Volume 11, 1999

University of Bialystok

Copyright (c) 1999 Association of Mizar Users

### The abstract of the Mizar article:

### Homeomorphism between [:$\cal E^i_\rm T, \cal E^j_\rm T$:] and $\cal E^i+j_\rm T$

**by****Artur Kornilowicz**- Received February 21, 1999
- MML identifier: TOPREAL7

- [ Mizar article, MML identifier index ]

environ vocabulary FINSEQ_1, METRIC_1, SQUARE_1, RELAT_1, ARYTM_1, FUNCT_1, FINSEQ_2, RVSUM_1, ABSVALUE, EUCLID, PRE_TOPC, MCART_1, RELAT_2, PCOMPS_1, TARSKI, SETFAM_1, BORSUK_1, T_0TOPSP, SUBSET_1, ARYTM_3, COMPLEX1, RLVECT_1, ORDINAL2, TOPREAL7; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2, BINOP_1, MCART_1, DOMAIN_1, NUMBERS, XREAL_0, REAL_1, ABSVALUE, SQUARE_1, NAT_1, FINSEQ_1, RVSUM_1, FINSEQOP, STRUCT_0, PRE_TOPC, TOPS_2, PCOMPS_1, BORSUK_1, METRIC_1, EUCLID, FINSEQ_2, T_0TOPSP, GOBOARD1; constructors ABSVALUE, DOMAIN_1, FINSEQOP, GOBOARD1, INT_1, NAT_1, REAL_1, SEQ_1, SQUARE_1, T_0TOPSP, TOPS_2, BORSUK_1, MEMBERED; clusters SUBSET_1, INT_1, METRIC_1, PCOMPS_1, RELSET_1, STRUCT_0, EUCLID, MEMBERED; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin reserve i, j, n for Nat, f, g, h, k for FinSequence of REAL, M, N for non empty MetrSpace; theorem :: TOPREAL7:1 for a, b being Real st max(a,b) <= a holds max(a,b) = a; theorem :: TOPREAL7:2 for a, b, c, d being Real holds max(a+c,b+d) <= max(a,b) + max(c,d); theorem :: TOPREAL7:3 for a, b, c, d, e, f being Real st a <= b+c & d <= e+f holds max(a,d) <= max(b,e) + max(c,f); theorem :: TOPREAL7:4 for f, g being FinSequence holds dom g c= dom (f^g); theorem :: TOPREAL7:5 for f, g being FinSequence st len f < i & i <= len f + len g holds i - len f in dom g; theorem :: TOPREAL7:6 for f, g, h, k being FinSequence st f^g = h^k & len f = len h & len g = len k holds f = h & g = k; theorem :: TOPREAL7:7 len f = len g or dom f = dom g implies len (f+g) = len f & dom (f+g) = dom f; theorem :: TOPREAL7:8 len f = len g or dom f = dom g implies len (f-g) = len f & dom (f-g) = dom f; theorem :: TOPREAL7:9 len f = len sqr f & dom f = dom sqr f; theorem :: TOPREAL7:10 len f = len abs f & dom f = dom abs f; theorem :: TOPREAL7:11 sqr (f^g) = sqr f ^ sqr g; theorem :: TOPREAL7:12 abs (f^g) = abs f ^ abs g; theorem :: TOPREAL7:13 len f = len h & len g = len k implies sqr (f^g + h^k) = sqr (f+h) ^ sqr (g+k ); theorem :: TOPREAL7:14 len f = len h & len g = len k implies abs (f^g + h^k) = abs (f+h) ^ abs (g+k ); theorem :: TOPREAL7:15 len f = len h & len g = len k implies sqr (f^g - h^k) = sqr (f-h) ^ sqr (g-k ); theorem :: TOPREAL7:16 len f = len h & len g = len k implies abs (f^g - h^k) = abs (f-h) ^ abs (g-k); theorem :: TOPREAL7:17 len f = n implies f in the carrier of Euclid n; theorem :: TOPREAL7:18 len f = n implies f in the carrier of TOP-REAL n; theorem :: TOPREAL7:19 for f being FinSequence st f in the carrier of Euclid n holds len f = n; definition let M, N be non empty MetrStruct; func max-Prod2(M,N) -> strict MetrStruct means :: TOPREAL7:def 1 the carrier of it = [:the carrier of M,the carrier of N:] & for x, y being Point of it ex x1, y1 being Point of M, x2, y2 being Point of N st x = [x1,x2] & y = [y1,y2] & (the distance of it).(x,y) = max ((the distance of M).(x1,y1),(the distance of N).(x2,y2)); end; definition let M, N be non empty MetrStruct; cluster max-Prod2(M,N) -> non empty; end; definition let M, N be non empty MetrStruct, x be Point of M, y be Point of N; redefine func [x,y] -> Element of max-Prod2(M,N); end; definition let M, N be non empty MetrStruct, x be Point of max-Prod2(M,N); redefine func x`1 -> Element of M; redefine func x`2 -> Element of N; end; theorem :: TOPREAL7:20 for M, N being non empty MetrStruct, m1, m2 being Point of M, n1, n2 being Point of N holds dist([m1,n1],[m2,n2]) = max (dist(m1,m2),dist(n1,n2)); theorem :: TOPREAL7:21 for M, N being non empty MetrStruct, m, n being Point of max-Prod2(M,N) holds dist(m,n) = max (dist(m`1,n`1),dist(m`2,n`2)); theorem :: TOPREAL7:22 for M, N being Reflexive (non empty MetrStruct) holds max-Prod2(M,N) is Reflexive; definition let M, N be Reflexive (non empty MetrStruct); cluster max-Prod2(M,N) -> Reflexive; end; theorem :: TOPREAL7:23 for M, N being symmetric (non empty MetrStruct) holds max-Prod2(M,N) is symmetric; definition let M, N be symmetric (non empty MetrStruct); cluster max-Prod2(M,N) -> symmetric; end; theorem :: TOPREAL7:24 for M, N being triangle (non empty MetrStruct) holds max-Prod2(M,N) is triangle; definition let M, N be triangle (non empty MetrStruct); cluster max-Prod2(M,N) -> triangle; end; definition let M, N be non empty MetrSpace; cluster max-Prod2(M,N) -> discerning; end; theorem :: TOPREAL7:25 [:TopSpaceMetr M,TopSpaceMetr N:] = TopSpaceMetr max-Prod2(M,N); theorem :: TOPREAL7:26 the carrier of M = the carrier of N & (for m being Point of M, n being Point of N, r being Real st r > 0 & m = n ex r1 being Real st r1 > 0 & Ball(n,r1) c= Ball(m,r)) & (for m being Point of M, n being Point of N, r being Real st r > 0 & m = n ex r1 being Real st r1 > 0 & Ball(m,r1) c= Ball(n,r)) implies TopSpaceMetr M = TopSpaceMetr N; theorem :: TOPREAL7:27 [:TOP-REAL i,TOP-REAL j:], TOP-REAL (i+j) are_homeomorphic;

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