (LSeg(p11,p2) \/ LSeg( p10,p2))) by A214,TOPREAL1:5 .= LSeg(p1,p00) \/ L3 \/ (LSeg(p10,p2) \/ (LSeg(p1,p01) \/ L2 \/ LSeg( p11,p2))) by XBOOLE_1:4 .= P1 \/ P2 by XBOOLE_1:4; A237: p1 in LSeg(p1,p01) by RLTOPSP1:68; p1 in LSeg(p1,p00) by RLTOPSP1:68; then p1 in LSeg(p1,p00) /\ LSeg(p1,p01) by A237,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p00) /\ LSeg(p1,p01) by ZFMISC_1:31; then A238: LSeg(p1,p00) /\ LSeg(p1,p01) = {p1} by A219,XBOOLE_0:def 10; A239: LSeg(p1,p00) /\ LSeg(p11,p2) = {} by A4,A234,Lm3,XBOOLE_1:3,27; A240: LSeg(p10,p2) c= L4 by A214,Lm25,TOPREAL1:6; then A241: LSeg(p10,p2) /\ LSeg(p1,p01) = {} by A13,Lm3,XBOOLE_1:3,27; A242: P1 /\ P2 = (LSeg(p1,p00) \/ L3) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11, p2)) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) \/ (L3 /\ ( LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ (LSeg(p1,p01) \/ L2)) \/ (LSeg(p1,p00) /\ LSeg(p11 ,p2)) \/ (L3 /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg (p1,p01) \/ L2 \/ LSeg(p11,p2))) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ LSeg(p1,p01)) \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ ( LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by A239,XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ (LSeg(p1,p01) \/ L2)) \/ ( L3 /\ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2))) by A238,XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ L2) \/ (L3 /\ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2 \/ LSeg(p11,p2 ))) by XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2)) \/ (LSeg(p10,p2) /\ LSeg(p11,p2))) by Lm2,XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L2)) \/ {p2}) by A221,A228 ,XBOOLE_0:def 10 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p01)) \/ (LSeg(p10,p2) /\ L2) \/ { p2}) by XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ (L3 /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ L2) \/ {p2}) by A241; A243: now per cases; suppose A244: p2 = p11; then L3 /\ LSeg(p11,p2) = L3 /\ {p11} by RLTOPSP1:70 .= {} by Lm1,Lm19; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ LSeg(p1,p01)) \/ {p2} by A242,A244,TOPREAL1:18; end; suppose A245: p2 = p10; then LSeg(p10,p2) /\ L2 = {p10} /\ L2 by RLTOPSP1:70 .= {} by Lm1,Lm17; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ {p2} \/ {p2}) by A242,A245,TOPREAL1:16,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ ((L3 /\ LSeg(p1,p01)) \/ ({p2} \/ {p2})) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ LSeg(p1,p01)) \/ {p2} by XBOOLE_1:4; end; suppose A246: p2 <> p10 & p2 <> p11; now assume p11 in LSeg(p10,p2) /\ L2; then A247: p11 in LSeg(p10,p2) by XBOOLE_0:def 4; p10`2 <= p2`2 by A222,EUCLID:52; then p11`2 <= p2`2 by A247,TOPREAL1:4; then p11`2 = p2`2 by A222,Lm11,XXREAL_0:1; then p2 = |[p11`1,p11`2]| by A222,Lm10,EUCLID:53 .= p11 by EUCLID:53; hence contradiction by A246; end; then A248: {p11} <> LSeg(p10,p2) /\ L2 by ZFMISC_1:31; LSeg(p10,p2) /\ L2 c= L4 /\ L2 by A240,XBOOLE_1:27; then A249: LSeg(p10,p2) /\ L2 = {} by A248,TOPREAL1:18,ZFMISC_1:33; now assume p10 in L3 /\ LSeg(p11,p2); then A250: p10 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`2 <= p11`2 by A222,EUCLID:52; then p2`2 = p10`2 by A222,A250,Lm9,TOPREAL1:4; then p2 = |[p10`1,p10`2]| by A222,Lm8,EUCLID:53 .= p10 by EUCLID:53; hence contradiction by A246; end; then A251: L3 /\ LSeg(p11,p2) <> {p10} by ZFMISC_1:31; L3 /\ LSeg(p11,p2) c= {p10} by A234,TOPREAL1:16,XBOOLE_1:27; then L3 /\ LSeg(p11,p2) = {} by A251,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ L2) \/ (L3 /\ LSeg(p1,p01)) \/ {p2} by A242,A249; end; end; now per cases; suppose A252: p1 = p01; then L3 /\ LSeg(p1,p01) = L3 /\ {p01} by RLTOPSP1:70 .= {} by Lm1,Lm14; hence thesis by A243,A252,ENUMSET1:1,TOPREAL1:15; end; suppose A253: p1 <> p01 & p1 <> p00; now assume p01 in LSeg(p1,p00) /\ L2; then A254: p01 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`2 <= p1`2 by A15,A18,EUCLID:52; then p01`2 <= p1`2 by A254,TOPREAL1:4; then p1`2 = p01`2 by A15,A17,Lm7,XXREAL_0:1; then p1 = |[p01`1,p01`2]| by A15,A16,Lm6,EUCLID:53 .= p01 by EUCLID:53; hence contradiction by A253; end; then A255: {p01} <> LSeg(p1,p00) /\ L2 by ZFMISC_1:31; LSeg(p1,p00) /\ L2 c= {p01} by A4,TOPREAL1:15,XBOOLE_1:27; then A256: LSeg(p1,p00) /\ L2 = {} by A255,ZFMISC_1:33; now assume p00 in L3 /\ LSeg(p1,p01); then p00 in LSeg(p1,p01) by XBOOLE_0:def 4; then p1`2 = p00`2 by A15,A17,A18,Lm5,Lm7,TOPREAL1:4; then p1 = |[p00`1,p00`2]| by A15,A16,Lm4,EUCLID:53 .= p00 by EUCLID:53; hence contradiction by A253; end; then A257: {p00} <> L3 /\ LSeg(p1,p01) by ZFMISC_1:31; L3 /\ LSeg(p1,p01) c= L3 /\ L1 by A13,XBOOLE_1:27; then L3 /\ LSeg(p1,p01) = {} by A257,TOPREAL1:17,ZFMISC_1:33; hence thesis by A243,A256,ENUMSET1:1; end; suppose A258: p1 = p00; then LSeg(p1,p00) /\ L2 = {p00} /\ L2 by RLTOPSP1:70 .= {} by Lm1,Lm13; hence thesis by A243,A258,ENUMSET1:1,TOPREAL1:17; end; end; hence thesis; end; end; Lm31: p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg(p01, p11) implies ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof assume that A1: p1 <> p2 and A2: p2 in R^2-unit_square and A3: p1 in LSeg(p01, p11); A4: p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:def 2,XBOOLE_0:def 3; A5: LSeg(p01,p1) /\ L1 c= L2 /\ L1 by A3,Lm23,TOPREAL1:6,XBOOLE_1:26; p11 in LSeg(p1,p11) by RLTOPSP1:68; then A6: LSeg(p1,p11) /\ L4 <> {} by Lm27,XBOOLE_0:def 4; p01 in LSeg(p01,p1) by RLTOPSP1:68; then A7: LSeg(p01,p1) /\ L1 <> {} by Lm22,XBOOLE_0:def 4; A8: LSeg(p1,p11) c= L2 by A3,Lm26,TOPREAL1:6; then A9: LSeg(p1,p11) /\ L3 = {} by Lm2,XBOOLE_1:3,26; A10: LSeg(p1,p11) /\ L4 c= {p11} by A3,Lm26,TOPREAL1:6,18,XBOOLE_1:26; A11: LSeg(p01,p1) c= L2 by A3,Lm23,TOPREAL1:6; then A12: LSeg(p1,p01) /\ L3 = {} by Lm2,XBOOLE_1:3,26; consider q1 such that A13: q1 = p1 and A14: q1`1 <= 1 and A15: q1`1 >= 0 and A16: q1`2 = 1 by A3,TOPREAL1:13; per cases by A4,XBOOLE_0:def 3; suppose A17: p2 in L1; then A18: LSeg(p01,p2) c= L1 by Lm22,TOPREAL1:6; LSeg(p1,p01) c= L2 by A3,Lm23,TOPREAL1:6; then A19: LSeg(p1,p01) /\ LSeg(p01,p2) c= L2 /\ L1 by A18,XBOOLE_1:27; take P1 = LSeg(p1,p01) \/ LSeg(p01,p2),P2 = LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2)); A20: p01 in LSeg(p01,p2) by RLTOPSP1:68; p00 in LSeg(p00,p2) by RLTOPSP1:68; then A21: {} <> L3 /\ LSeg(p00,p2) by Lm21,XBOOLE_0:def 4; p01 in LSeg(p1,p01) by RLTOPSP1:68; then LSeg(p1,p01) /\ LSeg(p01,p2) <> {} by A20,XBOOLE_0:def 4; then A22: LSeg(p1,p01) /\ LSeg(p01,p2) = {p01} by A19,TOPREAL1:15,ZFMISC_1:33; p1 <> p01 or p2 <> p01 by A1; hence P1 is_an_arc_of p1,p2 by A22,TOPREAL1:12; A23: LSeg(p1,p01) \/ LSeg(p1,p11) = L2 by A3,TOPREAL1:5; A24: L4 is_an_arc_of p11,p10 by Lm9,Lm11,TOPREAL1:9; L3 is_an_arc_of p10,p00 by Lm4,Lm8,TOPREAL1:9; then A25: L3 \/ L4 is_an_arc_of p11,p00 by A24,TOPREAL1:2,16; A26: L3 /\ LSeg(p00,p2) c= {p00} by A17,Lm20,TOPREAL1:6,17,XBOOLE_1:26; A27: LSeg(p00,p2) \/ LSeg(p01,p2) = L1 by A17,TOPREAL1:5; A28: LSeg(p01,p2) /\ LSeg(p00,p2) = {p2} by A17,TOPREAL1:8; A29: LSeg(p00,p2) c= L1 by A17,Lm20,TOPREAL1:6; then A30: L4 /\ LSeg(p00,p2) = {} by Lm3,XBOOLE_1:3,27; A31: ex q2 st q2 = p2 & q2`1 = 0 & q2`2 <= 1 & q2`2 >= 0 by A17,TOPREAL1:13; A32: now A33: p00`2 <= p2`2 by A31,EUCLID:52; assume A34: p01 in LSeg(p1,p11) /\ LSeg(p00,p2); then A35: p01 in LSeg(p1,p11) by XBOOLE_0:def 4; p01 in LSeg(p00,p2) by A34,XBOOLE_0:def 4; then p01`2 <= p2`2 by A33,TOPREAL1:4; then A36: p01`2 = p2`2 by A31,Lm7,XXREAL_0:1; p1`1 <= p11`1 by A13,A14,EUCLID:52; then p01`1 = p1`1 by A13,A15,A35,Lm6,TOPREAL1:3; then p1 = |[p01`1,p01`2]| by A13,A16,Lm7,EUCLID:53 .= p2 by A31,A36,Lm6,EUCLID:53; hence contradiction by A1; end; (L3 \/ L4) /\ LSeg(p00,p2) = (L3 /\ LSeg(p00,p2)) \/ (L4 /\ LSeg(p00, p2)) by XBOOLE_1:23 .= {p00} by A26,A21,A30,ZFMISC_1:33; then A37: L3 \/ L4 \/ LSeg(p00,p2) is_an_arc_of p11,p2 by A25,TOPREAL1:10; LSeg(p1,p11) /\ LSeg(p00,p2) c= L2 /\ L1 by A8,A29,XBOOLE_1:27; then A38: LSeg(p1,p11) /\ LSeg(p00,p2) = {p01} or LSeg(p1,p11) /\ LSeg(p00,p2) = {} by TOPREAL1:15,ZFMISC_1:33; A39: LSeg(p2,p01) c= L1 by A17,Lm22,TOPREAL1:6; then A40: LSeg(p01,p2) /\ L4 = {} by Lm3,XBOOLE_1:3,27; LSeg(p1,p11) /\ (L3 \/ L4 \/ LSeg(p00,p2)) = (LSeg(p1,p11) /\ (L3 \/ L4)) \/ (LSeg(p1,p11) /\ LSeg(p00,p2)) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ L4) by A38,A32,XBOOLE_1:23 ,ZFMISC_1:31 .= {p11} by A9,A6,A10,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A37,TOPREAL1:11; thus P1 \/ P2 = LSeg(p01,p2) \/ (LSeg(p1,p01) \/ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2)))) by XBOOLE_1:4 .= LSeg(p01,p2) \/ (L2 \/ (L3 \/ L4 \/ LSeg(p00,p2))) by A23,XBOOLE_1:4 .= LSeg(p01,p2) \/ (L2 \/ (L3 \/ L4) \/ LSeg(p00,p2)) by XBOOLE_1:4 .= LSeg(p00,p2) \/ LSeg(p01,p2) \/ (L2 \/ (L3 \/ L4)) by XBOOLE_1:4 .= R^2-unit_square by A27,TOPREAL1:def 2,XBOOLE_1:4; A41: {p1} = LSeg(p1,p01) /\ LSeg(p1,p11) by A3,TOPREAL1:8; A42: P1 /\ P2 = (LSeg(p1,p01) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2 )))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p01) /\ LSeg(p1,p11)) \/ (LSeg(p1,p01) /\ (L3 \/ L4 \/ LSeg(p00,p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ (L3 \/ L4)) \/ (LSeg(p1,p01) /\ LSeg(p00, p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg(p00,p2)))) by A41, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4) \/ (LSeg(p1, p01) /\ LSeg(p00,p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p11) \/ (L3 \/ L4 \/ LSeg( p00,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L3) \/ (LSeg(p1,p01) /\ L4) \/ (LSeg(p1, p01) /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p11)) \/ (LSeg(p01,p2) /\ ( L3 \/ L4 \/ LSeg(p00,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L4) \/ (LSeg(p1,p01) /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p11)) \/ ((LSeg(p01,p2) /\ (L3 \/ L4)) \/ {p2})) by A12,A28,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L4) \/ (LSeg(p1,p01) /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p11)) \/ ((LSeg(p01,p2) /\ L3) \/ (LSeg(p01,p2) /\ L4 ) \/ {p2})) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p01) /\ L4) \/ (LSeg(p1,p01) /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ LSeg(p1,p11)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2})) by A40; A43: now per cases; suppose A44: p1 = p01; then p1 in LSeg(p01,p2) by RLTOPSP1:68; then A45: LSeg(p01,p2) /\ LSeg(p1,p11) <> {} by A44,Lm23,XBOOLE_0:def 4; LSeg (p01,p2) /\ LSeg(p1,p11) c= {p1} by A39,A44,TOPREAL1:15 ,XBOOLE_1:27; then A46: LSeg(p01,p2) /\ LSeg(p1,p11) = {p1} by A45,ZFMISC_1:33; LSeg(p1,p01) /\ L4 = {p1} /\ L4 by A44,RLTOPSP1:70; then LSeg(p1,p01) /\ L4 = {} by A44,Lm1,Lm15; hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2))) \/ (( LSeg(p01,p2) /\ L3) \/ {p2}) by A42,A46,XBOOLE_1:4 .= {p1} \/ {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2 ) /\ L3) \/ {p2}) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}); end; suppose A47: p1 = p11; then A48: LSeg(p01,p2) /\ LSeg(p1,p11) = LSeg(p01,p2) /\ {p1} by RLTOPSP1:70; not p1 in LSeg(p01,p2) by A31,A47,Lm6,Lm10,TOPREAL1:3; then LSeg(p01,p2) /\ LSeg(p1,p11) = {} by A48,Lm1; hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2)) \/ (( LSeg(p01,p2) /\ L3) \/ {p2}) by A42,A47,TOPREAL1:18,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}); end; suppose A49: p1 <> p11 & p1 <> p01; now assume p01 in LSeg(p01,p2) /\ LSeg(p1,p11); then A50: p01 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`1 <= p11`1 by A13,A14,EUCLID:52; then p1`1 = 0 by A13,A15,A50,Lm6,TOPREAL1:3; hence contradiction by A13,A16,A49,EUCLID:53; end; then A51: {p01} <> LSeg(p01,p2) /\ LSeg(p1,p11) by ZFMISC_1:31; LSeg (p01,p2) /\ LSeg(p1,p11) c= {p01} by A8,A39,TOPREAL1:15 ,XBOOLE_1:27; then A52: LSeg(p01,p2) /\ LSeg(p1,p11) = {} by A51,ZFMISC_1:33; now assume p11 in LSeg(p1,p01) /\ L4; then A53: p11 in LSeg(p01,p1) by XBOOLE_0:def 4; p01`1 <= p1`1 by A13,A15,EUCLID:52; then p11`1 <= p1`1 by A53,TOPREAL1:3; then p1`1 = 1 by A13,A14,Lm10,XXREAL_0:1; hence contradiction by A13,A16,A49,EUCLID:53; end; then A54: {p11} <> LSeg(p1,p01) /\ L4 by ZFMISC_1:31; LSeg(p1,p01) /\ L4 c= {p11} by A11,TOPREAL1:18,XBOOLE_1:27; then LSeg(p1,p01) /\ L4 = {} by A54,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p01) /\ LSeg(p00,p2)) \/ ((LSeg(p01, p2) /\ L3) \/ {p2}) by A42,A52; end; end; now per cases; suppose A55: p2 <> p00 & p2 <> p01; now assume p01 in LSeg(p1,p01) /\ LSeg(p00,p2); then A56: p01 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`2 <= p2`2 by A31,EUCLID:52; then p01`2 <= p2`2 by A56,TOPREAL1:4; then p2`2 = 1 by A31,Lm7,XXREAL_0:1; hence contradiction by A31,A55,EUCLID:53; end; then A57: {p01} <> LSeg(p1,p01) /\ LSeg(p00,p2) by ZFMISC_1:31; LSeg(p1,p01) /\ LSeg(p00,p2) c= L2 /\ L1 by A11,A29,XBOOLE_1:27; then A58: LSeg(p1,p01) /\ LSeg(p00,p2) = {} by A57,TOPREAL1:15,ZFMISC_1:33; now assume p00 in LSeg(p01,p2) /\ L3; then A59: p00 in LSeg(p2,p01) by XBOOLE_0:def 4; p2`2 <= p01`2 by A31,EUCLID:52; then 0 = p2`2 by A31,A59,Lm5,TOPREAL1:4; hence contradiction by A31,A55,EUCLID:53; end; then A60: {p00} <> LSeg(p01,p2) /\ L3 by ZFMISC_1:31; LSeg(p01,p2) /\ L3 c= {p00} by A39,TOPREAL1:17,XBOOLE_1:27; then LSeg(p01,p2) /\ L3 = {} by A60,ZFMISC_1:33; hence thesis by A43,A58,ENUMSET1:1; end; suppose A61: p2 = p00; then A62: LSeg(p1,p01) /\ LSeg(p00,p2) = LSeg(p1,p01) /\ {p00} by RLTOPSP1:70; not p00 in LSeg(p1,p01) by A11,Lm5,Lm7,Lm11,TOPREAL1:4; then LSeg(p1,p01) /\ LSeg(p00,p2) = {} by A62,Lm1; hence thesis by A43,A61,ENUMSET1:1,TOPREAL1:17; end; suppose A63: p2 = p01; then p2 in LSeg(p1,p01) by RLTOPSP1:68; then A64: {} <> LSeg(p1,p01) /\ LSeg(p00,p2) by A63,Lm22,XBOOLE_0:def 4; LSeg(p01,p2) /\ L3 = {p01} /\ L3 by A63,RLTOPSP1:70; then A65: LSeg(p01,p2) /\ L3 = {} by Lm1,Lm14; LSeg(p1,p01) /\ LSeg(p00,p2) c= L2 /\ L1 by A11,A29,XBOOLE_1:27; then LSeg(p1,p01) /\ LSeg(p00,p2) = {p2} by A63,A64,TOPREAL1:15 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A43,A65,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; end; hence thesis; end; suppose A66: p2 in L2; A67: q1 = |[q1`1,q1`2]| by EUCLID:53; A68: LSeg(p1,p2) c= L2 by A3,A66,TOPREAL1:6; consider q such that A69: q = p2 and A70: q`1 <= 1 and A71: q`1 >= 0 and A72: q`2 = 1 by A66,TOPREAL1:13; A73: q = |[q`1,q`2]| by EUCLID:53; now per cases by A1,A13,A16,A69,A72,A67,A73,XXREAL_0:1; suppose A74: q1`1 < q`1; A75: LSeg(p1,p2) /\ LSeg(p11,p2) c= {p2} proof let a be object; assume A76: a in LSeg(p1,p2) /\ LSeg(p11,p2); then reconsider p = a as Point of TOP-REAL 2; A77: p in LSeg(p2,p11) by A76,XBOOLE_0:def 4; p2`1 <= p11`1 by A69,A70,EUCLID:52; then A78: p2`1 <= p`1 by A77,TOPREAL1:3; A79: p in LSeg(p1,p2) by A76,XBOOLE_0:def 4; then A80: p`2 <= p2`2 by A13,A16,A69,A72,TOPREAL1:4; p`1 <= p2`1 by A13,A69,A74,A79,TOPREAL1:3; then A81: p2`1 = p`1 by A78,XXREAL_0:1; p1`2 <= p`2 by A13,A16,A69,A72,A79,TOPREAL1:4; then p`2 = 1 by A13,A16,A69,A72,A80,XXREAL_0:1; then p = |[ p2`1, 1]| by A81,EUCLID:53 .= p2 by A69,A72,EUCLID:53; hence thesis by TARSKI:def 1; end; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A82: LSeg(p1,p01) /\ L3 = {} by A11,XBOOLE_1:3,26; A83: now set a = the Element of LSeg(p1,p01) /\ LSeg(p11,p2); assume A84: LSeg(p1,p01) /\ LSeg(p11,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A85: p in LSeg(p01,p1) by A84,XBOOLE_0:def 4; A86: p in LSeg(p2,p11) by A84,XBOOLE_0:def 4; p2`1 <= p11 `1 by A69,A70,EUCLID:52; then A87: p2`1 <= p`1 by A86,TOPREAL1:3; p01`1 <= p1`1 by A13,A15,EUCLID:52; then p`1 <= p1`1 by A85,TOPREAL1:3; hence contradiction by A13,A69,A74,A87,XXREAL_0:2; end; A88: (L1 \/ L3) /\ L4 = L1 /\ L4 \/ L3 /\ L4 by XBOOLE_1:23 .= {p10} by Lm3,TOPREAL1:16; L1 \/ L3 is_an_arc_of p01,p10 by Lm5,Lm7,TOPREAL1:9,10,17; then A89: L1 \/ L3 \/ L4 is_an_arc_of p01,p11 by A88,TOPREAL1:10; A90: LSeg(p1,p2) /\ LSeg(p1,p01) c= {p1} proof let a be object; assume A91: a in LSeg(p1,p2) /\ LSeg(p1,p01); then reconsider p = a as Point of TOP-REAL 2; A92: p in LSeg(p01,p1) by A91,XBOOLE_0:def 4; p01`1 <= p1`1 by A13,A15,EUCLID:52; then A93: p`1 <= p1`1 by A92,TOPREAL1:3; A94: p in LSeg(p1,p2) by A91,XBOOLE_0:def 4; then A95: p`2 <= p2`2 by A13,A16,A69,A72,TOPREAL1:4; p1`1 <= p`1 by A13,A69,A74,A94,TOPREAL1:3; then A96: p1`1 = p`1 by A93,XXREAL_0:1; p1`2 <= p`2 by A13,A16,A69,A72,A94,TOPREAL1:4; then p`2 = 1 by A13,A16,A69,A72,A95,XXREAL_0:1; then p = |[p1`1, 1]| by A96,EUCLID:53 .= p1 by A13,A16,EUCLID:53; hence thesis by TARSKI:def 1; end; A97: LSeg(p1,p2) /\ L1 c= L2 /\ L1 by A3,A66,TOPREAL1:6,XBOOLE_1:26; now assume p11 in LSeg(p1,p01) /\ L4; then A98: p11 in LSeg(p01,p1) by XBOOLE_0:def 4; p01`1 <= p1`1 by A13,A15,EUCLID:52; then p11`1 <= p1`1 by A98,TOPREAL1:3; hence contradiction by A13,A14,A70,A74,Lm10,XXREAL_0:1; end; then A99: {p11} <> LSeg(p1,p01) /\ L4 by ZFMISC_1:31; LSeg(p1,p01) /\ L4 c= {p11} by A3,Lm23,TOPREAL1:6,18,XBOOLE_1:26; then A100: LSeg(p1,p01) /\ L4 = {} by A99,ZFMISC_1:33; p01 in LSeg(p1,p01) by RLTOPSP1:68; then A101: LSeg(p1,p01) /\ L1 <> {} by Lm22,XBOOLE_0:def 4; now assume p01 in L1 /\ LSeg(p11,p2); then A102: p01 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`1 <= p11`1 by A69,A70,EUCLID:52; hence contradiction by A15,A69,A74,A102,Lm6,TOPREAL1:3; end; then A103: {p01} <> L1 /\ LSeg(p11,p2) by ZFMISC_1:31; L1 /\ LSeg(p11,p2) c= {p01} by A66,Lm26,TOPREAL1:6,15,XBOOLE_1:26; then A104: L1 /\ LSeg(p11,p2) = {} by A103,ZFMISC_1:33; take P1 = LSeg(p1,p2),P2 = LSeg(p1,p01) \/ (L1 \/ L3 \/ L4 \/ LSeg(p11 ,p2)); A105: p1 in LSeg(p1,p01) by RLTOPSP1:68; A106: LSeg(p1,p01) /\ L1 c= L2 /\ L1 by A3,Lm23,TOPREAL1:6,XBOOLE_1:26; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A107: L4 /\ LSeg(p11,p2) <> {} by Lm27,XBOOLE_0:def 4; L4 /\ LSeg(p11,p2) c= L4 /\ L2 by A66,Lm26,TOPREAL1:6,XBOOLE_1:26; then A108: L4 /\ LSeg(p11,p2) = {p11} by A107,TOPREAL1:18,ZFMISC_1:33; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A109: L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; L3 /\ LSeg(p11,p2) c= L3 /\ L2 by A66,Lm26,TOPREAL1:6,XBOOLE_1:26; then A110: L3 /\ LSeg(p11,p2) = {} by A109,XBOOLE_1:3; (L1 \/ L3 \/ L4) /\ LSeg(p11,p2) = (L1 \/ L3) /\ LSeg(p11,p2) \/ L4 /\ LSeg(p11,p2) by XBOOLE_1:23 .= (L1 /\ LSeg(p11,p2)) \/ (L3 /\ LSeg(p11,p2)) \/ {p11} by A108, XBOOLE_1:23 .= {p11} by A104,A110; then A111: L1 \/ L3 \/ L4 \/ LSeg(p11,p2) is_an_arc_of p01,p2 by A89,TOPREAL1:10; LSeg(p1,p01) /\ (L1 \/ L3 \/ L4 \/ LSeg(p11,p2)) = LSeg(p1,p01) /\ (L1 \/ L3 \/ L4) \/ (LSeg(p1,p01) /\ LSeg(p11,p2)) by XBOOLE_1:23 .= LSeg(p1,p01) /\ (L1 \/ L3) \/ (LSeg(p1,p01) /\ L4) by A83, XBOOLE_1:23 .= LSeg(p1,p01) /\ L1 \/ (LSeg(p1,p01) /\ L3) by A100,XBOOLE_1:23 .= {p01} by A82,A106,A101,TOPREAL1:15,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A111,TOPREAL1:11; thus P1 \/ P2 = LSeg(p01,p1) \/ LSeg(p1,p2) \/ (L1 \/ L3 \/ L4 \/ LSeg (p11,p2)) by XBOOLE_1:4 .= LSeg(p01,p1) \/ LSeg(p1,p2) \/ LSeg(p2,p11) \/ (L1 \/ L3 \/ L4) by XBOOLE_1:4 .= L2 \/ (L1 \/ L3 \/ L4) by A3,A66,TOPREAL1:7 .= L2 \/ (L1 \/ (L3 \/ L4)) by XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A112: p2 in LSeg(p11,p2) by RLTOPSP1:68; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p11,p2) by A112,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p11,p2) by ZFMISC_1:31; then A113: LSeg(p1,p2) /\ LSeg(p11,p2) = {p2} by A75,XBOOLE_0:def 10; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A114: LSeg(p1,p2) /\ L3 = {} by A68,XBOOLE_1:3,26; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p01) by A105,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p01) by ZFMISC_1:31; then A115: LSeg(p1,p2) /\ LSeg(p1,p01) = {p1} by A90,XBOOLE_0:def 10; A116: P1 /\ P2 = (LSeg(p1,p2) /\ LSeg(p1,p01)) \/ (LSeg(p1,p2) /\ (L1 \/ L3 \/ L4 \/ LSeg(p11,p2))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L3 \/ L4)) \/ {p2}) by A115,A113, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L3)) \/ (LSeg(p1,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ (LSeg(p1,p2) /\ L3) \/ (LSeg(p1 ,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2})) by A114,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by XBOOLE_1:4; A117: now per cases; suppose A118: p1 = p01; p1 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L1 <> {} by A118,Lm22,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L1 = {p1} by A97,A118,TOPREAL1:15,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A116; end; suppose A119: p1 <> p01; now assume p01 in LSeg(p1,p2) /\ L1; then p01 in LSeg(p1,p2) by XBOOLE_0:def 4; then p1`1 = 0 by A13,A15,A69,A74,Lm6,TOPREAL1:3; hence contradiction by A13,A16,A119,EUCLID:53; end; then {p01} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:31; then LSeg(p1,p2) /\ L1 = {} by A97,TOPREAL1:15,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A116; end; end; A120: LSeg(p1,p2) /\ L4 c= {p11} by A3,A66,TOPREAL1:6,18,XBOOLE_1:26; now per cases; suppose A121: p2 = p11; p2 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L4 <> {} by A121,Lm27,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L4 = {p2} by A120,A121,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A117,ENUMSET1:1; end; suppose A122: p2 <> p11; now assume p11 in LSeg(p1,p2) /\ L4; then p11 in LSeg(p1,p2) by XBOOLE_0:def 4; then p11`1 <= p2`1 by A13,A69,A74,TOPREAL1:3; then p2`1 = 1 by A69,A70,Lm10,XXREAL_0:1; hence contradiction by A69,A72,A122,EUCLID:53; end; then {p11} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:31; then LSeg(p1,p2) /\ L4 = {} by A120,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A117,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; suppose A123: q`1 < q1`1; A124: LSeg(p1,p2) /\ LSeg(p01,p2) c= {p2} proof let a be object; assume A125: a in LSeg(p1,p2) /\ LSeg(p01,p2); then reconsider p = a as Point of TOP-REAL 2; A126: p in LSeg(p01,p2) by A125,XBOOLE_0:def 4; p01`1 <= p2`1 by A69,A71,EUCLID:52; then A127: p`1 <= p2`1 by A126,TOPREAL1:3; A128: p in LSeg(p2,p1) by A125,XBOOLE_0:def 4; then A129: p`2 <= p1`2 by A13,A16,A69,A72,TOPREAL1:4; p2`1 <= p`1 by A13,A69,A123,A128,TOPREAL1:3; then A130: p2`1 = p`1 by A127,XXREAL_0:1; p2`2 <= p`2 by A13,A16,A69,A72,A128,TOPREAL1:4; then p`2 = 1 by A13,A16,A69,A72,A129,XXREAL_0:1; then p = |[ p2`1, 1]| by A130,EUCLID:53 .= p2 by A69,A72,EUCLID:53; hence thesis by TARSKI:def 1; end; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A131: LSeg(p1,p11) /\ L3 = {} by A8,XBOOLE_1:3,26; A132: now set a = the Element of LSeg(p1,p11) /\ LSeg(p01,p2); assume A133: LSeg(p1,p11) /\ LSeg(p01,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A134: p in LSeg(p1,p11) by A133,XBOOLE_0:def 4; A135: p in LSeg(p01,p2) by A133,XBOOLE_0:def 4; p01`1 <= p2 `1 by A69,A71,EUCLID:52; then A136: p`1 <= p2`1 by A135,TOPREAL1:3; p1`1 <= p11`1 by A13,A14,EUCLID:52; then p1`1 <= p`1 by A134,TOPREAL1:3; hence contradiction by A13,A69,A123,A136,XXREAL_0:2; end; A137: (L4 \/ L3) /\ L1 = L1 /\ L4 \/ L3 /\ L1 by XBOOLE_1:23 .= {p00} by Lm3,TOPREAL1:17; L4 \/ L3 is_an_arc_of p11,p00 by Lm9,Lm11,TOPREAL1:9,10,16; then A138: L4 \/ L3 \/ L1 is_an_arc_of p11,p01 by A137,TOPREAL1:10; now assume p11 in L4 /\ LSeg(p01,p2); then A139: p11 in LSeg(p01,p2) by XBOOLE_0:def 4; p01`1 <= p2`1 by A69,A71,EUCLID:52; then p11`1 <= p2`1 by A139,TOPREAL1:3; hence contradiction by A14,A69,A70,A123,Lm10,XXREAL_0:1; end; then A140: {p11} <> L4 /\ LSeg(p01,p2) by ZFMISC_1:31; L4 /\ LSeg(p01,p2) c= L4 /\ L2 by A66,Lm23,TOPREAL1:6,XBOOLE_1:26; then A141: L4 /\ LSeg(p01,p2) = {} by A140,TOPREAL1:18,ZFMISC_1:33; p11 in LSeg(p1,p11) by RLTOPSP1:68; then A142: LSeg(p1,p11) /\ L4 <> {} by Lm27,XBOOLE_0:def 4; now assume p01 in LSeg(p1,p11) /\ L1; then A143: p01 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`1 <= p11`1 by A13,A14,EUCLID:52; hence contradiction by A13,A71,A123,A143,Lm6,TOPREAL1:3; end; then A144: {p01} <> LSeg(p1,p11) /\ L1 by ZFMISC_1:31; LSeg(p1,p11) /\ L1 c= L2 /\ L1 by A3,Lm26,TOPREAL1:6,XBOOLE_1:26; then A145: LSeg(p1,p11) /\ L1 = {} by A144,TOPREAL1:15,ZFMISC_1:33; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A146: LSeg(p1,p2) /\ L3 = {} by A68,XBOOLE_1:3,26; A147: LSeg(p1,p2) /\ LSeg(p1,p11) c= {p1} proof let a be object; assume A148: a in LSeg(p1,p2) /\ LSeg(p1,p11); then reconsider p = a as Point of TOP-REAL 2; A149: p in LSeg(p1,p11) by A148,XBOOLE_0:def 4; p1`1 <= p11`1 by A13,A14,EUCLID:52; then A150: p1`1 <= p`1 by A149,TOPREAL1:3; A151: p in LSeg(p2,p1) by A148,XBOOLE_0:def 4; then A152: p`2 <= p1`2 by A13,A16,A69,A72,TOPREAL1:4; p`1 <= p1`1 by A13,A69,A123,A151,TOPREAL1:3; then A153: p1`1 = p`1 by A150,XXREAL_0:1; p2`2 <= p`2 by A13,A16,A69,A72,A151,TOPREAL1:4; then p`2 = 1 by A13,A16,A69,A72,A152,XXREAL_0:1; then p = |[p1`1, 1]| by A153,EUCLID:53 .= p1 by A13,A16,EUCLID:53; hence thesis by TARSKI:def 1; end; A154: LSeg(p1,p11) /\ L4 c= {p11} by A3,Lm26,TOPREAL1:6,18,XBOOLE_1:26; p01 in LSeg(p01,p2) by RLTOPSP1:68; then A155: L1 /\ LSeg(p01,p2) <> {} by Lm22,XBOOLE_0:def 4; L1 /\ LSeg(p01,p2) c= {p01} by A66,Lm23,TOPREAL1:6,15,XBOOLE_1:26; then A156: L1 /\ LSeg(p01,p2) = {p01} by A155,ZFMISC_1:33; take P1 = LSeg(p1,p2),P2 = LSeg(p1,p11) \/ (L4 \/ L3 \/ L1 \/ LSeg(p01 ,p2)); A157: p1 in LSeg(p1,p11) by RLTOPSP1:68; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A158: L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; L3 /\ LSeg(p01,p2) c= L3 /\ L2 by A66,Lm23,TOPREAL1:6,XBOOLE_1:26; then A159: L3 /\ LSeg(p01,p2) = {} by A158,XBOOLE_1:3; (L4 \/ L3 \/ L1) /\ LSeg(p01,p2) = (L4 \/ L3) /\ LSeg(p01,p2) \/ L1 /\ LSeg(p01,p2) by XBOOLE_1:23 .= (L4 /\ LSeg(p01,p2)) \/ (L3 /\ LSeg(p01,p2)) \/ {p01} by A156, XBOOLE_1:23 .= {p01} by A141,A159; then A160: L4 \/ L3 \/ L1 \/ LSeg(p01,p2) is_an_arc_of p11,p2 by A138,TOPREAL1:10; LSeg(p1,p11) /\ (L4 \/ L3 \/ L1 \/ LSeg(p01,p2)) = LSeg(p1,p11) /\ (L4 \/ L3 \/ L1) \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) by XBOOLE_1:23 .= LSeg(p1,p11) /\ (L4 \/ L3) \/ (LSeg(p1,p11) /\ L1) by A132, XBOOLE_1:23 .= LSeg(p1,p11) /\ L4 \/ (LSeg(p1,p11) /\ L3) by A145,XBOOLE_1:23 .= {p11} by A131,A154,A142,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A160,TOPREAL1:11; thus P1 \/ P2 = LSeg(p2,p1) \/ LSeg(p1,p11) \/ (L4 \/ L3 \/ L1 \/ LSeg (p01,p2)) by XBOOLE_1:4 .= LSeg(p01,p2) \/ (LSeg(p2,p1) \/ LSeg(p1,p11)) \/ (L4 \/ L3 \/ L1) by XBOOLE_1:4 .= L2 \/ (L4 \/ L3 \/ L1) by A3,A66,TOPREAL1:7 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A161: p2 in LSeg(p01,p2) by RLTOPSP1:68; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p01,p2) by A161,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p01,p2) by ZFMISC_1:31; then A162: LSeg(p1,p2) /\ LSeg(p01,p2) = {p2} by A124,XBOOLE_0:def 10; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p11) by A157,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p11) by ZFMISC_1:31; then LSeg(p1,p2) /\ LSeg(p1,p11) = {p1} by A147,XBOOLE_0:def 10; then A163: P1 /\ P2 = {p1} \/ LSeg(p1,p2) /\ (L4 \/ L3 \/ L1 \/ LSeg(p01,p2 )) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L3 \/ L1)) \/ {p2}) by A162, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L3)) \/ (LSeg(p1,p2) /\ L1) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ (LSeg(p1,p2) /\ L3) \/ (LSeg(p1 ,p2) /\ L1) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/ {p2})) by A146,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/ {p2}) by XBOOLE_1:4; A164: LSeg(p1,p2) /\ L1 c= L2 /\ L1 by A3,A66,TOPREAL1:6,XBOOLE_1:26; A165: now per cases; suppose A166: p2 = p01; p2 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L1 <> {} by A166,Lm22,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L1 = {p2} by A164,A166,TOPREAL1:15,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A163; end; suppose A167: p2 <> p01; now assume p01 in LSeg(p1,p2) /\ L1; then p01 in LSeg(p2,p1) by XBOOLE_0:def 4; then p2`1 = 0 by A13,A69,A71,A123,Lm6,TOPREAL1:3; hence contradiction by A69,A72,A167,EUCLID:53; end; then {p01} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:31; then LSeg(p1,p2) /\ L1 = {} by A164,TOPREAL1:15,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A163; end; end; A168: LSeg(p1,p2) /\ L4 c= {p11} by A3,A66,TOPREAL1:6,18,XBOOLE_1:26; now per cases; suppose A169: p1 = p11; p1 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L4 <> {} by A169,Lm27,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L4 = {p1} by A168,A169,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A165,ENUMSET1:1; end; suppose A170: p1 <> p11; now assume p11 in LSeg(p1,p2) /\ L4; then p11 in LSeg(p2,p1) by XBOOLE_0:def 4; then p11`1 <= p1`1 by A13,A69,A123,TOPREAL1:3; then p1`1 = 1 by A13,A14,Lm10,XXREAL_0:1; hence contradiction by A13,A16,A170,EUCLID:53; end; then {p11} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:31; then LSeg(p1,p2) /\ L4 = {} by A168,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A165,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; end; hence thesis; end; suppose A171: p2 in L3; p00 in LSeg(p00,p2) by RLTOPSP1:68; then A172: LSeg(p01,p00) /\ LSeg(p00,p2) <> {} by Lm20,XBOOLE_0:def 4; LSeg(p00,p2) c= L3 by A171,Lm21,TOPREAL1:6; then LSeg(p01,p00) /\ LSeg(p00,p2) c= {p00} by TOPREAL1:17,XBOOLE_1:27; then LSeg(p01,p00) /\ LSeg(p00,p2) = {p00} by A172,ZFMISC_1:33; then A173: L1 \/ LSeg(p00,p2) is_an_arc_of p01,p2 by Lm5,Lm7,TOPREAL1:12; LSeg(p2,p00) c= L3 by A171,Lm21,TOPREAL1:6; then A174: LSeg(p1,p01) /\ LSeg(p00,p2) = {} by A11,Lm2,XBOOLE_1:3,27; p10 in LSeg(p10,p2) by RLTOPSP1:68; then A175: p10 in LSeg(p11,p10) /\ LSeg(p10,p2) by Lm25,XBOOLE_0:def 4; LSeg(p10,p2) c= L3 by A171,Lm24,TOPREAL1:6; then LSeg(p11,p10) /\ LSeg(p10,p2) c= L4 /\ L3 by XBOOLE_1:27; then LSeg(p11,p10) /\ LSeg(p10,p2) = {p10} by A175,TOPREAL1:16,ZFMISC_1:33; then A176: L4 \/ LSeg(p10,p2) is_an_arc_of p11,p2 by Lm9,Lm11,TOPREAL1:12; take P1 = LSeg(p1,p11) \/ L4 \/ LSeg(p10,p2),P2 = LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2); A177: LSeg(p1,p11) \/ LSeg(p1,p01) = L2 by A3,TOPREAL1:5; A178: LSeg(p2,p10) c= L3 by A171,Lm24,TOPREAL1:6; then A179: LSeg(p1,p11) /\ LSeg(p10,p2) = {} by A8,Lm2,XBOOLE_1:3,27; A180: L2 /\ LSeg(p00,p2) c= L2 /\ L3 by A171,Lm21,TOPREAL1:6,XBOOLE_1:26; LSeg(p1,p11) /\ LSeg(p00,p2) c= L2 /\ LSeg(p00,p2) by A3,Lm26,TOPREAL1:6 ,XBOOLE_1:26; then A181: LSeg(p1,p11) /\ LSeg(p00,p2) = {} by A180,Lm2,XBOOLE_1:1,3; A182: LSeg(p10,p2) /\ LSeg(p1,p01) = {} by A11,A178,Lm2,XBOOLE_1:3,27; A183: LSeg(p10,p2) /\ LSeg(p00,p2) = {p2} by A171,TOPREAL1:8; LSeg(p1,p11) /\ (L4 \/ LSeg(p10,p2)) = (LSeg(p1,p11) /\ L4) \/ (LSeg (p1,p11) /\ LSeg(p10,p2)) by XBOOLE_1:23 .= {p11} by A6,A10,A179,ZFMISC_1:33; then LSeg(p1,p11) \/ (L4 \/ LSeg(p10,p2)) is_an_arc_of p1,p2 by A176,TOPREAL1:11; hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4; A184: ex q2 st q2 = p2 & q2`1 <= 1 & q2`1 >= 0 & q2`2 = 0 by A171,TOPREAL1:13; LSeg(p1,p01) /\ (L1 \/ LSeg(p00,p2)) = (LSeg(p01,p1) /\ L1) \/ (LSeg (p1,p01) /\ LSeg(p00,p2)) by XBOOLE_1:23 .= {p01} by A7,A5,A174,TOPREAL1:15,ZFMISC_1:33; then LSeg(p1,p01) \/ (L1 \/ LSeg(p00,p2)) is_an_arc_of p1,p2 by A173,TOPREAL1:11; hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4; LSeg(p10,p2) \/ LSeg(p00,p2) = L3 by A171,TOPREAL1:5; hence R^2-unit_square = L2 \/ (L4 \/ (LSeg(p10,p2) \/ LSeg(p00,p2)) \/ L1) by TOPREAL1:def 2,XBOOLE_1:4 .= L2 \/ (L4 \/ LSeg(p10,p2) \/ LSeg(p00,p2) \/ L1) by XBOOLE_1:4 .= L2 \/ (L4 \/ LSeg(p10,p2) \/ (L1 \/ LSeg(p00,p2))) by XBOOLE_1:4 .= LSeg(p1,p11) \/ ((L4 \/ LSeg(p10,p2) \/ (L1 \/ LSeg(p00,p2))) \/ LSeg(p1,p01)) by A177,XBOOLE_1:4 .= LSeg(p1,p11) \/ (L4 \/ LSeg(p10,p2) \/ (L1 \/ LSeg(p00,p2) \/ LSeg( p1,p01))) by XBOOLE_1:4 .= LSeg(p1,p11) \/ (L4 \/ LSeg(p10,p2)) \/ (L1 \/ LSeg(p00,p2) \/ LSeg (p1,p01)) by XBOOLE_1:4 .= LSeg(p1,p11) \/ L4 \/ LSeg(p10,p2) \/ (LSeg(p1,p01) \/ (L1 \/ LSeg( p00,p2))) by XBOOLE_1:4 .= P1 \/ P2 by XBOOLE_1:4; A185: LSeg(p1,p11) /\ LSeg(p1,p01) = {p1} by A3,TOPREAL1:8; A186: P1 /\ P2 = (LSeg(p1,p11) \/ L4) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00, p2)) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) \/ (L4 /\ ( LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ (LSeg(p1,p01) \/ L1)) \/ (LSeg(p1,p11) /\ LSeg(p00 ,p2)) \/ (L4 /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg (p1,p01) \/ L1 \/ LSeg(p00,p2))) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ LSeg(p1,p01)) \/ (LSeg(p1,p11) /\ L1) \/ (L4 /\ ( LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by A181,XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((L4 /\ (LSeg(p1,p01) \/ L1)) \/ ( L4 /\ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2))) by A185,XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((L4 /\ LSeg(p1,p01)) \/ (L1 /\ L4) \/ (L4 /\ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1 \/ LSeg(p00,p2 ))) by XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((L4 /\ LSeg(p1,p01)) \/ (L4 /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ (LSeg(p1,p01) \/ L1)) \/ {p2}) by A183,Lm3, XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((L4 /\ LSeg(p1,p01)) \/ (L4 /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p01)) \/ (LSeg(p10,p2) /\ L1) \/ { p2}) by XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((L4 /\ LSeg(p1,p01)) \/ (L4 /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A182; A187: now per cases; suppose A188: p1 = p01; then L4 /\ LSeg(p1,p01) = L4 /\ {p01} by RLTOPSP1:70 .= {} by Lm1,Lm15; hence P1 /\ P2 = {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A186,A188,TOPREAL1:15; end; suppose A189: p1 = p11; then LSeg(p1,p11) /\ L1 = {p11} /\ L1 by RLTOPSP1:70 .= {} by Lm1,Lm18; hence P1 /\ P2 = {p1} \/ {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A186,A189,TOPREAL1:18,XBOOLE_1:4 .= {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}); end; suppose A190: p1 <> p11 & p1 <> p01; now assume p11 in L4 /\ LSeg(p1,p01); then A191: p11 in LSeg(p01,p1) by XBOOLE_0:def 4; p01`1 <= p1`1 by A13,A15,EUCLID:52; then 1 <= p1`1 by A191,Lm10,TOPREAL1:3; then p1`1 = 1 by A13,A14,XXREAL_0:1; hence contradiction by A13,A16,A190,EUCLID:53; end; then A192: {p11} <> L4 /\ LSeg(p1,p01) by ZFMISC_1:31; L4 /\ LSeg(p1,p01) c= L4 /\ L2 by A3,Lm23,TOPREAL1:6,XBOOLE_1:26; then A193: L4 /\ LSeg(p1,p01) = {} by A192,TOPREAL1:18,ZFMISC_1:33; now assume p01 in LSeg(p1,p11) /\ L1; then A194: p01 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`1 <= p11`1 by A13,A14,EUCLID:52; then p1`1 = 0 by A13,A15,A194,Lm6,TOPREAL1:3; hence contradiction by A13,A16,A190,EUCLID:53; end; then A195: {p01} <> LSeg(p1,p11) /\ L1 by ZFMISC_1:31; LSeg(p1,p11) /\ L1 c= L2 /\ L1 by A3,Lm26,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p11) /\ L1 = {} by A195,TOPREAL1:15,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (L4 /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}) by A186,A193; end; end; now per cases; suppose A196: p2 = p00; then L4 /\ LSeg(p00,p2) = L4 /\ {p00} by RLTOPSP1:70 .= {} by Lm1,Lm12; hence thesis by A187,A196,ENUMSET1:1,TOPREAL1:17; end; suppose A197: p2 = p10; then LSeg(p10,p2) /\ L1 = {p10} /\ L1 by RLTOPSP1:70 .= {} by Lm1,Lm16; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A187,A197,TOPREAL1:16 ,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A198: p2 <> p10 & p2 <> p00; now assume p00 in LSeg(p10,p2) /\ L1; then A199: p00 in LSeg(p2,p10) by XBOOLE_0:def 4; p2`1 <= p10`1 by A184,EUCLID:52; then p2`1 = 0 by A184,A199,Lm4,TOPREAL1:3; hence contradiction by A184,A198,EUCLID:53; end; then A200: {p00} <> LSeg(p10,p2) /\ L1 by ZFMISC_1:31; LSeg(p10,p2) /\ L1 c= L3 /\ L1 by A171,Lm24,TOPREAL1:6,XBOOLE_1:26; then A201: LSeg(p10,p2) /\ L1 = {} by A200,TOPREAL1:17,ZFMISC_1:33; now assume p10 in L4 /\ LSeg(p00,p2); then A202: p10 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`1 <= p2`1 by A184,EUCLID:52; then 1 <= p2`1 by A202,Lm8,TOPREAL1:3; then p2`1 = 1 by A184,XXREAL_0:1; hence contradiction by A184,A198,EUCLID:53; end; then A203: {p10} <> L4 /\ LSeg(p00,p2) by ZFMISC_1:31; L4 /\ LSeg(p00,p2) c= L4 /\ L3 by A171,Lm21,TOPREAL1:6,XBOOLE_1:26; then L4 /\ LSeg(p00,p2) = {} by A203,TOPREAL1:16,ZFMISC_1:33; hence thesis by A187,A201,ENUMSET1:1; end; end; hence thesis; end; suppose A204: p2 in L4; then A205: LSeg(p11,p2) c= L4 by Lm27,TOPREAL1:6; LSeg(p1,p11) c= L2 by A3,Lm26,TOPREAL1:6; then A206: LSeg(p1,p11) /\ LSeg(p11,p2) c= L2 /\ L4 by A205,XBOOLE_1:27; take P1 = LSeg(p1,p11) \/ LSeg(p11,p2),P2 = LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2)); A207: p11 in LSeg(p11,p2) by RLTOPSP1:68; p10 in LSeg(p10,p2) by RLTOPSP1:68; then A208: L3 /\ LSeg(p10,p2) <> {} by Lm24,XBOOLE_0:def 4; p11 in LSeg(p1,p11) by RLTOPSP1:68; then LSeg(p1,p11) /\ LSeg(p11,p2) <> {} by A207,XBOOLE_0:def 4; then A209: LSeg(p1,p11) /\ LSeg(p11,p2) = {p11} by A206,TOPREAL1:18,ZFMISC_1:33; p1 <> p11 or p11 <> p2 by A1; hence P1 is_an_arc_of p1,p2 by A209,TOPREAL1:12; A210: L2 = LSeg(p1,p11) \/ LSeg(p1,p01) by A3,TOPREAL1:5; A211: L3 is_an_arc_of p00,p10 by Lm4,Lm8,TOPREAL1:9; L1 is_an_arc_of p01,p00 by Lm5,Lm7,TOPREAL1:9; then A212: L1 \/ L3 is_an_arc_of p01,p10 by A211,TOPREAL1:2,17; A213: LSeg(p11,p2) /\ LSeg(p10,p2) = {p2} by A204,TOPREAL1:8; A214: L4 = LSeg(p10,p2) \/ LSeg(p11,p2) by A204,TOPREAL1:5; A215: LSeg(p10,p2) c= L4 by A204,Lm25,TOPREAL1:6; then A216: L3 /\ LSeg(p10,p2) c= {p10} by TOPREAL1:16,XBOOLE_1:27; A217: ex q st q = p2 & q`1 = 1 & q`2 <= 1 & q`2 >= 0 by A204,TOPREAL1:13; now A218: p10`2 <= p2`2 by A217,EUCLID:52; assume A219: p11 in LSeg(p1,p01) /\ LSeg(p10,p2); then A220: p11 in LSeg(p01,p1) by XBOOLE_0:def 4; p11 in LSeg(p10,p2) by A219,XBOOLE_0:def 4; then p11`2 <= p2`2 by A218,TOPREAL1:4; then A221: p11`2 = p2`2 by A217,Lm11,XXREAL_0:1; p01`1 <= p1`1 by A13,A15,EUCLID:52; then p11`1 <= p1`1 by A220,TOPREAL1:3; then p11`1 = p1`1 by A13,A14,Lm10,XXREAL_0:1; then p1 = |[p11`1,p11`2]| by A13,A16,Lm11,EUCLID:53 .= p2 by A217,A221,Lm10,EUCLID:53; hence contradiction by A1; end; then A222: {p11} <> LSeg(p1,p01) /\ LSeg(p10,p2) by ZFMISC_1:31; A223: L1 /\ LSeg(p10,p2) = {} by A215,Lm3,XBOOLE_1:3,26; (L1 \/ L3) /\ LSeg(p10,p2) = (L1 /\ LSeg(p10,p2)) \/ (L3 /\ LSeg(p10 ,p2)) by XBOOLE_1:23 .= {p10} by A223,A216,A208,ZFMISC_1:33; then A224: L1 \/ L3 \/ LSeg(p10,p2) is_an_arc_of p01,p2 by A212,TOPREAL1:10; A225: LSeg(p2,p11) c= L4 by A204,Lm27,TOPREAL1:6; then A226: LSeg(p11,p2) /\ L1 = {} by Lm3,XBOOLE_1:3,27; LSeg(p1,p01) /\ LSeg(p10,p2) c= {p11} by A11,A215,TOPREAL1:18,XBOOLE_1:27; then A227: LSeg(p1,p01) /\ LSeg(p10,p2) = {} by A222,ZFMISC_1:33; LSeg(p1,p01) /\ (L1 \/ L3 \/ LSeg(p10,p2)) = (LSeg(p1,p01) /\ (L1 \/ L3)) \/ (LSeg(p1,p01) /\ LSeg(p10,p2)) by XBOOLE_1:23 .= (LSeg(p1,p01) /\ L1) \/ (LSeg(p1,p01) /\ L3) by A227,XBOOLE_1:23 .= {p01} by A12,A7,A5,TOPREAL1:15,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A224,TOPREAL1:11; thus P1 \/ P2 = LSeg(p11,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2)))) by XBOOLE_1:4 .= L2 \/ (L1 \/ L3 \/ LSeg(p10,p2)) \/ LSeg(p11,p2) by A210,XBOOLE_1:4 .= L2 \/ ((L1 \/ L3 \/ LSeg(p10,p2)) \/ LSeg(p11,p2)) by XBOOLE_1:4 .= L2 \/ (L1 \/ L3 \/ (LSeg(p10,p2) \/ LSeg(p11,p2))) by XBOOLE_1:4 .= L2 \/ (L1 \/ (L3 \/ L4)) by A214,XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A228: {p1} = LSeg(p1,p11) /\ LSeg(p1,p01) by A3,TOPREAL1:8; A229: P1 /\ P2 = (LSeg(p1,p11) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10, p2)))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ LSeg(p1,p01)) \/ (LSeg(p1,p11) /\ (L1 \/ L3 \/ LSeg(p10,p2))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ (L1 \/ L3)) \/ (LSeg(p1,p11) /\ LSeg(p10, p2))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg(p10,p2)))) by A228 ,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3) \/ (LSeg(p1, p11) /\ LSeg(p10,p2))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p01) \/ (L1 \/ L3 \/ LSeg( p10,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3) \/ (LSeg(p1, p11) /\ LSeg(p10,p2))) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/ (LSeg(p11,p2) /\ ( L1 \/ L3 \/ LSeg(p10,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3) \/ (LSeg(p1, p11) /\ LSeg(p10,p2))) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/ ((LSeg(p11,p2) /\ (L1 \/ L3)) \/ {p2})) by A213,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ L3) \/ (LSeg(p1, p11) /\ LSeg(p10,p2))) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/ ((LSeg(p11,p2) /\ L1) \/ (LSeg(p11,p2) /\ L3) \/ {p2})) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L1) \/ (LSeg(p1,p11) /\ LSeg(p10,p2))) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/ (LSeg(p11,p2) /\ L3 \/ {p2})) by A9,A226; A230: now per cases; suppose A231: p2 = p10; then A232: not p2 in LSeg(p1,p11) by A8,Lm7,Lm9,Lm11,TOPREAL1:4; LSeg(p1,p11) /\ LSeg(p10,p2) = LSeg(p1,p11) /\ {p2} by A231,RLTOPSP1:70 .= {} by A232,Lm1; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((LSeg(p11,p2) /\ LSeg (p1,p01)) \/ {p2}) by A229,A231,TOPREAL1:16; end; suppose A233: p2 = p11; then p2 in LSeg(p1,p11) by RLTOPSP1:68; then A234: LSeg(p1,p11) /\ LSeg(p10,p2) <> {} by A233,Lm27,XBOOLE_0:def 4; LSeg(p1,p11) /\ LSeg(p10,p2) c= {p2} by A8,A233,TOPREAL1:18,XBOOLE_1:27 ; then A235: LSeg(p1,p11) /\ LSeg(p10,p2) = {p2} by A234,ZFMISC_1:33; LSeg(p11,p2) /\ L3 = {p11} /\ L3 by A233,RLTOPSP1:70 .= {} by Lm1,Lm19; hence P1 /\ P2 = ({p1} \/ (LSeg(p1,p11) /\ L1)) \/ {p2} \/ ((LSeg(p11, p2) /\ LSeg(p1,p01)) \/ {p2}) by A229,A235,XBOOLE_1:4 .= ({p1} \/ (LSeg(p1,p11) /\ L1)) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01 )) \/ {p2} \/ {p2}) by XBOOLE_1:4 .= ({p1} \/ (LSeg(p1,p11) /\ L1)) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01 )) \/ ({p2} \/ {p2})) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((LSeg(p11,p2) /\ LSeg(p1,p01)) \/ {p2}); end; suppose A236: p2 <> p11 & p2 <> p10; now assume p11 in LSeg(p1,p11) /\ LSeg(p10,p2); then A237: p11 in LSeg(p10,p2) by XBOOLE_0:def 4; p10`2 <= p2`2 by A217,EUCLID:52; then p11`2 <= p2`2 by A237,TOPREAL1:4; then p2`2 = 1 by A217,Lm11,XXREAL_0:1; hence contradiction by A217,A236,EUCLID:53; end; then A238: {p11} <> LSeg(p1,p11) /\ LSeg(p10,p2) by ZFMISC_1:31; LSeg(p1,p11) /\ LSeg(p10,p2) c= {p11} by A8,A215,TOPREAL1:18 ,XBOOLE_1:27; then A239: LSeg(p1,p11) /\ LSeg(p10,p2) = {} by A238,ZFMISC_1:33; now assume p10 in LSeg(p11,p2) /\ L3; then A240: p10 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`2 <= p11`2 by A217,EUCLID:52; then p2`2 = 0 by A217,A240,Lm9,TOPREAL1:4; hence contradiction by A217,A236,EUCLID:53; end; then A241: {p10} <> LSeg(p11,p2) /\ L3 by ZFMISC_1:31; LSeg(p11,p2) /\ L3 c= L4 /\ L3 by A225,XBOOLE_1:27; then LSeg(p11,p2) /\ L3 = {} by A241,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p11) /\ L1) \/ ((LSeg(p11,p2) /\ LSeg (p1,p01)) \/ {p2}) by A229,A239; end; end; now per cases; suppose A242: p1 = p01; then A243: LSeg(p11,p2) /\ LSeg(p1,p01) = LSeg(p11,p2) /\ {p1} by RLTOPSP1:70; p1 in LSeg(p11,p2) implies contradiction by A225,A242,Lm6,Lm8,Lm10, TOPREAL1:3; then LSeg(p11,p2) /\ LSeg(p1,p01) = {} by A243,Lm1; hence thesis by A230,A242,ENUMSET1:1,TOPREAL1:15; end; suppose A244: p1 = p11; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A245: LSeg(p11,p2) /\ LSeg(p1,p01) <> {} by A244,Lm26,XBOOLE_0:def 4; LSeg(p1,p11) /\ L1 = {p1} /\ L1 by A244,RLTOPSP1:70; then A246: LSeg(p1,p11) /\ L1 = {} by A244,Lm1,Lm18; LSeg(p11,p2) /\ LSeg(p1,p01) c= L4 /\ L2 by A11,A225,XBOOLE_1:27; then LSeg(p11,p2) /\ LSeg(p1,p01) = {p1} by A244,A245,TOPREAL1:18 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ {p1} \/ {p2} by A230,A246,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A247: p1 <> p11 & p1 <> p01; now assume p11 in LSeg(p11,p2) /\ LSeg(p1,p01); then A248: p11 in LSeg(p01,p1) by XBOOLE_0:def 4; p01`1 <= p1`1 by A13,A15,EUCLID:52; then p11`1 <= p1`1 by A248,TOPREAL1:3; then p1`1 = 1 by A13,A14,Lm10,XXREAL_0:1; hence contradiction by A13,A16,A247,EUCLID:53; end; then A249: {p11} <> LSeg(p11,p2) /\ LSeg(p1,p01) by ZFMISC_1:31; LSeg(p11,p2) /\ LSeg(p1,p01) c= L4 /\ L2 by A11,A225,XBOOLE_1:27; then A250: LSeg(p11,p2) /\ LSeg(p1,p01) = {} by A249,TOPREAL1:18,ZFMISC_1:33; now assume p01 in LSeg(p1,p11) /\ L1; then A251: p01 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`1 <= p11`1 by A13,A14,EUCLID:52; then p1`1 = 0 by A13,A15,A251,Lm6,TOPREAL1:3; hence contradiction by A13,A16,A247,EUCLID:53; end; then A252: {p01} <> LSeg(p1,p11) /\ L1 by ZFMISC_1:31; LSeg(p1,p11) /\ L1 c= L2 /\ L1 by A8,XBOOLE_1:27; then LSeg(p1,p11) /\ L1 = {} by A252,TOPREAL1:15,ZFMISC_1:33; hence thesis by A230,A250,ENUMSET1:1; end; end; hence thesis; end; end; Lm32: p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg(p00, p10) implies ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof assume that A1: p1 <> p2 and A2: p2 in R^2-unit_square and A3: p1 in LSeg(p00, p10); A4: p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:def 2,XBOOLE_0:def 3; A5: LSeg(p10,p1) /\ L4 c= L3 /\ L4 by A3,Lm24,TOPREAL1:6,XBOOLE_1:26; p00 in LSeg(p1,p00) by RLTOPSP1:68; then A6: p00 in LSeg(p1,p00) /\ L1 by Lm20,XBOOLE_0:def 4; p10 in LSeg(p10,p1) by RLTOPSP1:68; then A7: LSeg(p10,p1) /\ L4 <> {} by Lm25,XBOOLE_0:def 4; A8: LSeg(p1,p00) /\ L1 c= L3 /\ L1 by A3,Lm21,TOPREAL1:6,XBOOLE_1:26; A9: LSeg(p1,p00) /\ LSeg(p1,p10) = {p1} by A3,TOPREAL1:8; A10: LSeg(p00,p1) c= L3 by A3,Lm21,TOPREAL1:6; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A11: LSeg(p1,p00) /\ L2 = {} by A10,XBOOLE_1:3,26; A12: LSeg(p10,p1) c= L3 by A3,Lm24,TOPREAL1:6; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A13: LSeg(p10,p1) /\ L2 = {} by A12,XBOOLE_1:3,26; consider p such that A14: p = p1 and A15: p`1 <= 1 and A16: p`1 >= 0 and A17: p`2 = 0 by A3,TOPREAL1:13; per cases by A4,XBOOLE_0:def 3; suppose A18: p2 in L1; A19: L2 is_an_arc_of p11,p01 by Lm6,Lm10,TOPREAL1:9; L4 is_an_arc_of p10,p11 by Lm9,Lm11,TOPREAL1:9; then A20: L4 \/ L2 is_an_arc_of p10,p01 by A19,TOPREAL1:2,18; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A21: LSeg(p1,p00) /\ L2 = {} by A10,XBOOLE_1:3,26; take P1 = LSeg(p1,p00) \/ LSeg(p00,p2),P2 = LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2)); A22: LSeg(p1,p00) \/ LSeg(p1,p10) = L3 by A3,TOPREAL1:5; p01 in LSeg(p01,p2) by RLTOPSP1:68; then A23: p01 in L2 /\ LSeg(p01,p2) by Lm23,XBOOLE_0:def 4; A24: p00 in LSeg(p00,p2) by RLTOPSP1:68; p00 in LSeg(p1,p00) by RLTOPSP1:68; then A25: p00 in LSeg(p1,p00) /\ LSeg(p00,p2) by A24,XBOOLE_0:def 4; A26: LSeg(p00,p2) c= L1 by A18,Lm20,TOPREAL1:6; then LSeg(p1,p00) /\ LSeg(p00,p2) c= L3 /\ L1 by A10,XBOOLE_1:27; then A27: LSeg(p1,p00) /\ LSeg(p00,p2) = {p00} by A25,TOPREAL1:17,ZFMISC_1:33; A28: ex q st q = p2 & q`1 = 0 & q`2 <= 1 & q`2 >= 0 by A18,TOPREAL1:13; now A29: p2`2 <= p01`2 by A28,EUCLID:52; assume A30: p00 in LSeg(p1,p10) /\ LSeg(p01,p2); then A31: p00 in LSeg(p1,p10) by XBOOLE_0:def 4; p00 in LSeg(p2,p01) by A30,XBOOLE_0:def 4; then A32: 0 = p2`2 by A28,A29,Lm5,TOPREAL1:4; p1`1 <= p10`1 by A14,A15,EUCLID:52; then 0 = p1`1 by A14,A16,A31,Lm4,TOPREAL1:3; then p1 = p00 by A14,A17,EUCLID:53 .= p2 by A28,A32,EUCLID:53; hence contradiction by A1; end; then A33: {p00} <> LSeg(p1,p10) /\ LSeg(p01,p2) by ZFMISC_1:31; p1 <> p00 or p00 <> p2 by A1; hence P1 is_an_arc_of p1,p2 by A27,TOPREAL1:12; A34: {p1} = LSeg(p1,p00) /\ LSeg(p1,p10) by A3,TOPREAL1:8; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A35: LSeg(p00,p2) /\ L4 = {} by A26,XBOOLE_1:3,26; A36: LSeg(p2,p01) c= L1 by A18,Lm22,TOPREAL1:6; then A37: L2 /\ LSeg(p01,p2) c= L2 /\ L1 by XBOOLE_1:27; A38: L4 /\ LSeg(p01,p2) = {} by A36,Lm3,XBOOLE_1:3,26; (L4 \/ L2) /\ LSeg(p01,p2) = (L4 /\ LSeg(p01,p2)) \/ (L2 /\ LSeg(p01, p2)) by XBOOLE_1:23 .= {p01} by A38,A37,A23,TOPREAL1:15,ZFMISC_1:33; then A39: L4 \/ L2 \/ LSeg(p01,p2) is_an_arc_of p10,p2 by A20,TOPREAL1:10; A40: {p2} = LSeg(p00,p2) /\ LSeg(p01,p2) by A18,TOPREAL1:8; A41: LSeg(p01,p2) \/ LSeg(p00,p2) = L1 by A18,TOPREAL1:5; LSeg(p1,p10) /\ LSeg(p01,p2) c= L3 /\ L1 by A12,A36,XBOOLE_1:27; then A42: LSeg(p1,p10) /\ LSeg(p01,p2) = {} by A33,TOPREAL1:17,ZFMISC_1:33; LSeg(p1,p10) /\ (L4 \/ L2 \/ LSeg(p01,p2)) = (LSeg(p1,p10) /\ (L4 \/ L2)) \/ (LSeg(p1,p10) /\ LSeg(p01,p2)) by XBOOLE_1:23 .= (LSeg(p1,p10) /\ L4) \/ (LSeg(p10,p1) /\ L2) by A42,XBOOLE_1:23 .= {p10} by A13,A5,A7,TOPREAL1:16,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A39,TOPREAL1:11; thus P1 \/ P2 = LSeg(p00,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2)))) by XBOOLE_1:4 .= LSeg(p00,p2) \/ (L3 \/ (L4 \/ L2 \/ LSeg(p01,p2))) by A22,XBOOLE_1:4 .= LSeg(p00,p2) \/ (L3 \/ (L4 \/ L2) \/ LSeg(p01,p2)) by XBOOLE_1:4 .= LSeg(p00,p2) \/ (L3 \/ L4 \/ L2 \/ LSeg(p01,p2)) by XBOOLE_1:4 .= LSeg(p00,p2) \/ (L3 \/ L4 \/ (L2 \/ LSeg(p01,p2))) by XBOOLE_1:4 .= (L2 \/ LSeg(p01,p2) \/ LSeg(p00,p2)) \/ (L3 \/ L4) by XBOOLE_1:4 .= R^2-unit_square by A41,TOPREAL1:def 2,XBOOLE_1:4; A43: P1 /\ P2 = (LSeg(p1,p00) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2 )))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ LSeg(p1,p10)) \/ (LSeg(p1,p00) /\ (L4 \/ L2 \/ LSeg(p01,p2))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ (L4 \/ L2)) \/ (LSeg(p1,p00) /\ LSeg(p01, p2))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg(p01,p2)))) by A34, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ L2) \/ (LSeg(p1, p00) /\ LSeg(p01,p2))) \/ (LSeg(p00,p2) /\ (LSeg(p1,p10) \/ (L4 \/ L2 \/ LSeg( p01,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2))) \/ ((LSeg(p00,p2) /\ LSeg(p1,p10)) \/ (LSeg(p00,p2) /\ (L4 \/ L2 \/ LSeg(p01,p2))) ) by A21,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2))) \/ ((LSeg(p00,p2) /\ (LSeg(p1,p10)) \/ ((LSeg(p00,p2) /\ (L4 \/ L2)) \/ {p2}))) by A40,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2))) \/ ((LSeg(p00,p2) /\ (LSeg(p1,p10)) \/ (((LSeg(p00,p2) /\ L4) \/ (LSeg(p00,p2) /\ L2)) \/ {p2}))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p00) /\ L4) \/ (LSeg(p1,p00) /\ LSeg(p01,p2))) \/ ((LSeg(p00,p2) /\ LSeg(p1,p10)) \/ ((LSeg(p00,p2) /\ L2) \/ {p2})) by A35; A44: now per cases; suppose A45: p1 = p00; A46: p1 in LSeg(p1,p10) by RLTOPSP1:68; p1 in LSeg(p00,p2) by A45,RLTOPSP1:68; then A47: LSeg(p00,p2) /\ LSeg(p1,p10) <> {} by A46,XBOOLE_0:def 4; LSeg(p1,p00) /\ L4 = {p00} /\ L4 by A45,RLTOPSP1:70; then A48: LSeg(p1,p00) /\ L4 = {} by Lm1,Lm12; LSeg(p00,p2) /\ LSeg(p1,p10) c= {p1} by A18,A45,Lm20,TOPREAL1:6,17 ,XBOOLE_1:26; then LSeg(p00,p2) /\ LSeg(p1,p10) = {p1} by A47,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p1} \/ LSeg(p1,p00) /\ LSeg(p01,p2)) \/ (( LSeg(p00,p2) /\ L2) \/ {p2}) by A43,A48,XBOOLE_1:4 .= {p1} \/ {p1} \/ LSeg(p1,p00) /\ LSeg(p01,p2) \/ ((LSeg(p00,p2) /\ L2) \/ {p2}) by XBOOLE_1:4 .= {p1} \/ LSeg(p1,p00) /\ LSeg(p01,p2) \/ ((LSeg(p00,p2) /\ L2) \/ {p2}); end; suppose A49: p1 = p10; then A50: LSeg(p00,p2) /\ LSeg(p1,p10) = LSeg(p00,p2) /\ {p10} by RLTOPSP1:70; not p10 in LSeg(p00,p2) by A26,Lm4,Lm6,Lm8,TOPREAL1:3; then LSeg(p00,p2) /\ LSeg(p1,p10) = {} by A50,Lm1; hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p00) /\ LSeg(p01,p2)) \/ (( LSeg(p00,p2) /\ L2) \/ {p2}) by A43,A49,TOPREAL1:16,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p00) /\ LSeg(p01,p2)) \/ ((LSeg(p00,p2) /\ L2) \/ {p2}); end; suppose A51: p1 <> p10 & p1 <> p00; now assume p00 in LSeg(p00,p2) /\ LSeg(p1,p10); then A52: p00 in LSeg(p1,p10) by XBOOLE_0:def 4; p1`1 <= p10`1 by A14,A15,EUCLID:52; then 0 = p1`1 by A14,A16,A52,Lm4,TOPREAL1:3; hence contradiction by A14,A17,A51,EUCLID:53; end; then A53: {p00} <> LSeg(p00,p2) /\ LSeg(p1,p10) by ZFMISC_1:31; LSeg(p00,p2) /\ LSeg(p1,p10) c= {p00} by A12,A26,TOPREAL1:17 ,XBOOLE_1:27; then A54: LSeg(p00,p2) /\ LSeg(p1,p10) = {} by A53,ZFMISC_1:33; now assume p10 in LSeg(p1,p00) /\ L4; then A55: p10 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`1 <= p1`1 by A14,A16,EUCLID:52; then p10`1 <= p1`1 by A55,TOPREAL1:3; then p1`1 = 1 by A14,A15,Lm8,XXREAL_0:1; hence contradiction by A14,A17,A51,EUCLID:53; end; then A56: {p10} <> LSeg(p1,p00) /\ L4 by ZFMISC_1:31; LSeg(p1,p00) /\ L4 c= {p10} by A3,Lm21,TOPREAL1:6,16,XBOOLE_1:26; then LSeg(p1,p00) /\ L4 = {} by A56,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p00) /\ LSeg(p01,p2)) \/ ((LSeg(p00, p2) /\ L2) \/ {p2}) by A43,A54; end; end; now per cases; suppose A57: p2 = p00; p00 in LSeg(p1,p00) by RLTOPSP1:68; then A58: LSeg(p1,p00) /\ LSeg(p01,p2) <> {} by A57,Lm20,XBOOLE_0:def 4; LSeg(p00,p2) /\ L2 = {p00} /\ L2 by A57,RLTOPSP1:70; then A59: LSeg(p00,p2) /\ L2 = {} by Lm1,Lm13; LSeg(p1,p00) /\ LSeg(p01,p2) c= L3 /\ L1 by A10,A36,XBOOLE_1:27; then LSeg(p1,p00) /\ LSeg(p01,p2) = {p2} by A57,A58,TOPREAL1:17 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A44,A59,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; hence thesis; end; suppose A60: p2 = p01; then A61: LSeg(p1,p00) /\ LSeg(p01,p2) = LSeg(p1,p00) /\ {p01} by RLTOPSP1:70; not p01 in LSeg(p1,p00) by A10,Lm5,Lm7,Lm9,TOPREAL1:4; then A62: LSeg(p1,p00) /\ LSeg(p01,p2) = {} by A61,Lm1; hence thesis by A44,A60,ENUMSET1:1,TOPREAL1:15; thus thesis by A44,A60,A62,ENUMSET1:1,TOPREAL1:15; end; suppose A63: p2 <> p01 & p2 <> p00; now assume p01 in LSeg(p00,p2) /\ L2; then A64: p01 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`2 <= p2`2 by A28,EUCLID:52; then p01`2 <= p2`2 by A64,TOPREAL1:4; then 1 = p2`2 by A28,Lm7,XXREAL_0:1; hence contradiction by A28,A63,EUCLID:53; end; then A65: {p01} <> LSeg(p00,p2) /\ L2 by ZFMISC_1:31; LSeg(p00,p2) /\ L2 c= {p01} by A18,Lm20,TOPREAL1:6,15,XBOOLE_1:26; then A66: LSeg(p00,p2) /\ L2 = {} by A65,ZFMISC_1:33; now assume p00 in LSeg(p1,p00) /\ LSeg(p01,p2); then A67: p00 in LSeg(p2,p01) by XBOOLE_0:def 4; p2`2 <= p01`2 by A28,EUCLID:52; then p2`2 = 0 by A28,A67,Lm5,TOPREAL1:4; hence contradiction by A28,A63,EUCLID:53; end; then A68: {p00} <> LSeg(p1,p00) /\ LSeg(p01,p2) by ZFMISC_1:31; LSeg(p1,p00) /\ LSeg(p01,p2) c= L3 /\ L1 by A10,A36,XBOOLE_1:27; then A69: LSeg(p1,p00) /\ LSeg(p01,p2) = {} by A68,TOPREAL1:17,ZFMISC_1:33; hence thesis by A44,A66,ENUMSET1:1; thus thesis by A44,A69,A66,ENUMSET1:1; end; end; hence thesis; end; suppose A70: p2 in L2; then A71: LSeg(p2,p11) c= L2 by Lm26,TOPREAL1:6; then A72: LSeg(p1,p10) /\ LSeg(p11,p2) = {} by A12,Lm2,XBOOLE_1:3,27; A73: LSeg(p2,p01) c= L2 by A70,Lm23,TOPREAL1:6; then A74: LSeg(p1,p00) /\ LSeg(p01,p2) = {} by A10,Lm2,XBOOLE_1:3,27; take P1 = LSeg(p1,p00) \/ L1 \/ LSeg(p01,p2),P2 = LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2); p01 in LSeg(p01,p2) by RLTOPSP1:68; then A75: L1 /\ LSeg(p01,p2) <> {} by Lm22,XBOOLE_0:def 4; L1 /\ LSeg(p01,p2) c= {p01} by A70,Lm23,TOPREAL1:6,15,XBOOLE_1:26; then L1 /\ LSeg(p01,p2) = {p01} by A75,ZFMISC_1:33; then A76: L1 \/ LSeg(p01,p2) is_an_arc_of p00,p2 by Lm5,Lm7,TOPREAL1:12; LSeg(p1,p00) /\ (L1 \/ LSeg(p01,p2)) = (LSeg(p1,p00) /\ L1) \/ (LSeg (p1,p00) /\ LSeg(p01,p2)) by XBOOLE_1:23 .= {p00} by A8,A6,A74,TOPREAL1:17,ZFMISC_1:33; then LSeg(p1,p00) \/ (L1 \/ LSeg(p01,p2)) is_an_arc_of p1,p2 by A76,TOPREAL1:11; hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A77: L4 /\ LSeg(p11,p2) <> {} by Lm27,XBOOLE_0:def 4; L4 /\ LSeg(p11,p2) c= L4 /\ L2 by A70,Lm26,TOPREAL1:6,XBOOLE_1:26; then L4 /\ LSeg(p11,p2) = {p11} by A77,TOPREAL1:18,ZFMISC_1:33; then A78: L4 \/ LSeg(p11,p2) is_an_arc_of p10,p2 by Lm9,Lm11,TOPREAL1:12; LSeg(p1,p10) /\ (L4 \/ LSeg(p11,p2)) = (LSeg(p1,p10) /\ L4) \/ (LSeg (p1,p10) /\ LSeg(p11,p2)) by XBOOLE_1:23 .= {p10} by A5,A7,A72,TOPREAL1:16,ZFMISC_1:33; then LSeg(p1,p10) \/ (L4 \/ LSeg(p11,p2)) is_an_arc_of p1,p2 by A78,TOPREAL1:11; hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4; thus R^2-unit_square = L1 \/ (LSeg(p01,p2) \/ LSeg(p11,p2)) \/ (L3 \/ L4) by A70,TOPREAL1:5,def 2 .= L1 \/ LSeg(p01,p2) \/ LSeg(p11,p2) \/ (L3 \/ L4) by XBOOLE_1:4 .= L1 \/ LSeg(p01,p2) \/ ((L3 \/ L4) \/ LSeg(p11,p2)) by XBOOLE_1:4 .= L1 \/ LSeg(p01,p2) \/ (L3 \/ (L4 \/ LSeg(p11,p2))) by XBOOLE_1:4 .= L1 \/ LSeg(p01,p2) \/ (LSeg(p1,p00) \/ LSeg(p1,p10) \/ (L4 \/ LSeg( p11,p2))) by A3,TOPREAL1:5 .= L1 \/ LSeg(p01,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p10) \/ (L4 \/ LSeg (p11,p2)))) by XBOOLE_1:4 .= L1 \/ LSeg(p01,p2) \/ (LSeg(p1,p00) \/ (LSeg(p1,p10) \/ L4 \/ LSeg( p11,p2))) by XBOOLE_1:4 .= (LSeg(p1,p00) \/ (L1 \/ LSeg(p01,p2))) \/ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)) by XBOOLE_1:4 .= P1 \/ P2 by XBOOLE_1:4; A79: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 1 by A70,TOPREAL1:13; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A80: LSeg(p1,p00) /\ LSeg(p11,p2) = {} by A10,A71,XBOOLE_1:3,27; A81: LSeg(p01,p2) /\ LSeg(p1,p10) = {} by A12,A73,Lm2,XBOOLE_1:3,27; A82: LSeg(p01,p2) /\ LSeg(p11,p2) = {p2} by A70,TOPREAL1:8; A83: P1 /\ P2 = (LSeg(p1,p00) \/ L1) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11, p2)) \/ (LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))) by XBOOLE_1:23 .= (LSeg(p1,p00) \/ L1) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)) \/ ( LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L4) \/ {p2}) by A82,XBOOLE_1:23 .= (LSeg(p1,p00) \/ L1) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2)) \/ (( LSeg(p01,p2) /\ LSeg(p1,p10)) \/ (LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))) \/ (L1 /\ ( LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A81, XBOOLE_1:23 .= ((LSeg(p1,p00) /\ (LSeg(p1,p10) \/ L4)) \/ (LSeg(p1,p00) /\ LSeg( p11,p2))) \/ (L1 /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= ({p1} \/ (LSeg(p1,p00) /\ L4)) \/ (L1 /\ (LSeg(p1,p10) \/ L4 \/ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A9,A80,XBOOLE_1:23 .= ({p1} \/ (LSeg(p1,p00) /\ L4)) \/ ((L1 /\ (LSeg(p1,p10) \/ L4)) \/ (L1 /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= ({p1} \/ (LSeg(p1,p00) /\ L4)) \/ ((L1 /\ LSeg(p1,p10)) \/ (L1 /\ L4) \/ (L1 /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= ({p1} \/ (LSeg(p1,p00) /\ L4)) \/ ((L1 /\ LSeg(p1,p10)) \/ (L1 /\ LSeg(p11,p2))) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by Lm3; A84: now per cases; suppose A85: p1 = p00; then LSeg(p1,p00) /\ L4 = {p00} /\ L4 by RLTOPSP1:70; then LSeg(p1,p00) /\ L4 = {} by Lm1,Lm12; hence P1 /\ P2 = {p1} \/ {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A83,A85,TOPREAL1:17,XBOOLE_1:4 .= {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}); end; suppose A86: p1 = p10; then L1 /\ LSeg(p1,p10) = L1 /\ {p10} by RLTOPSP1:70; then L1 /\ LSeg(p1,p10) = {} by Lm1,Lm16; hence P1 /\ P2 = {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A83,A86,TOPREAL1:16; end; suppose A87: p1 <> p10 & p1 <> p00; now assume p00 in L1 /\ LSeg(p1,p10); then A88: p00 in LSeg(p1,p10) by XBOOLE_0:def 4; p1`1 <= p10`1 by A14,A15,EUCLID:52; then 0 = p1`1 by A14,A16,A88,Lm4,TOPREAL1:3; hence contradiction by A14,A17,A87,EUCLID:53; end; then A89: {p00} <> L1 /\ LSeg(p1,p10) by ZFMISC_1:31; L1 /\ LSeg(p1,p10) c= {p00} by A3,Lm24,TOPREAL1:6,17,XBOOLE_1:26; then A90: L1 /\ LSeg(p1,p10) = {} by A89,ZFMISC_1:33; now assume p10 in LSeg(p1,p00) /\ L4; then A91: p10 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`1 <= p1`1 by A14,A16,EUCLID:52; then p10`1 <= p1`1 by A91,TOPREAL1:3; then 1 = p1`1 by A14,A15,Lm8,XXREAL_0:1; hence contradiction by A14,A17,A87,EUCLID:53; end; then A92: {p10} <> LSeg(p1,p00) /\ L4 by ZFMISC_1:31; LSeg(p1,p00) /\ L4 c= {p10} by A3,Lm21,TOPREAL1:6,16,XBOOLE_1:26; then LSeg(p1,p00) /\ L4 = {} by A92,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (L1 /\ LSeg(p11,p2)) \/ ((LSeg(p01,p2) /\ L4) \/ {p2}) by A83,A90; end; end; now per cases; suppose A93: p2 = p01; then LSeg(p01,p2) /\ L4 = {p01} /\ L4 by RLTOPSP1:70; then LSeg(p01,p2) /\ L4 = {} by Lm1,Lm15; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A84,A93,TOPREAL1:15 ,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A94: p2 = p11; then L1 /\ LSeg(p11,p2) = L1 /\ {p11} by RLTOPSP1:70; then L1 /\ LSeg(p11,p2) = {} by Lm1,Lm18; hence thesis by A84,A94,ENUMSET1:1,TOPREAL1:18; end; suppose A95: p2 <> p11 & p2 <> p01; now assume p11 in LSeg(p01,p2) /\ L4; then A96: p11 in LSeg(p01,p2) by XBOOLE_0:def 4; p01`1 <= p2`1 by A79,EUCLID:52; then p11`1 <= p2`1 by A96,TOPREAL1:3; then 1 = p2`1 by A79,Lm10,XXREAL_0:1; hence contradiction by A79,A95,EUCLID:53; end; then A97: {p11} <> LSeg(p01,p2) /\ L4 by ZFMISC_1:31; LSeg(p01,p2) /\ L4 c= {p11} by A70,Lm23,TOPREAL1:6,18,XBOOLE_1:26; then A98: LSeg(p01,p2) /\ L4 = {} by A97,ZFMISC_1:33; now assume p01 in L1 /\ LSeg(p11,p2); then A99: p01 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`1 <= p11`1 by A79,EUCLID:52; then p2`1 = 0 by A79,A99,Lm6,TOPREAL1:3; hence contradiction by A79,A95,EUCLID:53; end; then A100: {p01} <> L1 /\ LSeg(p11,p2) by ZFMISC_1:31; L1 /\ LSeg(p11,p2) c= {p01} by A70,Lm26,TOPREAL1:6,15,XBOOLE_1:26; then L1 /\ LSeg(p11,p2) = {} by A100,ZFMISC_1:33; hence thesis by A84,A98,ENUMSET1:1; end; end; hence thesis; end; suppose A101: p2 in L3; A102: p = |[p`1,p`2]| by EUCLID:53; A103: LSeg(p1,p2) c= L3 by A3,A101,TOPREAL1:6; consider q such that A104: q = p2 and A105: q`1 <= 1 and A106: q`1 >= 0 and A107: q`2 = 0 by A101,TOPREAL1:13; A108: q = |[q`1,q`2]| by EUCLID:53; now per cases by A1,A14,A17,A104,A107,A102,A108,XXREAL_0:1; suppose A109: p`1 < q`1; now assume p10 in LSeg(p1,p00) /\ L4; then A110: p10 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`1 <= p1`1 by A14,A16,EUCLID:52; then p10`1 <= p1`1 by A110,TOPREAL1:3; hence contradiction by A14,A15,A105,A109,Lm8,XXREAL_0:1; end; then A111: {p10} <> LSeg(p1,p00) /\ L4 by ZFMISC_1:31; LSeg(p1,p00) /\ L4 c= {p10} by A3,Lm21,TOPREAL1:6,16,XBOOLE_1:26; then A112: LSeg(p1,p00) /\ L4 = {} by A111,ZFMISC_1:33; p00 in LSeg(p1,p00) by RLTOPSP1:68; then A113: LSeg(p1,p00) /\ L1 <> {} by Lm20,XBOOLE_0:def 4; now assume p00 in L1 /\ LSeg(p10,p2); then A114: p00 in LSeg(p2,p10) by XBOOLE_0:def 4; p2`1 <= p10`1 by A104,A105,EUCLID:52; hence contradiction by A16,A104,A109,A114,Lm4,TOPREAL1:3; end; then A115: {p00} <> L1 /\ LSeg(p10,p2) by ZFMISC_1:31; L1 /\ LSeg(p10,p2) c= {p00} by A101,Lm24,TOPREAL1:6,17,XBOOLE_1:26; then A116: L1 /\ LSeg(p10,p2) = {} by A115,ZFMISC_1:33; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A117: LSeg(p1,p2) /\ L2 = {} by A103,XBOOLE_1:3,26; A118: LSeg(p1,p2) /\ LSeg(p1,p00) c= {p1} proof let a be object; assume A119: a in LSeg(p1,p2) /\ LSeg(p1,p00); then reconsider p = a as Point of TOP-REAL 2; A120: p in LSeg(p00,p1) by A119,XBOOLE_0:def 4; p00`1 <= p1`1 by A14,A16,EUCLID:52; then A121: p`1 <= p1`1 by A120,TOPREAL1:3; A122: p in LSeg(p1,p2) by A119,XBOOLE_0:def 4; then p1`1 <= p`1 by A14,A104,A109,TOPREAL1:3; then A123: p1`1 = p`1 by A121,XXREAL_0:1; p1`2 <= p`2 by A14,A17,A104,A107,A122,TOPREAL1:4; then p`2 = 0 by A14,A17,A104,A107,A122,TOPREAL1:4; then p = |[p1`1, 0]| by A123,EUCLID:53 .= p1 by A14,A17,EUCLID:53; hence thesis by TARSKI:def 1; end; A124: LSeg (p1,p00) /\ L1 c= L3 /\ L1 by A3,Lm21,TOPREAL1:6,XBOOLE_1:26; take P1 = LSeg(p1,p2),P2 = LSeg(p1,p00) \/ (L1 \/ L2 \/ L4 \/ LSeg(p10 ,p2)); A125: L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; L2 /\ LSeg(p10,p2) c= L2 /\ L3 by A101,Lm24,TOPREAL1:6,XBOOLE_1:26; then A126: L2 /\ LSeg(p10,p2) = {} by A125,XBOOLE_1:3; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A127: (L1 \/ L2) /\ L4 = L1 /\ L4 \/ L2 /\ L4 by XBOOLE_1:23 .= {p11} by Lm3,TOPREAL1:18; A128: LSeg(p1,p2) /\ LSeg(p10,p2) c= {p2} proof let a be object; assume A129: a in LSeg(p1,p2) /\ LSeg(p10,p2); then reconsider p = a as Point of TOP-REAL 2; A130: p in LSeg(p2,p10) by A129,XBOOLE_0:def 4; p2`1 <= p10`1 by A104,A105,EUCLID:52; then A131: p2`1 <= p`1 by A130,TOPREAL1:3; A132: p in LSeg(p1,p2) by A129,XBOOLE_0:def 4; then p`1 <= p2`1 by A14,A104,A109,TOPREAL1:3; then A133: p2`1 = p`1 by A131,XXREAL_0:1; p1`2 <= p`2 by A14,A17,A104,A107,A132,TOPREAL1:4; then p`2 = 0 by A14,A17,A104,A107,A132,TOPREAL1:4; then p = |[ p2`1, 0]| by A133,EUCLID:53 .= p2 by A104,A107,EUCLID:53; hence thesis by TARSKI:def 1; end; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A134: LSeg(p1,p00) /\ L2 = {} by A10,XBOOLE_1:3,26; A135: now set a = the Element of LSeg(p1,p00) /\ LSeg(p10,p2); assume A136: LSeg(p1,p00) /\ LSeg(p10,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A137: p in LSeg(p00,p1) by A136,XBOOLE_0:def 4; A138: p in LSeg(p2,p10) by A136,XBOOLE_0:def 4; p2`1 <= p10 `1 by A104,A105,EUCLID:52; then A139: p2`1 <= p`1 by A138,TOPREAL1:3; p00`1 <= p1`1 by A14,A16,EUCLID:52; then p`1 <= p1`1 by A137,TOPREAL1:3; hence contradiction by A14,A104,A109,A139,XXREAL_0:2; end; p10 in LSeg(p10,p2) by RLTOPSP1:68; then A140: L4 /\ LSeg(p10,p2) <> {} by Lm25,XBOOLE_0:def 4; L4 /\ LSeg(p10,p2) c= L4 /\ L3 by A101,Lm24,TOPREAL1:6,XBOOLE_1:26; then A141: L4 /\ LSeg(p10,p2) = {p10} by A140,TOPREAL1:16,ZFMISC_1:33; L1 \/ L2 is_an_arc_of p00,p11 by Lm5,Lm7,TOPREAL1:9,10,15; then A142: L1 \/ L2 \/ L4 is_an_arc_of p00,p10 by A127,TOPREAL1:10; (L1 \/ L2 \/ L4) /\ LSeg(p10,p2) = (L1 \/ L2) /\ LSeg(p10,p2) \/ L4 /\ LSeg(p10,p2) by XBOOLE_1:23 .= (L1 /\ LSeg(p10,p2)) \/ (L2 /\ LSeg(p10,p2)) \/ {p10} by A141, XBOOLE_1:23 .= {p10} by A116,A126; then A143: L1 \/ L2 \/ L4 \/ LSeg(p10,p2) is_an_arc_of p00,p2 by A142,TOPREAL1:10; LSeg(p1,p00) /\ (L1 \/ L2 \/ L4 \/ LSeg(p10,p2)) = LSeg(p1,p00) /\ (L1 \/ L2 \/ L4) \/ (LSeg(p1,p00) /\ LSeg(p10,p2)) by XBOOLE_1:23 .= LSeg(p1,p00) /\ (L1 \/ L2) \/ (LSeg(p1,p00) /\ L4) by A135, XBOOLE_1:23 .= LSeg(p1,p00) /\ L1 \/ (LSeg(p1,p00) /\ L2) by A112,XBOOLE_1:23 .= {p00} by A134,A124,A113,TOPREAL1:17,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A143,TOPREAL1:11; A144: p1 in LSeg(p1,p00) by RLTOPSP1:68; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p00) by A144,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p00) by ZFMISC_1:31; then A145: LSeg(p1,p2) /\ LSeg(p1,p00) = {p1} by A118,XBOOLE_0:def 10; thus P1 \/ P2 = LSeg(p00,p1) \/ LSeg(p1,p2) \/ (L1 \/ L2 \/ L4 \/ LSeg (p10,p2)) by XBOOLE_1:4 .= LSeg(p00,p1) \/ LSeg(p1,p2) \/ LSeg(p2,p10) \/ (L1 \/ L2 \/ L4) by XBOOLE_1:4 .= (L1 \/ L2 \/ L4) \/ L3 by A3,A101,TOPREAL1:7 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A146: p2 in LSeg(p10,p2) by RLTOPSP1:68; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p10,p2) by A146,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p10,p2) by ZFMISC_1:31; then A147: LSeg(p1,p2) /\ LSeg(p10,p2) = {p2} by A128,XBOOLE_0:def 10; A148: P1 /\ P2 = (LSeg(p1,p2) /\ LSeg(p1,p00)) \/ (LSeg(p1,p2) /\ (L1 \/ L2 \/ L4 \/ LSeg(p10,p2))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L2 \/ L4)) \/ {p2}) by A145,A147, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L1 \/ L2)) \/ (LSeg(p1,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ (LSeg(p1,p2) /\ L2) \/ (LSeg(p1 ,p2) /\ L4) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2})) by A117,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L1) \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by XBOOLE_1:4; A149: now per cases; suppose A150: p1 = p00; p1 in LSeg(p1,p2) by RLTOPSP1:68; then A151: LSeg(p1,p2) /\ L1 <> {} by A150,Lm20,XBOOLE_0:def 4; LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A3,A101,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p2) /\ L1 = {p1} by A150,A151,TOPREAL1:17,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A148; end; suppose A152: p1 <> p00; now assume p00 in LSeg(p1,p2) /\ L1; then p00 in LSeg(p1,p2) by XBOOLE_0:def 4; then p1`1 = 0 by A14,A16,A104,A109,Lm4,TOPREAL1:3; hence contradiction by A14,A17,A152,EUCLID:53; end; then A153: {p00} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:31; LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A3,A101,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p2) /\ L1 = {} by A153,TOPREAL1:17,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L4) \/ {p2}) by A148; end; end; now per cases; suppose A154: p2 = p10; p2 in LSeg(p1,p2) by RLTOPSP1:68; then A155: LSeg(p1,p2) /\ L4 <> {} by A154,Lm25,XBOOLE_0:def 4; LSeg(p1,p2) /\ L4 c= {p2} by A3,A101,A154,TOPREAL1:6,16,XBOOLE_1:26 ; then LSeg(p1,p2) /\ L4 = {p2} by A155,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A149,ENUMSET1:1; end; suppose A156: p2 <> p10; now assume p10 in LSeg(p1,p2) /\ L4; then p10 in LSeg(p1,p2) by XBOOLE_0:def 4; then p10`1 <= p2`1 by A14,A104,A109,TOPREAL1:3; then p2`1 = 1 by A104,A105,Lm8,XXREAL_0:1; hence contradiction by A104,A107,A156,EUCLID:53; end; then A157: {p10} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:31; LSeg(p1,p2) /\ L4 c= {p10} by A3,A101,TOPREAL1:6,16,XBOOLE_1:26; then LSeg(p1,p2) /\ L4 = {} by A157,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A149,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; suppose A158: q`1 < p`1; A159: LSeg(p1,p2) /\ LSeg(p00,p2) c= {p2} proof let a be object; assume A160: a in LSeg(p1,p2) /\ LSeg(p00,p2); then reconsider p = a as Point of TOP-REAL 2; A161: p in LSeg(p00,p2) by A160,XBOOLE_0:def 4; p00`1 <= p2`1 by A104,A106,EUCLID:52; then A162: p`1 <= p2`1 by A161,TOPREAL1:3; A163: p in LSeg(p2,p1) by A160,XBOOLE_0:def 4; then p2`1 <= p`1 by A14,A104,A158,TOPREAL1:3; then A164: p2`1 = p`1 by A162,XXREAL_0:1; p2`2 <= p`2 by A14,A17,A104,A107,A163,TOPREAL1:4; then p`2 = 0 by A14,A17,A104,A107,A163,TOPREAL1:4; then p = |[ p2`1, 0]| by A164,EUCLID:53 .= p2 by A104,A107,EUCLID:53; hence thesis by TARSKI:def 1; end; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A165: LSeg(p1,p10) /\ L4 <> {} by Lm25,XBOOLE_0:def 4; now assume p10 in L4 /\ LSeg(p00,p2); then A166: p10 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`1 <= p2`1 by A104,A106,EUCLID:52; then p10`1 <= p2`1 by A166,TOPREAL1:3; hence contradiction by A15,A104,A105,A158,Lm8,XXREAL_0:1; end; then A167: {p10} <> L4 /\ LSeg(p00,p2) by ZFMISC_1:31; L4 /\ LSeg(p00,p2) c= L4 /\ L3 by A101,Lm21,TOPREAL1:6,XBOOLE_1:26; then A168: L4 /\ LSeg(p00,p2) = {} by A167,TOPREAL1:16,ZFMISC_1:33; A169: (L4 \/ L2) /\ L1 = L1 /\ L4 \/ L2 /\ L1 by XBOOLE_1:23 .= {p01} by Lm3,TOPREAL1:15; L4 \/ L2 is_an_arc_of p10,p01 by Lm9,Lm11,TOPREAL1:9,10,18; then A170: L4 \/ L2 \/ L1 is_an_arc_of p10,p00 by A169,TOPREAL1:10; now assume p00 in LSeg(p1,p10) /\ L1; then A171: p00 in LSeg(p1,p10) by XBOOLE_0:def 4; p1`1 <= p10`1 by A14,A15,EUCLID:52; hence contradiction by A14,A106,A158,A171,Lm4,TOPREAL1:3; end; then A172: {p00} <> LSeg(p1,p10) /\ L1 by ZFMISC_1:31; LSeg (p1,p10) /\ L1 c= L3 /\ L1 by A3,Lm24,TOPREAL1:6,XBOOLE_1:26; then A173: LSeg(p1,p10) /\ L1 = {} by A172,TOPREAL1:17,ZFMISC_1:33; p00 in LSeg(p00,p2) by RLTOPSP1:68; then A174: L1 /\ LSeg(p00,p2) <> {} by Lm20,XBOOLE_0:def 4; L1 /\ LSeg(p00,p2) c= {p00} by A101,Lm21,TOPREAL1:6,17,XBOOLE_1:26; then A175: L1 /\ LSeg(p00,p2) = {p00} by A174,ZFMISC_1:33; A176: LSeg(p1,p2) /\ LSeg(p1,p10) c= {p1} proof let a be object; assume A177: a in LSeg(p1,p2) /\ LSeg(p1,p10); then reconsider p = a as Point of TOP-REAL 2; A178: p in LSeg(p1,p10) by A177,XBOOLE_0:def 4; p1`1 <= p10`1 by A14,A15,EUCLID:52; then A179: p1`1 <= p`1 by A178,TOPREAL1:3; A180: p in LSeg(p2,p1) by A177,XBOOLE_0:def 4; then p`1 <= p1`1 by A14,A104,A158,TOPREAL1:3; then A181: p1`1 = p`1 by A179,XXREAL_0:1; p2`2 <= p`2 by A14,A17,A104,A107,A180,TOPREAL1:4; then p`2 = 0 by A14,A17,A104,A107,A180,TOPREAL1:4; then p = |[p1`1, 0]| by A181,EUCLID:53 .= p1 by A14,A17,EUCLID:53; hence thesis by TARSKI:def 1; end; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A182: LSeg(p1,p10) /\ L2 = {} by A12,XBOOLE_1:3,26; A183: now set a = the Element of LSeg(p1,p10) /\ LSeg(p00,p2); assume A184: LSeg(p1,p10) /\ LSeg(p00,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A185: p in LSeg(p1,p10) by A184,XBOOLE_0:def 4; A186: p in LSeg(p00,p2) by A184,XBOOLE_0:def 4; p00`1 <= p2 `1 by A104,A106,EUCLID:52; then A187: p`1 <= p2`1 by A186,TOPREAL1:3; p1`1 <= p10`1 by A14,A15,EUCLID:52; then p1`1 <= p`1 by A185,TOPREAL1:3; hence contradiction by A14,A104,A158,A187,XXREAL_0:2; end; take P1 = LSeg(p1,p2),P2 = LSeg(p1,p10) \/ (L4 \/ L2 \/ L1 \/ LSeg(p00 ,p2)); A188: L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; L2 /\ LSeg(p00,p2) c= L2 /\ L3 by A101,Lm21,TOPREAL1:6,XBOOLE_1:26; then A189: L2 /\ LSeg(p00,p2) = {} by A188,XBOOLE_1:3; A190: LSeg(p1,p10) /\ L4 c= {p10} by A3,Lm24,TOPREAL1:6,16,XBOOLE_1:26; (L4 \/ L2 \/ L1) /\ LSeg(p00,p2) = (L4 \/ L2) /\ LSeg(p00,p2) \/ L1 /\ LSeg(p00,p2) by XBOOLE_1:23 .= (L4 /\ LSeg(p00,p2)) \/ (L2 /\ LSeg(p00,p2)) \/ {p00} by A175, XBOOLE_1:23 .= {p00} by A168,A189; then A191: L4 \/ L2 \/ L1 \/ LSeg(p00,p2) is_an_arc_of p10,p2 by A170,TOPREAL1:10; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A192: p2 in LSeg(p00,p2) by RLTOPSP1:68; LSeg(p1,p10) /\ (L4 \/ L2 \/ L1 \/ LSeg(p00,p2)) = LSeg(p1,p10) /\ (L4 \/ L2 \/ L1) \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) by XBOOLE_1:23 .= LSeg(p1,p10) /\ (L4 \/ L2) \/ (LSeg(p1,p10) /\ L1) by A183, XBOOLE_1:23 .= LSeg(p1,p10) /\ L4 \/ (LSeg(p1,p10) /\ L2) by A173,XBOOLE_1:23 .= {p10} by A182,A190,A165,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A191,TOPREAL1:11; A193: p1 in LSeg(p1,p10) by RLTOPSP1:68; thus P1 \/ P2 = LSeg(p2,p1) \/ LSeg(p1,p10) \/ (L4 \/ L2 \/ L1 \/ LSeg (p00,p2)) by XBOOLE_1:4 .= LSeg(p00,p2) \/ (LSeg(p2,p1) \/ LSeg(p1,p10)) \/ (L4 \/ L2 \/ L1) by XBOOLE_1:4 .= L3 \/ (L4 \/ L2 \/ L1) by A3,A101,TOPREAL1:7 .= L3 \/ (L4 \/ (L1 \/ L2)) by XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A194: LSeg(p1,p2) /\ L2 = {} by A103,XBOOLE_1:3,26; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p00,p2) by A192,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p00,p2) by ZFMISC_1:31; then A195: LSeg(p1,p2) /\ LSeg(p00,p2) = {p2} by A159,XBOOLE_0:def 10; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p10) by A193,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p10) by ZFMISC_1:31; then LSeg(p1,p2) /\ LSeg(p1,p10) = {p1} by A176,XBOOLE_0:def 10; then A196: P1 /\ P2 = {p1} \/ LSeg(p1,p2) /\ (L4 \/ L2 \/ L1 \/ LSeg(p00,p2 )) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L2 \/ L1)) \/ {p2}) by A195, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L4 \/ L2)) \/ (LSeg(p1,p2) /\ L1) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ (LSeg(p1,p2) /\ L2) \/ (LSeg(p1 ,p2) /\ L1) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/ {p2})) by A194,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L4) \/ ((LSeg(p1,p2) /\ L1) \/ {p2}) by XBOOLE_1:4; A197: now per cases; suppose A198: p2 = p00; p2 in LSeg(p1,p2) by RLTOPSP1:68; then A199: LSeg(p1,p2) /\ L1 <> {} by A198,Lm20,XBOOLE_0:def 4; LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A3,A101,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p2) /\ L1 = {p2} by A198,A199,TOPREAL1:17,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A196; end; suppose A200: p2 <> p00; now assume p00 in LSeg(p1,p2) /\ L1; then p00 in LSeg(p2,p1) by XBOOLE_0:def 4; then p2`1 = 0 by A14,A104,A106,A158,Lm4,TOPREAL1:3; hence contradiction by A104,A107,A200,EUCLID:53; end; then A201: {p00} <> LSeg(p1,p2) /\ L1 by ZFMISC_1:31; LSeg(p1,p2) /\ L1 c= L3 /\ L1 by A3,A101,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p2) /\ L1 = {} by A201,TOPREAL1:17,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p2) /\ L4) \/ {p2} by A196; end; end; now per cases; suppose A202: p1 = p10; p1 in LSeg(p1,p2) by RLTOPSP1:68; then A203: LSeg(p1,p2) /\ L4 <> {} by A202,Lm25,XBOOLE_0:def 4; LSeg(p1,p2) /\ L4 c= {p1} by A3,A101,A202,TOPREAL1:6,16,XBOOLE_1:26 ; then LSeg(p1,p2) /\ L4 = {p1} by A203,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A197,ENUMSET1:1; end; suppose A204: p1 <> p10; now assume p10 in LSeg(p1,p2) /\ L4; then p10 in LSeg(p2,p1) by XBOOLE_0:def 4; then p10`1 <= p1`1 by A14,A104,A158,TOPREAL1:3; then p1`1 = 1 by A14,A15,Lm8,XXREAL_0:1; hence contradiction by A14,A17,A204,EUCLID:53; end; then A205: {p10} <> LSeg(p1,p2) /\ L4 by ZFMISC_1:31; LSeg(p1,p2) /\ L4 c= {p10} by A3,A101,TOPREAL1:6,16,XBOOLE_1:26; then LSeg(p1,p2) /\ L4 = {} by A205,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A197,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; end; hence thesis; end; suppose A206: p2 in L4; then A207: ex q st q = p2 & q`1 = 1 & q`2 <= 1 & q`2 >= 0 by TOPREAL1:13; now assume A208: p10 in LSeg(p1,p00) /\ LSeg(p11,p2); then A209: p10 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`1 <= p1`1 by A14,A16,EUCLID:52; then p10`1 <= p1`1 by A209,TOPREAL1:3; then 1 = p1`1 by A14,A15,Lm8,XXREAL_0:1; then A210: p1 = p10 by A14,A17,EUCLID:53; A211: p2`2 <= p11`2 by A207,EUCLID:52; p10 in LSeg(p2,p11) by A208,XBOOLE_0:def 4; then 0 = p2`2 by A207,A211,Lm9,TOPREAL1:4; hence contradiction by A1,A207,A210,EUCLID:53; end; then A212: {p10} <> LSeg(p1,p00) /\ LSeg(p11,p2) by ZFMISC_1:31; A213: L2 is_an_arc_of p01,p11 by Lm6,Lm10,TOPREAL1:9; L1 is_an_arc_of p00,p01 by Lm5,Lm7,TOPREAL1:9; then A214: L1 \/ L2 is_an_arc_of p00,p11 by A213,TOPREAL1:2,15; take P1 = LSeg(p1,p10) \/ LSeg(p10,p2),P2 = LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2)); A215: L3 = LSeg(p1,p10) \/ LSeg(p1,p00) by A3,TOPREAL1:5; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A216: L2 /\ LSeg(p11,p2) <> {} by Lm26,XBOOLE_0:def 4; A217: p10 in LSeg(p10,p2) by RLTOPSP1:68; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A218: LSeg(p1,p10) /\ LSeg(p10,p2) <> {} by A217,XBOOLE_0:def 4; A219: LSeg(p2,p10) c= L4 by A206,Lm25,TOPREAL1:6; then LSeg(p1,p10) /\ LSeg(p10,p2) c= L3 /\ L4 by A12,XBOOLE_1:27; then A220: LSeg(p1,p10) /\ LSeg(p10,p2) = {p10} by A218,TOPREAL1:16,ZFMISC_1:33; p1 <> p10 or p2 <> p10 by A1; hence P1 is_an_arc_of p1,p2 by A220,TOPREAL1:12; A221: LSeg(p1,p10) /\ LSeg(p1,p00) = {p1} by A3,TOPREAL1:8; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A222: LSeg(p10,p2) /\ L1 = {} by A219,XBOOLE_1:3,26; A223: LSeg(p2,p11) c= L4 by A206,Lm27,TOPREAL1:6; then A224: L2 /\ LSeg(p11,p2) c= {p11} by TOPREAL1:18,XBOOLE_1:27; A225: L1 /\ LSeg(p11,p2) = {} by A223,Lm3,XBOOLE_1:3,26; (L1 \/ L2) /\ LSeg(p11,p2) = (L1 /\ LSeg(p11,p2)) \/ (L2 /\ LSeg(p11 ,p2)) by XBOOLE_1:23 .= {p11} by A225,A224,A216,ZFMISC_1:33; then A226: L1 \/ L2 \/ LSeg(p11,p2) is_an_arc_of p00,p2 by A214,TOPREAL1:10; A227: LSeg(p10,p2) /\ LSeg(p11,p2) = {p2} by A206,TOPREAL1:8; A228: L4 = LSeg(p11,p2) \/ LSeg(p10,p2) by A206,TOPREAL1:5; LSeg(p1,p00) /\ LSeg(p11,p2) c= {p10} by A10,A223,TOPREAL1:16,XBOOLE_1:27; then A229: LSeg(p1,p00) /\ LSeg(p11,p2) = {} by A212,ZFMISC_1:33; LSeg(p1,p00) /\ (L1 \/ L2 \/ LSeg(p11,p2)) = (LSeg(p1,p00) /\ (L1 \/ L2)) \/ (LSeg(p1,p00) /\ LSeg(p11,p2)) by XBOOLE_1:23 .= (LSeg(p1,p00) /\ L1) \/ (LSeg(p1,p00) /\ L2) by A229,XBOOLE_1:23 .= {p00} by A8,A6,A11,TOPREAL1:17,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A226,TOPREAL1:11; thus P1 \/ P2 = LSeg(p10,p2) \/ (LSeg(p1,p10) \/ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2)))) by XBOOLE_1:4 .= LSeg(p10,p2) \/ (L3 \/ (L1 \/ L2 \/ LSeg(p11,p2))) by A215,XBOOLE_1:4 .= (L1 \/ L2) \/ L3 \/ LSeg(p11,p2) \/ LSeg(p10,p2) by XBOOLE_1:4 .= (L1 \/ L2) \/ L3 \/ L4 by A228,XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A230: P1 /\ P2 = (LSeg(p1,p10) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11, p2)))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p10) /\ LSeg(p1,p00)) \/ (LSeg(p1,p10) /\ (L1 \/ L2 \/ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ (L1 \/ L2)) \/ (LSeg(p1,p10) /\ LSeg(p11, p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg(p11,p2)))) by A221 ,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p10,p1) /\ L2) \/ (LSeg(p1, p10) /\ LSeg(p11,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p00) \/ (L1 \/ L2 \/ LSeg( p11,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p00)) \/ (LSeg(p10,p2) /\ (L1 \/ L2 \/ LSeg(p11,p2))) ) by A13,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p10,p2) /\ (L1 \/ L2)) \/ {p2})) by A227,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p10,p2) /\ L1) \/ (LSeg(p10,p2) /\ L2 ) \/ {p2})) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L1) \/ (LSeg(p1,p10) /\ LSeg(p11,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p00)) \/ ((LSeg(p10,p2) /\ L2) \/ {p2})) by A222; A231: now per cases; suppose A232: p1 = p00; then A233: LSeg(p10,p2) /\ LSeg(p1,p00) = LSeg(p10,p2) /\ {p00} by RLTOPSP1:70; not p00 in LSeg(p10,p2) by A219,Lm4,Lm8,Lm10,TOPREAL1:3; then LSeg(p10,p2) /\ LSeg(p1,p00) = {} by A233,Lm1; hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) \/ (( LSeg(p10,p2) /\ L2) \/ {p2}) by A230,A232,TOPREAL1:17,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) \/ ((LSeg(p10,p2) /\ L2) \/ {p2}); end; suppose A234: p1 = p10; A235: p1 in LSeg(p1,p00) by RLTOPSP1:68; p1 in LSeg(p10,p2) by A234,RLTOPSP1:68; then A236: {} <> LSeg(p10,p2) /\ LSeg(p1,p00) by A235,XBOOLE_0:def 4; LSeg(p1,p10) /\ L1 = {p10} /\ L1 by A234,RLTOPSP1:70; then A237: LSeg(p1,p10) /\ L1 = {} by Lm1,Lm16; LSeg(p10,p2) /\ LSeg(p1,p00) c= L4 /\ L3 by A10,A219,XBOOLE_1:27; then LSeg(p10,p2) /\ LSeg(p1,p00) = {p1} by A234,A236,TOPREAL1:16 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2))) \/ (( LSeg(p10,p2) /\ L2) \/ {p2}) by A230,A237,XBOOLE_1:4 .= {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) \/ ((LSeg(p10,p2 ) /\ L2) \/ {p2}) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) \/ ((LSeg(p10,p2) /\ L2) \/ {p2}); end; suppose A238: p1 <> p10 & p1 <> p00; now assume p10 in LSeg(p10,p2) /\ LSeg(p1,p00); then A239: p10 in LSeg(p00,p1) by XBOOLE_0:def 4; p00`1 <= p1`1 by A14,A16,EUCLID:52; then p10`1 <= p1`1 by A239,TOPREAL1:3; then p1`1 = 1 by A14,A15,Lm8,XXREAL_0:1; hence contradiction by A14,A17,A238,EUCLID:53; end; then A240: {p10} <> LSeg(p10,p2) /\ LSeg(p1,p00) by ZFMISC_1:31; LSeg(p10,p2) /\ LSeg(p1,p00) c= L4 /\ L3 by A10,A219,XBOOLE_1:27; then A241: LSeg(p10,p2) /\ LSeg(p1,p00) = {} by A240,TOPREAL1:16,ZFMISC_1:33; now assume p00 in LSeg(p1,p10) /\ L1; then A242: p00 in LSeg(p1,p10) by XBOOLE_0:def 4; p1`1 <= p10`1 by A14,A15,EUCLID:52; then p1`1 = 0 by A14,A16,A242,Lm4,TOPREAL1:3; hence contradiction by A14,A17,A238,EUCLID:53; end; then A243: {p00} <> LSeg(p1,p10) /\ L1 by ZFMISC_1:31; LSeg (p1,p10) /\ L1 c= L3 /\ L1 by A3,Lm24,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p10) /\ L1 = {} by A243,TOPREAL1:17,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) \/ ((LSeg(p10, p2) /\ L2) \/ {p2}) by A230,A241; end; end; now per cases; suppose A244: p2 = p10; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A245: {} <> LSeg(p1,p10) /\ LSeg(p11,p2) by A244,Lm25,XBOOLE_0:def 4; LSeg(p10,p2) /\ L2 = {p10} /\ L2 by A244,RLTOPSP1:70; then A246: LSeg(p10,p2) /\ L2 = {} by Lm1,Lm17; LSeg(p1,p10) /\ LSeg(p11,p2) c= {p2} by A12,A244,TOPREAL1:16 ,XBOOLE_1:27; then LSeg(p1,p10) /\ LSeg(p11,p2) = {p2} by A245,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A231,A246,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A247: p2 = p11; then A248: LSeg(p1,p10) /\ LSeg(p11,p2) = LSeg(p1,p10) /\ {p11} by RLTOPSP1:70; not p11 in LSeg(p1,p10) by A12,Lm5,Lm9,Lm11,TOPREAL1:4; then LSeg(p1,p10) /\ LSeg(p11,p2) = {} by A248,Lm1; hence thesis by A231,A247,ENUMSET1:1,TOPREAL1:18; end; suppose A249: p2 <> p11 & p2 <> p10; now assume p11 in LSeg(p10,p2) /\ L2; then A250: p11 in LSeg(p10,p2) by XBOOLE_0:def 4; p10`2 <= p2`2 by A207,EUCLID:52; then p11`2 <= p2`2 by A250,TOPREAL1:4; then 1 = p2`2 by A207,Lm11,XXREAL_0:1; hence contradiction by A207,A249,EUCLID:53; end; then A251: {p11} <> LSeg(p10,p2) /\ L2 by ZFMISC_1:31; LSeg (p10,p2) /\ L2 c= L4 /\ L2 by A206,Lm25,TOPREAL1:6,XBOOLE_1:26; then A252: LSeg(p10,p2) /\ L2 = {} by A251,TOPREAL1:18,ZFMISC_1:33; now assume p10 in LSeg(p1,p10) /\ LSeg(p11,p2); then A253: p10 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`2 <= p11`2 by A207,EUCLID:52; then p2`2 = 0 by A207,A253,Lm9,TOPREAL1:4; hence contradiction by A207,A249,EUCLID:53; end; then A254: {p10} <> LSeg(p1,p10) /\ LSeg(p11,p2) by ZFMISC_1:31; LSeg(p1,p10) /\ LSeg(p11,p2) c= {p10} by A12,A223,TOPREAL1:16 ,XBOOLE_1:27; then LSeg(p1,p10) /\ LSeg(p11,p2) = {} by A254,ZFMISC_1:33; hence thesis by A231,A252,ENUMSET1:1; end; end; hence thesis; end; end; Lm33: p1 <> p2 & p2 in R^2-unit_square & p1 in LSeg(p10, p11) implies ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof assume that A1: p1 <> p2 and A2: p2 in R^2-unit_square and A3: p1 in LSeg(p10, p11); A4: p2 in L1 \/ L2 or p2 in L3 \/ L4 by A2,TOPREAL1:def 2,XBOOLE_0:def 3; A5: LSeg(p1,p11) c= L4 by A3,Lm27,TOPREAL1:6; p11 in LSeg(p1,p11) by RLTOPSP1:68; then A6: {} <> LSeg(p1,p11) /\ L2 by Lm26,XBOOLE_0:def 4; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A7: {} <> LSeg(p1,p10) /\ L3 by Lm24,XBOOLE_0:def 4; A8: LSeg(p1,p11) /\ L2 c= L4 /\ L2 by A3,Lm27,TOPREAL1:6,XBOOLE_1:26; A9: LSeg(p1,p10) /\ L3 c= L4 /\ L3 by A3,Lm25,TOPREAL1:6,XBOOLE_1:26; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A10: LSeg(p1,p11) /\ L1 = {} by A5,XBOOLE_1:3,26; A11: LSeg(p1,p10) c= L4 by A3,Lm25,TOPREAL1:6; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A12: LSeg(p10,p1) /\ L1 = {} by A11,XBOOLE_1:3,26; consider p such that A13: p = p1 and A14: p`1 = 1 and A15: p`2 <= 1 and A16: p`2 >= 0 by A3,TOPREAL1:13; per cases by A4,XBOOLE_0:def 3; suppose A17: p2 in L1; p00 in LSeg(p00,p2) by RLTOPSP1:68; then A18: L3 /\ LSeg(p00,p2) <> {} by Lm21,XBOOLE_0:def 4; L3 /\ LSeg(p00,p2) c= L3 /\ L1 by A17,Lm20,TOPREAL1:6,XBOOLE_1:26; then L3 /\ LSeg(p00,p2) = {p00} by A18,TOPREAL1:17,ZFMISC_1:33; then A19: L3 \/ LSeg(p00,p2) is_an_arc_of p10,p2 by Lm4,Lm8,TOPREAL1:9,10; A20: LSeg(p2,p00) c= L1 by A17,Lm20,TOPREAL1:6; then A21: LSeg(p1,p10) /\ LSeg(p00,p2) = {} by A11,Lm3,XBOOLE_1:3,27; A22: LSeg(p2,p01) c= L1 by A17,Lm22,TOPREAL1:6; then A23: LSeg(p1,p11) /\ LSeg(p01,p2) = {} by A5,Lm3,XBOOLE_1:3,27; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A24: LSeg(p1,p11) /\ LSeg(p00,p2) = {} by A5,A20,XBOOLE_1:3,27; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A25: LSeg(p01,p2) /\ LSeg(p1,p10) = {} by A11,A22,XBOOLE_1:3,27; A26: LSeg(p01,p2) /\ LSeg(p00,p2) = {p2} by A17,TOPREAL1:8; p01 in LSeg(p01,p2) by RLTOPSP1:68; then A27: p01 in L2 /\ LSeg(p01,p2) by Lm23,XBOOLE_0:def 4; L2 /\ LSeg(p01,p2) c= L2 /\ L1 by A17,Lm22,TOPREAL1:6,XBOOLE_1:26; then L2 /\ LSeg(p01,p2) = {p01} by A27,TOPREAL1:15,ZFMISC_1:33; then A28: L2 \/ LSeg(p01,p2) is_an_arc_of p11,p2 by Lm6,Lm10,TOPREAL1:9,10; take P1 = LSeg(p1,p11) \/ L2 \/ LSeg(p01,p2),P2 = LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2); A29: LSeg(p1,p11) \/ LSeg(p1,p10) = L4 by A3,TOPREAL1:5; LSeg(p1,p11) /\ (L2 \/ LSeg(p01,p2)) = (LSeg(p1,p11) /\ L2) \/ (LSeg( p1,p11) /\ LSeg(p01,p2)) by XBOOLE_1:23 .= {p11} by A8,A6,A23,TOPREAL1:18,ZFMISC_1:33; then LSeg(p1,p11) \/ (L2 \/ LSeg(p01,p2)) is_an_arc_of p1,p2 by A28, TOPREAL1:11; hence P1 is_an_arc_of p1,p2 by XBOOLE_1:4; A30: ex q st q = p2 & q`1 = 0 & q`2 <= 1 & q`2 >= 0 by A17,TOPREAL1:13; LSeg(p1,p10) /\ (L3 \/ LSeg(p00,p2)) = (LSeg(p1,p10) /\ L3) \/ (LSeg( p1,p10) /\ LSeg(p00,p2)) by XBOOLE_1:23 .= {p10} by A9,A7,A21,TOPREAL1:16,ZFMISC_1:33; then LSeg(p1,p10) \/ (L3 \/ LSeg(p00,p2)) is_an_arc_of p1,p2 by A19, TOPREAL1:11; hence P2 is_an_arc_of p1,p2 by XBOOLE_1:4; thus R^2-unit_square = LSeg(p00,p2) \/ LSeg(p01,p2) \/ L2 \/ (L3 \/ L4) by A17,TOPREAL1:5,def 2 .= LSeg(p00,p2) \/ (LSeg(p01,p2) \/ L2) \/ (L3 \/ L4) by XBOOLE_1:4 .= L2 \/ LSeg(p01,p2) \/ (L4 \/ L3 \/ LSeg(p00,p2)) by XBOOLE_1:4 .= L2 \/ LSeg(p01,p2) \/ (L4 \/ (L3 \/ LSeg(p00,p2))) by XBOOLE_1:4 .= L2 \/ LSeg(p01,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p10) \/ (L3 \/ LSeg (p00,p2)))) by A29,XBOOLE_1:4 .= L2 \/ LSeg(p01,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p10) \/ L3 \/ LSeg( p00,p2))) by XBOOLE_1:4 .= (LSeg(p1,p11) \/ (L2 \/ LSeg(p01,p2))) \/ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)) by XBOOLE_1:4 .= P1 \/ P2 by XBOOLE_1:4; A31: LSeg(p1,p11) /\ LSeg(p1,p10) = {p1} by A3,TOPREAL1:8; A32: P1 /\ P2 = ((LSeg(p1,p11) \/ L2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00, p2))) \/ (LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) by XBOOLE_1:23 .= ((LSeg(p1,p11) \/ L2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) \/ ( (LSeg(p01,p2) /\ (LSeg(p1,p10) \/ L3)) \/ {p2}) by A26,XBOOLE_1:23 .= ((LSeg(p1,p11) \/ L2) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) \/ ( (LSeg(p01,p2) /\ LSeg(p1,p10)) \/ (LSeg(p01,p2) /\ L3) \/ {p2}) by XBOOLE_1:23 .= ((LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) \/ (L2 /\ ( LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2)))) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A25, XBOOLE_1:23 .= (LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) \/ ((L2 /\ ( LSeg(p1,p10) \/ L3)) \/ (L2 /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3 \/ LSeg(p00,p2))) \/ ((L2 /\ LSeg(p1,p10)) \/ (L3 /\ L2) \/ (L2 /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by XBOOLE_1:23 .= ((LSeg(p1,p11) /\ (LSeg(p1,p10) \/ L3)) \/ (LSeg(p1,p11) /\ LSeg( p00,p2))) \/ ((L2 /\ LSeg(p1,p10)) \/ (L2 /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by Lm2,XBOOLE_1:23 .= {p1} \/ (LSeg(p1,p11) /\ L3) \/ ((L2 /\ LSeg(p1,p10)) \/ (L2 /\ LSeg(p00,p2))) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A24,A31,XBOOLE_1:23; A33: now per cases; suppose A34: p1 = p10; then L2 /\ LSeg(p1,p10) = L2 /\ {p10} by RLTOPSP1:70; then L2 /\ LSeg(p1,p10) = {} by Lm1,Lm17; hence P1 /\ P2 = {p1} \/ (L2 /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A32,A34,TOPREAL1:16; end; suppose A35: p1 = p11; then LSeg(p1,p11) /\ L3 = {p11} /\ L3 by RLTOPSP1:70; then LSeg(p1,p11) /\ L3 = {} by Lm1,Lm19; hence P1 /\ P2 = {p1} \/ {p1} \/ (L2 /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A32,A35,TOPREAL1:18,XBOOLE_1:4 .= {p1} \/ (L2 /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}); end; suppose A36: p1 <> p11 & p1 <> p10; now assume p11 in L2 /\ LSeg(p1,p10); then A37: p11 in LSeg(p10,p1) by XBOOLE_0:def 4; p10`2 <= p1`2 by A13,A16,EUCLID:52; then p11`2 <= p1`2 by A37,TOPREAL1:4; then 1 = p1`2 by A13,A15,Lm11,XXREAL_0:1; hence contradiction by A13,A14,A36,EUCLID:53; end; then A38: {p11} <> L2 /\ LSeg(p1,p10) by ZFMISC_1:31; L2 /\ LSeg(p1,p10) c= {p11} by A3,Lm25,TOPREAL1:6,18,XBOOLE_1:26; then A39: L2 /\ LSeg(p1,p10) = {} by A38,ZFMISC_1:33; now assume p10 in LSeg(p1,p11) /\ L3; then A40: p10 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`2 <= p11`2 by A13,A15,EUCLID:52; then p1`2 = 0 by A13,A16,A40,Lm9,TOPREAL1:4; hence contradiction by A13,A14,A36,EUCLID:53; end; then A41: {p10} <> LSeg(p1,p11) /\ L3 by ZFMISC_1:31; LSeg (p1,p11) /\ L3 c= L4 /\ L3 by A3,Lm27,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p11) /\ L3 = {} by A41,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (L2 /\ LSeg(p00,p2)) \/ ((LSeg(p01,p2) /\ L3) \/ {p2}) by A32,A39; end; end; now per cases; suppose A42: p2 = p00; then L2 /\ LSeg(p00,p2) = L2 /\ {p00} by RLTOPSP1:70; then L2 /\ LSeg(p00,p2) = {} by Lm1,Lm13; hence thesis by A33,A42,ENUMSET1:1,TOPREAL1:17; end; suppose A43: p2 = p01; then LSeg(p01,p2) /\ L3 = {p01} /\ L3 by RLTOPSP1:70; then LSeg(p01,p2) /\ L3 = {} by Lm1,Lm14; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A33,A43,TOPREAL1:15 ,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A44: p2 <> p01 & p2 <> p00; now assume p00 in LSeg(p01,p2) /\ L3; then A45: p00 in LSeg(p2,p01) by XBOOLE_0:def 4; p2`2 <= p01`2 by A30,EUCLID:52; then 0 = p2`2 by A30,A45,Lm5,TOPREAL1:4; hence contradiction by A30,A44,EUCLID:53; end; then A46: {p00} <> LSeg(p01,p2) /\ L3 by ZFMISC_1:31; LSeg(p01,p2) /\ L3 c= {p00} by A17,Lm22,TOPREAL1:6,17,XBOOLE_1:26; then A47: LSeg(p01,p2) /\ L3 = {} by A46,ZFMISC_1:33; now assume p01 in L2 /\ LSeg(p00,p2); then A48: p01 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`2 <= p2`2 by A30,EUCLID:52; then p01`2 <= p2`2 by A48,TOPREAL1:4; then p2`2 = 1 by A30,Lm7,XXREAL_0:1; hence contradiction by A30,A44,EUCLID:53; end; then A49: {p01} <> L2 /\ LSeg(p00,p2) by ZFMISC_1:31; L2 /\ LSeg(p00,p2) c= L2 /\ L1 by A17,Lm20,TOPREAL1:6,XBOOLE_1:26; then L2 /\ LSeg(p00,p2) = {} by A49,TOPREAL1:15,ZFMISC_1:33; hence thesis by A33,A47,ENUMSET1:1; end; end; hence thesis; end; suppose A50: p2 in L2; then A51: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 1 by TOPREAL1:13; now A52: p01`1 <= p2`1 by A51,EUCLID:52; assume A53: p11 in LSeg(p1,p10) /\ LSeg(p01,p2); then A54: p11 in LSeg(p10,p1) by XBOOLE_0:def 4; p11 in LSeg(p01,p2) by A53,XBOOLE_0:def 4; then p11`1 <= p2`1 by A52,TOPREAL1:3; then A55: 1 = p2`1 by A51,Lm10,XXREAL_0:1; p10`2 <= p1`2 by A13,A16,EUCLID:52; then p11`2 <= p1`2 by A54,TOPREAL1:4; then 1 = p1`2 by A13,A15,Lm11,XXREAL_0:1; then p1 = p11 by A13,A14,EUCLID:53 .= p2 by A51,A55,EUCLID:53; hence contradiction by A1; end; then A56: {p11} <> LSeg(p1,p10) /\ LSeg(p01,p2) by ZFMISC_1:31; A57: L1 is_an_arc_of p00,p01 by Lm5,Lm7,TOPREAL1:9; L3 is_an_arc_of p10,p00 by Lm4,Lm8,TOPREAL1:9; then A58: L3 \/ L1 is_an_arc_of p10,p01 by A57,TOPREAL1:2,17; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A59: LSeg(p1,p11) /\ L1 = {} by A5,XBOOLE_1:3,26; take P1 = LSeg(p1,p11) \/ LSeg(p11,p2),P2 = LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2)); A60: LSeg(p1,p11) \/ LSeg(p1,p10) = L4 by A3,TOPREAL1:5; p01 in LSeg(p01,p2) by RLTOPSP1:68; then A61: L1 /\ LSeg(p01,p2) <> {} by Lm22,XBOOLE_0:def 4; A62: p11 in LSeg(p11,p2) by RLTOPSP1:68; p11 in LSeg(p1,p11) by RLTOPSP1:68; then A63: p11 in LSeg(p1,p11) /\ LSeg(p11,p2) by A62,XBOOLE_0:def 4; A64: LSeg(p11,p2) c= L2 by A50,Lm26,TOPREAL1:6; then LSeg(p1,p11) /\ LSeg(p11,p2) c= L4 /\ L2 by A5,XBOOLE_1:27; then A65: LSeg(p1,p11) /\ LSeg(p11,p2) = {p11} by A63,TOPREAL1:18,ZFMISC_1:33; p1 <> p11 or p11 <> p2 by A1; hence P1 is_an_arc_of p1,p2 by A65,TOPREAL1:12; A66: {p1} = LSeg(p1,p11) /\ LSeg(p1,p10) by A3,TOPREAL1:8; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A67: LSeg(p11,p2) /\ L3 = {} by A64,XBOOLE_1:3,26; A68: LSeg(p2,p01) c= L2 by A50,Lm23,TOPREAL1:6; then A69: L1 /\ LSeg(p01,p2) c= {p01} by TOPREAL1:15,XBOOLE_1:27; A70: L3 /\ LSeg(p01,p2) = {} by A68,Lm2,XBOOLE_1:3,26; (L3 \/ L1) /\ LSeg(p01,p2) = (L3 /\ LSeg(p01,p2)) \/ (L1 /\ LSeg(p01 ,p2)) by XBOOLE_1:23 .= {p01} by A70,A69,A61,ZFMISC_1:33; then A71: L3 \/ L1 \/ LSeg(p01,p2) is_an_arc_of p10,p2 by A58,TOPREAL1:10; A72: {p2} = LSeg(p11,p2) /\ LSeg(p01,p2) by A50,TOPREAL1:8; A73: LSeg(p01,p2) \/ LSeg(p11,p2) = L2 by A50,TOPREAL1:5; LSeg(p1,p10) /\ LSeg(p01,p2) c= L4 /\ L2 by A11,A68,XBOOLE_1:27; then A74: LSeg(p1,p10) /\ LSeg(p01,p2) = {} by A56,TOPREAL1:18,ZFMISC_1:33; LSeg(p1,p10) /\ (L3 \/ L1 \/ LSeg(p01,p2)) = (LSeg(p1,p10) /\ (L3 \/ L1)) \/ (LSeg(p1,p10) /\ LSeg(p01,p2)) by XBOOLE_1:23 .= (LSeg(p1,p10) /\ L3) \/ (LSeg(p10,p1) /\ L1) by A74,XBOOLE_1:23 .= {p10} by A9,A7,A12,TOPREAL1:16,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A71,TOPREAL1:11; thus P1 \/ P2 = LSeg(p11,p2) \/ (LSeg(p1,p11) \/ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2)))) by XBOOLE_1:4 .= LSeg(p11,p2) \/ (L4 \/ (L3 \/ L1 \/ LSeg(p01,p2))) by A60,XBOOLE_1:4 .= LSeg(p11,p2) \/ (L4 \/ (L3 \/ L1) \/ LSeg(p01,p2)) by XBOOLE_1:4 .= LSeg(p11,p2) \/ (L3 \/ L4 \/ L1 \/ LSeg(p01,p2)) by XBOOLE_1:4 .= LSeg(p11,p2) \/ (L3 \/ L4 \/ (L1 \/ LSeg(p01,p2))) by XBOOLE_1:4 .= (L1 \/ LSeg(p01,p2) \/ LSeg(p11,p2)) \/ (L3 \/ L4) by XBOOLE_1:4 .= R^2-unit_square by A73,TOPREAL1:def 2,XBOOLE_1:4; A75: P1 /\ P2 = (LSeg(p1,p11) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01, p2)))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ LSeg(p1,p10)) \/ (LSeg(p1,p11) /\ (L3 \/ L1 \/ LSeg(p01,p2))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ (L3 \/ L1)) \/ (LSeg(p1,p11) /\ LSeg(p01, p2))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg(p01,p2)))) by A66, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ L1) \/ (LSeg(p1, p11) /\ LSeg(p01,p2))) \/ (LSeg(p11,p2) /\ (LSeg(p1,p10) \/ (L3 \/ L1 \/ LSeg( p01,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2))) \/ ((LSeg(p11,p2) /\ LSeg(p1,p10)) \/ (LSeg(p11,p2) /\ (L3 \/ L1 \/ LSeg(p01,p2))) ) by A59,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2))) \/ ((LSeg(p11,p2) /\ (LSeg(p1,p10)) \/ ((LSeg(p11,p2) /\ (L3 \/ L1)) \/ {p2}))) by A72,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2))) \/ ((LSeg(p11,p2) /\ (LSeg(p1,p10)) \/ (((LSeg(p11,p2) /\ L3) \/ (LSeg(p11,p2) /\ L1)) \/ {p2}))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p11) /\ L3) \/ (LSeg(p1,p11) /\ LSeg(p01,p2))) \/ ((LSeg(p11,p2) /\ LSeg(p1,p10)) \/ ((LSeg(p11,p2) /\ L1) \/ {p2})) by A67; A76: now per cases; suppose A77: p1 = p10; then A78: LSeg(p11,p2) /\ LSeg(p1,p10) = LSeg(p11,p2) /\ {p10} by RLTOPSP1:70; p10 in LSeg(p11,p2) implies contradiction by A64,Lm7,Lm9,Lm11, TOPREAL1:4; then A79: LSeg(p11,p2) /\ LSeg(p1,p10) = {} by A78,Lm1; thus P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ (( LSeg(p11,p2) /\ LSeg(p1,p10)) \/ ((LSeg(p11,p2) /\ L1) \/ {p2})) by A75,A77, TOPREAL1:16,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11,p2) /\ L1) \/ {p2}) by A79; end; suppose A80: p1 = p11; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A81: LSeg(p11,p2) /\ LSeg(p1,p10) <> {} by A80,Lm27,XBOOLE_0:def 4; LSeg(p11,p2) /\ LSeg(p1,p10) c= {p1} by A64,A80,TOPREAL1:18,XBOOLE_1:27 ; then A82: LSeg(p11,p2) /\ LSeg(p1,p10) = {p1} by A81,ZFMISC_1:33; LSeg(p1,p11) /\ L3 = {p11} /\ L3 by A80,RLTOPSP1:70; then LSeg(p1,p11) /\ L3 = {} by Lm1,Lm19; hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2))) \/ (( LSeg(p11,p2) /\ L1) \/ {p2}) by A75,A82,XBOOLE_1:4 .= {p1} \/ {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11,p2 ) /\ L1) \/ {p2}) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11,p2) /\ L1) \/ {p2}); end; suppose A83: p1 <> p11 & p1 <> p10; now assume p11 in LSeg(p11,p2) /\ LSeg(p1,p10); then A84: p11 in LSeg(p10,p1) by XBOOLE_0:def 4; p10`2 <= p1`2 by A13,A16,EUCLID:52; then p11`2 <= p1`2 by A84,TOPREAL1:4; then 1 = p1`2 by A13,A15,Lm11,XXREAL_0:1; hence contradiction by A13,A14,A83,EUCLID:53; end; then A85: {p11} <> LSeg(p11,p2) /\ LSeg(p1,p10) by ZFMISC_1:31; LSeg (p11,p2) /\ LSeg(p1,p10) c= {p11} by A11,A64,TOPREAL1:18 ,XBOOLE_1:27; then A86: LSeg(p11,p2) /\ LSeg(p1,p10) = {} by A85,ZFMISC_1:33; now assume p10 in LSeg(p1,p11) /\ L3; then A87: p10 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`2 <= p11`2 by A13,A15,EUCLID:52; then p1`2 = 0 by A13,A16,A87,Lm9,TOPREAL1:4; hence contradiction by A13,A14,A83,EUCLID:53; end; then A88: {p10} <> LSeg(p1,p11) /\ L3 by ZFMISC_1:31; LSeg(p1,p11) /\ L3 c= L4 /\ L3 by A3,Lm27,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p11) /\ L3 = {} by A88,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p11) /\ LSeg(p01,p2)) \/ ((LSeg(p11, p2) /\ L1) \/ {p2}) by A75,A86; end; end; now per cases; suppose A89: p2 = p01; then A90: LSeg(p1,p11) /\ LSeg(p01,p2) = LSeg(p1,p11) /\ {p01} by RLTOPSP1:70; not p01 in LSeg(p1,p11) by A5,Lm6,Lm8,Lm10,TOPREAL1:3; then LSeg(p1,p11) /\ LSeg(p01,p2) = {} by A90,Lm1; hence thesis by A76,A89,ENUMSET1:1,TOPREAL1:15; end; suppose A91: p2 = p11; p11 in LSeg(p1,p11) by RLTOPSP1:68; then A92: LSeg(p1,p11) /\ LSeg(p01,p2) <> {} by A91,Lm26,XBOOLE_0:def 4; LSeg(p11,p2) /\ L1 = {p11} /\ L1 by A91,RLTOPSP1:70; then A93: LSeg(p11,p2) /\ L1 = {} by Lm1,Lm18; LSeg(p1,p11) /\ LSeg(p01,p2) c= L4 /\ L2 by A5,A68,XBOOLE_1:27; then LSeg(p1,p11) /\ LSeg(p01,p2) = {p2} by A91,A92,TOPREAL1:18 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A76,A93,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A94: p2 <> p11 & p2 <> p01; now assume p01 in LSeg(p11,p2) /\ L1; then A95: p01 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`1 <= p11`1 by A51,EUCLID:52; then p2`1 = 0 by A51,A95,Lm6,TOPREAL1:3; hence contradiction by A51,A94,EUCLID:53; end; then A96: {p01} <> LSeg(p11,p2) /\ L1 by ZFMISC_1:31; LSeg (p11,p2) /\ L1 c= L2 /\ L1 by A50,Lm26,TOPREAL1:6,XBOOLE_1:26; then A97: LSeg(p11,p2) /\ L1 = {} by A96,TOPREAL1:15,ZFMISC_1:33; now assume p11 in LSeg(p1,p11) /\ LSeg(p01,p2); then A98: p11 in LSeg(p01,p2) by XBOOLE_0:def 4; p01`1 <= p2`1 by A51,EUCLID:52; then p11`1 <= p2`1 by A98,TOPREAL1:3; then 1 = p2`1 by A51,Lm10,XXREAL_0:1; hence contradiction by A51,A94,EUCLID:53; end; then A99: {p11} <> LSeg(p1,p11) /\ LSeg(p01,p2) by ZFMISC_1:31; LSeg(p1,p11) /\ LSeg(p01,p2) c= L4 /\ L2 by A5,A68,XBOOLE_1:27; then LSeg(p1,p11) /\ LSeg(p01,p2) = {} by A99,TOPREAL1:18,ZFMISC_1:33; hence thesis by A76,A97,ENUMSET1:1; end; end; hence thesis; end; suppose A100: p2 in L3; then A101: ex q st q = p2 & q`1 <= 1 & q`1 >= 0 & q`2 = 0 by TOPREAL1:13; now A102: p00`1 <= p2`1 by A101,EUCLID:52; assume A103: p10 in LSeg(p1,p11) /\ LSeg(p00,p2); then A104: p10 in LSeg(p1,p11) by XBOOLE_0:def 4; p10 in LSeg(p00,p2) by A103,XBOOLE_0:def 4; then p10`1 <= p2`1 by A102,TOPREAL1:3; then A105: 1 = p2`1 by A101,Lm8,XXREAL_0:1; p1`2 <= p11`2 by A13,A15,EUCLID:52; then 0 = p1`2 by A13,A16,A104,Lm9,TOPREAL1:4; then p1 = p10 by A13,A14,EUCLID:53; hence contradiction by A1,A101,A105,EUCLID:53; end; then A106: {p10} <> LSeg(p1,p11) /\ LSeg(p00,p2) by ZFMISC_1:31; A107: L1 is_an_arc_of p01,p00 by Lm5,Lm7,TOPREAL1:9; L2 is_an_arc_of p11,p01 by Lm6,Lm10,TOPREAL1:9; then A108: L2 \/ L1 is_an_arc_of p11,p00 by A107,TOPREAL1:2,15; take P1 = LSeg(p1,p10) \/ LSeg(p10,p2),P2 = LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2)); A109: LSeg(p1,p10) \/ LSeg(p1,p11) = L4 by A3,TOPREAL1:5; p00 in LSeg(p00,p2) by RLTOPSP1:68; then A110: L1 /\ LSeg(p00,p2) <> {} by Lm20,XBOOLE_0:def 4; A111: p10 in LSeg(p10,p2) by RLTOPSP1:68; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A112: LSeg(p1,p10) /\ LSeg(p10,p2) <> {} by A111,XBOOLE_0:def 4; A113: LSeg(p2,p10) c= L3 by A100,Lm24,TOPREAL1:6; then LSeg(p1,p10) /\ LSeg(p10,p2) c= L4 /\ L3 by A11,XBOOLE_1:27; then A114: LSeg(p1,p10) /\ LSeg(p10,p2) = {p10} by A112,TOPREAL1:16,ZFMISC_1:33; p1 <> p10 or p2 <> p10 by A1; hence P1 is_an_arc_of p1,p2 by A114,TOPREAL1:12; A115: LSeg(p1,p10) /\ LSeg(p1,p11) = {p1} by A3,TOPREAL1:8; L3 /\ L2 = {} by TOPREAL1:19,XBOOLE_0:def 7; then A116: LSeg(p10,p2) /\ L2 = {} by A113,XBOOLE_1:3,26; A117: LSeg(p2,p00) c= L3 by A100,Lm21,TOPREAL1:6; then A118: L1 /\ LSeg(p00,p2) c= {p00} by TOPREAL1:17,XBOOLE_1:27; A119: L2 /\ LSeg(p00,p2) = {} by A117,Lm2,XBOOLE_1:3,26; (L2 \/ L1) /\ LSeg(p00,p2) = (L2 /\ LSeg(p00,p2)) \/ (L1 /\ LSeg(p00 ,p2)) by XBOOLE_1:23 .= {p00} by A119,A118,A110,ZFMISC_1:33; then A120: L2 \/ L1 \/ LSeg(p00,p2) is_an_arc_of p11,p2 by A108,TOPREAL1:10; A121: LSeg(p10,p2) /\ LSeg(p00,p2) = {p2} by A100,TOPREAL1:8; A122: LSeg(p00,p2) \/ LSeg(p10,p2) = L3 by A100,TOPREAL1:5; LSeg(p1,p11) /\ LSeg(p00,p2) c= L4 /\ L3 by A5,A117,XBOOLE_1:27; then A123: LSeg(p1,p11) /\ LSeg(p00,p2) = {} by A106,TOPREAL1:16,ZFMISC_1:33; LSeg(p1,p11) /\ (L2 \/ L1 \/ LSeg(p00,p2)) = (LSeg(p1,p11) /\ (L2 \/ L1)) \/ (LSeg(p1,p11) /\ LSeg(p00,p2)) by XBOOLE_1:23 .= (LSeg(p1,p11) /\ L2) \/ (LSeg(p1,p11) /\ L1) by A123,XBOOLE_1:23 .= {p11} by A8,A6,A10,TOPREAL1:18,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A120,TOPREAL1:11; thus P1 \/ P2 = LSeg(p10,p2) \/ (LSeg(p1,p10) \/ (LSeg(p1,p11) \/ (L1 \/ L2 \/ LSeg(p00,p2)))) by XBOOLE_1:4 .= LSeg(p10,p2) \/ (L4 \/ (L1 \/ L2 \/ LSeg(p00,p2))) by A109,XBOOLE_1:4 .= (L1 \/ L2) \/ L4 \/ LSeg(p00,p2) \/ LSeg(p10,p2) by XBOOLE_1:4 .= (L1 \/ L2) \/ L4 \/ L3 by A122,XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A124: P1 /\ P2 = (LSeg(p1,p10) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00, p2)))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2)))) by XBOOLE_1:23 .= (LSeg(p1,p10) /\ LSeg(p1,p11)) \/ (LSeg(p1,p10) /\ (L2 \/ L1 \/ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2))) ) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ (L2 \/ L1)) \/ (LSeg(p1,p10) /\ LSeg(p00, p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg(p00,p2)))) by A115 ,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p10,p1) /\ L1) \/ (LSeg(p1, p10) /\ LSeg(p00,p2))) \/ (LSeg(p10,p2) /\ (LSeg(p1,p11) \/ (L2 \/ L1 \/ LSeg( p00,p2)))) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p11)) \/ (LSeg(p10,p2) /\ (L2 \/ L1 \/ LSeg(p00,p2))) ) by A12,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p11)) \/ ((LSeg(p10,p2) /\ (L2 \/ L1)) \/ {p2})) by A121,XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p11)) \/ ((LSeg(p10,p2) /\ L2) \/ (LSeg(p10,p2) /\ L1 ) \/ {p2})) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p10) /\ L2) \/ (LSeg(p1,p10) /\ LSeg(p00,p2))) \/ ((LSeg(p10,p2) /\ LSeg(p1,p11)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2})) by A116; A125: now per cases; suppose A126: p1 = p10; p10 in LSeg(p10,p2) by RLTOPSP1:68; then A127: LSeg(p10,p2) /\ LSeg(p1,p11) <> {} by A126,Lm25,XBOOLE_0:def 4; LSeg (p10,p2) /\ LSeg(p1,p11) c= {p1} by A113,A126,TOPREAL1:16 ,XBOOLE_1:27; then A128: LSeg(p10,p2) /\ LSeg(p1,p11) = {p1} by A127,ZFMISC_1:33; LSeg(p1,p10) /\ L2 = {p10} /\ L2 by A126,RLTOPSP1:70; then LSeg(p1,p10) /\ L2 = {} by Lm1,Lm17; hence P1 /\ P2 = {p1} \/ ({p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2))) \/ (( LSeg(p10,p2) /\ L1) \/ {p2}) by A124,A128,XBOOLE_1:4 .= {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2 ) /\ L1) \/ {p2}) by XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}); end; suppose A129: p1 = p11; then A130: LSeg(p10,p2) /\ LSeg(p1,p11) = LSeg(p10,p2) /\ {p11} by RLTOPSP1:70; not p11 in LSeg(p10,p2) by A113,Lm5,Lm9,Lm11,TOPREAL1:4; then LSeg(p10,p2) /\ LSeg(p1,p11) = {} by A130,Lm1; hence P1 /\ P2 = {p1} \/ {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ (( LSeg(p10,p2) /\ L1) \/ {p2}) by A124,A129,TOPREAL1:18,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10,p2) /\ L1) \/ {p2}); end; suppose A131: p1 <> p11 & p1 <> p10; now assume p10 in LSeg(p10,p2) /\ LSeg(p1,p11); then A132: p10 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`2 <= p11`2 by A13,A15,EUCLID:52; then p1`2 = 0 by A13,A16,A132,Lm9,TOPREAL1:4; hence contradiction by A13,A14,A131,EUCLID:53; end; then A133: {p10} <> LSeg(p10,p2) /\ LSeg(p1,p11) by ZFMISC_1:31; LSeg (p10,p2) /\ LSeg(p1,p11) c= {p10} by A5,A113,TOPREAL1:16 ,XBOOLE_1:27; then A134: LSeg(p10,p2) /\ LSeg(p1,p11) = {} by A133,ZFMISC_1:33; now assume p11 in LSeg(p1,p10) /\ L2; then A135: p11 in LSeg(p10,p1) by XBOOLE_0:def 4; p10`2 <= p1`2 by A13,A16,EUCLID:52; then p11`2 <= p1`2 by A135,TOPREAL1:4; then p1`2 = 1 by A13,A15,Lm11,XXREAL_0:1; hence contradiction by A13,A14,A131,EUCLID:53; end; then A136: LSeg(p1,p10) /\ L2 <> {p11} by ZFMISC_1:31; LSeg(p1,p10) /\ L2 c= L4 /\ L2 by A3,Lm25,TOPREAL1:6,XBOOLE_1:26; then LSeg(p1,p10) /\ L2 = {} by A136,TOPREAL1:18,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ (LSeg(p1,p10) /\ LSeg(p00,p2)) \/ ((LSeg(p10, p2) /\ L1) \/ {p2}) by A124,A134; end; end; now per cases; suppose A137: p2 = p00; then A138: LSeg(p1,p10) /\ LSeg(p00,p2) = LSeg(p1,p10) /\ {p00} by RLTOPSP1:70; not p00 in LSeg(p1,p10) by A11,Lm4,Lm8,Lm10,TOPREAL1:3; then LSeg(p1,p10) /\ LSeg(p00,p2) = {} by A138,Lm1; hence thesis by A125,A137,ENUMSET1:1,TOPREAL1:17; end; suppose A139: p2 = p10; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A140: LSeg(p1,p10) /\ LSeg(p00,p2) <> {} by A139,Lm24,XBOOLE_0:def 4; LSeg(p10,p2) /\ L1 = {p10} /\ L1 by A139,RLTOPSP1:70; then A141: LSeg(p10,p2) /\ L1 = {} by Lm1,Lm16; LSeg(p1,p10) /\ LSeg(p00,p2) c= L4 /\ L3 by A11,A117,XBOOLE_1:27; then LSeg(p1,p10) /\ LSeg(p00,p2) = {p2} by A139,A140,TOPREAL1:16 ,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ({p2} \/ {p2}) by A125,A141,XBOOLE_1:4 .= {p1,p2} by ENUMSET1:1; end; suppose A142: p2 <> p10 & p2 <> p00; now assume p00 in LSeg(p10,p2) /\ L1; then A143: p00 in LSeg(p2,p10) by XBOOLE_0:def 4; p2`1 <= p10`1 by A101,EUCLID:52; then p2`1 = 0 by A101,A143,Lm4,TOPREAL1:3; hence contradiction by A101,A142,EUCLID:53; end; then A144: {p00} <> LSeg(p10,p2) /\ L1 by ZFMISC_1:31; LSeg (p10,p2) /\ L1 c= L3 /\ L1 by A100,Lm24,TOPREAL1:6,XBOOLE_1:26; then A145: LSeg(p10,p2) /\ L1 = {} by A144,TOPREAL1:17,ZFMISC_1:33; now assume p10 in LSeg(p1,p10) /\ LSeg(p00,p2); then A146: p10 in LSeg(p00,p2) by XBOOLE_0:def 4; p00`1 <= p2`1 by A101,EUCLID:52; then p10`1 <= p2`1 by A146,TOPREAL1:3; then p2`1 = 1 by A101,Lm8,XXREAL_0:1; hence contradiction by A101,A142,EUCLID:53; end; then A147: {p10} <> LSeg(p1,p10) /\ LSeg(p00,p2) by ZFMISC_1:31; LSeg(p1,p10) /\ LSeg(p00,p2) c= L4 /\ L3 by A11,A117,XBOOLE_1:27; then LSeg(p1,p10) /\ LSeg(p00,p2) = {} by A147,TOPREAL1:16,ZFMISC_1:33; hence thesis by A125,A145,ENUMSET1:1; end; end; hence thesis; end; suppose A148: p2 in L4; A149: p = |[p`1,p`2]| by EUCLID:53; A150: LSeg(p1,p2) c= L4 by A3,A148,TOPREAL1:6; consider q such that A151: q = p2 and A152: q`1 = 1 and A153: q`2 <= 1 and A154: q`2 >= 0 by A148,TOPREAL1:13; A155: q = |[q`1,q`2]| by EUCLID:53; now per cases by A1,A13,A14,A151,A152,A149,A155,XXREAL_0:1; suppose A156: p`2 < q`2; A157: LSeg(p1,p2) /\ LSeg(p11,p2) c= {p2} proof let a be object; assume A158: a in LSeg(p1,p2) /\ LSeg(p11,p2); then reconsider p = a as Point of TOP-REAL 2; A159: p in LSeg(p2,p11) by A158,XBOOLE_0:def 4; p2`2 <= p11`2 by A151,A153,EUCLID:52; then A160: p2`2 <= p`2 by A159,TOPREAL1:4; A161: p in LSeg(p1,p2) by A158,XBOOLE_0:def 4; then A162: p1`1 <= p`1 by A13,A14,A151,A152,TOPREAL1:3; p`2 <= p2`2 by A13,A151,A156,A161,TOPREAL1:4; then A163: p2`2 = p`2 by A160,XXREAL_0:1; p`1 <= p2`1 by A13,A14,A151,A152,A161,TOPREAL1:3; then p`1 = 1 by A13,A14,A151,A152,A162,XXREAL_0:1; then p = |[ 1, p2`2]| by A163,EUCLID:53 .= p2 by A151,A152,EUCLID:53; hence thesis by TARSKI:def 1; end; p10 in LSeg(p1,p10) by RLTOPSP1:68; then A164: LSeg(p1,p10) /\ L3 <> {} by Lm24,XBOOLE_0:def 4; A165: now set a = the Element of LSeg(p1,p10) /\ LSeg(p11,p2); assume A166: LSeg(p1,p10) /\ LSeg(p11,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A167: p in LSeg(p10,p1) by A166,XBOOLE_0:def 4; A168: p in LSeg(p2,p11) by A166,XBOOLE_0:def 4; p2`2 <= p11 `2 by A151,A153,EUCLID:52; then A169: p2`2 <= p`2 by A168,TOPREAL1:4; p10`2 <= p1`2 by A13,A16,EUCLID:52; then p`2 <= p1`2 by A167,TOPREAL1:4; hence contradiction by A13,A151,A156,A169,XXREAL_0:2; end; A170: (L3 \/ L1) /\ L2 = L3 /\ L2 \/ L1 /\ L2 by XBOOLE_1:23 .= {p01} by Lm2,TOPREAL1:15; L3 \/ L1 is_an_arc_of p10,p01 by Lm4,Lm8,TOPREAL1:9,10,17; then A171: L3 \/ L1 \/ L2 is_an_arc_of p10,p11 by A170,TOPREAL1:10; now assume p11 in LSeg(p1,p10) /\ L2; then A172: p11 in LSeg(p10,p1) by XBOOLE_0:def 4; p10`2 <= p1`2 by A13,A16,EUCLID:52; then p11`2 <= p1`2 by A172,TOPREAL1:4; hence contradiction by A13,A15,A153,A156,Lm11,XXREAL_0:1; end; then A173: {p11} <> LSeg(p1,p10) /\ L2 by ZFMISC_1:31; LSeg(p1,p10) /\ L2 c= L4 /\ L2 by A3,Lm25,TOPREAL1:6,XBOOLE_1:26; then A174: LSeg(p1,p10) /\ L2 = {} by A173,TOPREAL1:18,ZFMISC_1:33; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A175: LSeg(p1,p10) /\ L1 = {} by A11,XBOOLE_1:3,26; now assume p10 in L3 /\ LSeg(p11,p2); then A176: p10 in LSeg(p2,p11) by XBOOLE_0:def 4; p2`2 <= p11`2 by A151,A153,EUCLID:52; hence contradiction by A16,A151,A156,A176,Lm9,TOPREAL1:4; end; then A177: {p10} <> L3 /\ LSeg(p11,p2) by ZFMISC_1:31; L3 /\ LSeg(p11,p2) c= {p10} by A148,Lm27,TOPREAL1:6,16,XBOOLE_1:26; then A178: L3 /\ LSeg(p11,p2) = {} by A177,ZFMISC_1:33; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A179: LSeg(p1,p2) /\ L1 = {} by A150,XBOOLE_1:3,26; A180: LSeg(p1,p2) /\ LSeg(p1,p10) c= {p1} proof let a be object; assume A181: a in LSeg(p1,p2) /\ LSeg(p1,p10); then reconsider p = a as Point of TOP-REAL 2; A182: p in LSeg(p10,p1) by A181,XBOOLE_0:def 4; p10`2 <= p1`2 by A13,A16,EUCLID:52; then A183: p`2 <= p1`2 by A182,TOPREAL1:4; A184: p in LSeg(p1,p2) by A181,XBOOLE_0:def 4; then A185: p1`1 <= p`1 by A13,A14,A151,A152,TOPREAL1:3; p1`2 <= p`2 by A13,A151,A156,A184,TOPREAL1:4; then A186: p1`2 = p`2 by A183,XXREAL_0:1; p`1 <= p2`1 by A13,A14,A151,A152,A184,TOPREAL1:3; then p`1 = 1 by A13,A14,A151,A152,A185,XXREAL_0:1; then p = |[ 1, p1`2]| by A186,EUCLID:53 .= p1 by A13,A14,EUCLID:53; hence thesis by TARSKI:def 1; end; A187: LSeg (p1,p10) /\ L3 c= L4 /\ L3 by A3,Lm25,TOPREAL1:6,XBOOLE_1:26; p11 in LSeg(p11,p2) by RLTOPSP1:68; then A188: L2 /\ LSeg(p11,p2) <> {} by Lm26,XBOOLE_0:def 4; L2 /\ LSeg(p11,p2) c= {p11} by A148,Lm27,TOPREAL1:6,18,XBOOLE_1:26; then A189: L2 /\ LSeg(p11,p2) = {p11} by A188,ZFMISC_1:33; take P1 = LSeg(p1,p2),P2 = LSeg(p1,p10) \/ (L3 \/ L1 \/ L2 \/ LSeg(p11 ,p2)); A190: p1 in LSeg(p1,p10) by RLTOPSP1:68; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A191: L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; L1 /\ LSeg(p11,p2) c= L1 /\ L4 by A148,Lm27,TOPREAL1:6,XBOOLE_1:26; then A192: L1 /\ LSeg(p11,p2) = {} by A191,XBOOLE_1:3; (L3 \/ L1 \/ L2) /\ LSeg(p11,p2) = (L3 \/ L1) /\ LSeg(p11,p2) \/ L2 /\ LSeg(p11,p2) by XBOOLE_1:23 .= (L3 /\ LSeg(p11,p2)) \/ (L1 /\ LSeg(p11,p2)) \/ {p11} by A189, XBOOLE_1:23 .= {p11} by A178,A192; then A193: L3 \/ L1 \/ L2 \/ LSeg(p11,p2) is_an_arc_of p10,p2 by A171,TOPREAL1:10; LSeg(p1,p10) /\ (L3 \/ L1 \/ L2 \/ LSeg(p11,p2)) = LSeg(p1,p10) /\ (L3 \/ L1 \/ L2) \/ (LSeg(p1,p10) /\ LSeg(p11,p2)) by XBOOLE_1:23 .= LSeg(p1,p10) /\ (L3 \/ L1) \/ (LSeg(p1,p10) /\ L2) by A165, XBOOLE_1:23 .= LSeg(p1,p10) /\ L3 \/ (LSeg(p1,p10) /\ L1) by A174,XBOOLE_1:23 .= {p10} by A187,A164,A175,TOPREAL1:16,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A193,TOPREAL1:11; thus P1 \/ P2 = (L3 \/ L1 \/ L2) \/ LSeg(p11,p2) \/ (LSeg(p1,p10) \/ LSeg(p1,p2)) by XBOOLE_1:4 .= (L3 \/ L1 \/ L2) \/ (LSeg(p10,p1) \/ LSeg(p1,p2) \/ LSeg(p2,p11 )) by XBOOLE_1:4 .= (L3 \/ L1 \/ L2) \/ L4 by A3,A148,TOPREAL1:7 .= (L3 \/ (L1 \/ L2)) \/ L4 by XBOOLE_1:4 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A194: p2 in LSeg(p11,p2) by RLTOPSP1:68; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p11,p2) by A194,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p11,p2) by ZFMISC_1:31; then A195: LSeg(p1,p2) /\ LSeg(p11,p2) = {p2} by A157,XBOOLE_0:def 10; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p10) by A190,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p10) by ZFMISC_1:31; then LSeg(p1,p2) /\ LSeg(p1,p10) = {p1} by A180,XBOOLE_0:def 10; then A196: P1 /\ P2 = {p1} \/ LSeg(p1,p2) /\ (L3 \/ L1 \/ L2 \/ LSeg(p11,p2 )) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L3 \/ L1 \/ L2)) \/ {p2}) by A195, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L3 \/ L1)) \/ (LSeg(p1,p2) /\ L2) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ (LSeg(p1,p2) /\ L1) \/ (LSeg(p1 ,p2) /\ L2) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2})) by A179,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L3) \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by XBOOLE_1:4; A197: LSeg(p1,p2) /\ L3 c= L4 /\ L3 by A3,A148,TOPREAL1:6,XBOOLE_1:26; A198: now per cases; suppose A199: p1 = p10; then p10 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L3 <> {} by Lm24,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L3 = {p1} by A197,A199,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A196; end; suppose A200: p1 <> p10; now assume p10 in LSeg(p1,p2) /\ L3; then p10 in LSeg(p1,p2) by XBOOLE_0:def 4; then p1`2 = 0 by A13,A16,A151,A156,Lm9,TOPREAL1:4; hence contradiction by A13,A14,A200,EUCLID:53; end; then {p10} <> LSeg(p1,p2) /\ L3 by ZFMISC_1:31; then LSeg(p1,p2) /\ L3 = {} by A197,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L2) \/ {p2}) by A196; end; end; A201: LSeg(p1,p2) /\ L2 c= L4 /\ L2 by A3,A148,TOPREAL1:6,XBOOLE_1:26; now per cases; suppose A202: p2 = p11; then p11 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L2 <> {} by Lm26,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L2 = {p2} by A201,A202,TOPREAL1:18,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A198,ENUMSET1:1; end; suppose A203: p2 <> p11; now assume p11 in LSeg(p1,p2) /\ L2; then p11 in LSeg(p1,p2) by XBOOLE_0:def 4; then p11`2 <= p2`2 by A13,A151,A156,TOPREAL1:4; then p2`2 = 1 by A151,A153,Lm11,XXREAL_0:1; hence contradiction by A151,A152,A203,EUCLID:53; end; then {p11} <> LSeg(p1,p2) /\ L2 by ZFMISC_1:31; then LSeg(p1,p2) /\ L2 = {} by A201,TOPREAL1:18,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A198,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; suppose A204: q`2 < p`2; A205: LSeg(p1,p2) /\ LSeg(p10,p2) c= {p2} proof let a be object; assume A206: a in LSeg(p1,p2) /\ LSeg(p10,p2); then reconsider p = a as Point of TOP-REAL 2; A207: p in LSeg(p10,p2) by A206,XBOOLE_0:def 4; p10`2 <= p2`2 by A151,A154,EUCLID:52; then A208: p`2 <= p2`2 by A207,TOPREAL1:4; A209: p in LSeg(p2,p1) by A206,XBOOLE_0:def 4; then A210: p2`1 <= p`1 by A13,A14,A151,A152,TOPREAL1:3; p2`2 <= p`2 by A13,A151,A204,A209,TOPREAL1:4; then A211: p2`2 = p`2 by A208,XXREAL_0:1; p`1 <= p1`1 by A13,A14,A151,A152,A209,TOPREAL1:3; then p`1 = 1 by A13,A14,A151,A152,A210,XXREAL_0:1; then p = |[ 1, p2`2]| by A211,EUCLID:53 .= p2 by A151,A152,EUCLID:53; hence thesis by TARSKI:def 1; end; p11 in LSeg(p1,p11) by RLTOPSP1:68; then A212: LSeg(p1,p11) /\ L2 <> {} by Lm26,XBOOLE_0:def 4; A213: now set a = the Element of LSeg(p1,p11) /\ LSeg(p10,p2); assume A214: LSeg(p1,p11) /\ LSeg(p10,p2) <> {}; then reconsider p = a as Point of TOP-REAL 2 by TARSKI:def 3; A215: p in LSeg(p1,p11) by A214,XBOOLE_0:def 4; A216: p in LSeg(p10,p2) by A214,XBOOLE_0:def 4; p10`2 <= p2 `2 by A151,A154,EUCLID:52; then A217: p`2 <= p2`2 by A216,TOPREAL1:4; p1`2 <= p11`2 by A13,A15,EUCLID:52; then p1`2 <= p`2 by A215,TOPREAL1:4; hence contradiction by A13,A151,A204,A217,XXREAL_0:2; end; A218: (L2 \/ L1) /\ L3 = L3 /\ L2 \/ L1 /\ L3 by XBOOLE_1:23 .= {p00} by Lm2,TOPREAL1:17; L2 \/ L1 is_an_arc_of p11,p00 by Lm6,Lm10,TOPREAL1:9,10,15; then A219: L2 \/ L1 \/ L3 is_an_arc_of p11,p10 by A218,TOPREAL1:10; now assume p11 in L2 /\ LSeg(p10,p2); then A220: p11 in LSeg(p10,p2) by XBOOLE_0:def 4; p10`2 <= p2`2 by A151,A154,EUCLID:52; then p11`2 <= p2`2 by A220,TOPREAL1:4; hence contradiction by A15,A151,A153,A204,Lm11,XXREAL_0:1; end; then A221: {p11} <> L2 /\ LSeg(p10,p2) by ZFMISC_1:31; L2 /\ LSeg(p10,p2) c= {p11} by A148,Lm25,TOPREAL1:6,18,XBOOLE_1:26; then A222: L2 /\ LSeg(p10,p2) = {} by A221,ZFMISC_1:33; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A223: LSeg(p1,p11) /\ L1 = {} by A5,XBOOLE_1:3,26; now assume p10 in LSeg(p1,p11) /\ L3; then A224: p10 in LSeg(p1,p11) by XBOOLE_0:def 4; p1`2 <= p11`2 by A13,A15,EUCLID:52; hence contradiction by A13,A154,A204,A224,Lm9,TOPREAL1:4; end; then A225: {p10} <> LSeg(p1,p11) /\ L3 by ZFMISC_1:31; LSeg(p1,p11) /\ L3 c= L4 /\ L3 by A3,Lm27,TOPREAL1:6,XBOOLE_1:26; then A226: LSeg(p1,p11) /\ L3 = {} by A225,TOPREAL1:16,ZFMISC_1:33; L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; then A227: LSeg(p1,p2) /\ L1 = {} by A150,XBOOLE_1:3,26; A228: LSeg(p1,p2) /\ LSeg(p1,p11) c= {p1} proof let a be object; assume A229: a in LSeg(p1,p2) /\ LSeg(p1,p11); then reconsider p = a as Point of TOP-REAL 2; A230: p in LSeg(p1,p11) by A229,XBOOLE_0:def 4; p1`2 <= p11`2 by A13,A15,EUCLID:52; then A231: p1`2 <= p`2 by A230,TOPREAL1:4; A232: p in LSeg(p2,p1) by A229,XBOOLE_0:def 4; then A233: p2`1 <= p`1 by A13,A14,A151,A152,TOPREAL1:3; p`2 <= p1`2 by A13,A151,A204,A232,TOPREAL1:4; then A234: p1`2 = p`2 by A231,XXREAL_0:1; p`1 <= p1`1 by A13,A14,A151,A152,A232,TOPREAL1:3; then p`1 = 1 by A13,A14,A151,A152,A233,XXREAL_0:1; then p = |[ 1, p1`2]| by A234,EUCLID:53 .= p1 by A13,A14,EUCLID:53; hence thesis by TARSKI:def 1; end; A235: LSeg(p1,p11) /\ L2 c= L4 /\ L2 by A3,Lm27,TOPREAL1:6,XBOOLE_1:26; p10 in LSeg(p10,p2) by RLTOPSP1:68; then A236: L3 /\ LSeg(p10,p2) <> {} by Lm24,XBOOLE_0:def 4; L3 /\ LSeg(p10,p2) c= {p10} by A148,Lm25,TOPREAL1:6,16,XBOOLE_1:26; then A237: L3 /\ LSeg(p10,p2) = {p10} by A236,ZFMISC_1:33; take P1 = LSeg(p1,p2),P2 = LSeg(p1,p11) \/ (L2 \/ L1 \/ L3 \/ LSeg(p10 ,p2)); A238: p1 in LSeg(p1,p11) by RLTOPSP1:68; thus P1 is_an_arc_of p1,p2 by A1,TOPREAL1:9; A239: L1 /\ L4 = {} by TOPREAL1:20,XBOOLE_0:def 7; L1 /\ LSeg(p10,p2) c= L1 /\ L4 by A148,Lm25,TOPREAL1:6,XBOOLE_1:26; then A240: L1 /\ LSeg(p10,p2) = {} by A239,XBOOLE_1:3; (L2 \/ L1 \/ L3) /\ LSeg(p10,p2) = (L2 \/ L1) /\ LSeg(p10,p2) \/ L3 /\ LSeg(p10,p2) by XBOOLE_1:23 .= (L2 /\ LSeg(p10,p2)) \/ (L1 /\ LSeg(p10,p2)) \/ {p10} by A237, XBOOLE_1:23 .= {p10} by A222,A240; then A241: L2 \/ L1 \/ L3 \/ LSeg(p10,p2) is_an_arc_of p11,p2 by A219,TOPREAL1:10; LSeg(p1,p11) /\ (L2 \/ L1 \/ L3 \/ LSeg(p10,p2)) = LSeg(p1,p11) /\ (L2 \/ L1 \/ L3) \/ (LSeg(p1,p11) /\ LSeg(p10,p2)) by XBOOLE_1:23 .= LSeg(p1,p11) /\ (L2 \/ L1) \/ (LSeg(p1,p11) /\ L3) by A213, XBOOLE_1:23 .= LSeg(p1,p11) /\ L2 \/ (LSeg(p1,p11) /\ L1) by A226,XBOOLE_1:23 .= {p11} by A235,A212,A223,TOPREAL1:18,ZFMISC_1:33; hence P2 is_an_arc_of p1,p2 by A241,TOPREAL1:11; thus P1 \/ P2 = (L2 \/ L1 \/ L3) \/ LSeg(p10,p2) \/ (LSeg(p1,p11) \/ LSeg(p1,p2)) by XBOOLE_1:4 .= (L2 \/ L1 \/ L3) \/ (LSeg(p10,p2) \/ (LSeg(p1,p2) \/ LSeg(p1, p11))) by XBOOLE_1:4 .= (L1 \/ L2 \/ L3) \/ L4 by A3,A148,TOPREAL1:7 .= R^2-unit_square by TOPREAL1:def 2,XBOOLE_1:4; A242: p2 in LSeg(p10,p2) by RLTOPSP1:68; p2 in LSeg(p1,p2) by RLTOPSP1:68; then p2 in LSeg(p1,p2) /\ LSeg(p10,p2) by A242,XBOOLE_0:def 4; then {p2} c= LSeg(p1,p2) /\ LSeg(p10,p2) by ZFMISC_1:31; then A243: LSeg(p1,p2) /\ LSeg(p10,p2) = {p2} by A205,XBOOLE_0:def 10; p1 in LSeg(p1,p2) by RLTOPSP1:68; then p1 in LSeg(p1,p2) /\ LSeg(p1,p11) by A238,XBOOLE_0:def 4; then {p1} c= LSeg(p1,p2) /\ LSeg(p1,p11) by ZFMISC_1:31; then LSeg(p1,p2) /\ LSeg(p1,p11) = {p1} by A228,XBOOLE_0:def 10; then A244: P1 /\ P2 = {p1} \/ LSeg(p1,p2) /\ (L2 \/ L1 \/ L3 \/ LSeg(p10,p2 )) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L2 \/ L1 \/ L3)) \/ {p2}) by A243, XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ (L2 \/ L1)) \/ (LSeg(p1,p2) /\ L3) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L2) \/ (LSeg(p1,p2) /\ L1) \/ (LSeg(p1 ,p2) /\ L3) \/ {p2}) by XBOOLE_1:23 .= {p1} \/ ((LSeg(p1,p2) /\ L2) \/ ((LSeg(p1,p2) /\ L3) \/ {p2})) by A227,XBOOLE_1:4 .= {p1} \/ (LSeg(p1,p2) /\ L2) \/ ((LSeg(p1,p2) /\ L3) \/ {p2}) by XBOOLE_1:4; A245: LSeg(p1,p2) /\ L2 c= L4 /\ L2 by A3,A148,TOPREAL1:6,XBOOLE_1:26; A246: now per cases; suppose A247: p1 = p11; then p11 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L2 <> {} by Lm26,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L2 = {p1} by A245,A247,TOPREAL1:18,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L3) \/ {p2}) by A244; end; suppose A248: p1 <> p11; now assume p11 in LSeg(p1,p2) /\ L2; then p11 in LSeg(p2,p1) by XBOOLE_0:def 4; then p11`2 <= p1`2 by A13,A151,A204,TOPREAL1:4; then p1`2 = 1 by A13,A15,Lm11,XXREAL_0:1; hence contradiction by A13,A14,A248,EUCLID:53; end; then {p11} <> LSeg(p1,p2) /\ L2 by ZFMISC_1:31; then LSeg(p1,p2) /\ L2 = {} by A245,TOPREAL1:18,ZFMISC_1:33; hence P1 /\ P2 = {p1} \/ ((LSeg(p1,p2) /\ L3) \/ {p2}) by A244; end; end; A249: LSeg(p1,p2) /\ L3 c= L4 /\ L3 by A3,A148,TOPREAL1:6,XBOOLE_1:26; now per cases; suppose A250: p2 = p10; then p10 in LSeg(p1,p2) by RLTOPSP1:68; then LSeg(p1,p2) /\ L3 <> {} by Lm24,XBOOLE_0:def 4; then LSeg(p1,p2) /\ L3 = {p2} by A249,A250,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A246,ENUMSET1:1; end; suppose A251: p2 <> p10; now assume p10 in LSeg(p1,p2) /\ L3; then p10 in LSeg(p2,p1) by XBOOLE_0:def 4; then p2`2 = 0 by A13,A151,A154,A204,Lm9,TOPREAL1:4; hence contradiction by A151,A152,A251,EUCLID:53; end; then {p10} <> LSeg(p1,p2) /\ L3 by ZFMISC_1:31; then LSeg(p1,p2) /\ L3 = {} by A249,TOPREAL1:16,ZFMISC_1:33; hence P1 /\ P2 = {p1,p2} by A246,ENUMSET1:1; end; end; hence P1 /\ P2 = {p1,p2}; end; end; hence thesis; end; end; theorem Th1: p1 <> p2 & p1 in R^2-unit_square & p2 in R^2-unit_square implies ex P1, P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & R^2-unit_square = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof assume that A1: p1 <> p2 and A2: p1 in R^2-unit_square and A3: p2 in R^2-unit_square; A4: p1 in L1 \/ L2 or p1 in L3 \/ L4 by A2,TOPREAL1:def 2,XBOOLE_0:def 3; per cases by A4,XBOOLE_0:def 3; suppose p1 in L1; hence thesis by A1,A3,Lm30; end; suppose p1 in L2; hence thesis by A1,A3,Lm31; end; suppose p1 in L3; hence thesis by A1,A3,Lm32; end; suppose p1 in L4; hence thesis by A1,A3,Lm33; end; end; theorem Th2: R^2-unit_square is compact proof A1: I[01] is compact by HEINE:4,TOPMETR:20; consider P1,P2 being non empty Subset of TOP-REAL 2 such that A2: P1 is being_S-P_arc and A3: P2 is being_S-P_arc and A4: R^2-unit_square = P1 \/ P2 by TOPREAL1:27; consider f being Function of I[01], (TOP-REAL 2)|P1 such that A5: f is being_homeomorphism by A2,TOPREAL1:29; A6: rng f = [#]((TOP-REAL 2)|P1) by A5; consider f0 being Function of I[01], (TOP-REAL 2)|P2 such that A7: f0 is being_homeomorphism by A3,TOPREAL1:29; A8: rng f0 = [#]((TOP-REAL 2)|P2) by A7; reconsider P2 as non empty Subset of TOP-REAL 2; f0 is continuous by A7; then (TOP-REAL 2)|P2 is compact by A1,A8,COMPTS_1:14; then A9: P2 is compact by COMPTS_1:3; reconsider P1 as non empty Subset of TOP-REAL 2; f is continuous by A5; then (TOP-REAL 2)|P1 is compact by A1,A6,COMPTS_1:14; then P1 is compact by COMPTS_1:3; hence thesis by A4,A9,COMPTS_1:10; end; theorem Th3: for Q, P being non empty Subset of TOP-REAL 2 for f being Function of (TOP-REAL 2)|Q, (TOP-REAL 2)|P st f is being_homeomorphism & Q is_an_arc_of q1,q2 holds for p1, p2 st p1 = f.q1 & p2 = f.q2 holds P is_an_arc_of p1,p2 proof let Q, P be non empty Subset of TOP-REAL 2; let f be Function of (TOP-REAL 2)|Q, (TOP-REAL 2)|P; assume that A1: f is being_homeomorphism and A2: Q is_an_arc_of q1,q2; let p1, p2 such that A3: p1 = f.q1 and A4: p2 = f.q2; reconsider f as Function of (TOP-REAL 2)|Q, (TOP-REAL 2)|P; consider f1 being Function of I[01], (TOP-REAL 2)|Q such that A5: f1 is being_homeomorphism and A6: f1.0 = q1 and A7: f1.1 = q2 by A2,TOPREAL1:def 1; set g1 = f*f1; A8: dom f1 = the carrier of I[01] by FUNCT_2:def 1; then 0 in dom f1 by BORSUK_1:40,XXREAL_1:1; then A9: g1.0 = p1 by A3,A6,FUNCT_1:13; 1 in dom f1 by A8,BORSUK_1:40,XXREAL_1:1; then A10: g1.1 = p2 by A4,A7,FUNCT_1:13; g1 is being_homeomorphism by A1,A5,TOPS_2:57; hence thesis by A9,A10,TOPREAL1:def 1; end; definition let P be Subset of TOP-REAL 2; attr P is being_simple_closed_curve means ex f being Function of ( TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P st f is being_homeomorphism; end; registration cluster R^2-unit_square -> being_simple_closed_curve; coherence proof set T = (TOP-REAL 2)|R^2-unit_square; take f = id T; thus dom f = [#]T by FUNCT_2:def 1; thus rng f = [#]T by RELAT_1:45; then f is onto one-to-one by FUNCT_2:def 3; then A1: f" = (f qua Function)" by TOPS_2:def 4 .= f by FUNCT_1:45; thus f is one-to-one; thus f is continuous by FUNCT_2:94; hence thesis by A1; end; end; registration cluster being_simple_closed_curve non empty for Subset of TOP-REAL 2; existence proof take R^2-unit_square; thus thesis; end; end; definition mode Simple_closed_curve is being_simple_closed_curve Subset of TOP-REAL 2; end; theorem Th4: for P being non empty Subset of TOP-REAL 2 st P is being_simple_closed_curve ex p1,p2 st p1 <> p2 & p1 in P & p2 in P proof reconsider RS = R^2-unit_square as non empty Subset of TOP-REAL 2; let P be non empty Subset of TOP-REAL 2; A1: p00`1 = 0 by EUCLID:52; A2: [#]((TOP-REAL 2)|P) c= [#] (TOP-REAL 2) by PRE_TOPC:def 4; A3: p11`1 = 1 by EUCLID:52; assume P is being_simple_closed_curve; then consider f being Function of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P such that A4: f is being_homeomorphism; A5: rng f = [#]((TOP-REAL 2)|P) by A4 .= P by PRE_TOPC:def 5; reconsider f as Function of (TOP-REAL 2)|RS, (TOP-REAL 2)|P; A6: dom f = [#]((TOP-REAL 2)|RS) by FUNCT_2:def 1 .= R^2-unit_square by PRE_TOPC:def 5; set p1 = f.p00, p2 = f.(p11); p00`2 = 0 by EUCLID:52; then A7: p00 in dom f by A1,A6,TOPREAL1:14; then A8: p1 in rng f by FUNCT_1:def 3; p11`2 = 1 by EUCLID:52; then A9: p11 in dom f by A3,A6,TOPREAL1:14; then A10: p2 in rng f by FUNCT_1:def 3; reconsider p1, p2 as Point of TOP-REAL 2 by A2,A8,A10; take p1, p2; f is one-to-one by A4; hence p1 <> p2 by A1,A3,A7,A9,FUNCT_1:def 4; thus thesis by A5,A7,A9,FUNCT_1:def 3; end; Lm34: for P, P1, P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1 ,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} holds P is being_simple_closed_curve proof reconsider RS = R^2-unit_square as non empty Subset of TOP-REAL 2; let P, P1, P2 be non empty Subset of TOP-REAL 2 such that A1: P1 is_an_arc_of p1,p2 and A2: P2 is_an_arc_of p1,p2 and A3: P = P1 \/ P2 and A4: P1 /\ P2 = {p1,p2}; reconsider P9 = P, P19 = P1, P29 = P2 as non empty Subset of TOP-REAL 2; A5: [#]((TOP-REAL 2)|P1) = P1 by PRE_TOPC:def 5; consider h1,h2 such that A6: h1 is being_S-Seq and A7: h2 is being_S-Seq and A8: R^2-unit_square = L~h1 \/ L~h2 and A9: L~h1 /\ L~h2 = {p00, p11} and A10: h1/.1 = p00 and A11: h1/.len h1=p11 and A12: h2/.1 = p00 and A13: h2/.len h2 = p11 by TOPREAL1:24; A14: len h2 >= 2 by A7,TOPREAL1:def 8; len h1 >= 2 by A6,TOPREAL1:def 8; then reconsider Lh1 = L~h1, Lh2 = L~h2 as non empty Subset of TOP-REAL 2 by A14,TOPREAL1:23; set T1 = (TOP-REAL 2)|Lh1, T2 = (TOP-REAL 2)|Lh2, T = (TOP-REAL 2)|RS; A15: [#] T = R^2-unit_square by PRE_TOPC:def 5; A16: [#] T2 = L~h2 by PRE_TOPC:def 5; then A17: T2 is SubSpace of T by A8,A15,TOPMETR:3,XBOOLE_1:7; A18: [#] T1 = L~h1 by PRE_TOPC:def 5; then A19: T1 is SubSpace of T by A8,A15,TOPMETR:3,XBOOLE_1:7; A20: [#]((TOP-REAL 2)|P) = P by PRE_TOPC:def 5; A21: [#]((TOP-REAL 2)|P2) = P2 by PRE_TOPC:def 5; then A22: (TOP-REAL 2)|P29 is SubSpace of (TOP-REAL 2)|P9 by A3,A20,TOPMETR:3 ,XBOOLE_1:7; consider f2 being Function of I[01], (TOP-REAL 2)|P2 such that A23: f2 is being_homeomorphism and A24: f2.0 = p1 and A25: f2.1 = p2 by A2,TOPREAL1:def 1; A26: dom f2 = the carrier of I[01] by FUNCT_2:def 1; P2 c= P by A3,XBOOLE_1:7; then rng f2 c= the carrier of (TOP-REAL 2)|P by A21,A20; then reconsider ff2=f2 as Function of I[01], (TOP-REAL 2)|P9 by A26, RELSET_1:4; A27: dom ff2 = the carrier of I[01] by FUNCT_2:def 1; then A28: 0 in dom ff2 by BORSUK_1:40,XXREAL_1:1; f2 is continuous by A23; then A29: ff2 is continuous by A22,PRE_TOPC:26; A30: 1 in dom ff2 by A27,BORSUK_1:40,XXREAL_1:1; A31: [#]((TOP-REAL 2)|P) = P by PRE_TOPC:def 5; then A32: (TOP-REAL 2)|P19 is SubSpace of (TOP-REAL 2)|P9 by A3,A5,TOPMETR:3 ,XBOOLE_1:7; consider f1 being Function of I[01], (TOP-REAL 2)|P1 such that A33: f1 is being_homeomorphism and A34: f1.0 = p1 and A35: f1.1 = p2 by A1,TOPREAL1:def 1; A36: dom f1 = the carrier of I[01] by FUNCT_2:def 1; P1 c= P by A3,XBOOLE_1:7; then rng f1 c= the carrier of (TOP-REAL 2)|P by A5,A31; then reconsider ff1=f1 as Function of I[01], (TOP-REAL 2)|P9 by A36, RELSET_1:4; A37: dom f1 = the carrier of I[01] by FUNCT_2:def 1; A38: I[01] is compact by HEINE:4,TOPMETR:20; f1 is continuous by A33; then A39: ff1 is continuous by A32,PRE_TOPC:26; A40: f1 is one-to-one by A33; reconsider L1 = L~h1, L2 = L~h2 as non empty Subset of TOP-REAL 2 by A9; L1 is_an_arc_of p00,p11 by A6,A10,A11,TOPREAL1:25; then consider g1 being Function of I[01], (TOP-REAL 2)|L1 such that A41: g1 is being_homeomorphism and A42: g1.0 = p00 and A43: g1.1 = p11 by TOPREAL1:def 1; L2 is_an_arc_of p00,p11 by A7,A12,A13,TOPREAL1:25; then consider g2 being Function of I[01], (TOP-REAL 2)|L2 such that A44: g2 is being_homeomorphism and A45: g2.0 = p00 and A46: g2.1 = p11 by TOPREAL1:def 1; R^2-unit_square = [#] (T) by PRE_TOPC:def 5 .= the carrier of T; then reconsider p00,p11 as Point of T by Lm28,Lm29,TOPREAL1:14; A47: T is T_2 by TOPMETR:2; set k1 = ff1*(g1"), k2 = ff2*(g2"); reconsider g1 as Function of I[01], (TOP-REAL 2)|Lh1; A48: g1 is one-to-one by A41; A49: dom g1 = the carrier of I[01] by FUNCT_2:def 1; A50: rng g1 = [#](T1) by A41; then g1 is onto by FUNCT_2:def 3; then A51: g1" = (g1 qua Function)" by A48,TOPS_2:def 4; then rng(g1") = dom g1 by A48,FUNCT_1:33; then A52: rng k1 = rng f1 by A37,A49,RELAT_1:28 .= P1 by A33,A5; A53: dom g1 = the carrier of I[01] by FUNCT_2:def 1; then A54: 0 in dom g1 by BORSUK_1:40,XXREAL_1:1; then A55: 0 = (g1").p00 by A42,A48,A51,FUNCT_1:32; A56: dom(g1") = rng g1 by A48,A51,FUNCT_1:32; then A57: p00 in dom (g1") by A42,A54,FUNCT_1:def 3; A58: 1 in dom g1 by A53,BORSUK_1:40,XXREAL_1:1; then A59: p11 in dom (g1") by A43,A56,FUNCT_1:def 3; reconsider g2 as Function of I[01], (TOP-REAL 2)|Lh2; A60: g2 is one-to-one by A44; A61: rng g2 = [#](T2) by A44; then g2 is onto by FUNCT_2:def 3; then A62: g2" = (g2 qua Function)" by A60,TOPS_2:def 4; g2 is continuous by A44; then A63: T2 is compact by A38,A61,COMPTS_1:14; A64: g2" is continuous by A44; g1 is continuous by A41; then A65: T1 is compact by A38,A50,COMPTS_1:14; A66: f2 is one-to-one by A23; A67: dom g2 = the carrier of I[01] by FUNCT_2:def 1; then A68: 0 in dom g2 by BORSUK_1:40,XXREAL_1:1; then A69: p00 in rng g2 by A45,FUNCT_1:def 3; then A70: p00 in dom (g2") by A60,A62,FUNCT_1:32; (g2").p00 in dom ff2 by A45,A60,A62,A53,A67,A27,A54,FUNCT_1:32; then A71: p00 in dom(ff2*(g2")) by A70,FUNCT_1:11; A72: dom ff1 = the carrier of I[01] by FUNCT_2:def 1; then (g1").p00 in dom ff1 by A42,A48,A51,A53,A54,FUNCT_1:32; then p00 in dom(ff1*(g1")) by A57,FUNCT_1:11; then A73: k1.p00 = ff1.((g1").p00) by FUNCT_1:12 .= p1 by A34,A42,A48,A51,A54,FUNCT_1:32; then A74: k1.p00 = ff2.((g2").p00) by A24,A45,A60,A62,A68,FUNCT_1:32 .= k2.p00 by A71,FUNCT_1:12; A75: 1 in dom g2 by A67,BORSUK_1:40,XXREAL_1:1; then A76: 1 = (g2").p11 by A46,A60,A62,FUNCT_1:32; A77: dom(g2") = rng g2 by A60,A62,FUNCT_1:32; then A78: p11 in dom (g2") by A46,A75,FUNCT_1:def 3; (g2").p11 in dom ff2 by A46,A60,A62,A53,A67,A27,A58,FUNCT_1:32; then A79: p11 in dom(ff2*(g2")) by A78,FUNCT_1:11; (g1").p11 in dom ff1 by A43,A48,A51,A53,A72,A58,FUNCT_1:32; then p11 in dom(ff1*(g1")) by A59,FUNCT_1:11; then A80: k1.p11 = ff1.((g1").p11) by FUNCT_1:12 .= p2 by A35,A43,A48,A51,A58,FUNCT_1:32; then A81: k1.p11 = ff2.((g2").p11) by A25,A46,A60,A62,A75,FUNCT_1:32 .= k2.p11 by A79,FUNCT_1:12; g1" is continuous by A41; then reconsider h = k1+*k2 as continuous Function of T,(TOP-REAL 2)|P by A8,A9,A39,A29,A18 ,A16,A15,A65,A63,A47,A64,A74,A81,A19,A17,COMPTS_1:21; A82: 1 = (g1").p11 by A43,A48,A51,A58,FUNCT_1:32; A83: rng(g2") = dom g2 by A60,A62,FUNCT_1:33; then A84: rng k2 = rng f2 by A67,A27,RELAT_1:28 .= [#] ((TOP-REAL 2)|P2) by A23 .= P2 by PRE_TOPC:def 5; A85: 0 = (g2").p00 by A45,A60,A62,A68,FUNCT_1:32; now let x1,x2 be set; assume that A86: x1 in dom k2 and A87: x2 in dom k1 \ dom k2; A88: x1 in dom(g2") by A86,FUNCT_1:11; A89: k2.x1 in P2 by A84,A86,FUNCT_1:def 3; A90: x2 in dom k1 by A87,XBOOLE_0:def 5; then A91: x2 in dom(g1") by FUNCT_1:11; assume A92: k2.x1 = k1.x2; then k2.x1 in P1 by A52,A90,FUNCT_1:def 3; then A93: k2.x1 in P1 /\ P2 by A89,XBOOLE_0:def 4; per cases by A4,A93,TARSKI:def 2; suppose A94: k2.x1 = p1; A95: (g1").x2 in dom ff1 by A90,FUNCT_1:11; p1 = ff1.((g1").x2) by A92,A90,A94,FUNCT_1:12; then A96: (g1").x2 = 0 by A34,A72,A28,A40,A95,FUNCT_1:def 4; A97: p00 in dom (g2") by A60,A62,A69,FUNCT_1:32; A98: (g2").x1 in dom ff2 by A86,FUNCT_1:11; p1 = ff2.((g2").x1) by A86,A94,FUNCT_1:12; then (g2").x1 = 0 by A24,A28,A66,A98,FUNCT_1:def 4; then A99: x1 = p00 by A60,A62,A85,A88,A97,FUNCT_1:def 4; p00 in dom(g1") by A42,A53,A28,A56,FUNCT_1:def 3; then x2 in dom k2 by A48,A51,A55,A86,A91,A99,A96,FUNCT_1:def 4; hence contradiction by A87,XBOOLE_0:def 5; end; suppose A100: k2.x1 = p2; A101: (g1").x2 in dom ff1 by A90,FUNCT_1:11; p2 = ff1.((g1").x2) by A92,A90,A100,FUNCT_1:12; then A102: (g1").x2 = 1 by A35,A72,A30,A40,A101,FUNCT_1:def 4; A103: p11 in dom (g2") by A46,A67,A77,A30,FUNCT_1:def 3; A104: (g2").x1 in dom ff2 by A86,FUNCT_1:11; p2 = ff2.((g2").x1) by A86,A100,FUNCT_1:12; then (g2").x1 = 1 by A25,A30,A66,A104,FUNCT_1:def 4; then A105: x1 = p11 by A60,A62,A76,A88,A103,FUNCT_1:def 4; p11 in dom(g1") by A43,A53,A56,A30,FUNCT_1:def 3; then x2 in dom k2 by A48,A51,A82,A86,A91,A105,A102,FUNCT_1:def 4; hence contradiction by A87,XBOOLE_0:def 5; end; end; then A106: h is one-to-one by A48,A60,A62,A51,A40,A66,TOPMETR2:1; A107: (TOP-REAL 2)|P9 is T_2 by TOPMETR:2; A108: dom k2 = dom(g2") by A27,A83,RELAT_1:27; k1.:(dom k1 /\ dom k2) c= rng k2 proof let a be object; A109: dom k2 = the carrier of T2 by FUNCT_2:def 1; assume a in k1.:(dom k1 /\ dom k2); then A110: ex x being object st x in dom k1 & x in dom k1 /\ dom k2 & a = k1.x by FUNCT_1:def 6; dom k1 = the carrier of T1 by FUNCT_2:def 1; then a = p1 or a = p2 by A9,A18,A16,A73,A80,A110,A109,TARSKI:def 2; hence thesis by A70,A73,A74,A78,A80,A81,A108,FUNCT_1:def 3; end; then A111: rng h = [#]((TOP-REAL 2)|P9) by A3,A31,A52,A84,TOPMETR2:2; reconsider h as Function of ((TOP-REAL 2)|R^2-unit_square),(TOP-REAL 2)|P; take h; T is compact by Th2,COMPTS_1:3; hence thesis by A107,A111,A106,COMPTS_1:17; end; theorem Th5: for P being non empty Subset of TOP-REAL 2 holds P is being_simple_closed_curve iff (ex p1,p2 st p1 <> p2 & p1 in P & p2 in P) & for p1,p2 st p1 <> p2 & p1 in P & p2 in P ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof let P be non empty Subset of TOP-REAL 2; thus P is being_simple_closed_curve implies (ex p1,p2 st p1 <> p2 & p1 in P & p2 in P) & for p1,p2 st p1 <> p2 & p1 in P & p2 in P ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof assume A1: P is being_simple_closed_curve; then consider f being Function of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P such that A2: f is being_homeomorphism; A3: dom f = [#]((TOP-REAL 2)|R^2-unit_square) by A2; A4: [#]((TOP-REAL 2)|P) c= [#](TOP-REAL 2) by PRE_TOPC:def 4; A5: f is continuous by A2; thus ex p1,p2 st p1 <> p2 & p1 in P & p2 in P by A1,Th4; set RS = R^2-unit_square; let p1,p2; assume that A6: p1 <> p2 and A7: p1 in P and A8: p2 in P; A9: [#]((TOP-REAL 2)|R^2-unit_square) = R^2-unit_square by PRE_TOPC:def 5; set q1 = (f").p1, q2 = (f").p2; A10: [#]((TOP-REAL 2)|RS) c= [#](TOP-REAL 2) by PRE_TOPC:def 4; A11: I[01] is compact by HEINE:4,TOPMETR:20; A12: f is one-to-one by A2; A13: rng f = [#]((TOP-REAL 2)|P) by A2; then f is onto by FUNCT_2:def 3; then A14: f" = (f qua Function)" by A12,TOPS_2:def 4; then A15: rng(f") = dom f by A12,FUNCT_1:33; A16: dom(f") = rng f by A12,A14,FUNCT_1:32; then A17: p1 in dom(f") by A7,A13,PRE_TOPC:def 5; A18: p2 in dom(f") by A8,A13,A16,PRE_TOPC:def 5; reconsider f as Function of (TOP-REAL 2)|RS, (TOP-REAL 2)|P; A19: q1 in rng(f") by A17,FUNCT_1:def 3; A20: q2 in rng(f") by A18,FUNCT_1:def 3; reconsider q1, q2 as Point of TOP-REAL 2 by A10,A19,A20; A21: q1 <> q2 by A6,A12,A14,A17,A18,FUNCT_1:def 4; A22: dom f = the carrier of (TOP-REAL 2)|R^2-unit_square by FUNCT_2:def 1; then A23: q2 in R^2-unit_square by A15,A18,A9,FUNCT_1:def 3; A24: p1 = f.q1 by A12,A14,A16,A17,FUNCT_1:35; q1 in R^2-unit_square by A15,A17,A22,A9,FUNCT_1:def 3; then consider Q1,Q2 being non empty Subset of TOP-REAL 2 such that A25: Q1 is_an_arc_of q1,q2 and A26: Q2 is_an_arc_of q1,q2 and A27: R^2-unit_square = Q1 \/ Q2 and A28: Q1 /\ Q2 = {q1,q2} by A21,A23,Th1; A29: Q2 c= dom f by A22,A9,A27,XBOOLE_1:7; set P1 = f.:Q1, P2 = f.:Q2; Q1 c= dom f by A22,A9,A27,XBOOLE_1:7; then reconsider P1, P2 as non empty Subset of TOP-REAL 2 by A29,A4, XBOOLE_1:1; A30: rng(f|Q1) = P1 by RELAT_1:115 .= [#]((TOP-REAL 2)|P1) by PRE_TOPC:def 5 .= the carrier of (TOP-REAL 2)|P1; dom(f|Q1) = R^2-unit_square /\ Q1 by A22,A9,RELAT_1:61 .= Q1 by A27,XBOOLE_1:21 .= [#]((TOP-REAL 2)|Q1) by PRE_TOPC:def 5; then reconsider F1 = f|Q1 as Function of (TOP-REAL 2)|Q1, (TOP-REAL 2)|P1 by A30, FUNCT_2:def 1,RELSET_1:4; A31: f"P1 c= Q1 by A12,FUNCT_1:82; [#]((TOP-REAL 2)|Q1) = Q1 by PRE_TOPC:def 5; then A32: (TOP-REAL 2)|Q1 is SubSpace of (TOP-REAL 2)|R^2-unit_square by A9,A27, TOPMETR:3,XBOOLE_1:7; Q1 c= f"P1 by A22,A9,A27,FUNCT_1:76,XBOOLE_1:7; then A33: f"P1 = Q1 by A31,XBOOLE_0:def 10; for R being Subset of (TOP-REAL 2)|P1 st R is closed holds F1"R is closed proof let R be Subset of (TOP-REAL 2)|P1; assume R is closed; then consider S1 being Subset of TOP-REAL 2 such that A34: S1 is closed and A35: R = S1 /\ [#]((TOP-REAL 2)|P1) by PRE_TOPC:13; S1 /\ rng f is Subset of (TOP-REAL 2)|P; then reconsider S2 = rng f /\ S1 as Subset of (TOP-REAL 2)|P; S2 is closed by A13,A34,PRE_TOPC:13; then A36: f"S2 is closed by A5; F1"R = Q1 /\ (f"R) by FUNCT_1:70 .= Q1 /\ ((f"S1) /\ f"([#]((TOP-REAL 2)|P1))) by A35,FUNCT_1:68 .= (f"S1) /\ Q1 /\ Q1 by A33,PRE_TOPC:def 5 .= (f"S1) /\ (Q1 /\ Q1) by XBOOLE_1:16 .= (f"S1) /\ [#]((TOP-REAL 2)|Q1) by PRE_TOPC:def 5 .= (f"(rng f /\ S1)) /\ [#]((TOP-REAL 2)|Q1) by RELAT_1:133; hence thesis by A32,A36,PRE_TOPC:13; end; then A37: F1 is continuous; reconsider Q19=Q1, Q29=Q2 as non empty Subset of TOP-REAL 2; consider ff being Function of I[01], (TOP-REAL 2)|Q1 such that A38: ff is being_homeomorphism and ff.0 = q1 and ff.1 = q2 by A25,TOPREAL1:def 1; A39: rng ff = [#]((TOP-REAL 2)|Q1) by A38; A40: rng(f|Q2) = P2 by RELAT_1:115 .= [#]((TOP-REAL 2)|P2) by PRE_TOPC:def 5 .= the carrier of (TOP-REAL 2)|P2; A41: p2 = f.q2 by A12,A14,A16,A18,FUNCT_1:35; dom(f|Q2) = R^2-unit_square /\ Q2 by A22,A9,RELAT_1:61 .= Q2 by A27,XBOOLE_1:21 .= [#]((TOP-REAL 2)|Q2) by PRE_TOPC:def 5; then reconsider F2 = f|Q2 as Function of (TOP-REAL 2)|Q2, (TOP-REAL 2)|P2 by A40, FUNCT_2:def 1,RELSET_1:4; A42: f"P2 c= Q2 by A12,FUNCT_1:82; [#]((TOP-REAL 2)|Q2) = Q2 by PRE_TOPC:def 5; then A43: (TOP-REAL 2)|Q2 is SubSpace of (TOP-REAL 2)|R^2-unit_square by A9,A27, TOPMETR:3,XBOOLE_1:7; Q2 c= f"P2 by A22,A9,A27,FUNCT_1:76,XBOOLE_1:7; then A44: f"P2 = Q2 by A42,XBOOLE_0:def 10; for R being Subset of (TOP-REAL 2)|P2 st R is closed holds F2"R is closed proof let R be Subset of (TOP-REAL 2)|P2; assume R is closed; then consider S1 being Subset of TOP-REAL 2 such that A45: S1 is closed and A46: R = S1 /\ [#]((TOP-REAL 2)|P2) by PRE_TOPC:13; S1 /\ rng f is Subset of (TOP-REAL 2)|P; then reconsider S2 = rng f /\ S1 as Subset of (TOP-REAL 2)|P; S2 is closed by A13,A45,PRE_TOPC:13; then A47: f"S2 is closed by A5; F2"R = Q2 /\ (f"R) by FUNCT_1:70 .= Q2 /\ ((f"S1) /\ f"([#]((TOP-REAL 2)|P2))) by A46,FUNCT_1:68 .= (f"S1) /\ Q2 /\ Q2 by A44,PRE_TOPC:def 5 .= (f"S1) /\ (Q2 /\ Q2) by XBOOLE_1:16 .= (f"S1) /\ [#]((TOP-REAL 2)|Q2) by PRE_TOPC:def 5 .= (f"(rng f /\ S1)) /\ [#]((TOP-REAL 2)|Q2) by RELAT_1:133; hence thesis by A43,A47,PRE_TOPC:13; end; then A48: F2 is continuous; A49: q2 in {q1,q2} by TARSKI:def 2; A50: q1 in {q1,q2} by TARSKI:def 2; A51: q1 in {q1,q2} by TARSKI:def 2; {q1,q2} c= Q1 by A28,XBOOLE_1:17; then A52: q1 in dom f /\ Q1 by A15,A19,A51,XBOOLE_0:def 4; take P1,P2; A53: (TOP-REAL 2)|P1 is T_2 by TOPMETR:2; A54: q2 in {q1,q2} by TARSKI:def 2; {q1,q2} c= Q1 by A28,XBOOLE_1:17; then A55: q2 in dom f /\ Q1 by A15,A20,A54,XBOOLE_0:def 4; A56: p2 = f.q2 by A12,A14,A16,A18,FUNCT_1:35 .= F1.q2 by A55,FUNCT_1:48; A57: rng F1 = [#]((TOP-REAL 2)|P1) by A30; ff is continuous by A38; then A58: (TOP-REAL 2)|Q19 is compact by A11,A39,COMPTS_1:14; A59: F1 is one-to-one by A12,FUNCT_1:52; p1 = f.q1 by A12,A14,A16,A17,FUNCT_1:35 .= F1.q1 by A52,FUNCT_1:48; hence P1 is_an_arc_of p1,p2 by A25,A57,A59,A37,A58,A53,A56,Th3, COMPTS_1:17; A60: (TOP-REAL 2)|P2 is T_2 by TOPMETR:2; consider ff being Function of I[01], (TOP-REAL 2)|Q2 such that A61: ff is being_homeomorphism and ff.0 = q1 and ff.1 = q2 by A26,TOPREAL1:def 1; A62: rng ff = [#]((TOP-REAL 2)|Q2) by A61; {q1,q2} c= Q2 by A28,XBOOLE_1:17; then q1 in dom f /\ Q2 by A15,A19,A50,XBOOLE_0:def 4; then A63: p1 = F2.q1 by A24,FUNCT_1:48; A64: F2 is one-to-one by A12,FUNCT_1:52; {q1,q2} c= Q2 by A28,XBOOLE_1:17; then q2 in dom f /\ Q2 by A15,A20,A49,XBOOLE_0:def 4; then A65: p2 = F2.q2 by A41,FUNCT_1:48; ff is continuous by A61; then A66: (TOP-REAL 2)|Q29 is compact by A11,A62,COMPTS_1:14; rng F2 = [#]((TOP-REAL 2)|P2) by A40; hence P2 is_an_arc_of p1,p2 by A26,A64,A48,A66,A60,A63,A65,Th3, COMPTS_1:17; [#]((TOP-REAL 2)|P) = P by PRE_TOPC:def 5; hence P = f.:(Q1 \/ Q2) by A13,A3,A9,A27,RELAT_1:113 .= P1 \/ P2 by RELAT_1:120; thus P1 /\ P2 = f.:(Q1 /\ Q2) by A12,FUNCT_1:62 .= f.:({q1} \/ {q2}) by A28,ENUMSET1:1 .= Im(f,q1) \/ Im(f,q2) by RELAT_1:120 .= {p1} \/ Im(f,q2) by A15,A19,A24,FUNCT_1:59 .= {p1} \/ {p2} by A15,A20,A41,FUNCT_1:59 .= {p1,p2} by ENUMSET1:1; end; given p1,p2 such that A67: p1 <> p2 and A68: p1 in P and A69: p2 in P; assume for p1,p2 st p1 <> p2 & p1 in P & p2 in P ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2}; then ex P1,P2 being non empty Subset of TOP-REAL 2 st P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by A67,A68,A69; hence thesis by Lm34; end; theorem for P being non empty Subset of TOP-REAL 2 holds P is being_simple_closed_curve iff ex p1,p2 being Point of TOP-REAL 2, P1,P2 being non empty Subset of TOP-REAL 2 st p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} proof let P be non empty Subset of TOP-REAL 2; hereby assume A1: P is being_simple_closed_curve; then consider p1,p2 such that A2: p1 <> p2 and A3: p1 in P and A4: p2 in P by Th5; consider P1,P2 being non empty Subset of TOP-REAL 2 such that A5: P1 is_an_arc_of p1,p2 and A6: P2 is_an_arc_of p1,p2 and A7: P = P1 \/ P2 and A8: P1 /\ P2 = {p1,p2} by A1,A2,A3,A4,Th5; take p1,p2,P1,P2; thus p1 <> p2 & p1 in P & p2 in P & P1 is_an_arc_of p1,p2 & P2 is_an_arc_of p1,p2 & P = P1 \/ P2 & P1 /\ P2 = {p1,p2} by A2,A3,A4,A5,A6,A7,A8; end; thus thesis by Lm34; end; Lm35: for S being 1-sorted, T being 1-sorted, f being Function of S, T st S is empty & rng f = [#]T holds T is empty proof let S be 1-sorted, T be 1-sorted, f be Function of S, T such that A1: S is empty and A2: rng f = [#]T; assume T is non empty; then reconsider T as non empty 1-sorted; consider y being object such that A3: y in the carrier of T by XBOOLE_0:def 1; ex x being object st x in dom f & f.x = y by A2,A3,FUNCT_1:def 3; hence contradiction by A1; end; Lm36: for S being 1-sorted, T being 1-sorted, f being Function of S, T st T is empty & dom f = [#]S holds S is empty proof let S be 1-sorted, T be 1-sorted, f be Function of S, T such that A1: T is empty and A2: dom f = [#]S; assume S is non empty; then reconsider S as non empty 1-sorted; consider x being object such that A3: x in the carrier of S by XBOOLE_0:def 1; f.x in rng f by A2,A3,FUNCT_1:def 3; hence thesis by A1; end; Lm37: for S, T being TopStruct st ex f being Function of S,T st f is being_homeomorphism holds S is empty iff T is empty by Lm35,Lm36; registration cluster being_simple_closed_curve -> non empty compact for Subset of TOP-REAL 2; coherence proof let P be Subset of TOP-REAL 2; given f being Function of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P such that A1: f is being_homeomorphism; thus P is non empty by A1,Lm37; A2: rng f = [#]((TOP-REAL 2)|P) by A1; reconsider R = P as non empty Subset of TOP-REAL 2 by A1,Lm37; A3: f is continuous by A1; (TOP-REAL 2)|R^2-unit_square is compact by Th2,COMPTS_1:3; then (TOP-REAL 2)|R is compact by A3,A2,COMPTS_1:14; hence thesis by COMPTS_1:3; end; end;