environ vocabulary ARYTM, RCOMP_1, BOOLE, PRE_TOPC, TOPMETR, PCOMPS_1, METRIC_1, SUBSET_1, ARYTM_1, FUNCT_1, ABSVALUE, TARSKI, SETFAM_1, SEQ_1, BORSUK_1, TMAP_1, RELAT_1, ORDINAL2, ARYTM_3, TOPS_2, RELAT_2, SEQ_2, SEQ_4, SQUARE_1, FUNCT_3, TREAL_1; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2, ABSVALUE, REAL_1, SQUARE_1, SEQ_4, RCOMP_1, PRE_TOPC, TOPS_2, CONNSP_1, METRIC_1, GRCAT_1, STRUCT_0, PCOMPS_1, TOPMETR, TSEP_1, TMAP_1, BORSUK_1; constructors ABSVALUE, REAL_1, SQUARE_1, SEQ_4, RCOMP_1, TOPS_2, CONNSP_1, GRCAT_1, TOPMETR, TMAP_1, PARTFUN1, MEMBERED; clusters SUBSET_1, FUNCT_1, PRE_TOPC, TOPMETR, BORSUK_1, STRUCT_0, XREAL_0, METRIC_1, RELSET_1, MEMBERED, ZFMISC_1; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin :: 1. Properties of Topological Intervals. reserve a,b,c,d for real number; theorem :: TREAL_1:1 a <= c & d <= b implies [.c,d.] c= [.a,b.]; theorem :: TREAL_1:2 a <= c & b <= d & c <= b implies [.a,b.] \/ [.c,d.] = [.a,d.]; theorem :: TREAL_1:3 a <= c & b <= d & c <= b implies [.a,b.] /\ [.c,d.] = [.c,b.]; theorem :: TREAL_1:4 for A being Subset of R^1 st A = [.a,b.] holds A is closed; theorem :: TREAL_1:5 a <= b implies Closed-Interval-TSpace(a,b) is closed SubSpace of R^1; theorem :: TREAL_1:6 a <= c & d <= b & c <= d implies Closed-Interval-TSpace(c,d) is closed SubSpace of Closed-Interval-TSpace(a,b); theorem :: TREAL_1:7 a <= c & b <= d & c <= b implies Closed-Interval-TSpace(a,d) = Closed-Interval-TSpace(a,b) union Closed-Interval-TSpace(c,d) & Closed-Interval-TSpace(c,b) = Closed-Interval-TSpace(a,b) meet Closed-Interval-TSpace(c,d); definition let a,b be real number; assume a <= b; func (#)(a,b) -> Point of Closed-Interval-TSpace(a,b) equals :: TREAL_1:def 1 a; func (a,b)(#) -> Point of Closed-Interval-TSpace(a,b) equals :: TREAL_1:def 2 b; end; theorem :: TREAL_1:8 0[01] = (#)(0,1) & 1[01] = (0,1)(#); theorem :: TREAL_1:9 a <= b & b <= c implies (#)(a,b) = (#)(a,c) & (b,c)(#) = (a,c)(#); begin :: 2. Continuous Mappings Between Topological Intervals. definition let a,b be real number such that a <= b; let t1,t2 be Point of Closed-Interval-TSpace(a,b); func L[01](t1,t2) -> map of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(a,b) means :: TREAL_1:def 3 for s being Point of Closed-Interval-TSpace(0,1), r,r1,r2 being real number st s=r & r1=t1 & r2=t2 holds it.s = (1-r)*r1 + r*r2; end; theorem :: TREAL_1:10 a <= b implies for t1,t2 being Point of Closed-Interval-TSpace(a,b) holds for s being Point of Closed-Interval-TSpace(0,1), r,r1,r2 being real number st s=r & r1=t1 & r2=t2 holds L[01](t1,t2).s = (r2 - r1)*r + r1; theorem :: TREAL_1:11 a <= b implies for t1,t2 being Point of Closed-Interval-TSpace(a,b) holds L[01](t1,t2) is continuous map of Closed-Interval-TSpace(0,1),Closed-Interval-TSpace(a,b); theorem :: TREAL_1:12 a <= b implies for t1,t2 being Point of Closed-Interval-TSpace(a,b) holds L[01](t1,t2).(#)(0,1) = t1 & L[01](t1,t2).(0,1)(#) = t2; theorem :: TREAL_1:13 L[01]((#)(0,1),(0,1)(#)) = id Closed-Interval-TSpace(0,1); definition let a,b be real number such that a < b; let t1,t2 be Point of Closed-Interval-TSpace(0,1); func P[01](a,b,t1,t2) -> map of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(0,1) means :: TREAL_1:def 4 for s being Point of Closed-Interval-TSpace(a,b), r,r1,r2 being real number st s=r & r1=t1 & r2=t2 holds it.s = ((b-r)*r1 + (r-a)*r2)/(b-a); end; theorem :: TREAL_1:14 a < b implies for t1,t2 being Point of Closed-Interval-TSpace(0,1) for s being Point of Closed-Interval-TSpace(a,b), r,r1,r2 being Real st s=r & r1=t1 & r2=t2 holds P[01](a,b,t1,t2).s = ((r2 - r1)/(b-a))*r + (b*r1 -a*r2)/(b-a); theorem :: TREAL_1:15 a < b implies for t1,t2 being Point of Closed-Interval-TSpace(0,1) holds P[01](a,b,t1,t2) is continuous map of Closed-Interval-TSpace(a,b),Closed-Interval-TSpace(0,1); theorem :: TREAL_1:16 a < b implies for t1,t2 being Point of Closed-Interval-TSpace(0,1) holds P[01](a,b,t1,t2).(#)(a,b) = t1 & P[01](a,b,t1,t2).(a,b)(#) = t2; theorem :: TREAL_1:17 P[01](0,1,(#)(0,1),(0,1)(#)) = id Closed-Interval-TSpace(0,1); theorem :: TREAL_1:18 a < b implies id Closed-Interval-TSpace(a,b) = L[01]((#)(a,b),(a,b)(#)) * P[01](a,b,(#)(0,1),(0,1)(#)) & id Closed-Interval-TSpace(0,1) = P[01](a,b,(#)(0,1),(0,1)(#)) * L[01]((#)(a,b),(a,b)(#)); theorem :: TREAL_1:19 a < b implies id Closed-Interval-TSpace(a,b) = L[01]((a,b)(#),(#)(a,b)) * P[01](a,b,(0,1)(#),(#)(0,1)) & id Closed-Interval-TSpace(0,1) = P[01](a,b,(0,1)(#),(#)(0,1)) * L[01]((a,b)(#),(#)(a,b)); theorem :: TREAL_1:20 a < b implies L[01]((#)(a,b),(a,b)(#)) is_homeomorphism & L[01]((#)(a,b),(a,b)(#))" = P[01](a,b,(#)(0,1),(0,1)(#)) & P[01](a,b,(#)(0,1),(0,1)(#)) is_homeomorphism & P[01](a,b,(#)(0,1),(0,1)(#))" = L[01]((#)(a,b),(a,b)(#)); theorem :: TREAL_1:21 a < b implies L[01]((a,b)(#),(#)(a,b)) is_homeomorphism & L[01]((a,b)(#),(#)(a,b))" = P[01](a,b,(0,1)(#),(#)(0,1)) & P[01](a,b,(0,1)(#),(#)(0,1)) is_homeomorphism & P[01](a,b,(0,1)(#),(#)(0,1))" = L[01]((a,b)(#),(#)(a,b)); begin :: 3. Connectedness of Intervals and Brouwer Fixed Point Theorem for Intervals. theorem :: TREAL_1:22 I[01] is connected; theorem :: TREAL_1:23 a <= b implies Closed-Interval-TSpace(a,b) is connected; theorem :: TREAL_1:24 for f being continuous map of I[01],I[01] ex x being Point of I[01] st f.x = x; theorem :: TREAL_1:25 a <= b implies for f being continuous map of Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(a,b) ex x being Point of Closed-Interval-TSpace(a,b) st f.x = x; theorem :: TREAL_1:26 for X, Y being non empty SubSpace of R^1, f being continuous map of X,Y holds (ex a,b being Real st a <= b & [.a,b.] c= the carrier of X & [.a,b.] c= the carrier of Y & f.:[.a,b.] c= [.a,b.]) implies ex x being Point of X st f.x = x; theorem :: TREAL_1:27 for X, Y being non empty SubSpace of R^1, f being continuous map of X,Y holds (ex a,b being Real st a <= b & [.a,b.] c= the carrier of X & f.:[.a,b.] c= [.a,b.]) implies ex x being Point of X st f.x = x;