environ
vocabularies HIDDEN, ARYTM_1, ARYTM_3, BROUWER, CARD_1, COMPLEX1, CONVEX1, EUCLID, FUNCOP_1, FUNCT_1, FUNCT_4, JGRAPH_4, MEMBERED, METRIC_1, NAT_1, NUMBERS, ORDINAL2, PCOMPS_1, PRE_TOPC, RCOMP_1, REAL_1, RELAT_1, STRUCT_0, SUBSET_1, SUPINF_2, TARSKI, TOPMETR, TOPS_1, TOPS_2, VALUED_1, XBOOLE_0, XXREAL_0, FUNCT_2, EUCLID_9, TOPREALC, FINSEQ_1, FINSEQ_2, CARD_3, PARTFUN3, SQUARE_1, XCMPLX_0, PARTFUN1, FINSET_1, VALUED_0, ORDINAL4, PRE_POLY, PSCOMP_1, RVSUM_1, VALUED_2, T_0TOPSP, WAYBEL23, TOPDIM_1, MATRTOP3, TOPREALB, RLTOPSP1, FUNCT_3, RLAFFIN1, RLVECT_5, MEASURE5, BORSUK_1, UNIALG_1, MSSUBFAM, MATRLIN, MESFUNC1, VECTSP_1;
notations HIDDEN, TARSKI, XBOOLE_0, FINSET_1, VALUED_0, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, COMPLEX1, ORDINAL1, NUMBERS, XXREAL_0, XREAL_0, RVSUM_1, VALUED_1, VALUED_2, XXREAL_3, XCMPLX_0, NAT_1, CARD_1, MEMBERED, FUNCOP_1, FINSEQ_1, FINSEQ_2, STRUCT_0, PRE_TOPC, TOPS_1, PSCOMP_1, METRIC_1, PCOMPS_1, TOPMETR, FUNCT_7, RLVECT_1, RLTOPSP1, EUCLID, TOPREAL9, PARTFUN3, TOPS_2, COMPTS_1, TBSP_1, JGRAPH_4, TMAP_1, BROUWER, EUCLID_9, CARD_3, SQUARE_1, PRE_POLY, JORDAN2B, WAYBEL23, T_0TOPSP, METRIZTS, RLAFFIN1, TOPREALB, FUNCT_4, DOMAIN_1, CONVEX1, TIETZE_2, RLVECT_5, BORSUK_1, VECTSP_1, TOPREALC, MATRTOP3, MATRIX_1, TOPDIM_1;
definitions TARSKI, XBOOLE_0;
theorems ABSVALUE, BORSUK_1, BROUWER, COMPLEX1, EUCLID, FUNCOP_1, FUNCT_1, FUNCT_2, FUNCT_4, GOBOARD6, JGRAPH_4, JORDAN24, NECKLACE, ORDINAL1, PRE_TOPC, RELAT_1, RLAFFIN1, RLAFFIN3, RLTOPSP1, RLVECT_1, SIMPLEX2, SUBSET_1, TARSKI, TBSP_1, TMAP_1, TOPMETR, TOPREAL9, TOPREALB, TOPREALC, TOPRNS_1, TOPS_1, TOPS_2, VALUED_2, XBOOLE_0, XBOOLE_1, XCMPLX_1, XREAL_0, XREAL_1, XXREAL_0, ZFMISC_1, EUCLID_9, FINSEQ_2, CARD_1, FINSEQ_1, PARTFUN3, YELLOW_8, FINSEQ_3, RVSUM_1, VALUED_1, PARTFUN1, FUNCT_7, TOPREAL7, TSEP_1, BROUWER2, NAT_1, REAL_NS1, SQUARE_1, TOPGRP_1, PRE_POLY, JORDAN2B, FINSEQ_4, JORDAN5A, VECTSP_1, JORDAN, ENUMSET1, TOPDIM_1, TOPDIM_2, MATRTOP3, TOPGEN_5, METRIZTS, T_0TOPSP, TIETZE_2;
schemes FINSEQ_1, FUNCT_2, FUNCT_1, NAT_1;
registrations BORSUK_1, BROUWER, EUCLID, EUCLID_9, FUNCOP_1, FINSEQ_1, FUNCT_1, FUNCT_2, JGRAPH_4, MEMBERED, NAT_1, PCOMPS_1, PRE_TOPC, RELAT_1, PARTFUN3, RVSUM_1, SIMPLEX2, STRUCT_0, SUBSET_1, TMAP_1, TOPGRP_1, TOPMETR, TOPS_1, VALUED_0, VALUED_2, XBOOLE_0, XREAL_0, XXREAL_0, RELSET_1, FINSET_1, MONOID_0, SQUARE_1, FUNCT_7, NUMBERS, JORDAN2B, FINSEQ_2, TOPREALC, PARTFUN4, COMPTS_1, TOPREALB, CARD_1, INT_1, TOPREAL1, TOPREAL9, METRIZTS, RLTOPSP1, RLAFFIN1, RLAFFIN3, XXREAL_3, TOPDIM_1, TOPDIM_2, REVROT_1, WAYBEL_2, XCMPLX_0, ZFMISC_1, VECTSP_1, PRE_POLY, VALUED_1, MATRTOP3;
constructors HIDDEN, TBSP_1, MONOID_0, CONVEX1, TOPS_1, COMPTS_1, FUNCSDOM, PARTFUN3, JGRAPH_4, TMAP_1, TOPREALC, BROUWER, EUCLID_9, SQUARE_1, JORDAN2B, PSCOMP_1, WAYBEL23, TOPDIM_1, METRIZTS, SIMPLEX0, RLAFFIN1, BORSUK_3, RLVECT_5, MATRTOP3, TIETZE_2, LAPLACE;
requirements HIDDEN, ARITHM, BOOLE, NUMERALS, SUBSET, REAL;
equalities FUNCOP_1, STRUCT_0, SQUARE_1, XCMPLX_0, ALGSTR_0, TOPREAL9, BROUWER, FINSEQ_1, ORDINAL1, EUCLID, FINSEQ_2;
expansions TARSKI, FUNCT_1, XBOOLE_0, FINSEQ_1;
Lm1:
for n being Nat
for V being Subset of (TopSpaceMetr (Euclid n)) st V is open holds
for e being Point of (Euclid n) st e in V holds
ex r being Real st
( r > 0 & OpenHypercube (e,r) c= V )
Lm2:
for n being Nat
for p being Point of (TOP-REAL n)
for r being Real st r > 0 holds
( not cl_Ball (p,r) is boundary & cl_Ball (p,r) is compact )
theorem Th1:
for
N being non
zero Nat for
F being
Element of
N -tuples_on (Funcs ( the carrier of (TOP-REAL (N + 1)), the carrier of R^1)) st ( for
i being
Nat st
i in dom F holds
F . i = PROJ (
(N + 1),
i) ) holds
( ( for
t being
Point of
(TOP-REAL (N + 1)) holds
<:F:> . t = t | N ) & ( for
Sp,
Sn being
Subset of
(TOP-REAL (N + 1)) st
Sp = { u where u is Point of (TOP-REAL (N + 1)) : ( u . (N + 1) >= 0 & |.u.| = 1 ) } &
Sn = { t where t is Point of (TOP-REAL (N + 1)) : ( t . (N + 1) <= 0 & |.t.| = 1 ) } holds
(
<:F:> .: Sp = cl_Ball (
(0. (TOP-REAL N)),1) &
<:F:> .: Sn = cl_Ball (
(0. (TOP-REAL N)),1) &
<:F:> .: (Sp /\ Sn) = Sphere (
(0. (TOP-REAL N)),1) & ( for
H being
Function of
((TOP-REAL (N + 1)) | Sp),
(Tdisk ((0. (TOP-REAL N)),1)) st
H = <:F:> | Sp holds
H is
being_homeomorphism ) & ( for
H being
Function of
((TOP-REAL (N + 1)) | Sn),
(Tdisk ((0. (TOP-REAL N)),1)) st
H = <:F:> | Sn holds
H is
being_homeomorphism ) ) ) )
Lm3:
for n being Nat
for A, B being Subset of (TOP-REAL n) st n = 0 holds
for h being Function of ((TOP-REAL n) | A),((TOP-REAL n) | B) st h is being_homeomorphism holds
for p being Point of (TOP-REAL n) holds
( ( p in Fr A implies h . p in Fr B ) & ( p in Int A implies h . p in Int B ) )
theorem Th15:
for
n being
Nat for
p,
q being
Point of
(TOP-REAL n) for
r being
Real holds
(
(transl (p,(TOP-REAL n))) .: (Ball (q,r)) = Ball (
(q + p),
r) &
(transl (p,(TOP-REAL n))) .: (cl_Ball (q,r)) = cl_Ball (
(q + p),
r) &
(transl (p,(TOP-REAL n))) .: (Sphere (q,r)) = Sphere (
(q + p),
r) )
theorem Th16:
for
n being
Nat for
p being
Point of
(TOP-REAL n) for
r,
s being
Real st
s > 0 holds
(
(mlt (s,(TOP-REAL n))) .: (Ball (p,r)) = Ball (
(s * p),
(r * s)) &
(mlt (s,(TOP-REAL n))) .: (cl_Ball (p,r)) = cl_Ball (
(s * p),
(r * s)) &
(mlt (s,(TOP-REAL n))) .: (Sphere (p,r)) = Sphere (
(s * p),
(r * s)) )
Lm4:
for r being Real
for n being non zero Element of NAT
for p being Point of (TOP-REAL (n + 1)) st r <= 1 & |.p.| >= 1 & |.p.| < 1 + r & p . (n + 1) > 0 & p | n = 0* n holds
ex c being Real ex h being Function of ((TOP-REAL (n + 1)) | ((Sphere ((0. (TOP-REAL (n + 1))),1)) /\ (cl_Ball (p,r)))),(Tdisk ((0. (TOP-REAL n)),c)) st
( c > 0 & h is being_homeomorphism & h .: ((Sphere ((0. (TOP-REAL (n + 1))),1)) /\ (Sphere (p,r))) = Sphere ((0. (TOP-REAL n)),c) )