environ
vocabularies HIDDEN, NUMBERS, PRE_TOPC, EUCLID, TOPREAL2, SUBSET_1, METRIC_1, XXREAL_0, ARYTM_1, CARD_1, PCOMPS_1, XBOOLE_0, RCOMP_1, WEIERSTR, FUNCT_1, VECTMETR, ORDINAL2, CONNSP_2, TOPS_1, TARSKI, RELAT_1, ARYTM_3, TOPS_2, REAL_1, FINSEQ_6, COMPLEX1, MCART_1, XCMPLX_0, INT_1, SIN_COS, COMPTRIG, STRUCT_0, SQUARE_1, JGRAPH_2, TOPGRP_1, FUNCT_2, RELAT_2, CONNSP_1, JORDAN1, JORDAN24;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, COMPLEX1, SQUARE_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, REAL_1, INT_1, STRUCT_0, PRE_TOPC, TOPS_1, TOPS_2, CONNSP_1, COMPTS_1, METRIC_1, PCOMPS_1, EUCLID, TOPREAL2, JORDAN1, TOPREAL6, WEIERSTR, VECTMETR, CONNSP_2, JGRAPH_2, COMPLEX2, COMPTRIG, SIN_COS, TOPGRP_1, TMAP_1;
definitions TARSKI, XBOOLE_0, FUNCT_2, CONNSP_1, BORSUK_1, VECTMETR, TOPS_2, SPRECT_1;
theorems TOPGRP_1, PRE_TOPC, EUCLID, SQUARE_1, XREAL_1, JORDAN1K, JGRAPH_7, TOPREAL2, TOPS_2, FUNCT_2, RELAT_1, FUNCT_1, XBOOLE_1, WEIERSTR, CONNSP_2, GOBOARD6, METRIC_1, VECTMETR, JGRAPH_2, COMPLEX2, XCMPLX_0, COMPLEX1, COMPTRIG, SIN_COS, XCMPLX_1, TMAP_1, JORDAN16, RFUNCT_2, TIETZE, XXREAL_0, XBOOLE_0, TOPREAL6, COMPTS_1;
schemes FUNCT_2;
registrations RELSET_1, FUNCT_2, XCMPLX_0, XXREAL_0, XREAL_0, SQUARE_1, STRUCT_0, PRE_TOPC, PCOMPS_1, EUCLID, TOPREAL2, TOPGRP_1, VECTMETR, INT_1, SIN_COS6, TMAP_1, RELAT_1, SIN_COS, ORDINAL1;
constructors HIDDEN, REAL_1, SQUARE_1, SIN_COS, COMPTRIG, COMPLEX2, TOPS_1, CONNSP_1, TOPS_2, COMPTS_1, TMAP_1, TOPREAL2, JORDAN1, WEIERSTR, TOPGRP_1, VECTMETR, TOPREAL6, JGRAPH_2, FUNCSDOM, NEWTON;
requirements HIDDEN, NUMERALS, BOOLE, SUBSET, ARITHM, REAL;
equalities XBOOLE_0, COMPLEX2, SQUARE_1, SUBSET_1, PCOMPS_1, STRUCT_0;
expansions XBOOLE_0, FUNCT_2, TOPS_2;
set rl = - 1;
set rp = 1;
set a = |[(- 1),0]|;
set b = |[1,0]|;
Lm1:
TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2)
by EUCLID:def 8;
definition
let a be
Real;
existence
ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st
for p being Point of (TOP-REAL 2) holds b1 . p = |[(Re (Rotate (((p `1) + ((p `2) * <i>)),a))),(Im (Rotate (((p `1) + ((p `2) * <i>)),a)))]|
uniqueness
for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for p being Point of (TOP-REAL 2) holds b1 . p = |[(Re (Rotate (((p `1) + ((p `2) * <i>)),a))),(Im (Rotate (((p `1) + ((p `2) * <i>)),a)))]| ) & ( for p being Point of (TOP-REAL 2) holds b2 . p = |[(Re (Rotate (((p `1) + ((p `2) * <i>)),a))),(Im (Rotate (((p `1) + ((p `2) * <i>)),a)))]| ) holds
b1 = b2
end;
theorem Th3:
for
p1,
p2 being
Point of
(TOP-REAL 2) for
P being
Subset of
(TOP-REAL 2) for
A,
B,
D being
Real st
p1,
p2 realize-max-dist-in P holds
(AffineMap (A,B,A,D)) . p1,
(AffineMap (A,B,A,D)) . p2 realize-max-dist-in (AffineMap (A,B,A,D)) .: P
Lm4:
for T1, T2 being TopSpace
for f being Function of T1,T2
for g being Function of TopStruct(# the carrier of T1, the topology of T1 #),TopStruct(# the carrier of T2, the topology of T2 #) st g = f & g is being_homeomorphism holds
f is being_homeomorphism
by PRE_TOPC:34;
::<DESC name="1.4.18 Twierdzenie o homeomorfizmie (cz/e/s/c): (a) iff (b)" monograph="topology"/>