environ
vocabularies HIDDEN, ALGSTR_0, AOFA_I00, ARYTM_1, ARYTM_3, CARD_1, CARD_3, CLASSES1, COMPLEX1, CONVEX1, EUCLID, EUCLID_9, FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSET_1, FUNCOP_1, FUNCSDOM, FUNCT_1, FUNCT_2, LMOD_7, MATRIX_1, MATRIX_3, MATRIX13, MATRLIN, MATRLIN2, MESFUNC1, METRIC_1, MONOID_0, NAT_1, NUMBERS, ORDINAL2, ORDINAL4, PARTFUN1, PCOMPS_1, PRE_TOPC, PROB_1, PRVECT_1, QC_LANG1, RCOMP_1, REAL_1, RELAT_1, RFINSEQ, RLAFFIN1, RLSUB_1, RLTOPSP1, RLVECT_1, RLVECT_2, RLVECT_3, RLVECT_5, RUSUB_4, SEMI_AF1, SETFAM_1, STRUCT_0, SUBSET_1, SUPINF_2, TARSKI, TOPMETR, TOPS_1, TOPS_2, VALUED_0, VALUED_1, VECTSP_1, XBOOLE_0, XCMPLX_0, XXREAL_0, XXREAL_1, ZFMISC_1;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ZFMISC_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, CARD_1, XREAL_0, REAL_1, FINSET_1, DOMAIN_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, FINSEQ_1, FUNCOP_1, STRUCT_0, FINSEQ_2, FINSEQOP, RVSUM_1, RLVECT_1, RLVECT_2, RUSUB_4, CONVEX1, RLAFFIN1, COMPLEX1, VALUED_0, METRIC_1, PRE_TOPC, PCOMPS_1, ALGSTR_0, FUNCSDOM, TOPS_2, RLTOPSP1, EUCLID, VFUNCT_1, FVSUM_1, RLSUB_1, RLVECT_3, RLVECT_5, CARD_3, RCOMP_1, EUCLID_9, VECTSP_1, MATRIX_0, MATRIX_1, NAT_D, MATRIX_3, MATRIX_6, VECTSP_6, VECTSP_7, MATRIX13, MATRLIN, RLSUB_2, RFINSEQ, TOPMETR, RLAFFIN2, SETFAM_1, PRVECT_1, TOPREAL9, MATRTOP1;
definitions RLTOPSP1, TARSKI, XBOOLE_0;
theorems BORSUK_5, CARD_1, CARD_2, CARD_3, COMPLEX1, CONVEX1, EUCLID, EUCLID_7, EUCLID_9, FINSEQ_1, FINSEQ_2, FINSEQ_3, FINSEQ_4, FUNCOP_1, FUNCT_1, FUNCT_2, FVSUM_1, GOBOARD6, JORDAN2B, JORDAN6, LAPLACE, MATRIX_6, MATRIX13, MATRLIN, MATRLIN2, MATRTOP1, MATRTOP2, METRIC_1, METRIZTS, NAT_1, PARTFUN1, PCOMPS_1, PRALG_3, PRE_TOPC, RELAT_1, RFINSEQ, RLAFFIN1, RLAFFIN2, RLSUB_1, RLSUB_2, RLTOPSP1, RLVECT_1, RLVECT_2, RLVECT_3, RLVECT_5, RUSUB_4, RVSUM_1, SETFAM_1, SPPOL_1, STIRL2_1, TARSKI, TOPGRP_1, TOPMETR, TOPREAL9, TOPS_1, TOPS_2, TREAL_1, TSEP_1, URYSOHN1, VECTSP_1, VECTSP_4, VECTSP_7, VECTSP_9, VFUNCT_1, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1, XREAL_0, XREAL_1, XXREAL_0, XXREAL_1, ZFMISC_1, NAT_D, TOPREAL3;
schemes FINSEQ_1, FUNCT_2, TREES_2;
registrations CARD_1, EUCLID, EUCLID_7, EUCLID_9, FINSEQ_1, FINSEQ_2, FINSET_1, FUNCT_1, JORDAN2B, MATRIX13, MATRTOP1, MONOID_0, NAT_1, NUMBERS, PRE_TOPC, PRVECT_1, RELSET_1, RLAFFIN1, RLTOPSP1, RLVECT_5, RVSUM_1, STRUCT_0, SUBSET_1, TOPMETR, TOPREAL1, TOPS_1, VALUED_0, VECTSP_1, VECTSP_9, XBOOLE_0, XREAL_0, XXREAL_0, ORDINAL1;
constructors HIDDEN, BINOP_2, CONVEX1, EUCLID_9, FUNCSDOM, FVSUM_1, MATRIX_6, MATRIX13, MATRTOP1, MONOID_0, NAT_D, RANKNULL, RCOMP_1, REAL_1, REALSET1, RFINSEQ, RLAFFIN1, RLAFFIN2, RLSUB_2, RLVECT_3, RLVECT_5, RUSUB_5, TOPMETR, TOPREAL9, TOPS_2, VECTSP_7, VFUNCT_1, WELLORD2, MATRIX_1;
requirements HIDDEN, REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
equalities FINSEQ_1, MATRIX13, MATRTOP1, STRUCT_0, SUBSET_1;
expansions FINSEQ_1, STRUCT_0, TARSKI, XBOOLE_0;
Lm1:
for n being Nat
for f being b1 -element real-valued FinSequence holds f is Point of (TOP-REAL n)
Lm2:
for V being RealLinearSpace
for A being Subset of V holds Lin A = Lin (A \ {(0. V)})
Lm3:
for n being Nat holds the carrier of (n -VectSp_over F_Real) = the carrier of (TOP-REAL n)
Lm4:
for V being non empty RLSStruct
for A being Subset of V
for r being Real holds card (r * A) c= card A
Lm5:
0 in REAL
by XREAL_0:def 1;
Lm6:
for k being Nat
for V being LinearTopSpace
for A being finite affinely-independent Subset of V
for E being Enumeration of A
for v being Point of V
for Ev being Enumeration of v + A st Ev = E + ((card A) |-> v) holds
for X being set holds (transl (v,V)) .: { u where u is Point of V : ( u in Affin A & (u |-- E) | k in X ) } = { w where w is Point of V : ( w in Affin (v + A) & (w |-- Ev) | k in X ) }
Lm7:
for n, k being Nat st k <= n holds
for A being non empty finite affinely-independent Subset of (TOP-REAL n) st card A = n + 1 holds
for E being Enumeration of A st E . (len E) = 0* n holds
for P being Subset of (TOP-REAL k)
for B being Subset of (TOP-REAL n) st B = { pn where pn is Point of (TOP-REAL n) : (pn |-- E) | k in P } holds
( P is open iff B is open )
Lm8:
for n, k being Nat
for A being non empty finite affinely-independent Subset of (TOP-REAL n) st k < card A holds
for E being Enumeration of A st E . (len E) = 0* n holds
for P being Subset of (TOP-REAL k)
for B being Subset of ((TOP-REAL n) | (Affin A)) st B = { v where v is Element of ((TOP-REAL n) | (Affin A)) : (v |-- E) | k in P } holds
( P is open iff B is open )
Lm9:
for n being Nat
for A being affinely-independent Subset of (TOP-REAL n) st card A = n + 1 holds
conv A is closed