:: On the Simple Closed Curve Property of the Circle and the
:: Fashoda Meet Theorem for It
:: by Yatsuka Nakamura
::
:: Received August 20, 2001
:: Copyright (c) 2001-2026 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SQUARE_1, ARYTM_3, XXREAL_0, CARD_1, RELAT_1, STRUCT_0,
EUCLID, PRE_TOPC, COMPLEX1, MCART_1, FUNCT_1, XBOOLE_0, TARSKI, SUBSET_1,
ORDINAL2, RCOMP_1, SUPINF_2, ARYTM_1, TOPMETR, REAL_1, XXREAL_1,
JORDAN2C, FUNCT_4, PARTFUN1, PCOMPS_1, METRIC_1, TOPS_2, TOPREAL2,
TOPREAL1, VALUED_1, BORSUK_1, JGRAPH_3, FUNCT_2;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0,
REAL_1, RLVECT_1, EUCLID, RELAT_1, TOPS_2, FUNCT_1, RELSET_1, PARTFUN1,
FUNCT_2, DOMAIN_1, STRUCT_0, TOPREAL1, TOPMETR, COMPTS_1, METRIC_1,
RCOMP_1, SQUARE_1, PCOMPS_1, PSCOMP_1, PRE_TOPC, FUNCT_4, TOPREAL2,
XXREAL_0;
constructors FUNCT_4, REAL_1, SQUARE_1, RCOMP_1, TOPS_2, COMPTS_1, TOPMETR,
TOPREAL1, TOPREAL2, PSCOMP_1, FUNCSDOM, PCOMPS_1;
registrations FUNCT_1, RELSET_1, FUNCT_2, NUMBERS, XXREAL_0, XREAL_0,
SQUARE_1, MEMBERED, STRUCT_0, PRE_TOPC, BORSUK_1, EUCLID, TOPMETR,
TOPREAL1, TOPREAL2, BORSUK_3, ORDINAL1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions TARSKI, FUNCT_1, XBOOLE_0;
equalities XBOOLE_0, SQUARE_1, STRUCT_0;
expansions TARSKI, FUNCT_1, XBOOLE_0;
theorems TARSKI, RELAT_1, SUBSET_1, FUNCT_1, FUNCT_2, FUNCT_4, TOPS_1, TOPS_2,
PARTFUN1, PRE_TOPC, REVROT_1, FRECHET, TOPMETR, JORDAN6, EUCLID,
JGRAPH_1, SQUARE_1, TOPREAL3, PSCOMP_1, METRIC_1, SPPOL_2, JGRAPH_2,
COMPTS_1, ZFMISC_1, GOBOARD6, TOPREAL1, TOPREAL2, COMPLEX1, XBOOLE_0,
XBOOLE_1, XREAL_0, TOPRNS_1, XCMPLX_1, XREAL_1, XXREAL_0, XXREAL_1,
RELSET_1;
schemes FUNCT_2, DOMAIN_1, JGRAPH_2, CLASSES1;
begin :: Preliminaries
reserve x for Real;
Lm1: x^2+1 > 0
proof
x^2 >= 0 by XREAL_1:63;
hence thesis;
end;
Lm2: dom proj1=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
Lm3: dom proj2=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
theorem
for p being Point of TOP-REAL 2 holds |.p.| = sqrt((p`1)^2+(p`2)^2) &
|.p.|^2 = (p`1)^2+(p`2)^2 by JGRAPH_1:29,30;
theorem
for f being Function,B,C being set holds (f|B).:C = f.:(C /\ B)
proof
let f be Function,B,C be set;
thus (f|B).:C c=f.:(C /\ B)
proof
let x be object;
assume x in (f|B).:C;
then consider y being object such that
A1: y in dom (f|B) and
A2: y in C and
A3: x=(f|B).y by FUNCT_1:def 6;
A4: (f|B).y=f.y by A1,FUNCT_1:47;
A5: dom (f|B)=(dom f)/\ B by RELAT_1:61;
then y in B by A1,XBOOLE_0:def 4;
then
A6: y in C /\ B by A2,XBOOLE_0:def 4;
y in dom f by A1,A5,XBOOLE_0:def 4;
hence thesis by A3,A6,A4,FUNCT_1:def 6;
end;
let x be object;
assume x in f.:(C /\ B);
then consider y being object such that
A7: y in dom f and
A8: y in C /\ B and
A9: x=f.y by FUNCT_1:def 6;
A10: y in C by A8,XBOOLE_0:def 4;
y in B by A8,XBOOLE_0:def 4;
then y in (dom f)/\ B by A7,XBOOLE_0:def 4;
then
A11: y in dom (f|B) by RELAT_1:61;
then (f|B).y=f.y by FUNCT_1:47;
hence thesis by A9,A10,A11,FUNCT_1:def 6;
end;
theorem Th3:
for X,Y being non empty TopSpace, p0 being Point of X, D being
non empty Subset of X, E being non empty Subset of Y, f being Function of X,Y
st D`={p0} & E`={f.p0} & X is T_2 & Y is T_2 & (for p being Point of X|D holds
f.p<>f.p0)& f|D is continuous Function of X|D,Y|E & (for V being Subset of Y st
f.p0 in V & V is open ex W being Subset of X st p0 in W & W is open & f.:W c= V
) holds f is continuous
proof
let X,Y be non empty TopSpace, p0 be Point of X, D be non empty Subset of X,
E be non empty Subset of Y, f be Function of X,Y;
assume that
A1: D`={p0} and
A2: E`={f.p0} and
A3: X is T_2 and
A4: Y is T_2 and
A5: for p being Point of X|D holds f.p<>f.p0 and
A6: f|D is continuous Function of X|D,Y|E and
A7: for V being Subset of Y st f.p0 in V & V is open ex W being Subset
of X st p0 in W & W is open & f.:W c= V;
for p being Point of X,V being Subset of Y st f.p in V & V is open holds
ex W being Subset of X st p in W & W is open & f.:W c= V
proof
A8: the carrier of X|D=D by PRE_TOPC:8;
let p be Point of X,V be Subset of Y;
assume that
A9: f.p in V and
A10: V is open;
per cases;
suppose
p=p0;
hence thesis by A7,A9,A10;
end;
suppose
A11: p<>p0;
then not p in D` by A1,TARSKI:def 1;
then p in (the carrier of X)\D` by XBOOLE_0:def 5;
then
A12: p in D`` by SUBSET_1:def 4;
then f.p<>f.p0 by A5,A8;
then consider G1,G2 being Subset of Y such that
A13: G1 is open and
G2 is open and
A14: f.p in G1 and
f.p0 in G2 and
G1 misses G2 by A4,PRE_TOPC:def 10;
A15: [#](X|D)=D by PRE_TOPC:def 5;
then reconsider p22=p as Point of X|D by A12;
consider h being Function of X|D,Y|E such that
A16: h=f|D and
A17: h is continuous by A6;
A18: h.p=f.p by A12,A16,FUNCT_1:49;
A19: [#](Y|E)=E by PRE_TOPC:def 5;
then reconsider V20=(G1 /\ V) /\ E as Subset of Y|E by XBOOLE_1:17;
G1 /\ V is open by A10,A13,TOPS_1:11;
then
A20: V20 is open by A19,TOPS_2:24;
f.p<>f.p0 by A5,A12,A15;
then not f.p in E` by A2,TARSKI:def 1;
then not f.p in (the carrier of Y) \E by SUBSET_1:def 4;
then
A21: h.p22 in E by A18,XBOOLE_0:def 5;
h.p22 in G1 /\ V by A9,A14,A18,XBOOLE_0:def 4;
then h.p22 in V20 by A21,XBOOLE_0:def 4;
then consider W2 being Subset of X|D such that
A22: p22 in W2 and
A23: W2 is open and
A24: h.:W2 c= V20 by A17,A20,JGRAPH_2:10;
consider W3b being Subset of X such that
A25: W3b is open and
A26: W2=W3b /\ [#](X|D) by A23,TOPS_2:24;
consider H1,H2 being Subset of X such that
A27: H1 is open and
H2 is open and
A28: p in H1 and
A29: p0 in H2 and
A30: H1 misses H2 by A3,A11,PRE_TOPC:def 10;
p22 in W3b by A22,A26,XBOOLE_0:def 4;
then
A31: p in H1 /\ W3b by A28,XBOOLE_0:def 4;
reconsider W3=H1 /\ W3b as Subset of X;
A32: W3 c= W3b by XBOOLE_1:17;
A33: f.:W3 c= h.:W2
proof
let xx be object;
assume xx in f.:W3;
then consider yy being object such that
A34: yy in dom f and
A35: yy in W3 and
A36: xx=f.yy by FUNCT_1:def 6;
H2 c= H1` by A30,SUBSET_1:23;
then D` c= H1` by A1,A29,ZFMISC_1:31;
then W3 c= H1 & H1 c= D by SUBSET_1:12,XBOOLE_1:17;
then
A37: W3 c= D;
then
A38: yy in W2 by A15,A26,A32,A35,XBOOLE_0:def 4;
dom h=dom f /\ D by A16,RELAT_1:61;
then
A39: yy in dom h by A34,A35,A37,XBOOLE_0:def 4;
then h.yy=f.yy by A16,FUNCT_1:47;
hence thesis by A36,A39,A38,FUNCT_1:def 6;
end;
(G1 /\ V) /\ E c= G1 /\ V by XBOOLE_1:17;
then G1 /\ V c= V & h.:W2 c= G1 /\ V by A24,XBOOLE_1:17;
then
A40: h.:W2 c= V;
H1 /\ W3b is open by A25,A27,TOPS_1:11;
hence thesis by A31,A33,A40,XBOOLE_1:1;
end;
end;
hence thesis by JGRAPH_2:10;
end;
begin :: The Circle is a Simple Closed Curve
reserve p,q for Point of TOP-REAL 2;
definition
func Sq_Circ -> Function of the carrier of TOP-REAL 2, the carrier of
TOP-REAL 2 means
:Def1:
for p being Point of TOP-REAL 2 holds (p=0.TOP-REAL 2
implies it.p=p) & ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.
TOP-REAL 2 implies it.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)& (
not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies it.p
=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|);
existence
proof
defpred P[set,set] means (for p being Point of TOP-REAL 2 st p=$1 holds (p
=0.TOP-REAL 2 implies $2=p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p
<>0.TOP-REAL 2 implies $2=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)&
(not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies $2=
|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|));
set BP= the carrier of TOP-REAL 2;
A1: for x being Element of BP ex y being Element of BP st P[x,y]
proof
let x be Element of BP;
set q=x;
per cases;
suppose
q=0.TOP-REAL 2;
then for p being Point of TOP-REAL 2 st p=x holds (p=0.TOP-REAL 2
implies 0.TOP-REAL 2=p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.
TOP-REAL 2 implies 0.TOP-REAL 2=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)
^2)]|)& (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2
implies 0.TOP-REAL 2=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|);
hence thesis;
end;
suppose
A2: (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1) & q<>0.TOP-REAL 2;
set r= |[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|;
for p being Point of TOP-REAL 2 st p=x holds (p=0.TOP-REAL 2
implies r=p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2
implies r=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)& (not(p`2<=p`1 &
-p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies r=|[p`1/sqrt(1+(p`1
/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|) by A2;
hence thesis;
end;
suppose
A3: not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1) & q<>0. TOP-REAL 2;
set r= |[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|;
for p being Point of TOP-REAL 2 st p=x holds (p=0.TOP-REAL 2
implies r=p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2
implies r=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)& (not(p`2<=p`1 &
-p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies r=|[p`1/sqrt(1+(p`1
/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|) by A3;
hence thesis;
end;
end;
ex h being Function of BP, BP st for x being Element of BP holds P[x,h
.x] from FUNCT_2:sch 3(A1);
then consider h being Function of BP,BP such that
A4: for x being Element of BP holds for p being Point of TOP-REAL 2 st
p=x holds (p=0.TOP-REAL 2 implies h.x=p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 &
p`2<=-p`1)& p<>0.TOP-REAL 2 implies h.x=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p
`2/p`1)^2)]|)& (not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.
TOP-REAL 2 implies h.x=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|);
for p being Point of TOP-REAL 2 holds (p=0.TOP-REAL 2 implies h.p=p)&
((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies h.p=|[
p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)& (not(p`2<=p`1 & -p`1<=p`2
or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies h.p=|[p`1/sqrt(1+(p`1/p`2)^2)
,p`2/sqrt(1+(p`1/p`2)^2)]|) by A4;
hence thesis;
end;
uniqueness
proof
let h1,h2 be Function of the carrier of TOP-REAL 2, the carrier of
TOP-REAL 2;
assume that
A5: for p being Point of TOP-REAL 2 holds (p=0.TOP-REAL 2 implies h1.
p=p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies
h1.p= |[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)& (not(p`2<=p`1 & -p
`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies h1.p=|[p`1/sqrt(1+(p
`1/p`2)^2 ),p`2/sqrt(1+(p`1/p`2)^2)]|) and
A6: for p being Point of TOP-REAL 2 holds (p=0.TOP-REAL 2 implies h2.
p =p)& ((p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies
h2.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)& (not(p`2<=p`1 & -p`1
<=p`2 or p`2>=p`1 & p`2<=-p`1)& p<>0.TOP-REAL 2 implies h2.p=|[p`1/sqrt(1+(p`1/
p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|);
for x being object st x in (the carrier of TOP-REAL 2) holds h1.x=h2.x
proof
let x be object;
assume x in the carrier of TOP-REAL 2;
then reconsider q=x as Point of TOP-REAL 2;
per cases;
suppose
A7: q=0.TOP-REAL 2;
then h1.q=q by A5;
hence thesis by A6,A7;
end;
suppose
A8: (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1)& q<>0.TOP-REAL 2;
then
h1.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A5;
hence thesis by A6,A8;
end;
suppose
A9: not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1) & q<>0. TOP-REAL 2;
then
h1.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]| by A5;
hence thesis by A6,A9;
end;
end;
hence h1=h2 by FUNCT_2:12;
end;
end;
theorem Th4:
for p being Point of TOP-REAL 2 st p<>0.TOP-REAL 2 holds ((p`1<=
p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2)implies Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2
)^2),p`2/sqrt(1+(p`1/p`2)^2)]|) & (not(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=
-p`2) implies Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|)
proof
let p be Point of TOP-REAL 2;
A1: -p`2<p`1 implies --p`2>-p`1 by XREAL_1:24;
assume
A2: p<>0.TOP-REAL 2;
hereby
assume
A3: p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2;
now
per cases by A3;
case
A4: p`1<=p`2 & -p`2<=p`1;
now
assume
A5: p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
A6: now
per cases by A5;
case
p`2<=p`1 & -p`1<=p`2;
hence p`1=p`2 or p`1=-p`2 by A4,XXREAL_0:1;
end;
case
p`2>=p`1 & p`2<=-p`1;
then -p`2>=--p`1 by XREAL_1:24;
hence p`1=p`2 or p`1=-p`2 by A4,XXREAL_0:1;
end;
end;
now
per cases by A6;
case
p`1=p`2;
hence
Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)
]| by A2,A5,Def1;
end;
case
A7: p`1=-p`2;
then p`1<>0 & -p`1=p`2 by A2,EUCLID:53,54;
then
A8: p`2/p`1=-1 by XCMPLX_1:197;
p`2<>0 by A2,A7,EUCLID:53,54;
then p`1/p`2=-1 by A7,XCMPLX_1:197;
hence
Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)
]| by A2,A5,A8,Def1;
end;
end;
hence Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|;
end;
hence Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]| by
A2,Def1;
end;
case
A9: p`1>=p`2 & p`1<=-p`2;
now
assume
A10: p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
A11: now
per cases by A10;
case
p`2<=p`1 & -p`1<=p`2;
then --p`1>=-p`2 by XREAL_1:24;
hence p`1=p`2 or p`1=-p`2 by A9,XXREAL_0:1;
end;
case
p`2>=p`1 & p`2<=-p`1;
hence p`1=p`2 or p`1=-p`2 by A9,XXREAL_0:1;
end;
end;
now
per cases by A11;
case
p`1=p`2;
hence
Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)
]| by A2,A10,Def1;
end;
case
A12: p`1=-p`2;
then p`1<>0 & -p`1=p`2 by A2,EUCLID:53,54;
then
A13: p`2/p`1=-1 by XCMPLX_1:197;
p`2<>0 by A2,A12,EUCLID:53,54;
then p`1/p`2=-1 by A12,XCMPLX_1:197;
hence
Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)
]| by A2,A10,A13,Def1;
end;
end;
hence Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|;
end;
hence Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]| by
A2,Def1;
end;
end;
hence Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|;
end;
A14: -p`2>p`1 implies --p`2<-p`1 by XREAL_1:24;
assume not(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2);
hence thesis by A2,A1,A14,Def1;
end;
theorem Th5:
for X being non empty TopSpace, f1 being Function of X,R^1 st f1
is continuous & (for q being Point of X ex r being Real st f1.q=r & r>=0
) holds ex g being Function of X,R^1 st (for p being Point of X,r1 being Real
st f1.p=r1 holds g.p=sqrt(r1)) & g is continuous
proof
let X being non empty TopSpace,f1 be Function of X,R^1;
assume that
A1: f1 is continuous and
A2: for q being Point of X ex r being Real st f1.q=r & r>=0;
defpred P[set,set] means (for r11 being Real st f1.$1=r11 holds $2=
sqrt(r11));
A3: for x being Element of X ex y being Element of REAL st P[x,y]
proof
let x be Element of X;
reconsider r1=f1.x as Element of REAL by TOPMETR:17;
reconsider y = sqrt(r1) as Element of REAL by XREAL_0:def 1;
take y;
thus thesis;
end;
ex f being Function of the carrier of X,REAL st for x2 being Element of
X holds P[x2,f.x2] from FUNCT_2:sch 3(A3);
then consider f being Function of the carrier of X,REAL such that
A4: for x2 being Element of X holds for r11 being Real st f1.x2=
r11 holds f.x2=sqrt(r11);
reconsider g0=f as Function of X,R^1 by TOPMETR:17;
for p being Point of X,V being Subset of R^1 st g0.p in V & V is open
holds ex W being Subset of X st p in W & W is open & g0.:W c= V
proof
let p be Point of X,V be Subset of R^1;
reconsider r=g0.p as Real;
reconsider r1=f1.p as Real;
assume g0.p in V & V is open;
then consider r01 being Real such that
A5: r01>0 and
A6: ].r-r01,r+r01.[ c= V by FRECHET:8;
set r0=min(r01,1);
A7: r0>0 by A5,XXREAL_0:21;
A8: r0>0 by A5,XXREAL_0:21;
r0<=r01 by XXREAL_0:17;
then r-r01<=r-r0 & r+r0<=r+r01 by XREAL_1:6,10;
then ].r-r0,r+r0.[ c= ].r-r01,r+r01.[ by XXREAL_1:46;
then
A9: ].r-r0,r+r0.[ c= V by A6;
A10: ex r8 being Real st f1.p=r8 & r8>=0 by A2;
A11: r=sqrt(r1) by A4;
then
A12: r1=r^2 by A10,SQUARE_1:def 2;
A13: r>=0 by A10,A11,SQUARE_1:17,26;
then
A14: 2*r*r0+r0^2>0+0 by A8,SQUARE_1:12,XREAL_1:8;
per cases;
suppose
A15: r-r0>0;
set r4=r0*(r-r0);
reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:17;
A16: r1<r1+r4 by A8,A15,XREAL_1:29,129;
then r1-r4<r1 by XREAL_1:19;
then
A17: f1.p in G1 by A16,XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A18: p in W1 & W1 is open and
A19: f1.:W1 c= G1 by A1,A17,JGRAPH_2:10;
set W=W1;
A20: (r-(1/2)*r0)^2>=0 & r0^2>=0 by XREAL_1:63;
now
assume r1=0;
then r=0 by A4,SQUARE_1:17;
hence contradiction by A7,A15;
end;
then 0<r by A10,A11,SQUARE_1:25;
then
A21: r0*r>0 by A8,XREAL_1:129;
then 0+r*r0< r*r0+r*r0 by XREAL_1:8;
then r0*r-r0*r0< 2*r*r0-r0*r0 by XREAL_1:14;
then -r4 >-(2*r*r0-r0^2) by XREAL_1:24;
then r1+-r4 >r^2+-(2*r*r0-r0^2) by A12,XREAL_1:8;
then sqrt(r1-r4)>sqrt((r-r0)^2) by SQUARE_1:27,XREAL_1:63;
then
A22: sqrt(r1-r4)>r-r0 by A15,SQUARE_1:22;
0+r*r0< r*r0+r*r0 by A21,XREAL_1:8;
then r0*r+0< 2*r*r0+2*(r0*r0) by A8,XREAL_1:8;
then r0*r-r0*r0+r0*r0< 2*r*r0+r0*r0+r0*r0;
then r0*r-r0*r0< 2*r*r0+r0*r0 by XREAL_1:7;
then r1+r4 <r^2+(2*r*r0+r0^2) by A12,XREAL_1:8;
then sqrt(r1+r4)<sqrt((r+r0)^2) by A10,A8,A15,SQUARE_1:27;
then
A23: r+r0>sqrt(r1+r4) by A13,A7,SQUARE_1:22;
A24: r1-r4=r^2-(r0*r-r0*r0)by A10,A11,SQUARE_1:def 2
.=(r-(1/2)*r0)^2+(3/4)*r0^2;
g0.:W c= ].r-r0,r+r0.[
proof
let x be object;
assume x in g0.:W;
then consider z being object such that
A25: z in dom g0 and
A26: z in W and
A27: g0.z=x by FUNCT_1:def 6;
reconsider pz=z as Point of X by A25;
reconsider aa1=f1.pz as Real;
A28: ex r9 being Real st f1.pz=r9 & r9>=0 by A2;
pz in the carrier of X;
then pz in dom f1 by FUNCT_2:def 1;
then
A29: f1.pz in f1.:W1 by A26,FUNCT_1:def 6;
then aa1<r1+r4 by A19,XXREAL_1:4;
then sqrt(aa1)<sqrt(r1+r4) by A28,SQUARE_1:27;
then
A30: sqrt(aa1)< r+r0 by A23,XXREAL_0:2;
A31: r1-r4<aa1 by A19,A29,XXREAL_1:4;
A32: now
per cases;
case
0<=r1-r4;
then sqrt(r1-r4)<=sqrt(aa1) by A31,SQUARE_1:26;
hence r-r0< sqrt(aa1) by A22,XXREAL_0:2;
end;
case
0>r1-r4;
hence contradiction by A24,A20;
end;
end;
x=sqrt(aa1) by A4,A27;
hence thesis by A30,A32,XXREAL_1:4;
end;
hence thesis by A9,A18,XBOOLE_1:1;
end;
suppose
A33: r-r0<=0;
set r4=(2*r*r0+r0^2)/3;
reconsider G1=].r1-r4,r1+r4.[ as Subset of R^1 by TOPMETR:17;
(2*r*r0+r0^2)/3>0 by A14,XREAL_1:139;
then
A34: r1<r1+r4 by XREAL_1:29;
then r1-r4<r1 by XREAL_1:19;
then
A35: f1.p in G1 by A34,XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A36: p in W1 & W1 is open and
A37: f1.:W1 c= G1 by A1,A35,JGRAPH_2:10;
set W=W1;
(2*r*r0+r0^2)/3<(2*r*r0+r0^2) by A14,XREAL_1:221;
then r1+r4 <r^2+(2*r*r0+r0^2) by A12,XREAL_1:8;
then sqrt(r1+r4)<=sqrt((r+r0)^2) by A10,A13,A8,SQUARE_1:26;
then
A38: r+r0>=sqrt(r1+r4) by A13,A7,SQUARE_1:22;
g0.:W c= ].r-r0,r+r0.[
proof
let x be object;
assume x in g0.:W;
then consider z being object such that
A39: z in dom g0 and
A40: z in W and
A41: g0.z=x by FUNCT_1:def 6;
reconsider pz=z as Point of X by A39;
reconsider aa1=f1.pz as Real;
A42: ex r9 being Real st f1.pz=r9 & r9>=0 by A2;
pz in the carrier of X;
then pz in dom f1 by FUNCT_2:def 1;
then
A43: f1.pz in f1.:W1 by A40,FUNCT_1:def 6;
then aa1<r1+r4 by A37,XXREAL_1:4;
then sqrt(aa1)<sqrt(r1+r4) by A42,SQUARE_1:27;
then
A44: sqrt(aa1)< r+r0 by A38,XXREAL_0:2;
A45: r1-r4<aa1 by A37,A43,XXREAL_1:4;
A46: now
per cases by A33;
case
r-r0=0;
hence r-r0<sqrt(aa1) by A12,A45,SQUARE_1:17,27;
end;
case
r-r0<0;
hence r-r0<sqrt(aa1) by A42,SQUARE_1:17,26;
end;
end;
x=sqrt(aa1) by A4,A41;
hence thesis by A44,A46,XXREAL_1:4;
end;
hence thesis by A9,A36,XBOOLE_1:1;
end;
end;
then
A47: g0 is continuous by JGRAPH_2:10;
for p being Point of X,r11 being Real st f1.p=r11 holds g0.p=sqrt
(r11) by A4;
hence thesis by A47;
end;
theorem Th6:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g.p=(r1/r2)^2) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1;
assume
f1 is continuous & f2 is continuous & for q being Point of X holds f2.q <>0;
then consider g2 being Function of X,R^1 such that
A1: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=r1/r2 and
A2: g2 is continuous by JGRAPH_2:27;
consider g3 being Function of X,R^1 such that
A3: for p being Point of X,r1 being Real st g2.p=r1 holds g3.p=r1
*r1 and
A4: g3 is continuous by A2,JGRAPH_2:22;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=(r1/r2)^2
proof
let p be Point of X,r1,r2 be Real;
assume f1.p=r1 & f2.p=r2;
then g2.p=r1/r2 by A1;
hence thesis by A3;
end;
hence thesis by A4;
end;
theorem Th7:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g.p=1+(r1/r2)^2) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1;
assume
f1 is continuous & f2 is continuous & for q being Point of X holds f2.q <>0;
then consider g2 being Function of X,R^1 such that
A1: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=(r1/r2)^2 and
A2: g2 is continuous by Th6;
consider g3 being Function of X,R^1 such that
A3: for p being Point of X,r1 being Real st g2.p=r1 holds g3.p=r1 +1 and
A4: g3 is continuous by A2,JGRAPH_2:24;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=1+(r1/r2)^2
proof
let p be Point of X,r1,r2 be Real;
assume f1.p=r1 & f2.p=r2;
then g2.p=(r1/r2)^2 by A1;
hence thesis by A3;
end;
hence thesis by A4;
end;
theorem Th8:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g.p=sqrt(1+(r1/r2)^2)) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1;
assume
f1 is continuous & f2 is continuous & for q being Point of X holds f2.q <>0;
then consider g2 being Function of X,R^1 such that
A1: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=1+(r1/r2)^2 and
A2: g2 is continuous by Th7;
for q being Point of X ex r being Real st g2.q=r & r>=0
proof
let q be Point of X;
reconsider r1=f1.q,r2=f2.q as Real;
1+(r1/r2)^2>0 by Lm1;
hence thesis by A1;
end;
then consider g3 being Function of X,R^1 such that
A3: for p being Point of X,r1 being Real st g2.p=r1 holds g3.p=
sqrt( r1) and
A4: g3 is continuous by A2,Th5;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=sqrt(1+(r1/r2)^2)
proof
let p be Point of X,r1,r2 be Real;
assume f1.p=r1 & f2.p=r2;
then g2.p=1+(r1/r2)^2 by A1;
hence thesis by A3;
end;
hence thesis by A4;
end;
theorem Th9:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g.p=r1/sqrt(1+(r1/r2)^2)) & g is continuous
proof
let X be non empty TopSpace,f1,f2 be Function of X,R^1;
assume that
A1: f1 is continuous and
A2: f2 is continuous & for q being Point of X holds f2.q<>0;
consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=sqrt(1+(r1/r2)^2) and
A4: g2 is continuous by A1,A2,Th8;
for q being Point of X holds g2.q<>0
proof
let q be Point of X;
reconsider r1=f1.q,r2=f2.q as Real;
sqrt(1+(r1/r2)^2)>0 by Lm1,SQUARE_1:25;
hence thesis by A3;
end;
then consider g3 being Function of X,R^1 such that
A5: for p being Point of X,r1,r0 being Real st f1.p=r1 & g2.p=r0
holds g3.p=r1/r0 and
A6: g3 is continuous by A1,A4,JGRAPH_2:27;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=r1/sqrt(1+(r1/r2)^2)
proof
let p be Point of X,r1,r2 be Real;
assume that
A7: f1.p=r1 and
A8: f2.p=r2;
g2.p=sqrt(1+(r1/r2)^2) by A3,A7,A8;
hence thesis by A5,A7;
end;
hence thesis by A6;
end;
theorem Th10:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g.p=r2/sqrt(1+(r1/r2)^2)) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1;
assume that
A1: f1 is continuous and
A2: f2 is continuous and
A3: for q being Point of X holds f2.q<>0;
consider g2 being Function of X,R^1 such that
A4: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=sqrt(1+(r1/r2)^2) and
A5: g2 is continuous by A1,A2,A3,Th8;
for q being Point of X holds g2.q<>0
proof
let q be Point of X;
reconsider r1=f1.q,r2=f2.q as Real;
sqrt(1+(r1/r2)^2)>0 by Lm1,SQUARE_1:25;
hence thesis by A4;
end;
then consider g3 being Function of X,R^1 such that
A6: for p being Point of X,r2,r0 being Real st f2.p=r2 & g2.p=r0
holds g3.p=r2/r0 and
A7: g3 is continuous by A2,A5,JGRAPH_2:27;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=r2/sqrt(1+(r1/r2)^2)
proof
let p be Point of X,r1,r2 be Real;
assume that
A8: f1.p=r1 and
A9: f2.p=r2;
g2.p=sqrt(1+(r1/r2)^2) by A4,A8,A9;
hence thesis by A6,A9;
end;
hence thesis by A7;
end;
Lm4: for K1 being non empty Subset of TOP-REAL 2 holds for q being Point of (
TOP-REAL 2)|K1 holds (proj2|K1).q=proj2.q
proof
let K1 be non empty Subset of TOP-REAL 2;
let q be Point of (TOP-REAL 2)|K1;
the carrier of (TOP-REAL 2)|K1=K1 & q in the carrier of (TOP-REAL 2)|K1
by PRE_TOPC:8;
then q in dom proj2 /\ K1 by Lm3,XBOOLE_0:def 4;
hence thesis by FUNCT_1:48;
end;
Lm5: for K1 being non empty Subset of TOP-REAL 2 holds proj2|K1 is continuous
Function of (TOP-REAL 2)|K1,R^1
proof
let K1 be non empty Subset of TOP-REAL 2;
reconsider g2=proj2|K1 as Function of (TOP-REAL 2)|K1,R^1 by TOPMETR:17;
for q be Point of (TOP-REAL 2)|K1 holds g2.q=proj2.q by Lm4;
hence thesis by JGRAPH_2:30;
end;
Lm6: for K1 being non empty Subset of TOP-REAL 2 holds for q being Point of (
TOP-REAL 2)|K1 holds (proj1|K1).q=proj1.q
proof
let K1 be non empty Subset of TOP-REAL 2;
let q be Point of (TOP-REAL 2)|K1;
the carrier of (TOP-REAL 2)|K1=K1 & q in the carrier of (TOP-REAL 2)|K1
by PRE_TOPC:8;
then q in dom proj1 /\ K1 by Lm2,XBOOLE_0:def 4;
hence thesis by FUNCT_1:48;
end;
Lm7: for K1 being non empty Subset of TOP-REAL 2 holds proj1|K1 is continuous
Function of (TOP-REAL 2)|K1,R^1
proof
let K1 be non empty Subset of TOP-REAL 2;
reconsider g2=proj1|K1 as Function of (TOP-REAL 2)|K1,R^1 by TOPMETR:17;
for q be Point of (TOP-REAL 2)|K1 holds g2.q=proj1.q by Lm6;
hence thesis by JGRAPH_2:29;
end;
theorem Th11:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`1/sqrt(1+(p`2/p`1)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)| K1 holds f.p=p`1/sqrt(1+(p`2/p`1)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
now
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g1.q=proj1.q by Lm6
.=q2`1 by PSCOMP_1:def 5;
hence g1.q<>0 by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q
=r1 & g1.q=r2 holds g3.q=r2/sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th10;
A6: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`1/sqrt(1+(r`2/r`1)^2) by A1,A7;
hence thesis by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
theorem Th12:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`2/sqrt(1+(p`2/p`1)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`2/sqrt(1+(p`2/p`1)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
now
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g1.q=proj1.q by Lm6
.=q2`1 by PSCOMP_1:def 5;
hence g1.q<>0 by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q
=r1 & g1.q=r2 holds g3.q=r1/sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th9;
A6: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`2/sqrt(1+(r`2/r`1)^2) by A1,A7;
hence thesis by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
theorem Th13:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`2/sqrt(1+(p`1/p`2)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`2/sqrt(1+(p`1/p`2)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
now
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g2.q=proj2.q by Lm4
.=q2`2 by PSCOMP_1:def 6;
hence g2.q<>0 by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g1.q
=r1 & g2.q=r2 holds g3.q=r2/sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th10;
A6: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`2/sqrt(1+(r`1/r`2)^2) by A1,A7;
hence thesis by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
theorem Th14:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`1/sqrt(1+(p`1/p`2)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`1/sqrt(1+(p`1/p`2)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
now
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g2.q=proj2.q by Lm4
.=q2`2 by PSCOMP_1:def 6;
hence g2.q<>0 by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g1.q
=r1 & g2.q=r2 holds g3.q=r1/sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th9;
A6: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`1/sqrt(1+(r`1/r`2)^2) by A1,A7;
hence thesis by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
Lm8: 0.REAL 2 = 0.TOP-REAL 2 by EUCLID:66;
Lm9: (1.REAL 2)`2<=(1.REAL 2)`1 & -(1.REAL 2)`1<=(1.REAL 2)`2 or (1.REAL 2) `2
>=(1.REAL 2)`1 & (1.REAL 2)`2<=-(1.REAL 2)`1 by JGRAPH_2:5;
Lm10: (1.REAL 2)<>0.TOP-REAL 2 by Lm8,REVROT_1:19;
Lm11: for K1 being non empty Subset of TOP-REAL 2 holds dom ((proj2)*(Sq_Circ|
K1)) = the carrier of (TOP-REAL 2)|K1
proof
let K1 be non empty Subset of TOP-REAL 2;
A1: dom (Sq_Circ|K1) c= dom ((proj2)*(Sq_Circ|K1))
proof
let x be object;
assume
A2: x in dom (Sq_Circ|K1);
then x in dom Sq_Circ /\ K1 by RELAT_1:61;
then x in dom Sq_Circ by XBOOLE_0:def 4;
then
A3: Sq_Circ.x in rng Sq_Circ by FUNCT_1:3;
(Sq_Circ|K1).x=Sq_Circ.x by A2,FUNCT_1:47;
hence thesis by A2,A3,Lm3,FUNCT_1:11;
end;
dom ((proj2)*(Sq_Circ|K1)) c= dom (Sq_Circ|K1) by RELAT_1:25;
hence dom ((proj2)*(Sq_Circ|K1)) = dom (Sq_Circ|K1) by A1
.=dom Sq_Circ /\ K1 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28
.=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8;
end;
Lm12: for K1 being non empty Subset of TOP-REAL 2 holds dom ((proj1)*(Sq_Circ|
K1)) = the carrier of (TOP-REAL 2)|K1
proof
let K1 be non empty Subset of TOP-REAL 2;
A1: dom (Sq_Circ|K1) c= dom ((proj1)*(Sq_Circ|K1))
proof
let x be object;
assume
A2: x in dom (Sq_Circ|K1);
then x in dom Sq_Circ /\ K1 by RELAT_1:61;
then x in dom Sq_Circ by XBOOLE_0:def 4;
then
A3: Sq_Circ.x in rng Sq_Circ by FUNCT_1:3;
(Sq_Circ|K1).x=Sq_Circ.x by A2,FUNCT_1:47;
hence thesis by A2,A3,Lm2,FUNCT_1:11;
end;
dom ((proj1)*(Sq_Circ|K1)) c= dom (Sq_Circ|K1) by RELAT_1:25;
hence dom ((proj1)*(Sq_Circ|K1)) =dom (Sq_Circ|K1) by A1
.=dom Sq_Circ /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28
.=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8;
end;
Lm13: NonZero TOP-REAL 2 <> {} by JGRAPH_2:9;
theorem Th15:
for K0,B0 being Subset of TOP-REAL 2,f being Function of (
TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=Sq_Circ|K0 & B0=NonZero TOP-REAL 2 & K0={p:
(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} holds f is
continuous
proof
let K0,B0 be Subset of TOP-REAL 2,f be Function of (TOP-REAL 2)|K0,(TOP-REAL
2)|B0;
assume
A1: f=Sq_Circ|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`2<=p`1 & -p`1<=p`2
or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2};
then 1.REAL 2 in K0 by Lm9,Lm10;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
dom ((proj1)*(Sq_Circ|K1)) = the carrier of (TOP-REAL 2)|K1 & rng ((
proj1)*( Sq_Circ|K1)) c= the carrier of R^1 by Lm12,TOPMETR:17;
then reconsider
g1=(proj1)*(Sq_Circ|K1) as Function of (TOP-REAL 2)|K1,R^1 by FUNCT_2:2;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g1.p=p`1/sqrt(1+(p`2/p`1)^2)
proof
let p be Point of TOP-REAL 2;
A2: dom (Sq_Circ|K1)=dom Sq_Circ /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A4: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A3;
then
A5: Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2), p`2/sqrt(1+(p`2/p`1)^2)]| by Def1;
(Sq_Circ|K1).p=Sq_Circ.p by A4,A3,FUNCT_1:49;
then g1.p=(proj1).(|[p`1/sqrt(1+(p`2/p`1)^2), p`2/sqrt(1+(p`2/p`1)^2)]|)
by A4,A2,A3,A5,FUNCT_1:13
.=(|[p`1/sqrt(1+(p`2/p`1)^2), p`2/sqrt(1+(p`2/p`1)^2)]|)`1 by
PSCOMP_1:def 5
.=p`1/sqrt(1+(p`2/p`1)^2) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A6: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=p`1/sqrt(1+(p`2/p`1)^2);
dom ((proj2)*(Sq_Circ|K1)) = the carrier of (TOP-REAL 2)|K1 & rng ((
proj2)*( Sq_Circ|K1)) c= the carrier of R^1 by Lm11,TOPMETR:17;
then reconsider
g2=(proj2)*(Sq_Circ|K1) as Function of (TOP-REAL 2)|K1,R^1 by FUNCT_2:2;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g2.p=p`2/sqrt(1+(p`2/p`1)^2)
proof
let p be Point of TOP-REAL 2;
A7: dom (Sq_Circ|K1)=dom Sq_Circ /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A8: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A9: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A8;
then
A10: Sq_Circ.p =|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]| by Def1;
(Sq_Circ|K1).p=Sq_Circ.p by A9,A8,FUNCT_1:49;
then
g2.p=(proj2).(|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|) by A9,A7
,A8,A10,FUNCT_1:13
.=(|[p`1/sqrt(1+(p`2/p`1)^2), p`2/sqrt(1+(p`2/p`1)^2)]|)`2 by
PSCOMP_1:def 6
.=p`2/sqrt(1+(p`2/p`1)^2) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A11: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=p`2/sqrt(1+(p`2/p`1)^2);
A12: now
let q be Point of TOP-REAL 2;
A13: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
A14: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`2<=p3`1 & - p3`1<=p3`2 or
p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A13;
now
assume
A15: q`1=0;
then q`2=0 by A14;
hence contradiction by A14,A15,EUCLID:53,54;
end;
hence q`1<>0;
end;
then
A16: f1 is continuous by A6,Th11;
A17: for x,y,r,s being Real st |[x,y]| in K1 & r=f1.(|[x,y]|) & s=f2.
(|[x,y]|) holds f.(|[x,y]|)=|[r,s]|
proof
let x,y,r,s be Real;
assume that
A18: |[x,y]| in K1 and
A19: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
set p99=|[x,y]|;
A20: ex p3 being Point of TOP-REAL 2 st p99=p3 &( p3`2<=p3`1 & -p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A18;
A21: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A22: f1.p99=p99`1/sqrt(1+(p99`2/p99`1)^2) by A6,A18;
(Sq_Circ|K0).(|[x,y]|)=(Sq_Circ).(|[x,y]|) by A18,FUNCT_1:49
.= |[p99`1/sqrt(1+(p99`2/p99`1)^2), p99`2/sqrt(1+(p99`2/p99`1)^2)]| by
A20,Def1
.=|[r,s]| by A11,A18,A19,A21,A22;
hence thesis by A1;
end;
f2 is continuous by A12,A11,Th12;
hence thesis by A1,A16,A17,Lm13,JGRAPH_2:35;
end;
Lm14: (1.REAL 2)`1<=(1.REAL 2)`2 & -(1.REAL 2)`2<=(1.REAL 2)`1 or (1.REAL 2 )
`1>=(1.REAL 2)`2 & (1.REAL 2)`1<=-(1.REAL 2)`2 by JGRAPH_2:5;
Lm15: (1.REAL 2)<>0.TOP-REAL 2 by Lm8,REVROT_1:19;
theorem Th16:
for K0,B0 being Subset of TOP-REAL 2,f being Function of (
TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=Sq_Circ|K0 & B0=NonZero TOP-REAL 2 & K0={p:
(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} holds f is
continuous
proof
let K0,B0 be Subset of TOP-REAL 2,f be Function of (TOP-REAL 2)|K0,(TOP-REAL
2)|B0;
assume
A1: f=Sq_Circ|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`1<=p`2 & -p`2<=p`1
or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2};
then 1.REAL 2 in K0 by Lm14,Lm15;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
dom ((proj2)*(Sq_Circ|K1))=the carrier of (TOP-REAL 2)|K1 & rng ((proj2)
*( Sq_Circ|K1)) c= the carrier of R^1 by Lm11,TOPMETR:17;
then reconsider
g1=(proj2)*(Sq_Circ|K1) as Function of (TOP-REAL 2)|K1,R^1 by FUNCT_2:2;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g1.p=p`2/sqrt(1+(p`1/p`2)^2)
proof
let p be Point of TOP-REAL 2;
A2: dom (Sq_Circ|K1)=dom Sq_Circ /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A4: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`1<=p3`2 & - p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A3;
then
A5: Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2), p`2/sqrt(1+(p`1/p`2)^2)]| by Th4;
(Sq_Circ|K1).p=Sq_Circ.p by A4,A3,FUNCT_1:49;
then
g1.p=(proj2).(|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|) by A4,A2
,A3,A5,FUNCT_1:13
.=(|[p`1/sqrt(1+(p`1/p`2)^2), p`2/sqrt(1+(p`1/p`2)^2)]|)`2 by
PSCOMP_1:def 6
.=p`2/sqrt(1+(p`1/p`2)^2) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A6: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=p`2/sqrt(1+(p`1/p`2)^2);
dom ((proj1)*(Sq_Circ|K1))=the carrier of (TOP-REAL 2)|K1 & rng ((proj1)
*( Sq_Circ|K1)) c= the carrier of R^1 by Lm12,TOPMETR:17;
then reconsider
g2=(proj1)*(Sq_Circ|K1) as Function of (TOP-REAL 2)|K1,R^1 by FUNCT_2:2;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g2.p=p`1/sqrt(1+(p`1/p`2)^2)
proof
let p be Point of TOP-REAL 2;
A7: dom (Sq_Circ|K1)=dom Sq_Circ /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A8: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A9: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`1<=p3`2 & - p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A8;
then
A10: Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2), p`2/sqrt(1+(p`1/p`2)^2)]| by Th4;
(Sq_Circ|K1).p=Sq_Circ.p by A9,A8,FUNCT_1:49;
then
g2.p=(proj1).(|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|) by A9,A7
,A8,A10,FUNCT_1:13
.=(|[p`1/sqrt(1+(p`1/p`2)^2), p`2/sqrt(1+(p`1/p`2)^2)]|)`1 by
PSCOMP_1:def 5
.=p`1/sqrt(1+(p`1/p`2)^2) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A11: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=p`1/sqrt(1+(p`1/p`2)^2);
A12: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<>0
proof
let q be Point of TOP-REAL 2;
A13: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
A14: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`1<=p3`2 & - p3`2<=p3`1 or
p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A13;
now
assume
A15: q`2=0;
then q`1=0 by A14;
hence contradiction by A14,A15,EUCLID:53,54;
end;
hence thesis;
end;
then
A16: f1 is continuous by A6,Th13;
A17: now
let x,y,s,r be Real;
assume that
A18: |[x,y]| in K1 and
A19: s=f2.(|[x,y]|) & r=f1.(|[x,y]|);
set p99=|[x,y]|;
A20: ex p3 being Point of TOP-REAL 2 st p99=p3 &( p3`1<=p3`2 & -p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A18;
A21: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A22: f1.p99=p99`2/sqrt(1+(p99`1/p99`2)^2) by A6,A18;
(Sq_Circ|K0).(|[x,y]|)=(Sq_Circ).(|[x,y]|) by A18,FUNCT_1:49
.= |[p99`1/sqrt(1+(p99`1/p99`2)^2), p99`2/sqrt(1+(p99`1/p99`2)^2)]| by
A20,Th4
.=|[s,r]| by A11,A18,A19,A21,A22;
hence f.(|[x,y]|)=|[s,r]| by A1;
end;
f2 is continuous by A12,A11,Th14;
hence thesis by A1,A16,A17,Lm13,JGRAPH_2:35;
end;
scheme
TopIncl { P[set] } : { p: P[p] & p<>0.TOP-REAL 2 } c= NonZero TOP-REAL 2
proof
let x be object;
assume x in { p: P[p] & p<>0.TOP-REAL 2 };
then
A1: ex p8 being Point of TOP-REAL 2 st x=p8 &( P[p8])& p8<> 0.TOP-REAL 2;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,XBOOLE_0:def 5;
end;
scheme
TopInter { P[set] } : { p: P[p] & p<>0.TOP-REAL 2 } = { p7 where p7 is Point
of TOP-REAL 2 : P[p7]} /\ (NonZero TOP-REAL 2) proof
set B0 = NonZero TOP-REAL 2;
set K1 = { p7 where p7 is Point of TOP-REAL 2 : P[p7]};
set K0 = { p: P[p] & p<>0.TOP-REAL 2 };
A1: K1 /\ B0 c= K0
proof
let x be object;
assume
A2: x in K1 /\ B0;
then x in B0 by XBOOLE_0:def 4;
then not x in {0.TOP-REAL 2} by XBOOLE_0:def 5;
then
A3: x <> 0.TOP-REAL 2 by TARSKI:def 1;
x in K1 by A2,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & P[p7];
hence thesis by A3;
end;
K0 c= K1 /\ B0
proof
let x be object;
assume x in K0;
then
A4: ex p being Point of TOP-REAL 2 st x=p &( P[p])& p<>0. TOP-REAL 2;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
then
A5: x in B0 by A4,XBOOLE_0:def 5;
x in K1 by A4;
hence thesis by A5,XBOOLE_0:def 4;
end;
hence thesis by A1;
end;
theorem Th17:
for B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2
)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st f=Sq_Circ|K0
& B0=NonZero TOP-REAL 2 & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1)
& p<>0.TOP-REAL 2} holds f is continuous & K0 is closed
proof
reconsider K5={p7 where p7 is Point of TOP-REAL 2:p7`2<=-p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:47;
reconsider K4={p7 where p7 is Point of TOP-REAL 2: p7`1<=p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
reconsider K3={p7 where p7 is Point of TOP-REAL 2: -p7`1<=p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:47;
reconsider K2={p7 where p7 is Point of TOP-REAL 2: p7`2<=p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
defpred P[Point of TOP-REAL 2] means ($1`2<=$1`1 & -$1`1<=$1`2 or $1`2>=$1`1
& $1`2<=-$1`1);
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0,f being
Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
assume
A1: f=Sq_Circ|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`2<=p`1 & -p`1<=p`2
or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2};
the carrier of (TOP-REAL 2)|B0 = B0 by PRE_TOPC:8;
then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1;
{p:P[p] & p<>0.TOP-REAL 2} c= NonZero TOP-REAL 2 from TopIncl;
then
A2: ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by A1,PRE_TOPC:7;
defpred P[Point of TOP-REAL 2] means ($1`2<=$1`1 & -$1`1<=$1`2 or $1`2>=$1`1
& $1`2<=-$1`1);
reconsider K1={p7 where p7 is Point of TOP-REAL 2: P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`2<=$1`1 & -$1`1<=$1`2 or $1`2>=$1`1
& $1`2<=-$1`1);
{p: P[p] & p<>0.TOP-REAL 2} = {p7 where p7 is Point of TOP-REAL 2: P[p7
]} /\ (NonZero TOP-REAL 2) from TopInter;
then
A3: K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,PRE_TOPC:def 5;
A4: K2 /\ K3 \/ K4 /\ K5 c= K1
proof
let x be object;
assume
A5: x in K2 /\ K3 \/ K4 /\ K5;
per cases by A5,XBOOLE_0:def 3;
suppose
A6: x in K2 /\ K3;
then x in K3 by XBOOLE_0:def 4;
then
A7: ex p8 being Point of TOP-REAL 2 st p8=x & -p8`1<=p8`2;
x in K2 by A6,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`2<=(p7`1);
hence thesis by A7;
end;
suppose
A8: x in K4 /\ K5;
then x in K5 by XBOOLE_0:def 4;
then
A9: ex p8 being Point of TOP-REAL 2 st p8=x & p8`2<= -p8`1;
x in K4 by A8,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`2>=(p7`1);
hence thesis by A9;
end;
end;
A10: K2 /\ K3 is closed & K4 /\ K5 is closed by TOPS_1:8;
K1 c= K2 /\ K3 \/ K4 /\ K5
proof
let x be object;
assume x in K1;
then ex p being Point of TOP-REAL 2 st p=x &( p`2<=p`1 & -p`1 <=p`2 or p`2
>=p`1 & p`2<=-p`1);
then x in K2 & x in K3 or x in K4 & x in K5;
then x in K2 /\ K3 or x in K4 /\ K5 by XBOOLE_0:def 4;
hence thesis by XBOOLE_0:def 3;
end;
then K1=K2 /\ K3 \/ K4 /\ K5 by A4;
then K1 is closed by A10,TOPS_1:9;
hence thesis by A1,A2,A3,Th15,PRE_TOPC:13;
end;
theorem Th18:
for B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2
)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st f=Sq_Circ|K0
& B0=NonZero TOP-REAL 2 & K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2)
& p<>0.TOP-REAL 2} holds f is continuous & K0 is closed
proof
reconsider K5={p7 where p7 is Point of TOP-REAL 2: p7`1<=-p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:48;
reconsider K4={p7 where p7 is Point of TOP-REAL 2: p7`2<=p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
reconsider K3={p7 where p7 is Point of TOP-REAL 2: -p7`2<=p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:48;
reconsider K2={p7 where p7 is Point of TOP-REAL 2: p7`1<=p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
defpred P[Point of TOP-REAL 2] means ($1`1<=$1`2 & -$1`2<=$1`1 or $1`1>=$1`2
& $1`1<=-$1`2);
set b0 = NonZero TOP-REAL 2;
defpred P0[Point of TOP-REAL 2] means ($1`1<=$1`2 & -$1`2<=$1`1 or $1`1>=$1
`2 & $1`1<=-$1`2);
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0,f being
Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
assume
A1: f=Sq_Circ|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`1<=p`2 & -p`2<=p`1
or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2};
the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8;
then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1;
{p:P[p] & p<>0.TOP-REAL 2} c= NonZero TOP-REAL 2 from TopIncl;
then
A2: ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by A1,PRE_TOPC:7;
defpred P[Point of TOP-REAL 2] means ($1`1<=$1`2 & -$1`2<=$1`1 or $1`1>=$1`2
& $1`1<=-$1`2);
reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
A3: K2 /\ K3 \/ K4 /\ K5 c= K1
proof
let x be object;
assume
A4: x in K2 /\ K3 \/ K4 /\ K5;
per cases by A4,XBOOLE_0:def 3;
suppose
A5: x in K2 /\ K3;
then x in K3 by XBOOLE_0:def 4;
then
A6: ex p8 being Point of TOP-REAL 2 st p8=x & -p8`2<=p8`1;
x in K2 by A5,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`1<=(p7`2);
hence thesis by A6;
end;
suppose
A7: x in K4 /\ K5;
then x in K5 by XBOOLE_0:def 4;
then
A8: ex p8 being Point of TOP-REAL 2 st p8=x & p8`1<= -p8`2;
x in K4 by A7,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`1>=(p7`2);
hence thesis by A8;
end;
end;
set k0 = {p:P0[p] & p<>0.TOP-REAL 2};
A9: K2 /\ K3 is closed & K4 /\ K5 is closed by TOPS_1:8;
K1 c= K2 /\ K3 \/ K4 /\ K5
proof
let x be object;
assume x in K1;
then ex p being Point of TOP-REAL 2 st p=x &( p`1<=p`2 & -p`2 <=p`1 or p`1
>=p`2 & p`1<=-p`2);
then x in K2 & x in K3 or x in K4 & x in K5;
then x in K2 /\ K3 or x in K4 /\ K5 by XBOOLE_0:def 4;
hence thesis by XBOOLE_0:def 3;
end;
then K1=K2 /\ K3 \/ K4 /\ K5 by A3;
then
A10: K1 is closed by A9,TOPS_1:9;
k0 = {p7 where p7 is Point of TOP-REAL 2:P0[p7]} /\ b0 from TopInter;
then K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,PRE_TOPC:def 5;
hence thesis by A1,A2,A10,Th16,PRE_TOPC:13;
end;
theorem Th19:
for D being non empty Subset of TOP-REAL 2 st D`={0.TOP-REAL 2}
holds ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=Sq_Circ|D & h
is continuous
proof
set Y1=|[-1,1]|;
let D be non empty Subset of TOP-REAL 2;
A1: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
dom Sq_Circ=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1;
then
A2: dom (Sq_Circ|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|D) by A1,XBOOLE_1:28;
assume
A3: D`={0.TOP-REAL 2};
then
A4: D = {0.TOP-REAL 2}` .=(NonZero TOP-REAL 2) by SUBSET_1:def 4;
A5: {p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} c=
the carrier of (TOP-REAL 2)|D
proof
let x be object;
assume
x in {p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0. TOP-REAL 2};
then
A6: ex p st x=p &( p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p `1)& p<>0.
TOP-REAL 2;
now
assume not x in D;
then x in (the carrier of TOP-REAL 2) \ D by A6,XBOOLE_0:def 5;
then x in D` by SUBSET_1:def 4;
hence contradiction by A3,A6,TARSKI:def 1;
end;
hence thesis by PRE_TOPC:8;
end;
1.REAL 2 in {p where p is Point of TOP-REAL 2: (p`2<=p`1 & -p`1<=p`2 or
p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} by Lm9,Lm10;
then reconsider K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.
TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A5;
A7: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8;
A8: {p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} c=
the carrier of (TOP-REAL 2)|D
proof
let x be object;
assume
x in {p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0. TOP-REAL 2};
then
A9: ex p st x=p &( p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p `2)& p<>0.
TOP-REAL 2;
now
assume not x in D;
then x in (the carrier of TOP-REAL 2) \ D by A9,XBOOLE_0:def 5;
then x in D` by SUBSET_1:def 4;
hence contradiction by A3,A9,TARSKI:def 1;
end;
hence thesis by PRE_TOPC:8;
end;
Y1`1=-1 & Y1`2=1 by EUCLID:52;
then
Y1 in {p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1>=
p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} by JGRAPH_2:3;
then reconsider K1={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.
TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A8;
A10: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8;
A11: D c= K0 \/ K1
proof
let x be object;
assume
A12: x in D;
then reconsider px=x as Point of TOP-REAL 2;
not x in {0.TOP-REAL 2} by A4,A12,XBOOLE_0:def 5;
then (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.
TOP-REAL 2 or (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.
TOP-REAL 2 by TARSKI:def 1,XREAL_1:26;
then x in K0 or x in K1;
hence thesis by XBOOLE_0:def 3;
end;
A13: the carrier of ((TOP-REAL 2)|D) =[#](((TOP-REAL 2)|D))
.=(NonZero TOP-REAL 2) by A4,PRE_TOPC:def 5;
A14: the carrier of ((TOP-REAL 2)|D) =D by PRE_TOPC:8;
A15: rng (Sq_Circ|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
reconsider K00=K0 as Subset of TOP-REAL 2 by A14,XBOOLE_1:1;
let y be object;
A16: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|
K00 holds q`1<>0
proof
let q be Point of TOP-REAL 2;
A17: the carrier of (TOP-REAL 2)|K00=K0 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K00;
then
A18: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A17;
now
assume
A19: q`1=0;
then q`2=0 by A18;
hence contradiction by A18,A19,EUCLID:53,54;
end;
hence thesis;
end;
assume y in rng (Sq_Circ|K0);
then consider x being object such that
A20: x in dom (Sq_Circ|K0) and
A21: y=(Sq_Circ|K0).x by FUNCT_1:def 3;
A22: x in (dom Sq_Circ) /\ K0 by A20,RELAT_1:61;
then
A23: x in K0 by XBOOLE_0:def 4;
K0 c= the carrier of TOP-REAL 2 by A14,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A23;
K00=the carrier of ((TOP-REAL 2)|K00) by PRE_TOPC:8;
then p in the carrier of ((TOP-REAL 2)|K00) by A22,XBOOLE_0:def 4;
then
A24: p`1<>0 by A16;
A25: ex px being Point of TOP-REAL 2 st x=px &( px`2<=px`1 & - px`1<=px`2
or px`2>=px`1 & px`2<=-px`1)& px<>0.TOP-REAL 2 by A23;
then
A26: Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2), p`2/sqrt(1+(p`2/p`1)^2)]| by Def1;
A27: sqrt(1+(p`2/p`1)^2)>0 by Lm1,SQUARE_1:25;
then
p`2/sqrt(1+(p`2/p`1)^2)<=p`1/sqrt(1+(p`2/p`1)^2) & (-p`1)/sqrt(1+(p`2
/p`1)^2)<=p`2/sqrt(1+(p`2/p`1)^2) or p`2/sqrt(1+(p`2/p`1)^2)>=p`1/sqrt(1+(p`2/p
`1)^2) & p`2/sqrt(1+(p`2/p`1)^2)<=(-p`1)/sqrt(1+(p`2/p`1)^2) by A25,
XREAL_1:72;
then
A28: p`2/sqrt(1+(p`2/p`1)^2)<=p`1/sqrt(1+(p`2/p`1)^2) & -(p`1/sqrt(1+(p`2/
p`1)^2))<=p`2/sqrt(1+(p`2/p`1)^2) or p`2/sqrt(1+(p`2/p`1)^2)>=p`1/sqrt(1+(p`2/p
`1)^2) & p`2/sqrt(1+(p`2/p`1)^2)<=-(p`1/sqrt(1+(p`2/p`1)^2)) by
XCMPLX_1:187;
set p9=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]|;
A29: p9`1=p`1/sqrt(1+(p`2/p`1)^2) & p9`2=p`2/sqrt(1+(p`2/p`1)^2) by EUCLID:52;
A30: p9`1=p`1/sqrt(1+(p`2/p`1)^2) by EUCLID:52;
A31: now
assume p9=0.TOP-REAL 2;
then 0 *sqrt(1+(p`2/p`1)^2)=p`1/sqrt(1+(p`2/p`1)^2)*sqrt(1+(p`2/p`1)^2)
by A30,EUCLID:52,54;
hence contradiction by A24,A27,XCMPLX_1:87;
end;
Sq_Circ.p=y by A21,A23,FUNCT_1:49;
then y in K0 by A31,A26,A28,A29;
then y in [#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
hence thesis;
end;
A32: K0 c= the carrier of TOP-REAL 2
proof
let z be object;
assume z in K0;
then ex p8 being Point of TOP-REAL 2 st p8=z &( p8`2<=p8`1 & - p8`1<=p8`2
or p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2;
hence thesis;
end;
dom (Sq_Circ|K0)= dom (Sq_Circ) /\ K0 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1
.=K0 by A32,XBOOLE_1:28;
then reconsider
f=Sq_Circ|K0 as Function of ((TOP-REAL 2)|D)|K0, (TOP-REAL 2)|D
by A7,A15,FUNCT_2:2,XBOOLE_1:1;
A33: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
A34: K1 c= the carrier of TOP-REAL 2
proof
let z be object;
assume z in K1;
then ex p8 being Point of TOP-REAL 2 st p8=z &( p8`1<=p8`2 & - p8`2<=p8`1
or p8`1>=p8`2 & p8`1<=-p8`2)& p8<>0.TOP-REAL 2;
hence thesis;
end;
A35: rng (Sq_Circ|K1) c= the carrier of ((TOP-REAL 2)|D)|K1
proof
reconsider K10=K1 as Subset of TOP-REAL 2 by A34;
let y be object;
A36: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|
K10 holds q`2<>0
proof
let q be Point of TOP-REAL 2;
A37: the carrier of (TOP-REAL 2)|K10=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K10;
then
A38: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`1<=p3`2 & - p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A37;
now
assume
A39: q`2=0;
then q`1=0 by A38;
hence contradiction by A38,A39,EUCLID:53,54;
end;
hence thesis;
end;
assume y in rng (Sq_Circ|K1);
then consider x being object such that
A40: x in dom (Sq_Circ|K1) and
A41: y=(Sq_Circ|K1).x by FUNCT_1:def 3;
A42: x in (dom Sq_Circ) /\ K1 by A40,RELAT_1:61;
then
A43: x in K1 by XBOOLE_0:def 4;
then reconsider p=x as Point of TOP-REAL 2 by A34;
K10=the carrier of ((TOP-REAL 2)|K10) by PRE_TOPC:8;
then p in the carrier of ((TOP-REAL 2)|K10) by A42,XBOOLE_0:def 4;
then
A44: p`2<>0 by A36;
set p9=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|;
A45: p9`2=p`2/sqrt(1+(p`1/p`2)^2) & p9`1=p`1/sqrt(1+(p`1/p`2)^2) by EUCLID:52;
A46: ex px being Point of TOP-REAL 2 st x=px &( px`1<=px`2 & - px`2<=px`1
or px`1>=px`2 & px`1<=-px`2)& px<>0.TOP-REAL 2 by A43;
then
A47: Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2), p`2/sqrt(1+(p`1/p`2)^2)]| by Th4;
A48: sqrt(1+(p`1/p`2)^2)>0 by Lm1,SQUARE_1:25;
then
p`1/sqrt(1+(p`1/p`2)^2)<=p`2/sqrt(1+(p`1/p`2)^2) & (-p`2)/sqrt(1+(p`1
/p`2)^2)<=p`1/sqrt(1+(p`1/p`2)^2) or p`1/sqrt(1+(p`1/p`2)^2)>=p`2/sqrt(1+(p`1/p
`2)^2) & p`1/sqrt(1+(p`1/p`2)^2)<=(-p`2)/sqrt(1+(p`1/p`2)^2) by A46,
XREAL_1:72;
then
A49: p`1/sqrt(1+(p`1/p`2)^2)<=p`2/sqrt(1+(p`1/p`2)^2) & -(p`2/sqrt(1+(p`1/
p`2)^2))<=p`1/sqrt(1+(p`1/p`2)^2) or p`1/sqrt(1+(p`1/p`2)^2)>=p`2/sqrt(1+(p`1/p
`2)^2) & p`1/sqrt(1+(p`1/p`2)^2)<=-(p`2/sqrt(1+(p`1/p`2)^2)) by
XCMPLX_1:187;
A50: p9`2=p`2/sqrt(1+(p`1/p`2)^2) by EUCLID:52;
A51: now
assume p9=0.TOP-REAL 2;
then 0 *sqrt(1+(p`1/p`2)^2)=p`2/sqrt(1+(p`1/p`2)^2)*sqrt(1+(p`1/p`2)^2)
by A50,EUCLID:52,54;
hence contradiction by A44,A48,XCMPLX_1:87;
end;
Sq_Circ.p=y by A41,A43,FUNCT_1:49;
then y in K1 by A51,A47,A49,A45;
hence thesis by PRE_TOPC:8;
end;
dom (Sq_Circ|K1)= dom (Sq_Circ) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1
.=K1 by A34,XBOOLE_1:28;
then reconsider
g=Sq_Circ|K1 as Function of ((TOP-REAL 2)|D)|K1, ((TOP-REAL 2)|D)
by A10,A35,FUNCT_2:2,XBOOLE_1:1;
A52: dom g=K1 by A10,FUNCT_2:def 1;
g=Sq_Circ|K1;
then
A53: K1 is closed by A4,Th18;
A54: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
A55: for x be object st x in ([#](((TOP-REAL 2)|D)|K0)) /\ ([#] (((TOP-REAL 2)
|D)|K1)) holds f.x = g.x
proof
let x be object;
assume
A56: x in ([#](((TOP-REAL 2)|D)|K0)) /\ [#] (((TOP-REAL 2)|D)|K1);
then x in K0 by A54,XBOOLE_0:def 4;
then f.x=Sq_Circ.x by FUNCT_1:49;
hence thesis by A33,A56,FUNCT_1:49;
end;
f=Sq_Circ|K0;
then
A57: K0 is closed by A4,Th17;
A58: dom f=K0 by A7,FUNCT_2:def 1;
D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
then
A59: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#]((
TOP-REAL 2)|D) by A54,A33,A11;
A60: f is continuous & g is continuous by A4,Th17,Th18;
then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that
A61: h= f+*g and
h is continuous by A54,A33,A59,A57,A53,A55,JGRAPH_2:1;
K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
then
A62: f tolerates g by A55,A58,A52,PARTFUN1:def 4;
A63: for x being object st x in dom h holds h.x=(Sq_Circ|D).x
proof
let x be object;
assume
A64: x in dom h;
then reconsider p=x as Point of TOP-REAL 2 by A13,XBOOLE_0:def 5;
not x in {0.TOP-REAL 2} by A13,A64,XBOOLE_0:def 5;
then
A65: x <>0.TOP-REAL 2 by TARSKI:def 1;
x in (the carrier of TOP-REAL 2)\D` by A3,A13,A64;
then
A66: x in D`` by SUBSET_1:def 4;
per cases;
suppose
A67: x in K0;
A68: Sq_Circ|D.p=Sq_Circ.p by A66,FUNCT_1:49
.=f.p by A67,FUNCT_1:49;
h.p=(g+*f).p by A61,A62,FUNCT_4:34
.=f.p by A58,A67,FUNCT_4:13;
hence thesis by A68;
end;
suppose
not x in K0;
then not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) by A65;
then p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2 by XREAL_1:26;
then
A69: x in K1 by A65;
Sq_Circ|D.p=Sq_Circ.p by A66,FUNCT_1:49
.=g.p by A69,FUNCT_1:49;
hence thesis by A61,A52,A69,FUNCT_4:13;
end;
end;
dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1;
then f+*g=Sq_Circ|D by A61,A2,A63;
hence thesis by A54,A33,A59,A57,A60,A53,A55,JGRAPH_2:1;
end;
theorem Th20:
for D being non empty Subset of TOP-REAL 2 st D=NonZero TOP-REAL
2 holds D`= {0.TOP-REAL 2}
proof
let D be non empty Subset of TOP-REAL 2;
assume
A1: D=NonZero TOP-REAL 2;
A2: D` c= {0.TOP-REAL 2}
proof
let x be object;
assume
A3: x in D`;
then x in (the carrier of TOP-REAL 2)\D by SUBSET_1:def 4;
then not x in D by XBOOLE_0:def 5;
hence thesis by A1,A3,XBOOLE_0:def 5;
end;
{0.TOP-REAL 2} c= D`
proof
let x be object;
assume
A4: x in {0.TOP-REAL 2};
then not x in D by A1,XBOOLE_0:def 5;
then x in (the carrier of TOP-REAL 2)\D by A4,XBOOLE_0:def 5;
hence thesis by SUBSET_1:def 4;
end;
hence thesis by A2;
end;
Lm16: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
theorem Th21:
ex h being Function of TOP-REAL 2, TOP-REAL 2 st h=Sq_Circ & h is continuous
proof
reconsider D=NonZero TOP-REAL 2 as non empty Subset of TOP-REAL 2 by
JGRAPH_2:9;
reconsider f=Sq_Circ as Function of (TOP-REAL 2),(TOP-REAL 2);
A1: for p being Point of (TOP-REAL 2)|D holds f.p<>f.(0.TOP-REAL 2)
proof
let p be Point of (TOP-REAL 2)|D;
A2: [#]((TOP-REAL 2)|D)=D by PRE_TOPC:def 5;
then reconsider q=p as Point of TOP-REAL 2 by XBOOLE_0:def 5;
not p in {0.TOP-REAL 2} by A2,XBOOLE_0:def 5;
then
A3: not p=0.TOP-REAL 2 by TARSKI:def 1;
per cases;
suppose
A4: not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then A5: q`2<>0;
set q9=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|;
A6: q9`2=q`2/sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A7: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A8: now
assume q9=0.TOP-REAL 2;
then
0 *sqrt(1+(q`1/q`2)^2)=q`2/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2
) by A6,EUCLID:52,54;
hence contradiction by A5,A7,XCMPLX_1:87;
end;
Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]| by A3,A4
,Def1;
hence thesis by A8,Def1;
end;
suppose
A9: q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1;
A10: now
assume
A11: q`1=0;
then q`2=0 by A9;
hence contradiction by A3,A11,EUCLID:53,54;
end;
set q9=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|;
A12: q9`1=q`1/sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A13: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A14: now
assume q9=0.TOP-REAL 2;
then
0 *sqrt(1+(q`2/q`1)^2)=q`1/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2
) by A12,EUCLID:52,54;
hence contradiction by A10,A13,XCMPLX_1:87;
end;
Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A3,A9
,Def1;
hence thesis by A14,Def1;
end;
end;
A15: f.(0.TOP-REAL 2)=0.TOP-REAL 2 by Def1;
A16: for V being Subset of TOP-REAL 2 st f.(0.TOP-REAL 2) in V & V is open
ex W being Subset of TOP-REAL 2 st 0.TOP-REAL 2 in W & W is open & f.:W c= V
proof
reconsider u0=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67;
let V be Subset of (TOP-REAL 2);
reconsider VV=V as Subset of TopSpaceMetr Euclid 2 by Lm16;
assume that
A17: f.(0.TOP-REAL 2) in V and
A18: V is open;
VV is open by A18,Lm16,PRE_TOPC:30;
then consider r being Real such that
A19: r>0 and
A20: Ball(u0,r) c= V by A15,A17,TOPMETR:15;
reconsider r as Real;
reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2 by EUCLID:67;
A21: W1 is open by GOBOARD6:3;
A22: f.:W1 c= W1
proof
let z be object;
assume z in f.:W1;
then consider y being object such that
A23: y in dom f and
A24: y in W1 and
A25: z=f.y by FUNCT_1:def 6;
z in rng f by A23,A25,FUNCT_1:def 3;
then reconsider qz=z as Point of TOP-REAL 2;
reconsider pz=qz as Point of Euclid 2 by EUCLID:67;
reconsider q=y as Point of TOP-REAL 2 by A23;
reconsider qy=q as Point of Euclid 2 by EUCLID:67;
dist(u0,qy)<r by A24,METRIC_1:11;
then |.(0.TOP-REAL 2) - q.|<r by JGRAPH_1:28;
then sqrt((((0.TOP-REAL 2) - q)`1)^2+(((0.TOP-REAL 2) - q)`2)^2)<r by
JGRAPH_1:30;
then sqrt(((0.TOP-REAL 2)`1 - q`1)^2+(((0.TOP-REAL 2) - q)`2)^2)<r by
TOPREAL3:3;
then
A26: sqrt(((0.TOP-REAL 2)`1 - q`1)^2+((0.TOP-REAL 2)`2 - q`2)^2)<r by
TOPREAL3:3;
per cases;
suppose
q=0.TOP-REAL 2;
hence thesis by A24,A25,Def1;
end;
suppose
A27: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q `1);
A28: (q`2)^2>=0 by XREAL_1:63;
(q`2/q`1)^2 >=0 by XREAL_1:63;
then 1+(q`2/q`1)^2>=1+0 by XREAL_1:7;
then
A29: sqrt(1+(q`2/q`1)^2)>=1 by SQUARE_1:18,26;
then (sqrt(1+(q`2/q`1)^2))^2>=sqrt(1+(q`2/q`1)^2) by XREAL_1:151;
then
A30: 1<=(sqrt(1+(q`2/q`1)^2))^2 by A29,XXREAL_0:2;
A31: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2 ) ]|
by A27,Def1;
then (qz`2)^2=(q`2/sqrt(1+(q`2/q`1)^2))^2 by A25,EUCLID:52
.=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by XCMPLX_1:76;
then
A32: (qz`2)^2<=(q`2)^2/1 by A30,A28,XREAL_1:118;
A33: (q`1)^2>=0 by XREAL_1:63;
(qz`1)^2=(q`1/sqrt(1+(q`2/q`1)^2))^2 by A25,A31,EUCLID:52
.=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2 by XCMPLX_1:76;
then (qz`1)^2<=(q`1)^2/1 by A30,A33,XREAL_1:118;
then
A34: (qz`1)^2+(qz`2)^2<=(q`1)^2+(q`2)^2 by A32,XREAL_1:7;
(qz`1)^2>=0 & (qz`2)^2>=0 by XREAL_1:63;
then
A35: sqrt((qz`1)^2+(qz`2)^2) <= sqrt((q`1)^2+(q`2)^2) by A34,SQUARE_1:26;
A36: ((0.TOP-REAL 2) - qz)`2=(0.TOP-REAL 2)`2-qz`2 by TOPREAL3:3
.= -qz`2 by JGRAPH_2:3;
((0.TOP-REAL 2) - qz)`1=(0.TOP-REAL 2)`1-qz`1 by TOPREAL3:3
.= -qz`1 by JGRAPH_2:3;
then sqrt((((0.TOP-REAL 2) - qz)`1)^2+(((0.TOP-REAL 2) - qz)`2)^2)<r
by A26,A36,A35,JGRAPH_2:3,XXREAL_0:2;
then |.(0.TOP-REAL 2) - qz.|<r by JGRAPH_1:30;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
suppose
A37: q<>0.TOP-REAL 2 & not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2 <=-q`1);
A38: (q`2)^2>=0 by XREAL_1:63;
(q`1/q`2)^2 >=0 by XREAL_1:63;
then 1+(q`1/q`2)^2>=1+0 by XREAL_1:7;
then
A39: sqrt(1+(q`1/q`2)^2)>=1 by SQUARE_1:18,26;
then (sqrt(1+(q`1/q`2)^2))^2>=sqrt(1+(q`1/q`2)^2) by XREAL_1:151;
then
A40: 1<=(sqrt(1+(q`1/q`2)^2))^2 by A39,XXREAL_0:2;
A41: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2 ) ]|
by A37,Def1;
then (qz`2)^2=(q`2/sqrt(1+(q`1/q`2)^2))^2 by A25,EUCLID:52
.=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2 by XCMPLX_1:76;
then
A42: (qz`2)^2<=(q`2)^2/1 by A40,A38,XREAL_1:118;
A43: (q`1)^2>=0 by XREAL_1:63;
(qz`1)^2=(q`1/sqrt(1+(q`1/q`2)^2))^2 by A25,A41,EUCLID:52
.=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2 by XCMPLX_1:76;
then (qz`1)^2<=(q`1)^2/1 by A40,A43,XREAL_1:118;
then
A44: (qz`1)^2+(qz`2)^2<=(q`1)^2+(q`2)^2 by A42,XREAL_1:7;
(qz`1)^2>=0 & (qz`2)^2>=0 by XREAL_1:63;
then
A45: sqrt((qz`1)^2+(qz`2)^2) <= sqrt((q`1)^2+(q`2)^2) by A44,SQUARE_1:26;
A46: ((0.TOP-REAL 2) - qz)`2=(0.TOP-REAL 2)`2-qz`2 by TOPREAL3:3
.= -qz`2 by JGRAPH_2:3;
((0.TOP-REAL 2) - qz)`1=(0.TOP-REAL 2)`1-qz`1 by TOPREAL3:3
.= -qz`1 by JGRAPH_2:3;
then sqrt((((0.TOP-REAL 2) - qz)`1)^2+(((0.TOP-REAL 2) - qz)`2)^2)<r
by A26,A46,A45,JGRAPH_2:3,XXREAL_0:2;
then |.(0.TOP-REAL 2) - qz.|<r by JGRAPH_1:30;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
end;
u0 in W1 by A19,GOBOARD6:1;
hence thesis by A20,A21,A22,XBOOLE_1:1;
end;
A47: D`= {0.TOP-REAL 2} by Th20;
then
ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=Sq_Circ|D & h
is continuous by Th19;
hence thesis by A15,A47,A1,A16,Th3;
end;
theorem Th22:
Sq_Circ is one-to-one
proof
let x1,x2 be object;
assume that
A1: x1 in dom Sq_Circ and
A2: x2 in dom Sq_Circ and
A3: Sq_Circ.x1=Sq_Circ.x2;
reconsider p2=x2 as Point of TOP-REAL 2 by A2;
reconsider p1=x1 as Point of TOP-REAL 2 by A1;
set q=p1,p=p2;
per cases;
suppose
A4: q=0.TOP-REAL 2;
then
A5: Sq_Circ.q=0.TOP-REAL 2 by Def1;
now
per cases;
case
p=0.TOP-REAL 2;
hence thesis by A4;
end;
case
A6: p<>0.TOP-REAL 2 & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p `1);
(p`2/p`1)^2 >=0 by XREAL_1:63;
then 1+(p`2/p`1)^2>=1+0 by XREAL_1:7;
then
A7: sqrt(1+(p`2/p`1)^2)>=1 by SQUARE_1:18,26;
A8: Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]| by A6
,Def1;
then p`2/sqrt(1+(p`2/p`1)^2)=0 by A3,A5,EUCLID:52,JGRAPH_2:3;
then
A9: p`2= 0 *sqrt(1+(p`2/p`1)^2) by A7,XCMPLX_1:87
.=0;
p`1/sqrt(1+(p`2/p`1)^2)=0 by A3,A5,A8,EUCLID:52,JGRAPH_2:3;
then p`1= 0 *sqrt(1+(p`2/p`1)^2) by A7,XCMPLX_1:87
.=0;
hence contradiction by A6,A9,EUCLID:53,54;
end;
case
A10: p<>0.TOP-REAL 2 & not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2 <=-p`1);
(p`1/p`2)^2 >=0 by XREAL_1:63;
then 1+(p`1/p`2)^2>=1+0 by XREAL_1:7;
then
A11: sqrt(1+(p`1/p`2)^2)>=1 by SQUARE_1:18,26;
Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]| by A10
,Def1;
then p`2/sqrt(1+(p`1/p`2)^2)=0 by A3,A5,EUCLID:52,JGRAPH_2:3;
then p`2= 0 *sqrt(1+(p`1/p`2)^2) by A11,XCMPLX_1:87
.=0;
hence contradiction by A10;
end;
end;
hence thesis;
end;
suppose
A12: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
A13: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A14: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A12,Def1;
A15: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/sqrt(1+
(q `2/q`1)^2) by EUCLID:52;
A16: 1+(q`2/q`1)^2>0 by Lm1;
A17: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/sqrt(1+
(q `2/q`1)^2) by EUCLID:52;
now
per cases;
case
A18: p=0.TOP-REAL 2;
(q`2/q`1)^2 >=0 by XREAL_1:63;
then 1+(q`2/q`1)^2>=1+0 by XREAL_1:7;
then
A19: sqrt(1+(q`2/q`1)^2)>=1 by SQUARE_1:18,26;
A20: Sq_Circ.p=0.TOP-REAL 2 by A18,Def1;
then q`2/sqrt(1+(q`2/q`1)^2)=0 by A3,A14,EUCLID:52,JGRAPH_2:3;
then
A21: q`2= 0 *sqrt(1+(q`2/q`1)^2) by A19,XCMPLX_1:87
.=0;
q`1/sqrt(1+(q`2/q`1)^2)=0 by A3,A14,A20,EUCLID:52,JGRAPH_2:3;
then q`1= 0 *sqrt(1+(q`2/q`1)^2) by A19,XCMPLX_1:87
.=0;
hence contradiction by A12,A21,EUCLID:53,54;
end;
case
A22: p<>0.TOP-REAL 2 & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p `1);
now
assume
A23: p`1=0;
then p`2=0 by A22;
hence contradiction by A22,A23,EUCLID:53,54;
end;
then
A24: (p`1)^2>0 by SQUARE_1:12;
A25: sqrt(1+(p`2/p`1)^2)>0 by Lm1,SQUARE_1:25;
A26: 1+(p`2/p`1)^2>0 by Lm1;
A27: Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2 )]| by A22
,Def1;
then
A28: p
`2/sqrt(1+(p`2/p`1)^2)=q`2/sqrt(1+(q`2/q`1)^2) by A3,A14,A15,EUCLID:52;
then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76;
then (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A26,
SQUARE_1:def 2;
then
A29: (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(1+(q`2/q`1)^2) by A16,SQUARE_1:def 2;
A30: p`1/sqrt(1+(p`2/p`1)^2)=q`1/sqrt(1+(q`2/q`1)^2) by A3,A14,A17,A27,
EUCLID:52;
then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76;
then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2 by A26,
SQUARE_1:def 2;
then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(1+(q`2/q`1)^2) by A16,
SQUARE_1:def 2;
then (p`1)^2/(1+(p`2/p`1)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2)
by XCMPLX_1:48;
then (p`1)^2/(p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2)
by XCMPLX_1:48;
then 1/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2) by A24,
XCMPLX_1:60;
then
A31: 1/(1+(p`2/p`1)^2)*(1+(q`2/q`1)^2)=(q`1)^2/(p`1)^2 by A16,XCMPLX_1:87;
now
assume
A32: q`1=0;
then q`2=0 by A12;
hence contradiction by A12,A32,EUCLID:53,54;
end;
then
A33: (q`1)^2>0 by SQUARE_1:12;
now
per cases;
case
A34: p`2=0;
then (q`2)^2=0 by A16,A29,XCMPLX_1:50;
then
A35: q`2=0 by XCMPLX_1:6;
then p=|[q`1,0]|by A3,A14,A27,A34,EUCLID:53;
hence thesis by A35,EUCLID:53;
end;
case
p`2<>0;
then
A36: (p`2)^2>0 by SQUARE_1:12;
(p`2)^2/(1+(p`2/p`1)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`2/q`1)
^2) by A29,XCMPLX_1:48;
then (p`2)^2/(p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)
^2) by XCMPLX_1:48;
then 1/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2) by A36,
XCMPLX_1:60;
then 1/(1+(p`2/p`1)^2)*(1+(q`2/q`1)^2)=(q`2)^2/(p`2)^2 by A16,
XCMPLX_1:87;
then (q`1)^2/(q`1)^2/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A31,
XCMPLX_1:48;
then 1/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A33,XCMPLX_1:60;
then 1/(p`1)^2*(p`2)^2=(p`2)^2*((q`2)^2/(p`2)^2)/(q`1)^2 by
XCMPLX_1:74;
then 1/(p`1)^2*(p`2)^2=(q`2)^2/(q`1)^2 by A36,XCMPLX_1:87;
then (p`2)^2/(p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:99;
then (p`2/p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:76;
then
A37: (1+(p`2/p`1)^2)=(1+(q`2/q`1)^2) by XCMPLX_1:76;
then p`2=q`2/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A28,A25,
XCMPLX_1:87;
then
A38: p`2=q`2 by A13,XCMPLX_1:87;
p`1=q`1/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A30,A25,A37,
XCMPLX_1:87;
then p`1=q`1 by A13,XCMPLX_1:87;
then p=|[q`1,q`2]|by A38,EUCLID:53;
hence thesis by EUCLID:53;
end;
end;
hence thesis;
end;
case
A39: p<>0.TOP-REAL 2 & not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2 <=-p`1);
A40: 1+(p`1/p`2)^2>0 by Lm1;
A41: p<>0.TOP-REAL 2 & p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2 by A39,
JGRAPH_2:13;
p`2<>0 by A39;
then
A42: (p`2)^2>0 by SQUARE_1:12;
(|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|)`2 = p`2/
sqrt(1+(p `1/p`2)^2) by EUCLID:52;
then
A43: p`2/sqrt(1+(p`1/p`2)^2)=q`2/sqrt(1+(q`2/q`1)^2) by A3,A14,A15,A39,Def1;
then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76;
then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A40,
SQUARE_1:def 2;
then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(1+(q`2/q`1)^2) by A16,
SQUARE_1:def 2;
then (p`2)^2/(1+(p`1/p`2)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2)
by XCMPLX_1:48;
then (p`2)^2/(p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2)
by XCMPLX_1:48;
then 1/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`2/q`1)^2) by A42,
XCMPLX_1:60;
then
A44: 1/(1+(p`1/p`2)^2)*(1+(q`2/q`1)^2)=(q`2)^2/(p`2)^2 by A16,XCMPLX_1:87;
A45: sqrt(1+(p`1/p`2)^2)>0 by Lm1,SQUARE_1:25;
(|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]|)`1 = p`1/
sqrt(1+(p `1/p`2)^2) by EUCLID:52;
then
A46: p`1/sqrt(1+(p`1/p`2)^2)=q`1/sqrt(1+(q`2/q`1)^2) by A3,A14,A17,A39,Def1;
then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76;
then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76;
then (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(sqrt(1+(q`2/q`1)^2))^2 by A40,
SQUARE_1:def 2;
then
A47: (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(1+(q`2/q`1)^2) by A16,SQUARE_1:def 2;
A48: now
assume
A49: q`1=0;
then q`2=0 by A12;
hence contradiction by A12,A49,EUCLID:53,54;
end;
then
A50: (q`1)^2>0 by SQUARE_1:12;
now
per cases;
case
p`1=0;
then (q`1)^2=0 by A16,A47,XCMPLX_1:50;
then
A51: q`1=0 by XCMPLX_1:6;
then q`2=0 by A12;
hence contradiction by A12,A51,EUCLID:53,54;
end;
case
A52: p`1<>0;
set a=q`2/q`1;
(p`1)^2/(1+(p`1/p`2)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`2/q`1)
^2) by A47,XCMPLX_1:48;
then
A53: (p`1)^2/(p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)
^2) by XCMPLX_1:48;
A54: q`1*a<=q`1 & -q`1<=q`1*a or q`1*a>=q`1 & q`1*a<=-q`1 by A12,A48,
XCMPLX_1:87;
A55: now
per cases by A48;
case
A56: q`1>0;
then a*q`1/q`1<=q`1/q`1 & (-q`1)/q`1<=a*q`1/q`1 or a*q`1/q`1
>=q`1/q`1 & a*q`1/q`1<=(-q`1)/q`1 by A54,XREAL_1:72;
then
A57: a<=q`1/q`1 & (-q`1)/q`1<=a or a>=q`1/q`1 & a<=(-q`1)/q`1
by A56,XCMPLX_1:89;
q`1/q`1=1 by A56,XCMPLX_1:60;
hence a<=1 & -1<=a or a>=1 & a<=-1 by A57,XCMPLX_1:187;
end;
case
A58: q`1<0;
then
A59: q`1/q`1=1 & (-q`1)/q`1=-1 by XCMPLX_1:60,197;
a*q`1/q`1>=q`1/q`1 & (-q`1)/q`1>=a*q`1/q`1 or a*q`1/q`1
<=q`1/q`1 & a*q`1/q`1>=(-q`1)/q`1 by A54,A58,XREAL_1:73;
hence a<=1 & -1<=a or a>=1 & a<=-1 by A58,A59,XCMPLX_1:89;
end;
end;
(p`1)^2>0 by A52,SQUARE_1:12;
then 1/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`2/q`1)^2) by A53,
XCMPLX_1:60;
then 1/(1+(p`1/p`2)^2)*(1+(q`2/q`1)^2)=(q`1)^2/(p`1)^2 by A16,
XCMPLX_1:87;
then
(q`1)^2/(q`1)^2/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A44,XCMPLX_1:48;
then 1/(p`1)^2=(q`2)^2/(p`2)^2/(q`1)^2 by A50,XCMPLX_1:60;
then 1/(p`1)^2*(p`2)^2=(p`2)^2*((q`2)^2/(p`2)^2)/(q`1)^2 by
XCMPLX_1:74;
then 1/(p`1)^2*(p`2)^2=(q`2)^2/(q`1)^2 by A42,XCMPLX_1:87;
then (p`2)^2/(p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:99;
then (p`2/p`1)^2=(q`2)^2/(q`1)^2 by XCMPLX_1:76;
then
A60: (p`2/p`1)^2=(q`2/q`1)^2 by XCMPLX_1:76;
then
A61: p`2/p`1*p`1=a*p`1 or p`2/p`1*p`1=(-a)*p`1 by SQUARE_1:40;
A62: now
per cases by A52,A61,XCMPLX_1:87;
case
A63: p`2=a*p`1;
now
per cases by A52;
case
p`1>0;
then p`1/p`1<= a*p`1/p`1 & (-(a*p`1))/p`1<=p`1/p`1 or p`1
/p`1>=(a*p`1)/p`1 & p`1/p`1<=(-(a*p`1))/p`1 by A41,A63,XREAL_1:72;
then
A64: 1<= a*p`1/p`1 & (-(a*p`1))/p`1<=1 or 1>=(a*p`1)/p`1
& 1<=(-(a*p`1))/p`1 by A52,XCMPLX_1:60;
(a*p`1)/p`1=a by A52,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A64,XCMPLX_1:187;
end;
case
p`1<0;
then p`1/p`1>= a*p`1/p`1 & (-(a*p`1))/p`1>=p`1/p`1 or p`1
/p`1<=(a*p`1)/p`1 & p`1/p`1>=(-(a*p`1))/p`1 by A41,A63,XREAL_1:73;
then
A65: 1>= a*p`1/p`1 & (-(a*p`1))/p`1>=1 or 1<=(a*p`1)/p`1
& 1>=(-(a*p`1))/p`1 by A52,XCMPLX_1:60;
(a*p`1)/p`1=a by A52,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A65,XCMPLX_1:187;
end;
end;
then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
hence 1<=a or -1>=a;
end;
case
A66: p`2=(-a)*p`1;
now
per cases by A52;
case
p`1>0;
then p`1/p`1<= (-a)*p`1/p`1 & (-((-a)*p`1))/p`1<=p`1/p`1
or p`1/p`1>=((-a)*p`1)/p`1 & p`1/p`1<=(-((-a)*p`1))/p`1 by A41,A66,XREAL_1:72;
then 1<= (-a)*p`1/p`1 & (-((-a)*p`1))/p `1<=1 or 1>=((-a)
*p`1)/p`1 & 1<=(-((-a)*p`1))/p`1 by A52,XCMPLX_1:60;
then
A67: 1<= (-a) & -(((-a)*p`1)/p`1)<=1 or 1>=(-a) & 1<=-(((
-a)*p`1)/p`1) by A52,XCMPLX_1:89,187;
((-a)*p`1)/p`1=(-a) by A52,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A67;
end;
case
p`1<0;
then p`1/p`1>= (-a)*p`1/p`1 & (-((-a)*p`1))/p`1>=p`1/p`1
or p`1/p`1<=((-a)*p`1)/p`1 & p`1/p`1>=(-((-a)*p`1))/p`1 by A41,A66,XREAL_1:73;
then 1>= (-a)*p`1/p`1 & (-((-a)*p`1))/p `1>=1 or 1<=((-a)
*p`1)/p`1 & 1>=(-((-a)*p`1))/p`1 by A52,XCMPLX_1:60;
then
A68: 1>= (-a) & -(((-a)*p`1)/p`1)>=1 or 1<=(-a) & 1>=-(((
-a)*p`1)/p`1) by A52,XCMPLX_1:89,187;
((-a)*p`1)/p`1=(-a) by A52,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A68;
end;
end;
then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
hence 1<=a or -1>=a;
end;
end;
A69: now
per cases by A62,A55,XXREAL_0:1;
case
a=1;
then (p`2)^2/(p`1)^2=1 by A60,XCMPLX_1:76;
then
A70: (p`2)^2=(p`1)^2 by XCMPLX_1:58;
(p`1/p`2)^2=(p`1)^2/(p`2)^2 by XCMPLX_1:76;
hence (p`1/p`2)^2=(q`2/q`1)^2 by A60,A70,XCMPLX_1:76;
end;
case
a=-1;
then (p`2)^2/(p`1)^2=1 by A60,XCMPLX_1:76;
then
A71: (p`2)^2=(p`1)^2 by XCMPLX_1:58;
(p`1/p`2)^2=(p`1)^2/(p`2)^2 by XCMPLX_1:76;
hence (p`1/p`2)^2=(q`2/q`1)^2 by A60,A71,XCMPLX_1:76;
end;
end;
then p`2=q`2/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A43,A45,
XCMPLX_1:87;
then
A72: p`2=q`2 by A13,XCMPLX_1:87;
p`1=q`1/sqrt(1+(q`2/q`1)^2)*sqrt(1+(q`2/q`1)^2) by A46,A45,A69,
XCMPLX_1:87;
then p`1=q`1 by A13,XCMPLX_1:87;
then p=|[q`1,q`2]| by A72,EUCLID:53;
hence thesis by EUCLID:53;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
suppose
A73: q<>0.TOP-REAL 2 & not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=- q`1);
A74: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(1
+( q`1/q`2)^2) by EUCLID:52;
A75: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/sqrt(1
+( q`1/q`2)^2) by EUCLID:52;
A76: 1+(q`1/q`2)^2>0 by Lm1;
A77: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A78: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]| by A73,Def1;
A79: q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A73,JGRAPH_2:13;
now
per cases;
case
A80: p=0.TOP-REAL 2;
(q`1/q`2)^2 >=0 by XREAL_1:63;
then 1+(q`1/q`2)^2>=1+0 by XREAL_1:7;
then
A81: sqrt(1+(q`1/q`2)^2)>=1 by SQUARE_1:18,26;
Sq_Circ.p=0.TOP-REAL 2 by A80,Def1;
then q`2/sqrt(1+(q`1/q`2)^2)=0 by A3,A78,EUCLID:52,JGRAPH_2:3;
then q`2= 0 *sqrt(1+(q`1/q`2)^2) by A81,XCMPLX_1:87
.=0;
hence contradiction by A73;
end;
case
A82: p<>0.TOP-REAL 2 & (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=- p`1);
now
assume
A83: p`1=0;
then p`2=0 by A82;
hence contradiction by A82,A83,EUCLID:53,54;
end;
then
A84: (p`1)^2>0 by SQUARE_1:12;
A85: 1+(p`2/p`1)^2>0 by Lm1;
A86: Sq_Circ.p=|[p`1/sqrt(1+(p`2/p`1)^2),p`2/sqrt(1+(p`2/p`1)^2)]| by A82
,Def1;
then
A87: p`1/sqrt(1+(p`2/p`1)^2)=q`1/sqrt(1+(q`1/q`2)^2) by A3,A78,A75,EUCLID:52
;
then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
then (p`1)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2
by XCMPLX_1:76;
then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2 by A85,
SQUARE_1:def 2;
then (p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(1+(q`1/q`2)^2) by A76,
SQUARE_1:def 2;
then (p`1)^2/(1+(p`2/p`1)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2)
by XCMPLX_1:48;
then (p`1)^2/(p`1)^2/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2)
by XCMPLX_1:48;
then 1/(1+(p`2/p`1)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2) by A84,
XCMPLX_1:60;
then
A88: 1/(1+(p`2/p`1)^2)*(1+(q`1/q`2)^2)=(q`1)^2/(p`1)^2 by A76,XCMPLX_1:87;
A89: p`2/sqrt(1+(p`2/p`1)^2)=q`2/sqrt(1+(q`1/q`2)^2) by A3,A78,A74,A86,
EUCLID:52;
then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
then (p`2)^2/(sqrt(1+(p`2/p`1)^2))^2=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
by XCMPLX_1:76;
then (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2 by A85,
SQUARE_1:def 2;
then
A90: (p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(1+(q`1/q`2)^2) by A76,SQUARE_1:def 2;
A91: sqrt(1+(p`2/p`1)^2)>0 by Lm1,SQUARE_1:25;
A92: q`2<>0 by A73;
then
A93: (q`2)^2>0 by SQUARE_1:12;
now
per cases;
case
p`2=0;
then (q`2)^2=0 by A76,A90,XCMPLX_1:50;
then q`2=0 by XCMPLX_1:6;
hence contradiction by A73;
end;
case
A94: p`2<>0;
set a=q`1/q`2;
(p`2)^2/(1+(p`2/p`1)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`1/q`2)
^2) by A90,XCMPLX_1:48;
then
A95: (p`2)^2/(p`2)^2/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)
^2) by XCMPLX_1:48;
A96: q`2*a<=q`2 & -q`2<=q`2*a or q`2*a>=q`2 & q`2*a<=-q`2 by A79,A92,
XCMPLX_1:87;
A97: now
per cases by A73;
case
A98: q`2>0;
then
A99: q`2/q`2=1 & (-q`2)/q`2=-1 by XCMPLX_1:60,197;
a*q`2/q`2<=q`2/q`2 & (-q`2)/q`2<=a*q`2/q`2 or a*q`2/q`2
>=q`2/q`2 & a*q`2/q`2<=(-q`2)/q`2 by A96,A98,XREAL_1:72;
hence a<=1 & -1<=a or a>=1 & a<=-1 by A98,A99,XCMPLX_1:89;
end;
case
A100: q`2<0;
then a*q`2/q`2>=q`2/q`2 & (-q`2)/q`2>=a*q`2/q`2 or a*q`2/q`2
<=q`2/q`2 & a*q`2/q`2>=(-q`2)/q`2 by A96,XREAL_1:73;
then a>=q`2/q`2 & (-q`2)/q`2>=a or a<=q`2/q`2 & a>=(-q`2)/q`2
by A100,XCMPLX_1:89;
hence a<=1 & -1<=a or a>=1 & a<=-1 by A100,XCMPLX_1:60,197;
end;
end;
(p`2)^2>0 by A94,SQUARE_1:12;
then 1/(1+(p`2/p`1)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2) by A95,
XCMPLX_1:60;
then 1/(1+(p`2/p`1)^2)*(1+(q`1/q`2)^2)=(q`2)^2/(p`2)^2 by A76,
XCMPLX_1:87;
then
(q`2)^2/(q`2)^2/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A88,XCMPLX_1:48;
then 1/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A93,XCMPLX_1:60;
then 1/(p`2)^2*(p`1)^2=(p`1)^2*((q`1)^2/(p`1)^2)/(q`2)^2 by
XCMPLX_1:74;
then 1/(p`2)^2*(p`1)^2=(q`1)^2/(q`2)^2 by A84,XCMPLX_1:87;
then (p`1)^2/(p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:99;
then (p`1/p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:76;
then
A101: (p`1/p`2)^2=(q`1/q`2)^2 by XCMPLX_1:76;
then
A102: p`1/p`2=q`1/q`2 or p`1/p`2=-q`1/q`2 by SQUARE_1:40;
A103: now
per cases by A94,A102,XCMPLX_1:87;
case
A104: p`1=a*p`2;
now
per cases by A94;
case
p`2>0;
then p`2/p`2<= a*p`2/p`2 & (-(a*p`2))/p`2<=p`2/p`2 or p`2
/p`2>=(a*p`2)/p`2 & p`2/p`2<=(-(a*p`2))/p`2 by A82,A104,XREAL_1:72;
then
A105: 1<= a*p`2/p`2 & (-(a*p`2))/p`2<=1 or 1>=(a*p`2)/p`2
& 1<=(-(a*p`2))/p`2 by A94,XCMPLX_1:60;
(a*p`2)/p`2=a by A94,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A105,XCMPLX_1:187;
end;
case
p`2<0;
then p`2/p`2>= a*p`2/p`2 & (-(a*p`2))/p`2>=p`2/p`2 or p`2
/p`2<=(a*p`2)/p`2 & p`2/p`2>=(-(a*p`2))/p`2 by A82,A104,XREAL_1:73;
then
A106: 1>= a*p`2/p`2 & (-(a*p`2))/p`2>=1 or 1<=(a*p`2)/p`2
& 1>=(-(a*p`2))/p`2 by A94,XCMPLX_1:60;
(a*p`2)/p`2=a by A94,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A106,XCMPLX_1:187;
end;
end;
then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
hence 1<=a or -1>=a;
end;
case
A107: p`1=(-a)*p`2;
now
per cases by A94;
case
p`2>0;
then p`2/p`2<= (-a)*p`2/p`2 & (-((-a)*p`2))/p`2<=p`2/p`2
or p`2/p`2>=((-a)*p`2)/p`2 & p`2/p`2<=(-((-a)*p`2))/p`2 by A82,A107,XREAL_1:72;
then 1<= (-a)*p`2/p`2 & (-((-a)*p`2))/p `2<=1 or 1>=((-a)
*p`2)/p`2 & 1<=(-((-a)*p`2))/p`2 by A94,XCMPLX_1:60;
then
A108: 1<= (-a) & -(((-a)*p`2)/p`2)<=1 or 1>=(-a) & 1<=-(((
-a)*p`2)/p`2) by A94,XCMPLX_1:89,187;
((-a)*p`2)/p`2=(-a) by A94,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A108;
end;
case
p`2<0;
then p`2/p`2>= (-a)*p`2/p`2 & (-((-a)*p`2))/p`2>=p`2/p`2
or p`2/p`2<=((-a)*p`2)/p`2 & p`2/p`2>=(-((-a)*p`2))/p`2 by A82,A107,XREAL_1:73;
then 1>= (-a)*p`2/p`2 & (-((-a)*p`2))/p `2>=1 or 1<=((-a)
*p`2)/p`2 & 1>=(-((-a)*p`2))/p`2 by A94,XCMPLX_1:60;
then
A109: 1>= -a & -(((-a)*p`2)/p`2)>=1 or 1<=-a & 1>=-(((-a)*
p`2)/p`2) by A94,XCMPLX_1:89,187;
((-a)*p`2)/p`2=(-a) by A94,XCMPLX_1:89;
hence 1<=a & -a<=1 or 1>=a & 1<=-a by A109;
end;
end;
then 1<=a & -a<=1 or 1>=a & -1>=--a by XREAL_1:24;
hence 1<=a or -1>=a;
end;
end;
A110: now
per cases by A103,A97,XXREAL_0:1;
case
a=1;
then (p`1)^2/(p`2)^2=1 by A101,XCMPLX_1:76;
then
A111: (p`1)^2=(p`2)^2 by XCMPLX_1:58;
(p`2/p`1)^2=(p`2)^2/(p`1)^2 by XCMPLX_1:76;
hence (p`2/p`1)^2=(q`1/q`2)^2 by A101,A111,XCMPLX_1:76;
end;
case
a=-1;
then (p`1)^2/(p`2)^2=1 by A101,XCMPLX_1:76;
then
A112: (p`1)^2=(p`2)^2 by XCMPLX_1:58;
(p`2/p`1)^2=(p`2)^2/(p`1)^2 by XCMPLX_1:76;
hence (p`2/p`1)^2=(q`1/q`2)^2 by A101,A112,XCMPLX_1:76;
end;
end;
then p`1=q`1/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A87,A91,
XCMPLX_1:87;
then
A113: p`1=q`1 by A77,XCMPLX_1:87;
p`2=q`2/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A89,A91,A110,
XCMPLX_1:87;
then p`2=q`2 by A77,XCMPLX_1:87;
then p=|[q`1,q`2]|by A113,EUCLID:53;
hence thesis by EUCLID:53;
end;
end;
hence thesis;
end;
case
A114: p<>0.TOP-REAL 2 & not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p `2<=-p`1);
then p`2<>0;
then
A115: (p`2)^2>0 by SQUARE_1:12;
A116: sqrt(1+(p`1/p`2)^2)>0 by Lm1,SQUARE_1:25;
A117: 1+(p`1/p`2)^2>0 by Lm1;
A118: Sq_Circ.p=|[p`1/sqrt(1+(p`1/p`2)^2),p`2/sqrt(1+(p`1/p`2)^2)]| by A114
,Def1;
then
A119: p`1/sqrt(1+(p`1/p`2)^2)=q`1/sqrt(1+(q`1/q`2)^2) by A3,A78,A75,EUCLID:52
;
then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
then (p`1)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2
by XCMPLX_1:76;
then (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2 by A117,
SQUARE_1:def 2;
then
A120: (p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(1+(q`1/q`2)^2) by A76,SQUARE_1:def 2;
A121: p`2/sqrt(1+(p`1/p`2)^2)=q`2/sqrt(1+(q`1/q`2)^2) by A3,A78,A74,A118,
EUCLID:52;
then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76;
then (p`2)^2/(sqrt(1+(p`1/p`2)^2))^2=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2
by XCMPLX_1:76;
then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(sqrt(1+(q`1/q`2)^2))^2 by A117,
SQUARE_1:def 2;
then (p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(1+(q`1/q`2)^2) by A76,
SQUARE_1:def 2;
then (p`2)^2/(1+(p`1/p`2)^2)/(p`2)^2=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2)
by XCMPLX_1:48;
then (p`2)^2/(p`2)^2/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2)
by XCMPLX_1:48;
then 1/(1+(p`1/p`2)^2)=(q`2)^2/(p`2)^2/(1+(q`1/q`2)^2) by A115,
XCMPLX_1:60;
then
A122: 1/(1+(p`1/p`2)^2)*(1+(q`1/q`2)^2)=(q`2)^2/(p`2)^2 by A76,XCMPLX_1:87;
q`2<>0 by A73;
then
A123: (q`2)^2>0 by SQUARE_1:12;
now
per cases;
case
A124: p`1=0;
then (q`1)^2=0 by A76,A120,XCMPLX_1:50;
then
A125: q`1=0 by XCMPLX_1:6;
then p=|[0,q`2]|by A3,A78,A118,A124,EUCLID:53;
hence thesis by A125,EUCLID:53;
end;
case
p`1<>0;
then
A126: (p`1)^2>0 by SQUARE_1:12;
(p`1)^2/(1+(p`1/p`2)^2)/(p`1)^2=(q`1)^2/(p`1)^2/(1+(q`1/q`2)
^2) by A120,XCMPLX_1:48;
then (p`1)^2/(p`1)^2/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)
^2) by XCMPLX_1:48;
then 1/(1+(p`1/p`2)^2)=(q`1)^2/(p`1)^2/(1+(q`1/q`2)^2) by A126,
XCMPLX_1:60;
then 1/(1+(p`1/p`2)^2)*(1+(q`1/q`2)^2)=(q`1)^2/(p`1)^2 by A76,
XCMPLX_1:87;
then
(q`2)^2/(q`2)^2/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A122,XCMPLX_1:48;
then 1/(p`2)^2=(q`1)^2/(p`1)^2/(q`2)^2 by A123,XCMPLX_1:60;
then 1/(p`2)^2*(p`1)^2=(p`1)^2*((q`1)^2/(p`1)^2)/(q`2)^2 by
XCMPLX_1:74;
then 1/(p`2)^2*(p`1)^2=(q`1)^2/(q`2)^2 by A126,XCMPLX_1:87;
then (p`1)^2/(p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:99;
then (p`1/p`2)^2=(q`1)^2/(q`2)^2 by XCMPLX_1:76;
then
A127: (1+(p`1/p`2)^2)=(1+(q`1/q`2)^2) by XCMPLX_1:76;
then p`1=q`1/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A119,A116,
XCMPLX_1:87;
then
A128: p`1=q`1 by A77,XCMPLX_1:87;
p`2=q`2/sqrt(1+(q`1/q`2)^2)*sqrt(1+(q`1/q`2)^2) by A121,A116,A127,
XCMPLX_1:87;
then p`2=q`2 by A77,XCMPLX_1:87;
then p=|[q`1,q`2]|by A128,EUCLID:53;
hence thesis by EUCLID:53;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
end;
registration
cluster Sq_Circ -> one-to-one;
coherence by Th22;
end;
theorem Th23:
for Kb,Cb being Subset of TOP-REAL 2 st Kb={q: -1=q`1 & -1<=q`2
& q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1
<=q`1 & q`1<=1}& Cb={p2 where p2 is Point of TOP-REAL 2: |.p2.|=1} holds
Sq_Circ.:Kb=Cb
proof
let Kb,Cb be Subset of TOP-REAL 2;
assume
A1: Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q
`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1}& Cb={p2 where p2 is Point of
TOP-REAL 2: |.p2.|=1};
thus Sq_Circ.:Kb c= Cb
proof
let y be object;
assume y in Sq_Circ.:Kb;
then consider x being object such that
x in dom Sq_Circ and
A2: x in Kb and
A3: y=Sq_Circ.x by FUNCT_1:def 6;
consider q being Point of TOP-REAL 2 such that
A4: q=x and
A5: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 &
- 1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1 by A1,A2;
now
per cases;
case
q=0.TOP-REAL 2;
hence contradiction by A5,JGRAPH_2:3;
end;
case
A6: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q `1);
A7: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/
sqrt(1+(q `2/q `1)^2) & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2
= q`2/sqrt(1+ (q `2/q`1)^2) by EUCLID:52;
A8: 1+(q`2)^2>0 by Lm1;
A9: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2 )]| by A6
,Def1;
now
per cases by A5;
case
-1=q`1 & -1<=q`2 & q`2<=1;
then
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2 =
((-1)/sqrt(1+(q`2/(-1))^2))^2+(q`2/sqrt(1+(q`2/(-1))^2))^2 by A7,JGRAPH_1:29
.=(-1)^2/(sqrt(1+(q`2/(-1))^2))^2+(q`2/sqrt(1+(q`2/(-1))^2))^2
by XCMPLX_1:76
.=1/(sqrt(1+(-q`2)^2))^2+(q`2)^2/(sqrt(1+(-q`2)^2))^2 by
XCMPLX_1:76
.=1/(1+(q`2)^2)+(q`2)^2/(sqrt(1+(q`2)^2))^2 by A8,SQUARE_1:def 2
.=1/(1+(q`2)^2)+(q`2)^2/(1+(q`2)^2) by A8,SQUARE_1:def 2
.=(1+(q`2)^2)/(1+(q`2)^2) by XCMPLX_1:62
.=1 by A8,XCMPLX_1:60;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|=1
by SQUARE_1:18,22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A3,A4
,A9;
end;
case
q`1=1 & -1<=q`2 & q`2<=1;
then
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2 =
(1/sqrt(1+(q`2/(1))^2))^2+(q`2/sqrt(1+(q`2/(1))^2))^2 by A7,JGRAPH_1:29
.=1^2/(sqrt(1+(q`2/(1))^2))^2+(q`2/sqrt(1+(q`2/(1))^2))^2 by
XCMPLX_1:76
.=1/(sqrt(1+(q`2/(1))^2))^2+(q`2)^2/(sqrt(1+(q`2/(1))^2))^2 by
XCMPLX_1:76
.=1/(1+(q`2)^2)+(q`2)^2/(sqrt(1+(q`2)^2))^2 by A8,SQUARE_1:def 2
.=1/(1+(q`2)^2)+(q`2)^2/(1+(q`2)^2) by A8,SQUARE_1:def 2
.=(1+(q`2)^2)/(1+(q`2)^2) by XCMPLX_1:62
.=1 by A8,XCMPLX_1:60;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|=1
by SQUARE_1:18,22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A3,A4
,A9;
end;
case
A10: -1=q`2 & -1<=q`1 & q`1<=1;
then -1<=q`1 & q`1>=1 or -1>=q`1 & 1>=q`1 by A6,XREAL_1:24;
then
A11: q`1=1 or q`1=-1 by A10,XXREAL_0:1;
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2 =
((q`1)/sqrt(1+((-1)/(q`1))^2))^2+((-1)/sqrt(1+((-1)/(q`1))^2))^2 by A7,A10,
JGRAPH_1:29
.=((q`1)/sqrt(1+((-1)/(q`1))^2))^2+(-1)^2/(sqrt(1+((-1)/(q`1))
^2))^2 by XCMPLX_1:76
.=(q`1)^2/(sqrt(1+((-1)/(q`1))^2))^2+1/(sqrt(1+((-1)/(q`1))^2)
)^2 by XCMPLX_1:76
.=1/2+1/(sqrt(2))^2 by A11,SQUARE_1:def 2
.=1/2+1/2 by SQUARE_1:def 2
.=1;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|=1
by SQUARE_1:18,22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A3,A4
,A9;
end;
case
A12: 1=q`2 & -1<=q`1 & q`1<=1;
then 1<=q`1 & q`1>=-1 or 1>=q`1 & -1>=q`1 by A6,XREAL_1:25;
then
A13: q`1=1 or q`1=-1 by A12,XXREAL_0:1;
|.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|^2 =
((q`1)/sqrt(1+((1)/(q`1))^2))^2+((1)/sqrt(1+((1)/(q`1))^2))^2 by A7,A12,
JGRAPH_1:29
.=((q`1)/sqrt(1+((1)/(q`1))^2))^2+(1)^2/(sqrt(1+((1)/(q`1))^2)
)^2 by XCMPLX_1:76
.=1/(sqrt(1+1/1))^2+1/(sqrt(1+1/1))^2 by A13,XCMPLX_1:76
.=1/2+1/(sqrt(2))^2 by SQUARE_1:def 2
.=1/2+1/2 by SQUARE_1:def 2
.=1;
then |.(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|).|=1
by SQUARE_1:18,22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A3,A4
,A9;
end;
end;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1;
end;
case
A14: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2 <=-q`1);
A15: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q`1/
sqrt(1+(q `1/q `2)^2) & (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2
= q`2/sqrt(1+ (q `1/q`2)^2) by EUCLID:52;
A16: 1+(q`1)^2>0 by Lm1;
A17: Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2 )]| by A14
,Def1;
now
per cases by A5;
case
-1=q`2 & -1<=q`1 & q`1<=1;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).| ^2
=(q`1/sqrt(1+(q`1/(-1))^2))^2+((-1)/sqrt(1+(q`1/(-1))^2))^2 by A15,JGRAPH_1:29
.=(-1)^2/(sqrt(1+(q`1/(-1))^2))^2+(q`1/sqrt(1+(q`1/(-1))^2))^2
by XCMPLX_1:76
.=1/(sqrt(1+(-q`1)^2))^2+(q`1)^2/(sqrt(1+(-q`1)^2))^2 by
XCMPLX_1:76
.=1/(1+(q`1)^2)+(q`1)^2/(sqrt(1+(q`1)^2))^2 by A16,SQUARE_1:def 2
.=1/(1+(q`1)^2)+(q`1)^2/(1+(q`1)^2) by A16,SQUARE_1:def 2
.=(1+(q`1)^2)/(1+(q`1)^2) by XCMPLX_1:62
.=1 by A16,XCMPLX_1:60;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|=1
by SQUARE_1:18,22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A3,A4
,A17;
end;
case
q`2=1 & -1<=q`1 & q`1<=1;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).| ^2
=((1)/sqrt(1+(q`1/(1))^2))^2+(q`1/sqrt(1+(q`1/(1))^2))^2 by A15,JGRAPH_1:29
.=1^2/(sqrt(1+(q`1/(1))^2))^2+(q`1/sqrt(1+(q`1/(1))^2))^2 by
XCMPLX_1:76
.=1/(sqrt(1+(q`1/(1))^2))^2+(q`1)^2/(sqrt(1+(q`1/(1))^2))^2 by
XCMPLX_1:76
.=1/(1+(q`1)^2)+(q`1)^2/(sqrt(1+(q`1)^2))^2 by A16,SQUARE_1:def 2
.=1/(1+(q`1)^2)+(q`1)^2/(1+(q`1)^2) by A16,SQUARE_1:def 2
.=(1+(q`1)^2)/(1+(q`1)^2) by XCMPLX_1:62
.=1 by A16,XCMPLX_1:60;
then |.(|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|).|=1
by SQUARE_1:18,22;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A3,A4
,A17;
end;
case
-1=q`1 & -1<=q`2 & q`2<=1;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A14;
end;
case
1=q`1 & -1<=q`2 & q`2<=1;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1 by A14;
end;
end;
hence ex p2 being Point of TOP-REAL 2 st p2=y & |.p2.|=1;
end;
end;
hence thesis by A1;
end;
let y be object;
assume y in Cb;
then consider p2 being Point of TOP-REAL 2 such that
A18: p2=y and
A19: |.p2.|=1 by A1;
set q=p2;
now
per cases;
case
q=0.TOP-REAL 2;
hence contradiction by A19,TOPRNS_1:23;
end;
case
A20: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1 );
A21: |.q.|^2=q`1^2+q`2^2 by JGRAPH_1:29;
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A22: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A23: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
then
A24: q`2=q`2*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by XCMPLX_1:89
.=px`2/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
A25: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
then
A26: px`2/px`1=q`2/q`1 by A22,A23,XCMPLX_1:91;
then
A27: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A25,A23,XCMPLX_1:89;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2*sqrt(1+(q`2/q`1)^2) <= (-q`1
)*sqrt(1+(q`2/q`1)^2) by A20,A23,XREAL_1:64;
then
A28: q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1) ^2)
or px`2>=px`1 & px`2<=-px`1 by A22,A25,A23,XREAL_1:64;
A29: 1+(px`2/px`1)^2>0 by Lm1;
q`1=q`1*sqrt(1+(q`2/q`1)^2)/(sqrt(1+(q`2/q`1)^2))by A23,XCMPLX_1:89
.=px`1/(sqrt(1+(q`2/q`1)^2)) by EUCLID:52;
then (px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2/sqrt(1+(px`2/px`1)^2)) ^2
= 1 by A19,A26,A24,A21,XCMPLX_1:76;
then
(px`1)^2/(sqrt(1+(px`2/px`1)^2))^2+(px`2)^2/(sqrt(1+(px`2/px`1)^2))
^2=1 by XCMPLX_1:76;
then (px`1)^2/(1+(px`2/px`1)^2)+(px`2)^2/(sqrt(1+(px`2/px`1)^2))^2=1 by
A29,SQUARE_1:def 2;
then
1 *(1+(px`2/px`1)^2)= (1+(px`2/px`1)^2)*((px`1)^2/(1+(px`2/px`1)^2)
+ (px`2)^2/(1+(px`2/px`1)^2)) by A29,SQUARE_1:def 2
.= (px`1)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2) +(px`2)^2/(1+(px`2/
px`1)^2)*(1+(px`2/px`1)^2);
then (px`1)^2+(px`2)^2/(1+(px`2/px`1)^2)*(1+(px`2/px`1)^2)=1 *(1+(px`2/
px `1)^2) by A29,XCMPLX_1:87;
then
A30: (px`1)^2+(px`2)^2=1 *(1+(px`2/px`1)^2) by A29,XCMPLX_1:87
.=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
A31: now
assume that
A32: px`1=0 and
A33: px`2=0;
q`2*sqrt(1+(q`2/q`1)^2)=0 by A33,EUCLID:52;
then
A34: q`2=0 by A23,XCMPLX_1:6;
q`1*sqrt(1+(q`2/q`1)^2)=0 by A32,EUCLID:52;
then q`1=0 by A23,XCMPLX_1:6;
hence contradiction by A20,A34,EUCLID:53,54;
end;
then not px`1=0 by A22,A25,A23,A28,XREAL_1:64;
then ((px`1)^2+((px`2)^2-1))*(px`1)^2=(px`2)^2 by A30,XCMPLX_1:6,87;
then 0= ((px`1)^2-1)*((px`1)^2+(px`2)^2);
then
A35: (px`1)^2-1=0 or (px`1)^2+(px`2)^2=0 by XCMPLX_1:6;
now
per cases by A31,A35,COMPLEX1:1,SQUARE_1:41;
case
px`1=1;
hence -1=px`1 & -1<=px`2 & px`2<=1 or px`1=1 & -1<=px`2 & px`2<=1 or
-1=px`2 & -1<=px`1 & px`1<=1 or 1=px`2 & -1<=px`1 & px`1<=1 by A22,A25,A23,A28,
XREAL_1:64;
end;
case
px`1=-1;
hence -1=px`1 & -1<=px`2 & px`2<=1 or px`1=1 & -1<=px`2 & px`2<=1 or
-1=px`2 & -1<=px`1 & px`1<=1 or 1=px`2 & -1<=px`1 & px`1<=1 by A22,A23,A28,
XREAL_1:64;
end;
end;
then
A36: dom Sq_Circ=the carrier of TOP-REAL 2 & px in Kb by A1,FUNCT_2:def 1;
px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A22,A25,A23,A28,
XREAL_1:64;
then
A37: Sq_Circ.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/px`1 )^2)
]| by A31,Def1,JGRAPH_2:3;
px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A22,A23,A26,XCMPLX_1:89;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x by A18
,A37,A27,A36,EUCLID:53;
end;
case
A38: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=- q`1);
A39: |.q.|^2=q`2^2+q`1^2 by JGRAPH_1:29;
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A40: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A41: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
then
A42: q`1=px`1/(sqrt(1+(q`1/q`2)^2)) by A40,XCMPLX_1:89;
A43: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
then
A44: px`1/px`2=q`1/q`2 by A41,A40,XCMPLX_1:91;
then
A45: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A41,A40,XCMPLX_1:89;
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A38,JGRAPH_2:13;
then
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1*sqrt(1+(q`1/q`2)^2) <= (-q`2
)*sqrt(1+(q`1/q`2)^2) by A40,XREAL_1:64;
then
A46: q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2) ^2)
or px`1>=px`2 & px`1<=-px`2 by A43,A41,A40,XREAL_1:64;
A47: 1+(px`1/px`2)^2>0 by Lm1;
q`2=px`2/(sqrt(1+(q`1/q`2)^2)) by A43,A40,XCMPLX_1:89;
then (px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1/sqrt(1+(px`1/px`2)^2)) ^2
= 1 by A19,A44,A42,A39,XCMPLX_1:76;
then (px`2)^2/(sqrt(1+(px`1/px`2)^2))^2+(px`1)^2/(sqrt(1+(px`1/px`2)^2)
) ^2=1 by XCMPLX_1:76;
then (px`2)^2/(1+(px`1/px`2)^2)+(px`1)^2/(sqrt(1+(px`1/px`2)^2))^2=1 by
A47,SQUARE_1:def 2;
then 1 *(1+(px`1/px`2)^2) = (1+(px`1/px`2)^2)*( (px`2)^2/(1+(px`1/px`2)
^2) +(px`1)^2/(1+(px`1/px`2)^2)) by A47,SQUARE_1:def 2
.= (px`2)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2) +(px`1)^2/(1+(px`1/
px`2)^2)*(1+(px`1/px`2)^2);
then (px`2)^2+(px`1)^2/(1+(px`1/px`2)^2)*(1+(px`1/px`2)^2)=1 *(1+(px`1/
px `2)^2) by A47,XCMPLX_1:87;
then (px`2)^2+(px`1)^2=1 *(1+(px`1/px`2)^2) by A47,XCMPLX_1:87;
then
A48: (px`2)^2+(px`1)^2-1=(px`1)^2/(px`2)^2 by XCMPLX_1:76;
A49: now
assume that
A50: px`2=0 and
px`1=0;
q`2=0 by A43,A40,A50,XCMPLX_1:6;
hence contradiction by A38;
end;
then px`2<>0 by A43,A41,A40,A46,XREAL_1:64;
then ((px`2)^2+((px`1)^2-1))*(px`2)^2=(px`1)^2 by A48,XCMPLX_1:6,87;
then 0=((px`2)^2-1)*((px`2)^2+(px`1)^2);
then
A51: (px`2)^2-1=0 or (px`2)^2+(px`1)^2=0 by XCMPLX_1:6;
now
per cases by A49,A51,COMPLEX1:1,SQUARE_1:41;
case
px`2=1;
hence -1=px`2 & -1<=px`1 & px`1<=1 or px`2=1 & -1<=px`1 & px`1<=1 or
-1=px`1 & -1<=px`2 & px`2<=1 or 1=px`1 & -1<=px`2 & px`2<=1 by A43,A41,A40,A46,
XREAL_1:64;
end;
case
px`2=-1;
hence -1=px`2 & -1<=px`1 & px`1<=1 or px`2=1 & -1<=px`1 & px`1<=1 or
-1=px`1 & -1<=px`2 & px`2<=1 or 1=px`1 & -1<=px`2 & px`2<=1 by A43,A40,A46,
XREAL_1:64;
end;
end;
then
A52: dom Sq_Circ=the carrier of TOP-REAL 2 & px in Kb by A1,FUNCT_2:def 1;
px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2 by A43,A41,A40,A46,
XREAL_1:64;
then
A53: Sq_Circ.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/px`2 )^2
)]| by A49,Th4,JGRAPH_2:3;
px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A43,A40,A44,XCMPLX_1:89;
hence ex x being set st x in dom Sq_Circ & x in Kb & y=Sq_Circ.x by A18
,A53,A45,A52,EUCLID:53;
end;
end;
hence thesis by FUNCT_1:def 6;
end;
theorem Th24:
for P,Kb being Subset of TOP-REAL 2,f being Function of (
TOP-REAL 2)|Kb,(TOP-REAL 2)|P st Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1
<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1} & f is
being_homeomorphism holds P is being_simple_closed_curve
proof
set X=(TOP-REAL 2)|R^2-unit_square;
set b=1,a=0;
set v= |[1,0]|;
let P,Kb be Subset of TOP-REAL 2,f be Function of (TOP-REAL 2)|Kb,(TOP-REAL
2)|P;
assume
A1: Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q
`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1} & f is being_homeomorphism;
v`1=1 & v`2=0 by EUCLID:52;
then
A2: |[1,0]| in {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or
-1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1};
then reconsider Kbb=Kb as non empty Subset of TOP-REAL 2 by A1;
set A=2/(b-a),B=1-2*b/(b-a),C=2/(b-a),D=1-2*b/(b-a);
reconsider Kbd=Kbb as non empty Subset of TOP-REAL 2;
defpred P[object,object] means
(for t being Point of TOP-REAL 2 st t=$1 holds $2=
|[A*(t`1)+B,C*(t`2)+D]|);
A3: for x being object st x in the carrier of TOP-REAL 2
ex y being object st P[x, y]
proof
let x be object;
assume x in the carrier of TOP-REAL 2;
then reconsider t2=x as Point of TOP-REAL 2;
reconsider y2=|[A*(t2`1)+B,C*(t2`2)+D]| as set;
for t being Point of TOP-REAL 2 st t=x holds y2 =|[A*(t`1)+B,C*(t`2)+D ]|;
hence thesis;
end;
ex ff being Function st dom ff=the carrier of TOP-REAL 2 & for x being
object st x in the carrier of TOP-REAL 2 holds P[x,ff.x]
from CLASSES1:sch 1(A3);
then consider ff being Function such that
A4: dom ff=the carrier of TOP-REAL 2 and
A5: for x being object st x in the carrier of TOP-REAL 2 holds for t being
Point of TOP-REAL 2 st t=x holds ff.x=|[A*(t`1)+B,C*(t`2)+D]|;
A6: for t being Point of TOP-REAL 2 holds ff.t=|[A*(t`1)+B,C*(t`2)+D]| by A5;
for x being object st x in the carrier of TOP-REAL 2 holds ff.x in the
carrier of TOP-REAL 2
proof
let x be object;
assume x in the carrier of TOP-REAL 2;
then reconsider t=x as Point of TOP-REAL 2;
ff.t=|[A*(t`1)+B,C*(t`2)+D]| by A5;
hence thesis;
end;
then reconsider ff as Function of TOP-REAL 2,TOP-REAL 2 by A4,FUNCT_2:3;
reconsider f11=ff|(R^2-unit_square) as Function of (TOP-REAL 2)|
R^2-unit_square,(TOP-REAL 2) by PRE_TOPC:9;
A7: f11 is continuous by A6,JGRAPH_2:43,TOPMETR:7;
ff is one-to-one
proof
let x1,x2 be object;
assume that
A8: x1 in dom ff & x2 in dom ff and
A9: ff.x1=ff.x2;
reconsider p1=x1,p2=x2 as Point of TOP-REAL 2 by A8;
A10: ff.x1= |[A*(p1`1)+B,C*(p1`2)+D]| & ff.x2= |[A*(p2`1)+B,C*(p2`2)+D]| by A5;
then A*(p1`1)+B-B=A*(p2`1)+B-B by A9,SPPOL_2:1;
then A*(p1`1)/A=p2`1 by XCMPLX_1:89;
then
A11: p1`1=p2`1 by XCMPLX_1:89;
C*(p1`2)+D-D=C*(p2`2)+D-D by A9,A10,SPPOL_2:1;
then C*(p1`2)/C=p2`2 by XCMPLX_1:89;
hence thesis by A11,TOPREAL3:6,XCMPLX_1:89;
end;
then
A12: f11 is one-to-one by FUNCT_1:52;
A13: dom f11=(dom ff)/\ (R^2-unit_square) by RELAT_1:61
.= R^2-unit_square by A4,XBOOLE_1:28;
A14: Kbd c= rng f11
proof
let y be object;
assume
A15: y in Kbd;
then reconsider py=y as Point of TOP-REAL 2;
set t=|[(py`1-B)/2,(py`2-D)/2]|;
A16: ex q st py=q &( -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<= q`2 & q`2<=1
or -1=q`2 & - 1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1) by A1,A15;
now
per cases by A16;
case
A17: -1=py`1 & -1<=py`2 & py`2<=1;
then 2-1>=py`2;
then 2>= py`2+1 by XREAL_1:19;
then
A18: 2/2 >= (py`2-D)/2 by XREAL_1:72;
0-1<=py`2 by A17;
then 0<= py`2+1 by XREAL_1:20;
hence
t`1 = 0 & t`2 <= 1 & t`2 >= 0 or t`1 <= 1 & t`1 >= 0 & t`2 = 1 or
t`1 <= 1 & t`1 >= 0 & t`2 = 0 or t`1 = 1 & t`2 <= 1 & t`2 >= 0 by A17,A18,
EUCLID:52;
end;
case
A19: py`1=1 & -1<=py`2 & py`2<=1;
then 2-1>=py`2;
then 2>= py`2+1 by XREAL_1:19;
then
A20: 2/2 >= (py`2-D)/2 by XREAL_1:72;
0-1<=py`2 by A19;
then 0<= py`2+1 by XREAL_1:20;
hence
t`1 = 0 & t`2 <= 1 & t`2 >= 0 or t`1 <= 1 & t`1 >= 0 & t`2 = 1 or
t`1 <= 1 & t`1 >= 0 & t`2 = 0 or t`1 = 1 & t`2 <= 1 & t`2 >= 0 by A19,A20,
EUCLID:52;
end;
case
A21: -1=py`2 & -1<=py`1 & py`1<=1;
then 2-1>=py`1;
then 2>= py`1+1 by XREAL_1:19;
then
A22: 2/2 >= (py`1-B)/2 by XREAL_1:72;
0-1<=py`1 by A21;
then 0<= py`1+1 by XREAL_1:20;
hence
t`1 = 0 & t`2 <= 1 & t`2 >= 0 or t`1 <= 1 & t`1 >= 0 & t`2 = 1 or
t`1 <= 1 & t`1 >= 0 & t`2 = 0 or t`1 = 1 & t`2 <= 1 & t`2 >= 0 by A21,A22,
EUCLID:52;
end;
case
A23: 1=py`2 & -1<=py`1 & py`1<=1;
then 2-1>=py`1;
then 2>= py`1+1 by XREAL_1:19;
then
A24: 2/2 >= (py`1-B)/2 by XREAL_1:72;
0-1<=py`1 by A23;
then 0<= py`1+1 by XREAL_1:20;
hence
t`1 = 0 & t`2 <= 1 & t`2 >= 0 or t`1 <= 1 & t`1 >= 0 & t`2 = 1 or
t`1 <= 1 & t`1 >= 0 & t`2 = 0 or t`1 = 1 & t`2 <= 1 & t`2 >= 0 by A23,A24,
EUCLID:52;
end;
end;
then
A25: t in (R^2-unit_square) by TOPREAL1:14;
t`1=(py`1-B)/2 & t`2=(py`2-D)/2 by EUCLID:52;
then py=|[A*(t`1)+B,C*(t`2)+D]| by EUCLID:53;
then py=ff.t by A5
.=f11.t by A25,FUNCT_1:49;
hence thesis by A13,A25,FUNCT_1:def 3;
end;
rng f11 c= Kbd
proof
let y be object;
assume y in rng f11;
then consider x being object such that
A26: x in dom f11 and
A27: y=f11.x by FUNCT_1:def 3;
reconsider t=x as Point of TOP-REAL 2 by A13,A26;
A28: y=ff.t by A13,A26,A27,FUNCT_1:49
.= |[A*(t`1)+B,C*(t`2)+D]| by A5;
then reconsider qy=y as Point of TOP-REAL 2;
A29: ex p st t=p &( p`1 = 0 & p`2 <= 1 & p`2 >= 0 or p`1 <= 1 & p`1 >= 0 &
p`2 = 1 or p`1 <= 1 & p`1 >= 0 & p`2 = 0 or p`1 = 1 & p`2 <= 1 & p `2 >= 0) by
A13,A26,TOPREAL1:14;
now
per cases by A29;
suppose
A30: t`1 = 0 & t`2 <= 1 & t`2 >= 0;
A31: qy`2=2*(t`2)-1 by A28,EUCLID:52;
2*1>=2*t`2 by A30,XREAL_1:64;
then
A32: 1+1-1>=qy`2+1-1 by A31,XREAL_1:9;
0-1<=qy`2+1-1 by A30,A31,XREAL_1:9;
hence
-1=qy`1 & -1<=qy`2 & qy`2<=1 or qy`1=1 & -1<=qy`2 & qy`2<=1 or -1
=qy`2 & -1<=qy`1 & qy`1<=1 or 1=qy`2 & -1<=qy`1 & qy`1<=1 by A28,A30,A32,
EUCLID:52;
end;
suppose
A33: t`1 <= 1 & t`1 >= 0 & t`2 = 1;
A34: qy`1=2*(t`1)-1 by A28,EUCLID:52;
2*1>=2*t`1 by A33,XREAL_1:64;
then
A35: 1+1-1>=qy`1+1-1 by A34,XREAL_1:9;
0-1<=qy`1+1-1 by A33,A34,XREAL_1:9;
hence
-1=qy`1 & -1<=qy`2 & qy`2<=1 or qy`1=1 & -1<=qy`2 & qy`2<=1 or -1
=qy`2 & -1<=qy`1 & qy`1<=1 or 1=qy`2 & -1<=qy`1 & qy`1<=1 by A28,A33,A35,
EUCLID:52;
end;
suppose
A36: t`1 <= 1 & t`1 >= 0 & t`2 = 0;
A37: qy`1=2*(t`1)-1 by A28,EUCLID:52;
2*1>=2*t`1 by A36,XREAL_1:64;
then
A38: 1+1-1>=qy`1+1-1 by A37,XREAL_1:9;
0-1<=qy`1+1-1 by A36,A37,XREAL_1:9;
hence
-1=qy`1 & -1<=qy`2 & qy`2<=1 or qy`1=1 & -1<=qy`2 & qy`2<=1 or -1
=qy`2 & -1<=qy`1 & qy`1<=1 or 1=qy`2 & -1<=qy`1 & qy`1<=1 by A28,A36,A38,
EUCLID:52;
end;
suppose
A39: t`1 = 1 & t`2 <= 1 & t`2 >= 0;
A40: qy`2=2*(t`2)-1 by A28,EUCLID:52;
2*1>=2*t`2 by A39,XREAL_1:64;
then
A41: 1+1-1>=qy`2+1-1 by A40,XREAL_1:9;
0-1<=qy`2+1-1 by A39,A40,XREAL_1:9;
hence
-1=qy`1 & -1<=qy`2 & qy`2<=1 or qy`1=1 & -1<=qy`2 & qy`2<=1 or -1
=qy`2 & -1<=qy`1 & qy`1<=1 or 1=qy`2 & -1<=qy`1 & qy`1<=1 by A28,A39,A41,
EUCLID:52;
end;
end;
hence thesis by A1;
end;
then Kbd=rng f11 by A14;
then consider f1 being Function of X,((TOP-REAL 2)|Kbd) such that
f11=f1 and
A42: f1 is being_homeomorphism by A7,A12,JGRAPH_1:46;
dom f=[#]((TOP-REAL 2)|Kb) by A1,TOPS_2:def 5
.=Kb by PRE_TOPC:def 5;
then f.(|[1,0]|) in rng f by A1,A2,FUNCT_1:3;
then reconsider PP=P as non empty Subset of TOP-REAL 2;
reconsider g=f as Function of (TOP-REAL 2)|Kbb,(TOP-REAL 2)|PP;
reconsider g as Function of (TOP-REAL 2)|Kbb,(TOP-REAL 2)|PP;
reconsider f22=f1 as Function of X,((TOP-REAL 2)|Kbb);
reconsider h=g*f22 as Function of (TOP-REAL 2)|R^2-unit_square,(TOP-REAL 2)|
PP;
h is being_homeomorphism by A1,A42,TOPS_2:57;
hence thesis by TOPREAL2:def 1;
end;
theorem Th25:
for Kb being Subset of TOP-REAL 2 st Kb={q: -1=q`1 & -1<=q`2 & q
`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q
`1 & q`1<=1} holds Kb is being_simple_closed_curve & Kb is compact
proof
set v= |[1,0]|;
let Kb be Subset of TOP-REAL 2;
assume
A1: Kb={q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or -1=q
`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1};
v`1=1 & v`2=0 by EUCLID:52;
then
|[1,0]| in {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or
-1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1};
then reconsider Kbd=Kb as non empty Subset of TOP-REAL 2 by A1;
set P=Kb;
id ((TOP-REAL 2)|Kbd) is being_homeomorphism;
hence Kb is being_simple_closed_curve by A1,Th24;
then consider
f being Function of (TOP-REAL 2)|R^2-unit_square, (TOP-REAL 2)|P
such that
A2: f is being_homeomorphism by TOPREAL2:def 1;
per cases;
suppose
A3: P is empty;
Kbd <>{};
hence thesis by A3;
end;
suppose
P is non empty;
then reconsider R = P as non empty Subset of TOP-REAL 2;
f is continuous & rng f = [#]((TOP-REAL 2)|P) by A2,TOPS_2:def 5;
then (TOP-REAL 2)|R is compact by COMPTS_1:14;
hence thesis by COMPTS_1:3;
end;
end;
theorem
for Cb being Subset of TOP-REAL 2 st Cb={p where p is Point of
TOP-REAL 2: |.p.|=1} holds Cb is being_simple_closed_curve
proof
defpred P[Point of TOP-REAL 2] means -1=$1`1 & -1<=$1`2 & $1`2<=1 or $1`1=1
& -1<=$1`2 & $1`2<=1 or -1=$1`2 & -1<=$1`1 & $1`1<=1 or 1=$1`2 & -1<=$1`1 & $1
`1<=1;
A1: (|[1,0]|)`1=1 & (|[1,0]|)`2=0 by EUCLID:52;
A2: dom Sq_Circ = the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set v= |[1,0]|;
let Cb be Subset of TOP-REAL 2;
assume
A3: Cb={p where p is Point of TOP-REAL 2: |.p.|=1};
v`1=1 & v`2=0 by EUCLID:52;
then
A4: |[1,0]| in {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<=1 or
-1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1};
{q where q is Element of TOP-REAL 2: P[q]} is Subset of TOP-REAL 2 from
DOMAIN_1:sch 7;
then reconsider
Kb= {q: -1=q`1 & -1<=q`2 & q`2<=1 or q`1=1 & -1<=q`2 & q`2<= 1 or
-1=q`2 & -1<=q`1 & q`1<=1 or 1=q`2 & -1<=q`1 & q`1<=1} as non empty Subset of
TOP-REAL 2 by A4;
|.(|[1,0]|).|=sqrt(((|[1,0]|)`1)^2+((|[1,0]|)`2)^2) by JGRAPH_1:30
.=1 by A1;
then |[1,0]| in Cb by A3;
then reconsider Cbb=Cb as non empty Subset of TOP-REAL 2;
A5: the carrier of (TOP-REAL 2)|Kb=Kb by PRE_TOPC:8;
A6: dom (Sq_Circ|Kb)=(dom Sq_Circ)/\ Kb by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|Kb) by A5,A2,XBOOLE_1:28;
A7: rng (Sq_Circ|Kb) c= (Sq_Circ|Kb).:(the carrier of ((TOP-REAL 2)|Kb))
proof
let u be object;
assume u in rng (Sq_Circ|Kb);
then ex z being object st z in dom ((Sq_Circ|Kb)) & u=(Sq_Circ| Kb).z by
FUNCT_1:def 3;
hence thesis by A6,FUNCT_1:def 6;
end;
(Sq_Circ|Kb).: (the carrier of ((TOP-REAL 2)|Kb)) = Sq_Circ.:Kb by A5,
RELAT_1:129
.= Cb by A3,Th23
.=the carrier of (TOP-REAL 2)|Cbb by PRE_TOPC:8;
then reconsider
f0=Sq_Circ|Kb as Function of (TOP-REAL 2)|Kb, (TOP-REAL 2)|Cbb by A6,A7,
FUNCT_2:2;
rng (Sq_Circ|Kb) c= the carrier of (TOP-REAL 2);
then reconsider f00=f0 as Function of (TOP-REAL 2)|Kb,TOP-REAL 2 by A6,
FUNCT_2:2;
A8: f0 is one-to-one & Kb is compact by Th25,FUNCT_1:52;
rng f0 = (Sq_Circ|Kb).: (the carrier of ((TOP-REAL 2)|Kb)) by RELSET_1:22
.= Sq_Circ.:Kb by A5,RELAT_1:129
.= Cb by A3,Th23;
then
ex f1 being Function of (TOP-REAL 2)|Kb,(TOP-REAL 2)|Cbb st f00=f1 & f1
is being_homeomorphism by A8,Th21,JGRAPH_1:46,TOPMETR:7;
hence thesis by Th24;
end;
begin :: The Fashoda Meet Theorem for the Circle
theorem
for K0,C0 being Subset of TOP-REAL 2 st K0={p: -1<=p`1 & p`1<=1 & -1<=
p`2 & p`2<=1} & C0={p1 where p1 is Point of TOP-REAL 2: |.p1.|<=1} holds
Sq_Circ"(C0) c= K0
proof
let K0,C0 be Subset of TOP-REAL 2;
assume
A1: K0={p: -1<=p`1 & p`1<=1 & -1<=p`2 & p`2<=1} & C0={p1 where p1 is
Point of TOP-REAL 2: |.p1.|<=1};
let x be object;
assume
A2: x in Sq_Circ"(C0);
then reconsider px=x as Point of TOP-REAL 2;
set q=px;
A3: Sq_Circ.x in C0 by A2,FUNCT_1:def 7;
now
per cases;
case
q=0.TOP-REAL 2;
hence -1<=px`1 & px`1<=1 & -1<=px`2 & px`2<=1 by JGRAPH_2:3;
end;
case
A4: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
A5: now
assume ((px`1)^2+(px`2)^2)=0;
then px`1=0 & px`2=0 by COMPLEX1:1;
hence contradiction by A4,EUCLID:53,54;
end;
A6: (px`1)^2 >=0 by XREAL_1:63;
A7: now
assume
A8: px`1=0;
then px`2=0 by A4;
hence contradiction by A4,A8,EUCLID:53,54;
end;
A9: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/sqrt(1
+(q `2/q `1)^2) & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2
/sqrt(1+ (q `2/q`1)^2) by EUCLID:52;
consider p1 being Point of TOP-REAL 2 such that
A10: p1=Sq_Circ.q and
A11: |.p1.|<=1 by A1,A3;
(|.p1.|)^2<= |.p1.| by A11,SQUARE_1:42;
then
A12: (|.p1.|)^2<=1 by A11,XXREAL_0:2;
A13: 1+(q`2/q`1)^2>0 by Lm1;
Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A4,Def1;
then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))
^2 by A9,A10,JGRAPH_1:29
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A13,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A13,
SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then
((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)<=1 *(1+(q`2/q`1 )
^2) by A13,A12,XREAL_1:64;
then ((q`1)^2+(q`2)^2)<=(1+(q`2/q`1)^2) by A13,XCMPLX_1:87;
then (px`1)^2+(px`2)^2<=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
then (px`1)^2+(px`2)^2-1<=1+(px`2)^2/(px`1)^2-1 by XREAL_1:9;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2/(px`1)^2*(px`1)^2 by A6,
XREAL_1:64;
then (px`1)^2*(px`1)^2+((px`2)^2-1)*(px`1)^2<=(px`2)^2 by A7,XCMPLX_1:6
,87;
then (px`1)^2*(px`1)^2-(px`1)^2*1+(px`1)^2*(px`2)^2-1 *(px`2)^2<=0 by
XREAL_1:47;
then
A14: ((px`1)^2-1)*((px`1)^2+(px`2)^2)<=0;
(px`2)^2>=0 by XREAL_1:63;
then
A15: (px`1)^2-1<=0 by A6,A14,A5,XREAL_1:129;
then
A16: px`1<=1 by SQUARE_1:43;
A17: -1<=px`1 by A15,SQUARE_1:43;
then q`2<=1 & --q`1>=-q`2 or q`2>=-1 & -q`2>=--q`1 by A4,A16,XREAL_1:24
,XXREAL_0:2;
then q`2<=1 & 1>=-q`2 or q`2>=-1 & -q`2>=q`1 by A16,XXREAL_0:2;
then q`2<=1 & -1<=--q`2 or q`2>=-1 & -q`2>=-1 by A17,XREAL_1:24
,XXREAL_0:2;
hence -1<=px`1 & px`1<=1 & -1<=px`2 & px`2<=1 by A15,SQUARE_1:43
,XREAL_1:24;
end;
case
A18: q<>0.TOP-REAL 2 & not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<= -q`1);
A19: now
assume ((px`2)^2+(px`1)^2)=0;
then px`2=0 by COMPLEX1:1;
hence contradiction by A18;
end;
A20: (px`2)^2 >=0 by XREAL_1:63;
A21: px`2<>0 by A18;
A22: (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`2 = q`2/sqrt(
1+(q `1/q `2)^2) & (|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]|)`1 = q
`1/sqrt(1+ (q `1/q`2)^2) by EUCLID:52;
consider p1 being Point of TOP-REAL 2 such that
A23: p1=Sq_Circ.q and
A24: |.p1.|<=1 by A1,A3;
(|.p1.|)^2<= |.p1.| by A24,SQUARE_1:42;
then
A25: (|.p1.|)^2<=1 by A24,XXREAL_0:2;
A26: 1+(q`1/q`2)^2>0 by Lm1;
Sq_Circ.q=|[q`1/sqrt(1+(q`1/q`2)^2),q`2/sqrt(1+(q`1/q`2)^2)]| by A18,Def1
;
then (|.p1.|)^2= (q`1/sqrt(1+(q`1/q`2)^2))^2+(q`2/sqrt(1+(q`1/q`2)^2))
^2 by A22,A23,JGRAPH_1:29
.= (q`2)^2/(sqrt(1+(q`1/q`2)^2))^2+(q`1/sqrt(1+(q`1/q`2)^2))^2 by
XCMPLX_1:76
.= (q`2)^2/(sqrt(1+(q`1/q`2)^2))^2+(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2
by XCMPLX_1:76
.= (q`2)^2/(1+(q`1/q`2)^2)+(q`1)^2/(sqrt(1+(q`1/q`2)^2))^2 by A26,
SQUARE_1:def 2
.= (q`2)^2/(1+(q`1/q`2)^2)+(q`1)^2/(1+(q`1/q`2)^2) by A26,
SQUARE_1:def 2
.= ((q`2)^2+(q`1)^2)/(1+(q`1/q`2)^2) by XCMPLX_1:62;
then
((q`2)^2+(q`1)^2)/(1+(q`1/q`2)^2)*(1+(q`1/q`2)^2)<=1 *(1+(q`1/q`2 )
^2) by A26,A25,XREAL_1:64;
then ((q`2)^2+(q`1)^2)<=(1+(q`1/q`2)^2) by A26,XCMPLX_1:87;
then (px`2)^2+(px`1)^2<=1+(px`1)^2/(px`2)^2 by XCMPLX_1:76;
then (px`2)^2+(px`1)^2-1<=1+(px`1)^2/(px`2)^2-1 by XREAL_1:9;
then ((px`2)^2+(px`1)^2-1)*(px`2)^2<=(px`1)^2/(px`2)^2*(px`2)^2 by A20,
XREAL_1:64;
then (px`2)^2*(px`2)^2+((px`1)^2-1)*(px`2)^2<=(px`1)^2 by A21,XCMPLX_1:6
,87;
then (px`2)^2*(px`2)^2-(px`2)^2*1+(px`2)^2*(px`1)^2-1 *(px`1)^2<=0 by
XREAL_1:47;
then
A27: ((px`2)^2-1)*((px`2)^2+(px`1)^2)<=0;
(px`1)^2>=0 by XREAL_1:63;
then
A28: (px`2)^2-1<=0 by A20,A27,A19,XREAL_1:129;
then -1<=px`2 & px`2<=1 by SQUARE_1:43;
then q`1<=1 & 1>=-q`1 or q`1>=-1 & -q`1>=-1 by A18,XXREAL_0:2;
then q`1<=1 & -1<=--q`1 or q`1>=-1 & q`1<=1 by XREAL_1:24;
hence -1<=px`1 & px`1<=1 & -1<=px`2 & px`2<=1 by A28,SQUARE_1:43;
end;
end;
hence thesis by A1;
end;
theorem Th28:
for p holds (p=0.TOP-REAL 2 implies Sq_Circ".p=0.TOP-REAL 2) & (
(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2 implies
Sq_Circ".p=|[p`1*sqrt(1+(p`2/p`1)^2),p`2*sqrt(1+(p`2/p`1)^2)]|)& (not(p`2<=p`1
& -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) implies Sq_Circ".p=|[p`1*sqrt(1+(p`1/p`2)
^2),p`2*sqrt(1+(p`1/p`2)^2)]|)
proof
let p;
set q=p;
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A1: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A2: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
hereby
assume
A3: p=0.TOP-REAL 2;
then Sq_Circ.p=p by Def1;
hence Sq_Circ".p=0.TOP-REAL 2 by A2,A3,FUNCT_1:34;
end;
hereby
A4: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set q=p;
assume that
A5: p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1 and
A6: p<>0.TOP-REAL 2;
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A7: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A8: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A9: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
then
A10: px`2/px`1=q`2/q`1 by A7,A8,XCMPLX_1:91;
then
A11: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A9,A8,XCMPLX_1:89;
A12: now
assume px`1=0 & px`2=0;
then q`1=0 & q`2=0 by A7,A9,A8,XCMPLX_1:6;
hence contradiction by A6,EUCLID:53,54;
end;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2*sqrt(1+(q`2/q`1)^2) <= (-q`1)*
sqrt(1+(q`2/q`1)^2) by A5,A8,XREAL_1:64;
then q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1)^2) or
px`2>=px`1 & px`2<=-px`1 by A7,A9,A8,XREAL_1:64;
then q`2*sqrt(1+(q`2/q`1)^2) <= q`1*sqrt(1+(q`2/q`1)^2) & -px`1<=px`2 or
px`2>=px`1 & px`2<=-px`1 by A7,A8,EUCLID:52,XREAL_1:64;
then
A13: Sq_Circ.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/px`1)^2 )
]| by A7,A9,A12,Def1,JGRAPH_2:3;
px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A7,A8,A10,XCMPLX_1:89;
then q=Sq_Circ.px by A13,A11,EUCLID:53;
hence (Sq_Circ").p=|[p`1*sqrt(1+(p`2/p`1)^2),p`2*sqrt(1+(p`2/p`1)^2)]| by
A4,FUNCT_1:34;
end;
A14: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A15: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A16: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
then
A17: px`1/px`2=q`1/q`2 by A1,A15,XCMPLX_1:91;
then
A18: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A16,A15,XCMPLX_1:89;
assume
A19: not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1);
A20: now
assume that
A21: px`2=0 and
px`1=0;
q`2=0 by A1,A15,A21,XCMPLX_1:6;
hence contradiction by A19;
end;
p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2 by A19,JGRAPH_2:13;
then q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1*sqrt(1+(q`1/q`2)^2) <= (-q`2)*
sqrt(1+(q`1/q`2)^2) by A15,XREAL_1:64;
then q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2)^2) or px
`1>=px`2 & px`1<=-px`2 by A1,A16,A15,XREAL_1:64;
then q`1*sqrt(1+(q`1/q`2)^2) <= q`2*sqrt(1+(q`1/q`2)^2) & -px`2<=px`1 or px
`1>=px`2 & px`1<=-px`2 by A1,A15,EUCLID:52,XREAL_1:64;
then
A22: Sq_Circ.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/px`2)^2) ]|
by A1,A16,A20,Th4,JGRAPH_2:3;
px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A1,A15,A17,XCMPLX_1:89;
then q=Sq_Circ.px by A22,A18,EUCLID:53;
hence thesis by A14,FUNCT_1:34;
end;
theorem Th29:
Sq_Circ" is Function of TOP-REAL 2,TOP-REAL 2
proof
A1: the carrier of TOP-REAL 2 c= rng Sq_Circ
proof
let y be object;
assume y in the carrier of TOP-REAL 2;
then reconsider py=y as Point of TOP-REAL 2;
A2: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
now
per cases;
case
py=0.TOP-REAL 2;
then Sq_Circ.py=py by Def1;
hence ex x being set st x in dom Sq_Circ & y=Sq_Circ.x by A2;
end;
case
A3: (py`2<=py`1 & -py`1<=py`2 or py`2>=py`1 & py`2<=-py`1) & py<>
0.TOP-REAL 2;
set q=py;
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A4: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A5: now
assume that
A6: px`1=0 and
A7: px`2=0;
q`2*sqrt(1+(q`2/q`1)^2)=0 by A7,EUCLID:52;
then
A8: q`2=0 by A4,XCMPLX_1:6;
q`1*sqrt(1+(q`2/q`1)^2)=0 by A6,EUCLID:52;
then q`1=0 by A4,XCMPLX_1:6;
hence contradiction by A3,A8,EUCLID:53,54;
end;
A9: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A10: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A11: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
then
A12: px`2/px`1=q`2/q`1 by A10,A4,XCMPLX_1:91;
then
A13: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A11,A4,XCMPLX_1:89;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2*sqrt(1+(q`2/q`1)^2) <= (-q
`1)*sqrt(1+(q`2/q`1)^2) by A3,A4,XREAL_1:64;
then q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1)^2)
or px`2>=px`1 & px`2<=-px`1 by A10,A11,A4,XREAL_1:64;
then q`2*sqrt(1+(q`2/q`1)^2) <= q`1*sqrt(1+(q`2/q`1)^2) & -px`1<=px`2
or px`2>=px`1 & px`2<=-px`1 by A10,A4,EUCLID:52,XREAL_1:64;
then
A14: Sq_Circ.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+(px`2/ px`1)
^2) ]| by A10,A11,A5,Def1,JGRAPH_2:3;
px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A10,A4,A12,XCMPLX_1:89;
hence ex x being set st x in dom Sq_Circ & y=Sq_Circ.x by A14,A13,A9,
EUCLID:53;
end;
case
A15: not(py`2<=py`1 & -py`1<=py`2 or py`2>=py`1 & py`2<=-py`1)& py
<>0.TOP-REAL 2;
set q=py;
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A16: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A17: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A18: now
assume that
A19: px`2=0 and
px`1=0;
q`2=0 by A17,A16,A19,XCMPLX_1:6;
hence contradiction by A15;
end;
A20: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
then
A21: px`1/px`2=q`1/q`2 by A17,A16,XCMPLX_1:91;
then
A22: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A20,A16,XCMPLX_1:89;
py`1<=py`2 & -py`2<=py`1 or py`1>=py`2 & py`1<=-py`2 by A15,JGRAPH_2:13
;
then
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1*sqrt(1+(q`1/q`2)^2) <= (-q
`2)*sqrt(1+(q`1/q`2)^2) by A16,XREAL_1:64;
then q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2)^2)
or px`1>=px`2 & px`1<=-px`2 by A17,A20,A16,XREAL_1:64;
then q`1*sqrt(1+(q`1/q`2)^2) <= q`2*sqrt(1+(q`1/q`2)^2) & -px`2<=px`1
or px`1>=px`2 & px`1<=-px`2 by A17,A16,EUCLID:52,XREAL_1:64;
then
A23: Sq_Circ.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+(px`1/ px`2)
^2 )]| by A17,A20,A18,Th4,JGRAPH_2:3;
A24: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A17,A16,A21,XCMPLX_1:89;
hence ex x being set st x in dom Sq_Circ & y=Sq_Circ.x by A23,A22,A24,
EUCLID:53;
end;
end;
hence thesis by FUNCT_1:def 3;
end;
A25: rng (Sq_Circ")=dom (Sq_Circ) by FUNCT_1:33
.=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
dom (Sq_Circ")=rng (Sq_Circ) by FUNCT_1:33;
then dom (Sq_Circ")=the carrier of TOP-REAL 2 by A1;
hence thesis by A25,FUNCT_2:1;
end;
theorem Th30:
for p being Point of TOP-REAL 2 st p<>0.TOP-REAL 2 holds ((p`1<=
p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) implies (Sq_Circ").p=|[p`1*sqrt(1+(p`1
/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|) & (not(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p
`1<=-p`2) implies (Sq_Circ").p=|[p`1*sqrt(1+(p`2/p`1)^2),p`2*sqrt(1+(p`2/p`1)^2
)]|)
proof
let p be Point of TOP-REAL 2;
A1: -p`2<p`1 implies --p`2>-p`1 by XREAL_1:24;
assume
A2: p<>0.TOP-REAL 2;
hereby
assume
A3: p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2;
now
per cases by A3;
case
A4: p`1<=p`2 & -p`2<=p`1;
now
assume
A5: p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
A6: now
per cases by A5;
case
p`2<=p`1 & -p`1<=p`2;
hence p`1=p`2 or p`1=-p`2 by A4,XXREAL_0:1;
end;
case
p`2>=p`1 & p`2<=-p`1;
then -p`2>=--p`1 by XREAL_1:24;
hence p`1=p`2 or p`1=-p`2 by A4,XXREAL_0:1;
end;
end;
now
per cases by A6;
case
p`1=p`2;
hence
(Sq_Circ").p =|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2
)^2)]| by A2,A5,Th28;
end;
case
p`1=-p`2;
then p`1<>0 & -p`1=p`2 by A2,EUCLID:53,54;
then p`1/p`2=-1 & p`2/p`1=-1 by XCMPLX_1:197,198;
hence
(Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)
^2) ]| by A2,A5,Th28;
end;
end;
hence (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)
]|;
end;
hence (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|
by Th28;
end;
case
A7: p`1>=p`2 & p`1<=-p`2;
now
assume
A8: p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1;
A9: now
per cases by A8;
case
p`2<=p`1 & -p`1<=p`2;
then --p`1>=-p`2 by XREAL_1:24;
hence p`1=p`2 or p`1=-p`2 by A7,XXREAL_0:1;
end;
case
p`2>=p`1 & p`2<=-p`1;
hence p`1=p`2 or p`1=-p`2 by A7,XXREAL_0:1;
end;
end;
now
per cases by A9;
case
p`1=p`2;
hence
(Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)
^2) ]| by A2,A8,Th28;
end;
case
A10: p`1=-p`2;
then p`1<>0 & -p`1=p`2 by A2,EUCLID:53,54;
then
A11: p`2/p`1=-1 by XCMPLX_1:197;
p`2<>0 by A2,A10,EUCLID:53,54;
then p`1/p`2=-1 by A10,XCMPLX_1:197;
hence
(Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)
^2) ]| by A2,A8,A11,Th28;
end;
end;
hence (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)
]|;
end;
hence (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|
by Th28;
end;
end;
hence (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|;
end;
A12: -p`2>p`1 implies --p`2<-p`1 by XREAL_1:24;
assume not(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2);
hence thesis by A2,A1,A12,Th28;
end;
theorem Th31:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0)
holds ex g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g.p=r1*sqrt(1+(r1/r2)^2)) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1;
assume that
A1: f1 is continuous and
A2: f2 is continuous & for q being Point of X holds f2.q<>0;
consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=sqrt(1+(r1/r2)^2) and
A4: g2 is continuous by A1,A2,Th8;
consider g3 being Function of X,R^1 such that
A5: for p being Point of X,r1,r0 being Real st f1.p=r1 & g2.p=r0
holds g3.p=r1*r0 and
A6: g3 is continuous by A1,A4,JGRAPH_2:25;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=r1*sqrt(1+(r1/r2)^2)
proof
let p be Point of X,r1,r2 be Real;
assume that
A7: f1.p=r1 and
A8: f2.p=r2;
g2.p=sqrt(1+(r1/r2)^2) by A3,A7,A8;
hence thesis by A5,A7;
end;
hence thesis by A6;
end;
theorem Th32:
for X being non empty TopSpace, f1,f2 being Function of X,R^1 st
f1 is continuous & f2 is continuous & (for q being Point of X holds f2.q<>0) ex
g being Function of X,R^1 st (for p being Point of X,r1,r2 being Real st
f1.p=r1 & f2.p=r2 holds g.p=r2*sqrt(1+(r1/r2)^2)) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1;
assume that
A1: f1 is continuous and
A2: f2 is continuous and
A3: for q being Point of X holds f2.q<>0;
consider g2 being Function of X,R^1 such that
A4: for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g2.p=sqrt(1+(r1/r2)^2) and
A5: g2 is continuous by A1,A2,A3,Th8;
consider g3 being Function of X,R^1 such that
A6: for p being Point of X,r2,r0 being Real st f2.p=r2 & g2.p=r0
holds g3.p=r2*r0 and
A7: g3 is continuous by A2,A5,JGRAPH_2:25;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g3.p=r2*sqrt(1+(r1/r2)^2)
proof
let p be Point of X,r1,r2 be Real;
assume that
A8: f1.p=r1 and
A9: f2.p=r2;
g2.p=sqrt(1+(r1/r2)^2) by A4,A8,A9;
hence thesis by A6,A9;
end;
hence thesis by A7;
end;
theorem Th33:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`1*sqrt(1+(p`2/p`1)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`1*sqrt(1+(p`2/p`1)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0
proof
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g1.q=proj1.q by Lm6
.=q2`1 by PSCOMP_1:def 5;
hence thesis by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q
=r1 & g1.q=r2 holds g3.q=r2*sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th32;
A6: now
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`1*sqrt(1+(r`2/r`1)^2) by A1,A7;
hence f.x=g3.x by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
theorem Th34:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`2*sqrt(1+(p`2/p`1)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`1<>0) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`2*sqrt(1+(p`2/p`1)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0
proof
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g1.q=proj1.q by Lm6
.=q2`1 by PSCOMP_1:def 5;
hence thesis by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q
=r1 & g1.q=r2 holds g3.q=r1*sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th31;
A6: now
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`2*sqrt(1+(r`2/r`1)^2) by A1,A7;
hence f.x=g3.x by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
theorem Th35:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`2*sqrt(1+(p`1/p`2)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`2*sqrt(1+(p`1/p`2)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
for q being Point of (TOP-REAL 2)|K1 holds g2.q<>0
proof
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g2.q=proj2.q by Lm4
.=q2`2 by PSCOMP_1:def 6;
hence thesis by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g1.q
=r1 & g2.q=r2 holds g3.q=r2*sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th32;
A6: now
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`2*sqrt(1+(r`1/r`2)^2) by A1,A7;
hence f.x=g3.x by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
theorem Th36:
for K1 being non empty Subset of TOP-REAL 2, f being Function of
(TOP-REAL 2)|K1,R^1 st (for p being Point of TOP-REAL 2 st p in the carrier of
(TOP-REAL 2)|K1 holds f.p=p`1*sqrt(1+(p`1/p`2)^2)) & (for q being Point of
TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1 holds q`2<>0 ) holds f is
continuous
proof
let K1 be non empty Subset of TOP-REAL 2, f be Function of (TOP-REAL 2)|K1,
R^1;
reconsider g1=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm7;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm5;
assume that
A1: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|
K1 holds f.p=p`1*sqrt(1+(p`1/p`2)^2) and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<>0;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
for q being Point of (TOP-REAL 2)|K1 holds g2.q<>0
proof
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then reconsider q2=q as Point of TOP-REAL 2 by A3;
g2.q=proj2.q by Lm4
.=q2`2 by PSCOMP_1:def 6;
hence thesis by A2;
end;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A4: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g1.q
=r1 & g2.q=r2 holds g3.q=r1*sqrt(1+(r1/r2)^2) and
A5: g3 is continuous by Th31;
A6: now
let x be object;
assume
A7: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in the carrier of (TOP-REAL 2)|K1 by A7;
then x in K1 by PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A8: proj2.r=r`2 & proj1.r=r`1 by PSCOMP_1:def 5,def 6;
A9: g2.s=proj2.s & g1.s=proj1.s by Lm4,Lm6;
f.r=r`1*sqrt(1+(r`1/r`2)^2) by A1,A7;
hence f.x=g3.x by A4,A9,A8;
end;
dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then dom f=dom g3 by FUNCT_2:def 1;
hence thesis by A5,A6,FUNCT_1:2;
end;
Lm17: for K1 being non empty Subset of TOP-REAL 2 holds (proj2)*((Sq_Circ")|K1
) is Function of (TOP-REAL 2)|K1,R^1
proof
let K1 be non empty Subset of TOP-REAL 2;
A1: rng ((proj2)*((Sq_Circ")|K1)) c= rng (proj2) by RELAT_1:26;
A2: dom ((Sq_Circ")|K1) c= dom ((proj2)*((Sq_Circ")|K1))
proof
let x be object;
A3: rng (Sq_Circ") c= the carrier of TOP-REAL 2 by Th29,RELAT_1:def 19;
assume
A4: x in dom ((Sq_Circ")|K1);
then x in dom (Sq_Circ") /\ K1 by RELAT_1:61;
then x in dom (Sq_Circ") by XBOOLE_0:def 4;
then
A5: (Sq_Circ").x in rng (Sq_Circ") by FUNCT_1:3;
((Sq_Circ")|K1).x=(Sq_Circ").x by A4,FUNCT_1:47;
hence thesis by A4,A5,A3,Lm3,FUNCT_1:11;
end;
dom ((proj2)*((Sq_Circ")|K1)) c= dom ((Sq_Circ")|K1) by RELAT_1:25;
then dom ((proj2)*((Sq_Circ")|K1))=dom ((Sq_Circ")|K1) by A2
.=dom (Sq_Circ") /\ K1 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K1 by Th29,FUNCT_2:def 1
.=K1 by XBOOLE_1:28
.=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8;
hence thesis by A1,FUNCT_2:2,TOPMETR:17,XBOOLE_1:1;
end;
Lm18: for K1 being non empty Subset of TOP-REAL 2 holds (proj1)*((Sq_Circ")|K1
) is Function of (TOP-REAL 2)|K1,R^1
proof
let K1 be non empty Subset of TOP-REAL 2;
A1: rng ((proj1)*((Sq_Circ")|K1)) c= rng (proj1) by RELAT_1:26;
A2: dom ((Sq_Circ")|K1) c= dom ((proj1)*((Sq_Circ")|K1))
proof
let x be object;
A3: rng (Sq_Circ") c= the carrier of TOP-REAL 2 by Th29,RELAT_1:def 19;
assume
A4: x in dom ((Sq_Circ")|K1);
then x in dom (Sq_Circ") /\ K1 by RELAT_1:61;
then x in dom (Sq_Circ") by XBOOLE_0:def 4;
then
A5: (Sq_Circ").x in rng (Sq_Circ") by FUNCT_1:3;
((Sq_Circ")|K1).x=(Sq_Circ").x by A4,FUNCT_1:47;
hence thesis by A4,A5,A3,Lm2,FUNCT_1:11;
end;
dom ((proj1)*((Sq_Circ")|K1)) c= dom ((Sq_Circ")|K1) by RELAT_1:25;
then dom ((proj1)*((Sq_Circ")|K1)) =dom ((Sq_Circ")|K1) by A2
.=dom (Sq_Circ") /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by Th29,FUNCT_2:def 1
.=K1 by XBOOLE_1:28
.=the carrier of (TOP-REAL 2)|K1 by PRE_TOPC:8;
hence thesis by A1,FUNCT_2:2,TOPMETR:17,XBOOLE_1:1;
end;
theorem Th37:
for K0,B0 being Subset of TOP-REAL 2,f being Function of (
TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=(Sq_Circ")|K0 & B0=NonZero TOP-REAL 2 & K0=
{p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} holds f is
continuous
proof
let K0,B0 be Subset of TOP-REAL 2,f be Function of (TOP-REAL 2)|K0,(TOP-REAL
2)|B0;
assume
A1: f=(Sq_Circ")|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`2<=p`1 & -p`1<=p
`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2};
then 1.REAL 2 in K0 by Lm9,Lm10;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
reconsider g1=(proj1)*((Sq_Circ")|K1) as Function of (TOP-REAL 2)|K1,R^1 by
Lm18;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g1.p=p`1*sqrt(1+(p`2/p`1)^2)
proof
A2: dom ((Sq_Circ")|K1)=dom (Sq_Circ") /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by Th29,FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
let p be Point of TOP-REAL 2;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A4: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A3;
then
A5: (Sq_Circ").p=|[p`1*sqrt(1+(p`2/p`1)^2), p`2*sqrt(1+(p`2/p`1)^2)]| by Th28;
((Sq_Circ")|K1).p=(Sq_Circ").p by A4,A3,FUNCT_1:49;
then g1.p=(proj1).(|[p`1*sqrt(1+(p`2/p`1)^2), p`2*sqrt(1+(p`2/p`1)^2)]|)
by A4,A2,A3,A5,FUNCT_1:13
.=(|[p`1*sqrt(1+(p`2/p`1)^2), p`2*sqrt(1+(p`2/p`1)^2)]|)`1 by
PSCOMP_1:def 5
.=p`1*sqrt(1+(p`2/p`1)^2) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A6: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=p`1*sqrt(1+(p`2/p`1)^2);
reconsider g2=(proj2)*((Sq_Circ")|K1) as Function of (TOP-REAL 2)|K1,R^1 by
Lm17;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g2.p=p`2*sqrt(1+(p`2/p`1)^2)
proof
A7: dom ((Sq_Circ")|K1)=dom (Sq_Circ") /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by Th29,FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
let p be Point of TOP-REAL 2;
A8: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A9: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A8;
then
A10: (Sq_Circ").p =|[p`1*sqrt(1+(p`2/p`1)^2),p`2*sqrt(1+(p`2/p`1)^2)]| by Th28;
((Sq_Circ")|K1).p=(Sq_Circ").p by A9,A8,FUNCT_1:49;
then
g2.p=(proj2).(|[p`1*sqrt(1+(p`2/p`1)^2),p`2*sqrt(1+(p`2/p`1)^2)]|) by A9,A7
,A8,A10,FUNCT_1:13
.=(|[p`1*sqrt(1+(p`2/p`1)^2), p`2*sqrt(1+(p`2/p`1)^2)]|)`2 by
PSCOMP_1:def 6
.=p`2*sqrt(1+(p`2/p`1)^2) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A11: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=p`2*sqrt(1+(p`2/p`1)^2);
A12: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1<>0
proof
let q be Point of TOP-REAL 2;
A13: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
A14: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`2<=p3`1 & - p3`1<=p3`2 or
p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A13;
now
assume
A15: q`1=0;
then q`2=0 by A14;
hence contradiction by A14,A15,EUCLID:53,54;
end;
hence thesis;
end;
then
A16: f1 is continuous by A6,Th33;
A17: now
let x,y,r,s be Real;
assume that
A18: |[x,y]| in K1 and
A19: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
set p99=|[x,y]|;
A20: ex p3 being Point of TOP-REAL 2 st p99=p3 &( p3`2<=p3`1 & -p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A1,A18;
A21: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A22: f1.p99=p99`1*sqrt(1+(p99`2/p99`1)^2) by A6,A18;
((Sq_Circ")|K0).(|[x,y]|)=((Sq_Circ")).(|[x,y]|) by A18,FUNCT_1:49
.= |[p99`1*sqrt(1+(p99`2/p99`1)^2), p99`2*sqrt(1+(p99`2/p99`1)^2)]| by
A20,Th28
.=|[r,s]| by A11,A18,A19,A21,A22;
hence f.(|[x,y]|)=|[r,s]| by A1;
end;
f2 is continuous by A12,A11,Th34;
hence thesis by A1,A16,A17,Lm13,JGRAPH_2:35;
end;
theorem Th38:
for K0,B0 being Subset of TOP-REAL 2,f being Function of (
TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st f=(Sq_Circ")|K0 & B0=NonZero TOP-REAL 2 & K0=
{p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} holds f is
continuous
proof
let K0,B0 be Subset of TOP-REAL 2,f be Function of (TOP-REAL 2)|K0,(TOP-REAL
2)|B0;
assume
A1: f=(Sq_Circ")|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`1<=p`2 & -p`2<=p
`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2};
then 1.REAL 2 in K0 by Lm14,Lm15;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
reconsider g1=(proj2)*((Sq_Circ")|K1) as Function of (TOP-REAL 2)|K1,R^1 by
Lm17;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g1.p=p`2*sqrt(1+(p`1/p`2)^2)
proof
A2: dom ((Sq_Circ")|K1)=dom (Sq_Circ") /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by Th29,FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
let p be Point of TOP-REAL 2;
A3: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A4: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`1<=p3`2 & - p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A3;
then
A5: (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2), p`2*sqrt(1+(p`1/p`2)^2)]| by Th30;
((Sq_Circ")|K1).p=(Sq_Circ").p by A4,A3,FUNCT_1:49;
then
g1.p=(proj2).(|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|) by A4,A2
,A3,A5,FUNCT_1:13
.=(|[p`1*sqrt(1+(p`1/p`2)^2), p`2*sqrt(1+(p`1/p`2)^2)]|)`2 by
PSCOMP_1:def 6
.=p`2*sqrt(1+(p`1/p`2)^2) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A6: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=p`2*sqrt(1+(p`1/p`2)^2);
reconsider g2=(proj1)*((Sq_Circ")|K1) as Function of (TOP-REAL 2)|K1,R^1 by
Lm18;
for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)|K1
holds g2.p=p`1*sqrt(1+(p`1/p`2)^2)
proof
A7: dom ((Sq_Circ")|K1)=dom (Sq_Circ") /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by Th29,FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
let p be Point of TOP-REAL 2;
A8: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A9: p in the carrier of (TOP-REAL 2)|K1;
then ex p3 being Point of TOP-REAL 2 st p=p3 &( p3`1<=p3`2 & - p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A8;
then
A10: (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2), p`2*sqrt(1+(p`1/p`2)^2)]| by Th30;
((Sq_Circ")|K1).p=(Sq_Circ").p by A9,A8,FUNCT_1:49;
then
g2.p=(proj1).(|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|) by A9,A7
,A8,A10,FUNCT_1:13
.=(|[p`1*sqrt(1+(p`1/p`2)^2), p`2*sqrt(1+(p`1/p`2)^2)]|)`1 by
PSCOMP_1:def 5
.=p`1*sqrt(1+(p`1/p`2)^2) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A11: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=p`1*sqrt(1+(p`1/p`2)^2);
A12: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<>0
proof
let q be Point of TOP-REAL 2;
A13: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
A14: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`1<=p3`2 & - p3`2<=p3`1 or
p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A13;
now
assume
A15: q`2=0;
then q`1=0 by A14;
hence contradiction by A14,A15,EUCLID:53,54;
end;
hence thesis;
end;
then
A16: f1 is continuous by A6,Th35;
A17: for x,y,s,r being Real st |[x,y]| in K1 & s=f2.(|[x,y]|) & r=f1.
(|[x,y]|) holds f.(|[x,y]|)=|[s,r]|
proof
let x,y,s,r be Real;
assume that
A18: |[x,y]| in K1 and
A19: s=f2.(|[x,y]|) & r=f1.(|[x,y]|);
set p99=|[x,y]|;
A20: ex p3 being Point of TOP-REAL 2 st p99=p3 &( p3`1<=p3`2 & -p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A1,A18;
A21: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A22: f1.p99=p99`2*sqrt(1+(p99`1/p99`2)^2) by A6,A18;
((Sq_Circ")|K0).(|[x,y]|)=((Sq_Circ")).(|[x,y]|) by A18,FUNCT_1:49
.= |[p99`1*sqrt(1+(p99`1/p99`2)^2), p99`2*sqrt(1+(p99`1/p99`2)^2)]| by
A20,Th30
.=|[s,r]| by A11,A18,A19,A21,A22;
hence thesis by A1;
end;
f2 is continuous by A12,A11,Th36;
hence thesis by A1,A16,A17,Lm13,JGRAPH_2:35;
end;
theorem Th39:
for B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2
)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st f=(Sq_Circ")
|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p
`1) & p<>0.TOP-REAL 2} holds f is continuous & K0 is closed
proof
reconsider K5={p7 where p7 is Point of TOP-REAL 2:p7`2<=-p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:47;
reconsider K4={p7 where p7 is Point of TOP-REAL 2: p7`1<=p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
reconsider K3 = {p7 where p7 is Point of TOP-REAL 2:-p7`1<=p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:47;
reconsider K2={p7 where p7 is Point of TOP-REAL 2:p7`2<=p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
defpred P[Point of TOP-REAL 2] means ($1`2<=$1`1 & -$1`1<=$1`2 or $1`2>=$1`1
& $1`2<=-$1`1);
set b0 = NonZero TOP-REAL 2;
defpred P0[Point of TOP-REAL 2] means ($1`2<=$1`1 & -$1`1<=$1`2 or $1`2>=$1
`1 & $1`2<=-$1`1);
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0,f being
Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
set k0 = {p:P0[p] & p<>0.TOP-REAL 2};
assume that
A1: f=(Sq_Circ")|K0 and
A2: B0=b0 & K0=k0;
the carrier of (TOP-REAL 2)|B0 = B0 by PRE_TOPC:8;
then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1;
k0 c= NonZero TOP-REAL 2 from TopIncl;
then
A3: ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by A2,PRE_TOPC:7;
reconsider K1={p7 where p7 is Point of TOP-REAL 2: P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
A4: K2 /\ K3 \/ K4 /\ K5 c= K1
proof
let x be object;
assume
A5: x in K2 /\ K3 \/ K4 /\ K5;
per cases by A5,XBOOLE_0:def 3;
suppose
A6: x in K2 /\ K3;
then x in K3 by XBOOLE_0:def 4;
then
A7: ex p8 being Point of TOP-REAL 2 st p8=x & -p8`1<=p8`2;
x in K2 by A6,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`2<=(p7`1);
hence thesis by A7;
end;
suppose
A8: x in K4 /\ K5;
then x in K5 by XBOOLE_0:def 4;
then
A9: ex p8 being Point of TOP-REAL 2 st p8=x & p8`2<= -p8`1;
x in K4 by A8,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`2>=(p7`1);
hence thesis by A9;
end;
end;
A10: K2 /\ K3 is closed & K4 /\ K5 is closed by TOPS_1:8;
K1 c= K2 /\ K3 \/ K4 /\ K5
proof
let x be object;
assume x in K1;
then ex p being Point of TOP-REAL 2 st p=x &( p`2<=p`1 & -p`1 <=p`2 or p`2
>=p`1 & p`2<=-p`1);
then x in K2 & x in K3 or x in K4 & x in K5;
then x in K2 /\ K3 or x in K4 /\ K5 by XBOOLE_0:def 4;
hence thesis by XBOOLE_0:def 3;
end;
then K1=K2 /\ K3 \/ K4 /\ K5 by A4;
then
A11: K1 is closed by A10,TOPS_1:9;
k0={p7 where p7 is Point of TOP-REAL 2:P0[p7]} /\ b0 from TopInter;
then K0=K1 /\ [#]((TOP-REAL 2)|B0) by A2,PRE_TOPC:def 5;
hence thesis by A1,A2,A3,A11,Th37,PRE_TOPC:13;
end;
theorem Th40:
for B0 being Subset of TOP-REAL 2,K0 being Subset of (TOP-REAL 2
)|B0,f being Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0) st f=(Sq_Circ")
|K0 & B0=NonZero TOP-REAL 2 & K0={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p
`2) & p<>0.TOP-REAL 2} holds f is continuous & K0 is closed
proof
reconsider K5={p7 where p7 is Point of TOP-REAL 2:p7`1<=-p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:48;
reconsider K4={p7 where p7 is Point of TOP-REAL 2:p7`2<=p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
reconsider K3={p7 where p7 is Point of TOP-REAL 2:-p7`2<=p7`1 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:48;
reconsider K2={p7 where p7 is Point of TOP-REAL 2:p7`1<=p7`2 } as closed
Subset of TOP-REAL 2 by JGRAPH_2:46;
defpred P[Point of TOP-REAL 2] means ($1`1<=$1`2 & -$1`2<=$1`1 or $1`1>=$1`2
& $1`1<=-$1`2);
defpred P0[Point of TOP-REAL 2] means ($1`1<=$1`2 & -$1`2<=$1`1 or $1`1>=$1
`2 & $1`1<=-$1`2);
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0,f being
Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
set k0 = {p:P0[p] & p<>0.TOP-REAL 2}, b0=NonZero TOP-REAL 2;
assume that
A1: f=(Sq_Circ")|K0 and
A2: B0=b0 & K0=k0;
the carrier of (TOP-REAL 2)|B0= B0 by PRE_TOPC:8;
then reconsider K1=K0 as Subset of TOP-REAL 2 by XBOOLE_1:1;
{p:P[p] & p<>0.TOP-REAL 2} c= NonZero TOP-REAL 2 from TopIncl;
then
A3: ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by A2,PRE_TOPC:7;
set k1 = {p7 where p7 is Point of TOP-REAL 2: P0[p7]};
A4: K2 /\ K3 is closed & K4 /\ K5 is closed by TOPS_1:8;
reconsider K1=k1 as Subset of TOP-REAL 2 from JGRAPH_2:sch 1;
A5: K2 /\ K3 \/ K4 /\ K5 c= K1
proof
let x be object;
assume
A6: x in K2 /\ K3 \/ K4 /\ K5;
per cases by A6,XBOOLE_0:def 3;
suppose
A7: x in K2 /\ K3;
then x in K3 by XBOOLE_0:def 4;
then
A8: ex p8 being Point of TOP-REAL 2 st p8=x & -p8`2<=p8`1;
x in K2 by A7,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`1<=(p7`2);
hence thesis by A8;
end;
suppose
A9: x in K4 /\ K5;
then x in K5 by XBOOLE_0:def 4;
then
A10: ex p8 being Point of TOP-REAL 2 st p8=x & p8`1<= -p8`2;
x in K4 by A9,XBOOLE_0:def 4;
then ex p7 being Point of TOP-REAL 2 st p7=x & p7`1>=(p7`2);
hence thesis by A10;
end;
end;
K1 c= K2 /\ K3 \/ K4 /\ K5
proof
let x be object;
assume x in K1;
then ex p being Point of TOP-REAL 2 st p=x &( p`1<=p`2 & -p`2 <=p`1 or p`1
>=p`2 & p`1<=-p`2);
then x in K2 & x in K3 or x in K4 & x in K5;
then x in K2 /\ K3 or x in K4 /\ K5 by XBOOLE_0:def 4;
hence thesis by XBOOLE_0:def 3;
end;
then K1=K2 /\ K3 \/ K4 /\ K5 by A5;
then
A11: K1 is closed by A4,TOPS_1:9;
k0=k1 /\ b0 from TopInter;
then K0=K1 /\ [#]((TOP-REAL 2)|B0) by A2,PRE_TOPC:def 5;
hence thesis by A1,A2,A3,A11,Th38,PRE_TOPC:13;
end;
theorem Th41:
for D being non empty Subset of TOP-REAL 2 st D`={0.TOP-REAL 2}
holds ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=(Sq_Circ")|D &
h is continuous
proof
set Y1=|[-1,1]|;
set B0 = {0.TOP-REAL 2};
let D be non empty Subset of TOP-REAL 2;
A1: the carrier of ((TOP-REAL 2)|D) = D by PRE_TOPC:8;
dom (Sq_Circ")=(the carrier of (TOP-REAL 2)) by Th29,FUNCT_2:def 1;
then
A2: dom ((Sq_Circ")|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|D) by A1,XBOOLE_1:28;
assume
A3: D`={0.TOP-REAL 2};
then
A4: D = (B0)` .=(NonZero TOP-REAL 2) by SUBSET_1:def 4;
A5: {p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} c=
the carrier of (TOP-REAL 2)|D
proof
let x be object;
assume
x in {p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0. TOP-REAL 2};
then
A6: ex p st x=p &( p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p `1)& p<>0.
TOP-REAL 2;
now
assume not x in D;
then x in (the carrier of TOP-REAL 2) \ D by A6,XBOOLE_0:def 5;
then x in D` by SUBSET_1:def 4;
hence contradiction by A3,A6,TARSKI:def 1;
end;
hence thesis by PRE_TOPC:8;
end;
1.REAL 2 in {p where p is Point of TOP-REAL 2: (p`2<=p`1 & -p`1<=p`2 or
p`2>=p`1 & p`2<=-p`1) & p<>0.TOP-REAL 2} by Lm9,Lm10;
then reconsider K0={p:(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) & p<>0.
TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A5;
A7: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8;
A8: {p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} c=
the carrier of (TOP-REAL 2)|D
proof
let x be object;
assume
x in {p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0. TOP-REAL 2};
then
A9: ex p st x=p &( p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p `2)& p<>0.
TOP-REAL 2;
now
assume not x in D;
then x in (the carrier of TOP-REAL 2) \ D by A9,XBOOLE_0:def 5;
then x in D` by SUBSET_1:def 4;
hence contradiction by A3,A9,TARSKI:def 1;
end;
hence thesis by PRE_TOPC:8;
end;
Y1`1=-1 & Y1`2=1 by EUCLID:52;
then
Y1 in {p where p is Point of TOP-REAL 2: (p`1<=p`2 & -p`2<=p`1 or p`1>=
p`2 & p`1<=-p`2) & p<>0.TOP-REAL 2} by JGRAPH_2:3;
then reconsider K1={p:(p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2) & p<>0.
TOP-REAL 2} as non empty Subset of (TOP-REAL 2)|D by A8;
A10: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8;
A11: D c= K0 \/ K1
proof
let x be object;
assume
A12: x in D;
then reconsider px=x as Point of TOP-REAL 2;
not x in {0.TOP-REAL 2} by A4,A12,XBOOLE_0:def 5;
then (px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1) & px<>0.
TOP-REAL 2 or (px`1<=px`2 & -px`2<=px`1 or px`1>=px`2 & px`1<=-px`2) & px<>0.
TOP-REAL 2 by TARSKI:def 1,XREAL_1:26;
then x in K0 or x in K1;
hence thesis by XBOOLE_0:def 3;
end;
A13: the carrier of ((TOP-REAL 2)|D)=[#](((TOP-REAL 2)|D))
.=(NonZero TOP-REAL 2) by A4,PRE_TOPC:def 5;
A14: K0 c= the carrier of TOP-REAL 2
proof
let z be object;
assume z in K0;
then ex p8 being Point of TOP-REAL 2 st p8=z &( p8`2<=p8`1 & - p8`1<=p8`2
or p8`2>=p8`1 & p8`2<=-p8`1)& p8<>0.TOP-REAL 2;
hence thesis;
end;
A15: rng ((Sq_Circ")|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
reconsider K00=K0 as Subset of TOP-REAL 2 by A14;
let y be object;
A16: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|
K00 holds q`1<>0
proof
let q be Point of TOP-REAL 2;
A17: the carrier of (TOP-REAL 2)|K00=K0 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K00;
then
A18: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`2<=p3`1 & - p3`1<=p3`2
or p3`2>=p3`1 & p3`2<=-p3`1)& p3<>0.TOP-REAL 2 by A17;
now
assume
A19: q`1=0;
then q`2=0 by A18;
hence contradiction by A18,A19,EUCLID:53,54;
end;
hence thesis;
end;
assume y in rng ((Sq_Circ")|K0);
then consider x being object such that
A20: x in dom ((Sq_Circ")|K0) and
A21: y=((Sq_Circ")|K0).x by FUNCT_1:def 3;
A22: x in (dom (Sq_Circ")) /\ K0 by A20,RELAT_1:61;
then
A23: x in K0 by XBOOLE_0:def 4;
then reconsider p=x as Point of TOP-REAL 2 by A14;
K00=the carrier of ((TOP-REAL 2)|K00) by PRE_TOPC:8;
then p in the carrier of ((TOP-REAL 2)|K00) by A22,XBOOLE_0:def 4;
then
A24: p`1<>0 by A16;
set p9=|[p`1*sqrt(1+(p`2/p`1)^2),p`2*sqrt(1+(p`2/p`1)^2)]|;
A25: p9`1=p`1*sqrt(1+(p`2/p`1)^2) & p9`2=p`2*sqrt(1+(p`2/p`1)^2) by EUCLID:52;
A26: ex px being Point of TOP-REAL 2 st x=px &( px`2<=px`1 & - px`1<=px`2
or px`2>=px`1 & px`2<=-px`1)& px<>0.TOP-REAL 2 by A23;
then
A27: (Sq_Circ").p=|[p`1*sqrt(1+(p`2/p`1)^2), p`2*sqrt(1+(p`2/p`1)^2)]| by Th28;
A28: sqrt(1+(p`2/p`1)^2)>0 by Lm1,SQUARE_1:25;
then
p`2*sqrt(1+(p`2/p`1)^2)<=p`1*sqrt(1+(p`2/p`1)^2) & (-p`1)*sqrt(1+(p`2
/p`1)^2)<=p`2*sqrt(1+(p`2/p`1)^2) or p`2*sqrt(1+(p`2/p`1)^2)>=p`1*sqrt(1+(p`2/p
`1)^2) & p`2*sqrt(1+(p`2/p`1)^2)<=(-p`1)*sqrt(1+(p`2/p`1)^2) by A26,
XREAL_1:64;
then
A29: p`2*sqrt(1+(p`2/p`1)^2)<=p`1*sqrt(1+(p`2/p`1)^2) & -(p`1*sqrt(1+(p`2/
p`1)^2))<=p`2*sqrt(1+(p`2/p`1)^2) or p`2*sqrt(1+(p`2/p`1)^2)>=p`1*sqrt(1+(p`2/p
`1)^2) & p`2*sqrt(1+(p`2/p`1)^2)<=-(p`1*sqrt(1+(p`2/p`1)^2));
A30: p9`1=p`1*sqrt(1+(p`2/p`1)^2) by EUCLID:52;
A31: now
assume p9=0.TOP-REAL 2;
then 0/sqrt(1+(p`2/p`1)^2)=p`1*sqrt(1+(p`2/p`1)^2)/sqrt(1+(p`2/p`1)^2)
by A30,EUCLID:52,54;
hence contradiction by A24,A28,XCMPLX_1:89;
end;
(Sq_Circ").p=y by A21,A23,FUNCT_1:49;
then y in K0 by A31,A27,A29,A25;
hence thesis by PRE_TOPC:8;
end;
dom ((Sq_Circ")|K0)= dom ((Sq_Circ")) /\ K0 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K0 by Th29,FUNCT_2:def 1
.=K0 by A14,XBOOLE_1:28;
then reconsider
f=(Sq_Circ")|K0 as Function of ((TOP-REAL 2)|D)|K0, (TOP-REAL 2)|
D by A7,A15,FUNCT_2:2,XBOOLE_1:1;
A32: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
A33: K1 c= the carrier of TOP-REAL 2
proof
let z be object;
assume z in K1;
then ex p8 being Point of TOP-REAL 2 st p8=z &( p8`1<=p8`2 & - p8`2<=p8`1
or p8`1>=p8`2 & p8`1<=-p8`2)& p8<>0.TOP-REAL 2;
hence thesis;
end;
A34: rng ((Sq_Circ")|K1) c= the carrier of ((TOP-REAL 2)|D)|K1
proof
reconsider K10=K1 as Subset of TOP-REAL 2 by A33;
let y be object;
A35: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|
K10 holds q`2<>0
proof
let q be Point of TOP-REAL 2;
A36: the carrier of (TOP-REAL 2)|K10=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K10;
then
A37: ex p3 being Point of TOP-REAL 2 st q=p3 &( p3`1<=p3`2 & - p3`2<=p3`1
or p3`1>=p3`2 & p3`1<=-p3`2)& p3<>0.TOP-REAL 2 by A36;
now
assume
A38: q`2=0;
then q`1=0 by A37;
hence contradiction by A37,A38,EUCLID:53,54;
end;
hence thesis;
end;
assume y in rng ((Sq_Circ")|K1);
then consider x being object such that
A39: x in dom ((Sq_Circ")|K1) and
A40: y=((Sq_Circ")|K1).x by FUNCT_1:def 3;
A41: x in (dom (Sq_Circ")) /\ K1 by A39,RELAT_1:61;
then
A42: x in K1 by XBOOLE_0:def 4;
then reconsider p=x as Point of TOP-REAL 2 by A33;
K10=the carrier of ((TOP-REAL 2)|K10) by PRE_TOPC:8;
then p in the carrier of ((TOP-REAL 2)|K10) by A41,XBOOLE_0:def 4;
then
A43: p`2<>0 by A35;
set p9=|[p`1*sqrt(1+(p`1/p`2)^2),p`2*sqrt(1+(p`1/p`2)^2)]|;
A44: p9`2=p`2*sqrt(1+(p`1/p`2)^2) & p9`1=p`1*sqrt(1+(p`1/p`2)^2) by EUCLID:52;
A45: ex px being Point of TOP-REAL 2 st x=px &( px`1<=px`2 & - px`2<=px`1
or px`1>=px`2 & px`1<=-px`2)& px<>0.TOP-REAL 2 by A42;
then
A46: (Sq_Circ").p=|[p`1*sqrt(1+(p`1/p`2)^2), p`2*sqrt(1+(p`1/p`2)^2)]| by Th30;
A47: sqrt(1+(p`1/p`2)^2)>0 by Lm1,SQUARE_1:25;
then
p`1*sqrt(1+(p`1/p`2)^2)<=p`2*sqrt(1+(p`1/p`2)^2) & (-p`2)*sqrt(1+(p`1
/p`2)^2)<=p`1*sqrt(1+(p`1/p`2)^2) or p`1*sqrt(1+(p`1/p`2)^2)>=p`2*sqrt(1+(p`1/p
`2)^2) & p`1*sqrt(1+(p`1/p`2)^2)<=(-p`2)*sqrt(1+(p`1/p`2)^2) by A45,
XREAL_1:64;
then
A48: p`1*sqrt(1+(p`1/p`2)^2)<=p`2*sqrt(1+(p`1/p`2)^2) & -(p`2*sqrt(1+(p`1/
p`2)^2))<=p`1*sqrt(1+(p`1/p`2)^2) or p`1*sqrt(1+(p`1/p`2)^2)>=p`2*sqrt(1+(p`1/p
`2)^2) & p`1*sqrt(1+(p`1/p`2)^2)<=-(p`2*sqrt(1+(p`1/p`2)^2));
A49: p9`2=p`2*sqrt(1+(p`1/p`2)^2) by EUCLID:52;
A50: now
assume p9=0.TOP-REAL 2;
then 0/sqrt(1+(p`1/p`2)^2)=p`2*sqrt(1+(p`1/p`2)^2)/sqrt(1+(p`1/p`2)^2)
by A49,EUCLID:52,54;
hence contradiction by A43,A47,XCMPLX_1:89;
end;
(Sq_Circ").p=y by A40,A42,FUNCT_1:49;
then y in K1 by A50,A46,A48,A44;
hence thesis by PRE_TOPC:8;
end;
dom ((Sq_Circ")|K1)= dom ((Sq_Circ")) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K1 by Th29,FUNCT_2:def 1
.=K1 by A33,XBOOLE_1:28;
then reconsider
g=(Sq_Circ")|K1 as Function of ((TOP-REAL 2)|D)|K1, ((TOP-REAL 2)
|D) by A10,A34,FUNCT_2:2,XBOOLE_1:1;
A51: dom g=K1 by A10,FUNCT_2:def 1;
g=(Sq_Circ")|K1;
then
A52: K1 is closed by A4,Th40;
A53: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
A54: now
let x be object;
assume
A55: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1) ));
then x in K0 by A53,XBOOLE_0:def 4;
then f.x=(Sq_Circ").x by FUNCT_1:49;
hence f.x = g.x by A32,A55,FUNCT_1:49;
end;
f=(Sq_Circ")|K0;
then
A56: K0 is closed by A4,Th39;
A57: dom f=K0 by A7,FUNCT_2:def 1;
D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
then
A58: ([#](((TOP-REAL 2)|D)|K0)) \/ [#](((TOP-REAL 2)|D)|K1) = [#]((TOP-REAL
2)|D) by A53,A32,A11;
A59: f is continuous & g is continuous by A4,Th39,Th40;
then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that
A60: h= f+*g and
h is continuous by A53,A32,A58,A56,A52,A54,JGRAPH_2:1;
K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
then
A61: f tolerates g by A54,A57,A51,PARTFUN1:def 4;
A62: for x being object st x in dom h holds h.x=((Sq_Circ")|D).x
proof
let x be object;
assume
A63: x in dom h;
then reconsider p=x as Point of TOP-REAL 2 by A13,XBOOLE_0:def 5;
not x in {0.TOP-REAL 2} by A13,A63,XBOOLE_0:def 5;
then
A64: x <>0.TOP-REAL 2 by TARSKI:def 1;
x in (the carrier of TOP-REAL 2)\D` by A3,A13,A63;
then
A65: x in D`` by SUBSET_1:def 4;
per cases;
suppose
A66: x in K0;
A67: (Sq_Circ")|D.p=(Sq_Circ").p by A65,FUNCT_1:49
.=f.p by A66,FUNCT_1:49;
h.p=(g+*f).p by A60,A61,FUNCT_4:34
.=f.p by A57,A66,FUNCT_4:13;
hence thesis by A67;
end;
suppose
not x in K0;
then not (p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1) by A64;
then p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2 by XREAL_1:26;
then
A68: x in K1 by A64;
(Sq_Circ")|D.p=(Sq_Circ").p by A65,FUNCT_1:49
.=g.p by A68,FUNCT_1:49;
hence thesis by A60,A51,A68,FUNCT_4:13;
end;
end;
dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1;
then f+*g=(Sq_Circ")|D by A60,A2,A62;
hence thesis by A53,A32,A58,A56,A59,A52,A54,JGRAPH_2:1;
end;
theorem Th42:
ex h being Function of TOP-REAL 2,TOP-REAL 2 st h=Sq_Circ" & h is continuous
proof
reconsider f=(Sq_Circ") as Function of (TOP-REAL 2),(TOP-REAL 2) by Th29;
reconsider D=NonZero TOP-REAL 2 as non empty Subset of TOP-REAL 2 by
JGRAPH_2:9;
A1: f.(0.TOP-REAL 2)=0.TOP-REAL 2 by Th28;
A2: for p being Point of (TOP-REAL 2)|D holds f.p<>f.(0.TOP-REAL 2)
proof
let p be Point of (TOP-REAL 2)|D;
A3: [#]((TOP-REAL 2)|D)=D by PRE_TOPC:def 5;
then reconsider q=p as Point of TOP-REAL 2 by XBOOLE_0:def 5;
not p in {0.TOP-REAL 2} by A3,XBOOLE_0:def 5;
then
A4: not p=0.TOP-REAL 2 by TARSKI:def 1;
per cases;
suppose
A5: not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1);
then A6: q`2<>0;
set q9=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A7: q9`2=q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A8: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
now
assume q9=0.TOP-REAL 2;
then 0 *q`2=q`2*sqrt(1+(q`1/q`2)^2) by A7,EUCLID:52,54;
then
0 *sqrt(1+(q`1/q`2)^2)=q`2*sqrt(1+(q`1/q`2)^2)/sqrt(1+(q`1/q`2)^2 );
hence contradiction by A6,A8,XCMPLX_1:89;
end;
hence thesis by A1,A5,Th28;
end;
suppose
A9: q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q`1;
A10: now
assume
A11: q`1=0;
then q`2=0 by A9;
hence contradiction by A4,A11,EUCLID:53,54;
end;
set q9=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A12: q9`1=q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A13: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
now
assume q9=0.TOP-REAL 2;
then
0/sqrt(1+(q`2/q`1)^2)=q`1*sqrt(1+(q`2/q`1)^2)/sqrt(1+(q`2/q`1)^2)
by A12,EUCLID:52,54;
hence contradiction by A10,A13,XCMPLX_1:89;
end;
hence thesis by A1,A4,A9,Th28;
end;
end;
A14: for V being Subset of TOP-REAL 2 st f.(0.TOP-REAL 2) in V & V is open
ex W being Subset of TOP-REAL 2 st 0.TOP-REAL 2 in W & W is open & f.:W c= V
proof
reconsider u0=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67;
let V be Subset of TOP-REAL 2;
reconsider VV=V as Subset of TopSpaceMetr Euclid 2 by Lm16;
assume that
A15: f.(0.TOP-REAL 2) in V and
A16: V is open;
VV is open by A16,Lm16,PRE_TOPC:30;
then consider r being Real such that
A17: r>0 and
A18: Ball(u0,r) c= V by A1,A15,TOPMETR:15;
reconsider r as Real;
reconsider W1=Ball(u0,r), V1=Ball(u0,r/sqrt(2)) as Subset of TOP-REAL 2 by
EUCLID:67;
A19: f.:V1 c= W1
proof
let z be object;
A20: sqrt(2)>0 by SQUARE_1:25;
assume z in f.:V1;
then consider y being object such that
A21: y in dom f and
A22: y in V1 and
A23: z=f.y by FUNCT_1:def 6;
z in rng f by A21,A23,FUNCT_1:def 3;
then reconsider qz=z as Point of TOP-REAL 2;
reconsider pz=qz as Point of Euclid 2 by EUCLID:67;
reconsider q=y as Point of TOP-REAL 2 by A21;
reconsider qy=q as Point of Euclid 2 by EUCLID:67;
A24: (q`1)^2 >=0 by XREAL_1:63;
A25: (q`2)^2>=0 by XREAL_1:63;
dist(u0,qy)<r/sqrt(2) by A22,METRIC_1:11;
then |.(0.TOP-REAL 2) - q.|<r/sqrt(2) by JGRAPH_1:28;
then sqrt((((0.TOP-REAL 2) - q)`1)^2+(((0.TOP-REAL 2) - q)`2)^2) < r/
sqrt(2) by JGRAPH_1:30;
then
sqrt(((0.TOP-REAL 2)`1 - q`1)^2+ (((0.TOP-REAL 2) - q)`2)^2)<r/sqrt
(2) by TOPREAL3:3;
then
sqrt(((0.TOP-REAL 2)`1 - q`1)^2+((0.TOP-REAL 2)`2 - q`2)^2)<r/sqrt(
2) by TOPREAL3:3;
then sqrt((q`1)^2+(q`2)^2)*sqrt(2)< r/sqrt(2)*sqrt(2) by A20,JGRAPH_2:3
,XREAL_1:68;
then sqrt(((q`1)^2+(q`2)^2)*2)< r/sqrt(2)*sqrt(2) by A24,A25,SQUARE_1:29;
then
A26: sqrt(((q`1)^2+(q`2)^2)*2)< r by A20,XCMPLX_1:87;
per cases;
suppose
q=0.TOP-REAL 2;
then z=0.TOP-REAL 2 by A23,Th28;
hence thesis by A17,GOBOARD6:1;
end;
suppose
A27: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=-q `1);
A28: now
assume (q`1)^2<=0;
then (q`1)^2=0 by XREAL_1:63;
then
A29: q`1=0 by XCMPLX_1:6;
then q`2=0 by A27;
hence contradiction by A27,A29,EUCLID:53,54;
end;
A30: (Sq_Circ").q=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1 ) ^2)
]| by A27,Th28;
then qz`1=q`1*sqrt(1+(q`2/q`1)^2) by A23,EUCLID:52;
then
A31: (qz`1)^2=(q`1)^2*(sqrt(1+(q`2/q`1)^2))^2;
qz`2=q`2*sqrt(1+(q`2/q`1)^2) by A23,A30,EUCLID:52;
then
A32: (qz`2)^2=(q`2)^2*(sqrt(1+(q`2/q`1)^2))^2;
A33: 1+(q`2/q`1)^2>0 by Lm1;
now
per cases by A27;
case
A34: q`2<=q`1 & -q`1<=q`2;
now
per cases;
case
0<=q`2;
hence (q`2)^2<=(q`1)^2 by A34,SQUARE_1:15;
end;
case
A35: 0>q`2;
--q`1>=-q`2 by A34,XREAL_1:24;
then (-q`2)^2<=(q`1)^2 by A35,SQUARE_1:15;
hence (q`2)^2<=(q`1)^2;
end;
end;
hence (q`2)^2<=(q`1)^2;
end;
case
A36: q`2>=q`1 & q`2<=-q`1;
now
per cases;
case
A37: 0>=q`2;
-q`2<=-q`1 by A36,XREAL_1:24;
then (-q`2)^2<=(-q`1)^2 by A37,SQUARE_1:15;
hence (q`2)^2<=(q`1)^2;
end;
case
0<q`2;
then (q`2)^2<=(-q`1)^2 by A36,SQUARE_1:15;
hence (q`2)^2<=(q`1)^2;
end;
end;
hence (q`2)^2<=(q`1)^2;
end;
end;
then (q`2)^2/(q`1)^2<=(q`1)^2/(q`1)^2 by A28,XREAL_1:72;
then (q`2/q`1)^2<=(q`1)^2/(q`1)^2 by XCMPLX_1:76;
then (q`2/q`1)^2 <=1 by A28,XCMPLX_1:60;
then
A38: 1+(q`2/q`1)^2<=1+1 by XREAL_1:7;
then (q`2)^2*(1+(q`2/q`1)^2)<=(q`2)^2*2 by A25,XREAL_1:64;
then
A39: (qz`2)^2<=(q`2)^2*2 by A33,A32,SQUARE_1:def 2;
(q`1)^2*(1+(q`2/q`1)^2)<=(q`1)^2*2 by A24,A38,XREAL_1:64;
then (qz`1)^2<=(q`1)^2*2 by A33,A31,SQUARE_1:def 2;
then
A40: (qz`1)^2+(qz`2)^2<=(q`1)^2*2+(q`2)^2*2 by A39,XREAL_1:7;
(qz`1)^2>=0 & (qz`2)^2>=0 by XREAL_1:63;
then
A41: sqrt((qz`1)^2+(qz`2)^2) <= sqrt((q`1)^2*2+(q`2)^2*2) by A40,SQUARE_1:26
;
A42: ((0.TOP-REAL 2) - qz)`2=(0.TOP-REAL 2)`2-qz`2 by TOPREAL3:3
.= -qz`2 by JGRAPH_2:3;
((0.TOP-REAL 2) - qz)`1=(0.TOP-REAL 2)`1-qz`1 by TOPREAL3:3
.= -qz`1 by JGRAPH_2:3;
then sqrt((((0.TOP-REAL 2) - qz)`1)^2+(((0.TOP-REAL 2) - qz)`2)^2)<r
by A26,A42,A41,XXREAL_0:2;
then |.(0.TOP-REAL 2) - qz.|<r by JGRAPH_1:30;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
suppose
A43: q<>0.TOP-REAL 2 & not (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2 <=-q`1);
A44: now
assume (q`2)^2<=0;
then (q`2)^2=0 by XREAL_1:63;
then q`2=0 by XCMPLX_1:6;
hence contradiction by A43;
end;
now
per cases by A43,JGRAPH_2:13;
case
A45: q`1<=q`2 & -q`2<=q`1;
now
per cases;
case
0<=q`1;
hence (q`1)^2<=(q`2)^2 by A45,SQUARE_1:15;
end;
case
A46: 0>q`1;
--q`2>=-q`1 by A45,XREAL_1:24;
then (-q`1)^2<=(q`2)^2 by A46,SQUARE_1:15;
hence (q`1)^2<=(q`2)^2;
end;
end;
hence (q`1)^2<=(q`2)^2;
end;
case
A47: q`1>=q`2 & q`1<=-q`2;
now
per cases;
case
A48: 0>=q`1;
-q`1<=-q`2 by A47,XREAL_1:24;
then (-q`1)^2<=(-q`2)^2 by A48,SQUARE_1:15;
hence (q`1)^2<=(q`2)^2;
end;
case
0<q`1;
then (q`1)^2<=(-q`2)^2 by A47,SQUARE_1:15;
hence (q`1)^2<=(q`2)^2;
end;
end;
hence (q`1)^2<=(q`2)^2;
end;
end;
then (q`1)^2/(q`2)^2<=(q`2)^2/(q`2)^2 by A44,XREAL_1:72;
then (q`1/q`2)^2<=(q`2)^2/(q`2)^2 by XCMPLX_1:76;
then (q`1/q`2)^2 <=1 by A44,XCMPLX_1:60;
then
A49: 1+(q`1/q`2)^2<=1+1 by XREAL_1:7;
then
A50: (q`2)^2*(1+(q`1/q`2)^2)<=(q`2)^2*2 by A25,XREAL_1:64;
1+(q`1/q`2)^2>0 by Lm1;
then
A51: (sqrt(1+(q`1/q`2)^2))^2=1+(q`1/q`2)^2 by SQUARE_1:def 2;
A52: (q`1)^2*(1+(q`1/q`2)^2)<=(q`1)^2*2 by A24,A49,XREAL_1:64;
A53: (Sq_Circ").q=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2 ) ^2)
]| by A43,Th28;
then qz`1=q`1*sqrt(1+(q`1/q`2)^2) by A23,EUCLID:52;
then
A54: (qz`1)^2<=(q`1)^2*2 by A52,A51,SQUARE_1:9;
qz`2=q`2*sqrt(1+(q`1/q`2)^2) by A23,A53,EUCLID:52;
then (qz`2)^2<=(q`2)^2*2 by A50,A51,SQUARE_1:9;
then
A55: (qz`2)^2+(qz`1)^2<=(q`2)^2*2+(q`1)^2*2 by A54,XREAL_1:7;
(qz`2)^2>=0 & (qz`1)^2>=0 by XREAL_1:63;
then
A56: sqrt((qz`2)^2+(qz`1)^2) <= sqrt((q`2)^2*2+(q`1)^2*2) by A55,SQUARE_1:26
;
A57: ((0.TOP-REAL 2) - qz)`2=(0.TOP-REAL 2)`2-qz`2 by TOPREAL3:3
.= -qz`2 by JGRAPH_2:3;
((0.TOP-REAL 2) - qz)`1=(0.TOP-REAL 2)`1-qz`1 by TOPREAL3:3
.= -qz`1 by JGRAPH_2:3;
then sqrt((((0.TOP-REAL 2) - qz)`2)^2+(((0.TOP-REAL 2) - qz)`1)^2)<r
by A26,A57,A56,XXREAL_0:2;
then |.(0.TOP-REAL 2) - qz.|<r by JGRAPH_1:30;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
end;
A58: V1 is open by GOBOARD6:3;
sqrt(2)>0 by SQUARE_1:25;
then u0 in V1 by A17,GOBOARD6:1,XREAL_1:139;
hence thesis by A18,A58,A19,XBOOLE_1:1;
end;
A59: D`= {0.TOP-REAL 2} by Th20;
then ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=(Sq_Circ")|D
& h is continuous by Th41;
hence thesis by A1,A59,A2,A14,Th3;
end;
theorem Th43:
Sq_Circ is Function of TOP-REAL 2,TOP-REAL 2 & rng Sq_Circ = the
carrier of TOP-REAL 2 & for f being Function of TOP-REAL 2,TOP-REAL 2 st f=
Sq_Circ holds f is being_homeomorphism
proof
thus Sq_Circ is Function of TOP-REAL 2,TOP-REAL 2;
A1: for f being Function of TOP-REAL 2,TOP-REAL 2 st f=Sq_Circ holds rng
Sq_Circ=the carrier of TOP-REAL 2 & f is being_homeomorphism
proof
let f be Function of TOP-REAL 2,TOP-REAL 2;
assume
A2: f=Sq_Circ;
reconsider g=f/" as Function of TOP-REAL 2,TOP-REAL 2;
A3: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
the carrier of TOP-REAL 2 c= rng f
proof
let y be object;
assume y in the carrier of TOP-REAL 2;
then reconsider p2=y as Point of TOP-REAL 2;
set q=p2;
now
per cases;
case
q=0.TOP-REAL 2;
then y=Sq_Circ.q by Def1;
hence ex x being set st x in dom Sq_Circ & y=Sq_Circ.x by A2,A3;
end;
case
A4: q<>0.TOP-REAL 2 & (q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2<=- q`1);
set px=|[q`1*sqrt(1+(q`2/q`1)^2),q`2*sqrt(1+(q`2/q`1)^2)]|;
A5: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A6: now
assume that
A7: px`1=0 and
A8: px`2=0;
q`2*sqrt(1+(q`2/q`1)^2 )=0 by A8,EUCLID:52;
then
A9: q`2=0 by A5,XCMPLX_1:6;
q`1*sqrt(1+(q`2/q`1)^2)=0 by A7,EUCLID:52;
then q`1=0 by A5,XCMPLX_1:6;
hence contradiction by A4,A9,EUCLID:53,54;
end;
A10: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A11: px`1 = q`1*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
A12: px`2 = q`2*sqrt(1+(q`2/q`1)^2) by EUCLID:52;
then
A13: px`2/px`1=q`2/q`1 by A11,A5,XCMPLX_1:91;
then
A14: px`2/sqrt(1+(px`2/px`1)^2)=q`2 by A12,A5,XCMPLX_1:89;
q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q`2*sqrt(1+(q`2/q`1)^2) <= (
-q`1)*sqrt(1+(q`2/q`1)^2) by A4,A5,XREAL_1:64;
then
q`2<=q`1 & (-q`1)*sqrt(1+(q`2/q`1)^2) <= q`2*sqrt(1+(q`2/q`1)^2
) or px`2>=px`1 & px`2<=-px`1 by A11,A12,A5,XREAL_1:64;
then q`2*sqrt(1+(q`2/q`1)^2) <= q`1*sqrt(1+(q`2/q`1)^2) & -px`1<=px
`2 or px`2>=px`1 & px`2<=-px`1 by A11,A5,EUCLID:52,XREAL_1:64;
then
A15: Sq_Circ.px=|[px`1/sqrt(1+(px`2/px`1)^2),px`2/sqrt(1+( px`2/px`1
)^2) ]| by A11,A12,A6,Def1,JGRAPH_2:3;
px`1/sqrt(1+(px`2/px`1)^2)=q`1 by A11,A5,A13,XCMPLX_1:89;
hence ex x being set st x in dom Sq_Circ & y=Sq_Circ.x by A15,A14,A10
,EUCLID:53;
end;
case
A16: q<>0.TOP-REAL 2 & not(q`2<=q`1 & -q`1<=q`2 or q`2>=q`1 & q `2<=-q`1);
set px=|[q`1*sqrt(1+(q`1/q`2)^2),q`2*sqrt(1+(q`1/q`2)^2)]|;
A17: sqrt(1+(q`1/q`2)^2)>0 by Lm1,SQUARE_1:25;
A18: now
assume that
A19: px`2=0 and
px`1=0;
q`2*sqrt(1+(q`1/q`2)^2)=0 by A19,EUCLID:52;
then q`2=0 by A17,XCMPLX_1:6;
hence contradiction by A16;
end;
A20: px`2 = q`2*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
A21: px`1 = q`1*sqrt(1+(q`1/q`2)^2) by EUCLID:52;
then
A22: px`1/px`2=q`1/q`2 by A20,A17,XCMPLX_1:91;
then
A23: px`1/sqrt(1+(px`1/px`2)^2)=q`1 by A21,A17,XCMPLX_1:89;
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1<=-q`2 by A16,JGRAPH_2:13;
then
q`1<=q`2 & -q`2<=q`1 or q`1>=q`2 & q`1*sqrt(1+(q`1/q`2)^2) <= (
-q`2)*sqrt(1+(q`1/q`2)^2) by A17,XREAL_1:64;
then
q`1<=q`2 & (-q`2)*sqrt(1+(q`1/q`2)^2) <= q`1*sqrt(1+(q`1/q`2)^2
) or px`1>=px`2 & px`1<=-px`2 by A20,A21,A17,XREAL_1:64;
then q`1*sqrt(1+(q`1/q`2)^2) <= q`2*sqrt(1+(q`1/q`2)^2) & -px`2<=px
`1 or px`1>=px`2 & px`1<=-px`2 by A20,A17,EUCLID:52,XREAL_1:64;
then
A24: Sq_Circ.px=|[px`1/sqrt(1+(px`1/px`2)^2),px`2/sqrt(1+( px`1/px`2
)^2 )]| by A20,A21,A18,Th4,JGRAPH_2:3;
A25: dom Sq_Circ=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
px`2/sqrt(1+(px`1/px`2)^2)=q`2 by A20,A17,A22,XCMPLX_1:89;
hence ex x being set st x in dom Sq_Circ & y=Sq_Circ.x by A24,A23,A25
,EUCLID:53;
end;
end;
hence thesis by A2,FUNCT_1:def 3;
end;
then
rng f=the carrier of TOP-REAL 2;
then
A26: f is onto by FUNCT_2:def 3;
A27: rng f=dom ((f qua Function)") by A2,FUNCT_1:33
.=dom (f/") by A2,A26,TOPS_2:def 4
.=[#](TOP-REAL 2) by FUNCT_2:def 1;
g=Sq_Circ" by A26,A2,TOPS_2:def 4;
hence thesis by A2,A3,A27,Th21,Th42,TOPS_2:def 5;
end;
hence rng Sq_Circ=the carrier of TOP-REAL 2;
thus thesis by A1;
end;
Lm19: now
let pz2, pz1 be Real;
assume ((pz2)^2+(pz1)^2-1)*(pz2)^2<=(pz1)^2;
then (pz2)^2*(pz2)^2+(pz2)^2*((pz1)^2-1)-(pz1)^2 <=(pz1)^2-(pz1)^2 by
XREAL_1:9;
hence ((pz2)^2-1)*((pz2)^2+(pz1)^2)<=0;
end;
Lm20: now
let px1 be Real;
assume (px1)^2-1=0;
then (px1-1)*(px1+1)=0;
then px1-1=0 or px1+1=0 by XCMPLX_1:6;
hence px1=1 or px1=-1;
end;
theorem
for f,g being Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN being
Subset of TOP-REAL 2, O,I being Point of I[01] st O=0 & I=1 & f is continuous
one-to-one & g is continuous one-to-one & C0={p: |.p.|<=1}& KXP={q1 where q1 is
Point of TOP-REAL 2: |.q1.|=1 & q1`2<=q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is
Point of TOP-REAL 2: |.q2.|=1 & q2`2>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is
Point of TOP-REAL 2: |.q3.|=1 & q3`2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is
Point of TOP-REAL 2: |.q4.|=1 & q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in
KXP & g.O in KYN & g.I in KYP & rng f c= C0 & rng g c= C0 holds rng f meets rng
g
proof
A1: dom (Sq_Circ")=the carrier of TOP-REAL 2 by Th29,FUNCT_2:def 1;
let f,g be Function of I[01],TOP-REAL 2, C0,KXP,KXN,KYP,KYN be Subset of
TOP-REAL 2, O,I be Point of I[01];
assume
A2: O=0 & I=1 & f is continuous one-to-one & g is continuous one-to-one
& C0={p: |.p.|<=1}& KXP={q1 where q1 is Point of TOP-REAL 2: |.q1.|=1 & q1`2<=
q1`1 & q1`2>=-q1`1} & KXN={q2 where q2 is Point of TOP-REAL 2: |.q2.|=1 & q2`2
>=q2`1 & q2`2<=-q2`1} & KYP={q3 where q3 is Point of TOP-REAL 2: |.q3.|=1 & q3
`2>=q3`1 & q3`2>=-q3`1} & KYN={q4 where q4 is Point of TOP-REAL 2: |.q4.|=1 &
q4`2<=q4`1 & q4`2<=-q4`1} & f.O in KXN & f.I in KXP & g.O in KYN & g.I in KYP &
rng f c= C0 & rng g c= C0;
then consider p1 being Point of TOP-REAL 2 such that
A3: f.O=p1 and
A4: |.p1.|=1 and
A5: p1`2>=p1`1 and
A6: p1`2<=-p1`1;
reconsider gg=Sq_Circ"*g as Function of I[01],TOP-REAL 2 by Th29,FUNCT_2:13;
A7: dom g=the carrier of I[01] by FUNCT_2:def 1;
reconsider ff=Sq_Circ"*f as Function of I[01],TOP-REAL 2 by Th29,FUNCT_2:13;
A8: dom gg=the carrier of I[01] by FUNCT_2:def 1;
A9: dom ff=the carrier of I[01] by FUNCT_2:def 1;
then
A10: (ff.O)=(Sq_Circ").(f.O) by FUNCT_1:12;
A11: dom f=the carrier of I[01] by FUNCT_2:def 1;
A12: for r being Point of I[01] holds -1<=(ff.r)`1 & (ff.r)`1<=1 & -1<=(gg.
r)`1 & (gg.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1
proof
let r be Point of I[01];
f.r in rng f by A11,FUNCT_1:3;
then f.r in C0 by A2;
then consider p1 being Point of TOP-REAL 2 such that
A13: f.r=p1 and
A14: |.p1.|<=1 by A2;
g.r in rng g by A7,FUNCT_1:3;
then g.r in C0 by A2;
then consider p2 being Point of TOP-REAL 2 such that
A15: g.r=p2 and
A16: |.p2.|<=1 by A2;
A17: (gg.r)=(Sq_Circ").(g.r) by A8,FUNCT_1:12;
A18: now
per cases;
case
p2=0.TOP-REAL 2;
hence
-1<=(gg.r)`1 & (gg.r)`1<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1 by A17,A15
,Th28,JGRAPH_2:3;
end;
case
A19: p2<>0.TOP-REAL 2 & (p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 &
p2`2<=-p2`1);
set px=gg.r;
A20: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2 /p2`1
)^2 )]| by A19,Th28;
then
A21: px`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A17,A15,EUCLID:52;
(|.p2.|)^2<=|.p2.| by A16,SQUARE_1:42;
then
A22: (|.p2.|)^2<=1 by A16,XXREAL_0:2;
A23: (px`2)^2>=0 by XREAL_1:63;
A24: (px`1)^2 >=0 by XREAL_1:63;
A25: px`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A17,A15,A20,EUCLID:52;
A26: sqrt(1+(p2`2/p2`1)^2)>0 by Lm1,SQUARE_1:25;
then p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2`1 & p2`2*sqrt(1+(p2`2/p2`1)
^2) <= (-p2`1)*sqrt(1+(p2`2/p2`1)^2) by A19,XREAL_1:64;
then
A27: p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/
p2`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A21,A25,A26,XREAL_1:64;
then
A28: p2`2*sqrt( 1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -px
`1<= px `2 or px`2>=px`1 & px`2<=-px`1 by A17,A15,A20,A21,A26,EUCLID:52
,XREAL_1:64;
A29: now
assume px`1=0 & px`2=0;
then p2`1=0 & p2`2=0 by A21,A25,A26,XCMPLX_1:6;
hence contradiction by A19,EUCLID:53,54;
end;
then
A30: px`1<>0 by A21,A25,A26,A27,XREAL_1:64;
set q=px;
A31: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/
sqrt(1+( q`2/q`1)^2) by EUCLID:52;
A32: 1+(q`2/q`1)^2>0 by Lm1;
A33: p2=Sq_Circ.px & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)
^2)]|)`1 = q `1/sqrt(1+( q`2/q`1)^2) by A17,A15,Th43,EUCLID:52,FUNCT_1:32;
Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A21
,A25,A29,A28,Def1,JGRAPH_2:3;
then (|.p2.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)
) ^2 by A33,A31,JGRAPH_1:29
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A32,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A32,
SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)<=1 *(1+(q`2/q
`1 )^2) by A32,A22,XREAL_1:64;
then ((q`1)^2+(q`2)^2)<=(1+(q`2/q`1)^2) by A32,XCMPLX_1:87;
then (px`1)^2+(px`2)^2<=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
then (px`1)^2+(px`2)^2-1<=(px`2)^2/(px`1)^2 by XREAL_1:20;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2/(px`1)^2*(px`1)^2 by A24,
XREAL_1:64;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2 by A30,XCMPLX_1:6,87;
then
A34: ((px`1)^2-1)*((px`1)^2+(px`2)^2)<=0 by Lm19;
((px`1)^2+(px`2)^2)<>0 by A29,COMPLEX1:1;
then
A35: ((px`1)^2-1)<=0 by A24,A34,A23,XREAL_1:129;
then
A36: px`1>=-1 by SQUARE_1:43;
A37: px`1<=1 by A35,SQUARE_1:43;
then q`2<=1 & --q`1>=-q`2 or q`2>=-1 & q`2<=-q`1 by A21,A25,A28,A36,
XREAL_1:24,XXREAL_0:2;
then q`2<=1 & q`1>=-q`2 or q`2>=-1 & -q`2>=--q`1 by XREAL_1:24;
then q`2<=1 & 1>=-q`2 or q`2>=-1 & -q`2>=q`1 by A37,XXREAL_0:2;
then q`2<=1 & -1<=--q`2 or q`2>=-1 & -q`2>=-1 by A36,XREAL_1:24
,XXREAL_0:2;
hence
-1<=(gg.r)`1 & (gg.r)`1<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1 by A35,
SQUARE_1:43,XREAL_1:24;
end;
case
A38: p2<>0.TOP-REAL 2 & not(p2`2<=p2`1 & -p2`1<=p2`2 or p2`2>=p2
`1 & p2`2<=-p2`1);
set pz=gg.r;
A39: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`1/p2`2)^2),p2`2*sqrt(1+(p2`1 /p2`2
)^2 )]| by A38,Th28;
then
A40: pz`2 = p2`2*sqrt(1+(p2`1/p2`2)^2) by A17,A15,EUCLID:52;
A41: pz`1 = p2`1*sqrt(1+(p2`1/p2`2)^2) by A17,A15,A39,EUCLID:52;
A42: sqrt(1+(p2`1/p2`2)^2)>0 by Lm1,SQUARE_1:25;
p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1<=-p2`2 by A38,JGRAPH_2:13
;
then p2`1<=p2`2 & -p2`2<=p2`1 or p2`1>=p2`2 & p2`1*sqrt(1+(p2`1/p2`2)
^2) <= (-p2`2)*sqrt(1+(p2`1/p2`2)^2) by A42,XREAL_1:64;
then
A43: p2`1<=p2`2 & (-p2`2)*sqrt(1+(p2`1/p2`2)^2) <= p2`1*sqrt(1+(p2`1/
p2`2) ^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A40,A41,A42,XREAL_1:64;
then
A44: p2`1*sqrt(1+(p2`1/p2`2)^2) <= p2`2*sqrt(1+(p2`1/p2`2)^2) & -pz
`2<=pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A17,A15,A39,A40,A42,EUCLID:52
,XREAL_1:64;
A45: now
assume that
A46: pz`2=0 and
pz`1=0;
p2`2=0 by A40,A42,A46,XCMPLX_1:6;
hence contradiction by A38;
end;
then
A47: pz`2<>0 by A40,A41,A42,A43,XREAL_1:64;
A48: p2=Sq_Circ.pz & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/
pz`2)^2)]|) `2 = pz`2/ sqrt(1+(pz`1/pz`2)^2) by A17,A15,Th43,EUCLID:52
,FUNCT_1:32;
A49: (pz`2)^2 >=0 by XREAL_1:63;
(|.p2.|)^2<=|.p2.| by A16,SQUARE_1:42;
then
A50: (|.p2.|)^2<=1 by A16,XXREAL_0:2;
A51: (pz`1)^2>=0 by XREAL_1:63;
A52: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 =
pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A53: 1+(pz`1/pz`2)^2>0 by Lm1;
Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)
^2)]| by A40,A41,A45,A44,Th4,JGRAPH_2:3;
then (|.p2.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz
`2)^2))^2 by A48,A52,JGRAPH_1:29
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))
^2 by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2
)^2))^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A53,SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A53,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2) <=1 *(1+
(pz`1/pz`2)^2) by A53,A50,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2)<=(1+(pz`1/pz`2)^2) by A53,XCMPLX_1:87;
then (pz`2)^2+(pz`1)^2<=1+(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
then (pz`2)^2+(pz`1)^2-1<=(pz`1)^2/(pz`2)^2 by XREAL_1:20;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2/(pz`2)^2*(pz`2)^2 by A49,
XREAL_1:64;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2 by A47,XCMPLX_1:6,87;
then
A54: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)<=0 by Lm19;
(pz`2)^2+(pz`1)^2<>0 by A45,COMPLEX1:1;
then
A55: (pz`2)^2-1<=0 by A49,A54,A51,XREAL_1:129;
then
A56: pz`2>=-1 by SQUARE_1:43;
A57: pz`2<=1 by A55,SQUARE_1:43;
then pz`1<=1 & --pz`2>=-pz`1 or pz`1>=-1 & pz`1<=-pz`2 by A40,A41,A44
,A56,XREAL_1:24,XXREAL_0:2;
then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=--pz`2 by A57,XREAL_1:24
,XXREAL_0:2;
then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=-1 by A56,XXREAL_0:2;
then pz`1<=1 & -1<=--pz`1 or pz`1>=-1 & pz`1<=1 by XREAL_1:24;
hence
-1<=(gg.r)`1 & (gg.r)`1<=1 & -1 <=(gg.r)`2 & (gg.r)`2<=1 by A55,
SQUARE_1:43;
end;
end;
A58: (ff.r)=(Sq_Circ").(f.r) by A9,FUNCT_1:12;
now
per cases;
case
p1=0.TOP-REAL 2;
hence
-1<=(ff.r)`1 & (ff.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 by A58,A13
,Th28,JGRAPH_2:3;
end;
case
A59: p1<>0.TOP-REAL 2 & (p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 &
p1`2<=-p1`1);
set px=ff.r;
Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2 /p1`1
)^2 ) ]| by A59,Th28;
then
A60: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) & px`2 = p1`2*sqrt(1+(p1`2/p1
`1)^2) by A58,A13,EUCLID:52;
A61: sqrt(1+(p1`2/p1`1)^2)>0 by Lm1,SQUARE_1:25;
then p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)
^2) <= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A59,XREAL_1:64;
then
A62: p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+ (p1`2
/p1`1 )^2) or px`2>=px`1 & px`2<=-px`1 by A60,A61,XREAL_1:64;
then
A63: px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A60,A61,
XREAL_1:64;
A64: now
assume px`1=0 & px`2=0;
then p1`1=0 & p1`2=0 by A60,A61,XCMPLX_1:6;
hence contradiction by A59,EUCLID:53,54;
end;
then
A65: px`1<>0 by A60,A61,A62,XREAL_1:64;
(|.p1.|)^2<=|.p1.| by A14,SQUARE_1:42;
then
A66: (|.p1.|)^2<=1 by A14,XXREAL_0:2;
A67: (px`1)^2 >=0 by XREAL_1:63;
A68: (px`2)^2>=0 by XREAL_1:63;
set q=px;
A69: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/
sqrt(1+( q`2/q`1)^2) by EUCLID:52;
A70: 1+(q`2/q`1)^2>0 by Lm1;
A71: p1=Sq_Circ.px & (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)
^2)]|)`1 = q `1/sqrt(1+( q`2/q`1)^2) by A58,A13,Th43,EUCLID:52,FUNCT_1:32;
Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A64
,A63,Def1,JGRAPH_2:3;
then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)
) ^2 by A71,A69,JGRAPH_1:29
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2
by XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A70,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A70,
SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)<=1 *(1+(q`2/q
`1 )^2) by A70,A66,XREAL_1:64;
then (q`1)^2+(q`2)^2<=1+(q`2/q`1)^2 by A70,XCMPLX_1:87;
then (px`1)^2+(px`2)^2<=1+(px`2)^2/(px`1)^2 by XCMPLX_1:76;
then (px`1)^2+(px`2)^2-1<=(px`2)^2/(px`1)^2 by XREAL_1:20;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2/(px`1)^2*(px`1)^2 by A67,
XREAL_1:64;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2<=(px`2)^2 by A65,XCMPLX_1:6,87;
then
A72: ((px`1)^2-1)*((px`1)^2+(px`2)^2)<=0 by Lm19;
((px`1)^2+(px`2)^2)<>0 by A64,COMPLEX1:1;
then
A73: (px`1)^2-1<=0 by A67,A72,A68,XREAL_1:129;
then
A74: px`1>=-1 by SQUARE_1:43;
A75: px`1<=1 by A73,SQUARE_1:43;
then q`2<=1 & --q`1>=-q`2 or q`2>=-1 & q`2<=-q`1 by A63,A74,XREAL_1:24
,XXREAL_0:2;
then q`2<=1 & q`1>=-q`2 or q`2>=-1 & -q`2>=--q`1 by XREAL_1:24;
then q`2<=1 & 1>=-q`2 or q`2>=-1 & -q`2>=q`1 by A75,XXREAL_0:2;
then q`2<=1 & -1<=--q`2 or q`2>=-1 & -q`2>=-1 by A74,XREAL_1:24
,XXREAL_0:2;
hence
-1<=(ff.r)`1 & (ff.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 by A73,
SQUARE_1:43,XREAL_1:24;
end;
case
A76: p1<>0.TOP-REAL 2 & not(p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1
`1 & p1`2<=-p1`1);
set pz=ff.r;
A77: Sq_Circ".p1=|[p1`1*sqrt(1+(p1`1/p1`2)^2),p1`2*sqrt(1+(p1`1 /p1`2
) ^2)]| by A76,Th28;
then
A78: pz`2 = p1`2*sqrt(1+(p1`1/p1`2)^2) by A58,A13,EUCLID:52;
A79: pz`1 = p1`1*sqrt(1+(p1`1/p1`2)^2) by A58,A13,A77,EUCLID:52;
A80: sqrt(1+(p1`1/p1`2)^2)>0 by Lm1,SQUARE_1:25;
p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1<=-p1`2 by A76,JGRAPH_2:13
;
then p1`1<=p1`2 & -p1`2<=p1`1 or p1`1>=p1`2 & p1`1*sqrt(1+(p1`1/p1`2)
^2) <= (-p1`2)*sqrt(1+(p1`1/p1`2)^2) by A80,XREAL_1:64;
then
A81: p1`1<=p1`2 & (-p1`2)*sqrt(1+(p1`1/p1`2)^2) <= p1`1*sqrt(1+(p1`1/
p1`2 )^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A78,A79,A80,XREAL_1:64;
then
A82: p1`1*sqrt(1+(p1`1/p1`2)^2) <= p1`2*sqrt(1+(p1`1/p1`2)^2) & -pz
`2<=pz `1 or pz`1>=pz`2 & pz`1<=-pz`2 by A58,A13,A77,A78,A80,EUCLID:52
,XREAL_1:64;
A83: now
assume that
A84: pz`2=0 and
pz`1=0;
p1`2=0 by A78,A80,A84,XCMPLX_1:6;
hence contradiction by A76;
end;
then
A85: pz`2<>0 by A78,A79,A80,A81,XREAL_1:64;
A86: p1=Sq_Circ.pz & (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/
pz`2)^2)]|) `2 = pz`2/ sqrt(1+(pz`1/pz`2)^2) by A58,A13,Th43,EUCLID:52
,FUNCT_1:32;
A87: (pz`2)^2 >=0 by XREAL_1:63;
(|.p1.|)^2<=|.p1.| by A14,SQUARE_1:42;
then
A88: (|.p1.|)^2<=1 by A14,XXREAL_0:2;
A89: (pz`1)^2>=0 by XREAL_1:63;
A90: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 =
pz`1/ sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A91: 1+(pz`1/pz`2)^2>0 by Lm1;
Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)
^2)]| by A78,A79,A83,A82,Th4,JGRAPH_2:3;
then (|.p1.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz
`2)^2))^2 by A86,A90,JGRAPH_1:29
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))
^2 by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2
)^2))^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2
by A91,SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A91,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2) <=1 *(1+
(pz`1/pz`2)^2) by A91,A88,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2)<=(1+(pz`1/pz`2)^2) by A91,XCMPLX_1:87;
then (pz`2)^2+(pz`1)^2<=1+(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
then (pz`2)^2+(pz`1)^2-1<=(pz`1)^2/(pz`2)^2 by XREAL_1:20;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2/(pz`2)^2*(pz`2)^2 by A87,
XREAL_1:64;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2<=(pz`1)^2 by A85,XCMPLX_1:6,87;
then
A92: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)<=0 by Lm19;
((pz`2)^2+(pz`1)^2)<>0 by A83,COMPLEX1:1;
then
A93: (pz`2)^2-1<=0 by A87,A92,A89,XREAL_1:129;
then
A94: pz`2>=-1 by SQUARE_1:43;
A95: pz`2<=1 by A93,SQUARE_1:43;
then pz`1<=1 & --pz`2>=-pz`1 or pz`1>=-1 & pz`1<=-pz`2 by A78,A79,A82
,A94,XREAL_1:24,XXREAL_0:2;
then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=--pz`2 by A95,XREAL_1:24
,XXREAL_0:2;
then pz`1<=1 & 1>=-pz`1 or pz`1>=-1 & -pz`1>=-1 by A94,XXREAL_0:2;
then pz`1<=1 & -1<=--pz`1 or pz`1>=-1 & pz`1<=1 by XREAL_1:24;
hence
-1<=(ff.r)`1 & (ff.r)`1<=1 & -1 <=(ff.r)`2 & (ff.r)`2<=1 by A93,
SQUARE_1:43;
end;
end;
hence thesis by A18;
end;
set y = the Element of rng ff /\ rng gg;
A96: p1<>0.TOP-REAL 2 by A4,TOPRNS_1:23;
then
A97: Sq_Circ".p1=|[p1`1*sqrt(1+(p1`2/p1`1)^2),p1`2*sqrt(1+(p1`2/p1`1)^2 )]|
by A5,A6,Th28;
(ff.O)`1=-1 & (ff.I)`1=1 & (gg.O)`2=-1 & (gg.I)`2=1
proof
set pz=gg.O;
set py=ff.I;
set px=ff.O;
set q=px;
A98: (|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`1 = q`1/sqrt(1+
(q `2/q`1)^2) by EUCLID:52;
set pu=gg.I;
A99: (|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`1 = py`1/
sqrt(1+(py`2/py`1)^2) by EUCLID:52;
A100: (|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`2 = pu`2
/ sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
A101: 1+(pu`1/pu`2)^2>0 by Lm1;
Sq_Circ".p1=q by A9,A3,FUNCT_1:12;
then
A102: p1=Sq_Circ.px by Th43,FUNCT_1:32;
consider p4 being Point of TOP-REAL 2 such that
A103: g.I=p4 and
A104: |.p4.|=1 and
A105: p4`2>=p4`1 and
A106: p4`2>=-p4`1 by A2;
A107: sqrt(1+(p4`1/p4`2)^2)>0 by Lm1,SQUARE_1:25;
A108: -p4`2<=--p4`1 by A106,XREAL_1:24;
then
A109: p4`1<=p4`2 & (-p4`2)*sqrt(1+(p4`1/p4`2)^2) <= p4`1*sqrt(1+(p4`1/p4`2
) ^2 ) or pu`1>=pu`2 & pu`1<=-pu`2 by A105,A107,XREAL_1:64;
A110: (gg.I)=(Sq_Circ").(g.I) by A8,FUNCT_1:12;
then
A111: p4=Sq_Circ.pu by A103,Th43,FUNCT_1:32;
A112: p4<>0.TOP-REAL 2 by A104,TOPRNS_1:23;
then
A113: Sq_Circ".p4=|[p4`1*sqrt(1+(p4`1/p4`2)^2),p4`2*sqrt(1+(p4`1/p4`2) ^2
)]| by A105,A108,Th30;
then
A114: pu`2 = p4`2*sqrt(1+(p4`1/p4`2)^2) by A110,A103,EUCLID:52;
A115: pu`1 = p4`1*sqrt(1+(p4`1/p4`2)^2) by A110,A103,A113,EUCLID:52;
A116: now
assume pu`2=0 & pu`1=0;
then p4`2=0 & p4`1=0 by A114,A115,A107,XCMPLX_1:6;
hence contradiction by A112,EUCLID:53,54;
end;
p4`1* sqrt(1+(p4`1/p4`2)^2) <= p4`2*sqrt(1+(p4`1/p4`2)^2) & -pu`2<=
pu `1 or pu`1>=pu`2 & pu`1<=-pu`2 by A110,A103,A113,A114,A107,A109,EUCLID:52
,XREAL_1:64;
then
A117: Sq_Circ.pu=|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|
by A114,A115,A116,Th4,JGRAPH_2:3;
(|[pu`1/sqrt(1+(pu`1/pu`2)^2),pu`2/sqrt(1+(pu`1/pu`2)^2)]|)`1 = pu`1
/ sqrt(1+(pu`1/pu`2)^2) by EUCLID:52;
then (|.p4.|)^2= (pu`2/sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)
^2 ) ) ^2 by A111,A117,A100,JGRAPH_1:29
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2+(pu`1/sqrt(1+(pu`1/pu`2)^2))^2
by XCMPLX_1:76
.= (pu`2)^2/(sqrt(1+(pu`1/pu`2)^2))^2 +(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2)
)^2 by XCMPLX_1:76
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(sqrt(1+(pu`1/pu`2)^2))^2 by A101,
SQUARE_1:def 2
.= (pu`2)^2/(1+(pu`1/pu`2)^2)+(pu`1)^2/(1+(pu`1/pu`2)^2) by A101,
SQUARE_1:def 2
.= ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2) by XCMPLX_1:62;
then ((pu`2)^2+(pu`1)^2)/(1+(pu`1/pu`2)^2)*(1+(pu`1/pu`2)^2)=1 *(1+(pu`1/
pu `2)^2) by A104;
then ((pu`2)^2+(pu`1)^2)=(1+(pu`1/pu`2)^2) by A101,XCMPLX_1:87;
then
A118: (pu`2)^2+(pu`1)^2-1=(pu`1)^2/(pu`2)^2 by XCMPLX_1:76;
pu`2<>0 by A114,A115,A107,A116,A109,XREAL_1:64;
then ((pu`2)^2+(pu`1)^2-1)*(pu`2)^2=(pu`1)^2 by A118,XCMPLX_1:6,87;
then
A119: ((pu`2)^2-1)*((pu`2)^2+(pu`1)^2)=0;
((pu`2)^2+(pu`1)^2)<>0 by A116,COMPLEX1:1;
then
A120: ((pu`2)^2-1)=0 by A119,XCMPLX_1:6;
A121: sqrt(1+(p1`2/p1`1)^2)>0 by Lm1,SQUARE_1:25;
A122: sqrt(1+(pu`1/pu`2)^2)>0 by Lm1,SQUARE_1:25;
A123: now
assume
A124: pu`2=-1;
then -p4`1<0 by A106,A111,A117,A100,A122,XREAL_1:141;
then --p4`1>-0;
hence contradiction by A105,A111,A117,A122,A124,EUCLID:52;
end;
A125: 1+(pz`1/pz`2)^2>0 by Lm1;
A126: px`1 = p1`1*sqrt(1+(p1`2/p1`1)^2) & px`2 = p1`2*sqrt(1+(p1`2/p1`1)^2)
by A10,A3,A97,EUCLID:52;
A127: now
assume px`1=0 & px`2=0;
then p1`1=0 & p1`2=0 by A126,A121,XCMPLX_1:6;
hence contradiction by A96,EUCLID:53,54;
end;
p1`2<=p1`1 & -p1`1<=p1`2 or p1`2>=p1`1 & p1`2*sqrt(1+(p1`2/p1`1)^2)
<= (-p1`1)*sqrt(1+(p1`2/p1`1)^2) by A5,A6,A121,XREAL_1:64;
then
A128: p1`2<=p1`1 & (-p1`1)*sqrt(1+(p1`2/p1`1)^2) <= p1`2*sqrt(1+(p1`2/ p1`1
)^2) or px`2>=px`1 & px`2<=-px`1 by A126,A121,XREAL_1:64;
then px`2<=px`1 & -px`1<=px`2 or px`2>=px`1 & px`2<=-px`1 by A126,A121,
XREAL_1:64;
then
A129: Sq_Circ.q=|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]| by A127
,Def1,JGRAPH_2:3;
A130: sqrt(1+(q`2/q`1)^2)>0 by Lm1,SQUARE_1:25;
A131: now
assume
A132: px`1=1;
-p1`2>=--p1`1 by A6,XREAL_1:24;
then -p1`2>0 by A102,A129,A98,A130,A132,XREAL_1:139;
then --p1`2<-0;
hence contradiction by A5,A102,A129,A130,A132,EUCLID:52;
end;
consider p2 being Point of TOP-REAL 2 such that
A133: f.I=p2 and
A134: |.p2.|=1 and
A135: p2`2<=p2`1 and
A136: p2`2>=-p2`1 by A2;
A137: (ff.I)=(Sq_Circ").(f.I) by A9,FUNCT_1:12;
then
A138: p2=Sq_Circ.py by A133,Th43,FUNCT_1:32;
A139: p2<>0.TOP-REAL 2 by A134,TOPRNS_1:23;
then
A140: Sq_Circ".p2=|[p2`1*sqrt(1+(p2`2/p2`1)^2),p2`2*sqrt(1+(p2`2/p2`1) ^2 )
]| by A135,A136,Th28;
then
A141: py`1 = p2`1*sqrt(1+(p2`2/p2`1)^2) by A137,A133,EUCLID:52;
A142: sqrt(1+(p2`2/p2`1)^2)>0 by Lm1,SQUARE_1:25;
A143: py`2 = p2`2*sqrt(1+(p2`2/p2`1)^2) by A137,A133,A140,EUCLID:52;
A144: now
assume py`1=0 & py`2=0;
then p2`1=0 & p2`2=0 by A141,A143,A142,XCMPLX_1:6;
hence contradiction by A139,EUCLID:53,54;
end;
A145: p2`2<=p2`1 & (-p2`1)*sqrt(1+(p2`2/p2`1)^2) <= p2`2*sqrt(1+(p2`2/p2`1
) ^2 ) or py`2>=py`1 & py`2<=-py`1 by A135,A136,A142,XREAL_1:64;
then p2`2*sqrt(1+(p2`2/p2`1)^2) <= p2`1*sqrt(1+(p2`2/p2`1)^2) & -py `1<=
py `2 or py`2>=py`1 & py`2<=-py`1 by A137,A133,A140,A141,A142,EUCLID:52
,XREAL_1:64;
then
A146: Sq_Circ.py=|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|
by A141,A143,A144,Def1,JGRAPH_2:3;
A147: sqrt(1+(py`2/py`1)^2)>0 by Lm1,SQUARE_1:25;
A148: now
assume
A149: py`1=-1;
-p2`2<=--p2`1 by A136,XREAL_1:24;
then -p2`2<0 by A138,A146,A99,A147,A149,XREAL_1:141;
then --p2`2>-0;
hence contradiction by A135,A138,A146,A147,A149,EUCLID:52;
end;
A150: 1+(py`2/py`1)^2>0 by Lm1;
(|[py`1/sqrt(1+(py`2/py`1)^2),py`2/sqrt(1+(py`2/py`1)^2)]|)`2 = py`2/
sqrt(1+(py`2/py`1)^2) by EUCLID:52;
then
(|.p2.|)^2= (py`1/sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2
) ) ^2 by A138,A146,A99,JGRAPH_1:29
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2+(py`2/sqrt(1+(py`2/py`1)^2))^2
by XCMPLX_1:76
.= (py`1)^2/(sqrt(1+(py`2/py`1)^2))^2 +(py`2)^2/(sqrt(1+(py`2/py`1)^2)
)^2 by XCMPLX_1:76
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(sqrt(1+(py`2/py`1)^2))^2 by A150,
SQUARE_1:def 2
.= (py`1)^2/(1+(py`2/py`1)^2)+(py`2)^2/(1+(py`2/py`1)^2) by A150,
SQUARE_1:def 2
.= ((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2) by XCMPLX_1:62;
then
((py`1)^2+(py`2)^2)/(1+(py`2/py`1)^2)*(1+(py`2/py`1)^2) =1 *(1+(py`2/
py`1)^2) by A134;
then ((py`1)^2+(py`2)^2)=(1+(py`2/py`1)^2) by A150,XCMPLX_1:87;
then
A151: (py`1)^2+(py`2)^2-1=(py`2)^2/(py`1)^2 by XCMPLX_1:76;
py`1<>0 by A141,A143,A142,A144,A145,XREAL_1:64;
then ((py`1)^2+(py`2)^2-1)*(py`1)^2=(py`2)^2 by A151,XCMPLX_1:6,87;
then
A152: ((py`1)^2-1)*((py`1)^2+(py`2)^2)=0;
((py`1)^2+(py`2)^2)<>0 by A144,COMPLEX1:1;
then
A153: ((py`1)^2-1)=0 by A152,XCMPLX_1:6;
A154: (|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`2 = pz`2/
sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
A155: 1+(q`2/q`1)^2>0 by Lm1;
(|[q`1/sqrt(1+(q`2/q`1)^2),q`2/sqrt(1+(q`2/q`1)^2)]|)`2 = q`2/sqrt(1+
(q `2/q`1)^2) by EUCLID:52;
then (|.p1.|)^2= (q`1/sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2)) ^2
by A102,A129,A98,JGRAPH_1:29
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2/sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(sqrt(1+(q`2/q`1)^2))^2+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by
XCMPLX_1:76
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(sqrt(1+(q`2/q`1)^2))^2 by A155,
SQUARE_1:def 2
.= (q`1)^2/(1+(q`2/q`1)^2)+(q`2)^2/(1+(q`2/q`1)^2) by A155,SQUARE_1:def 2
.= ((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2) by XCMPLX_1:62;
then
((q`1)^2+(q`2)^2)/(1+(q`2/q`1)^2)*(1+(q`2/q`1)^2)=1 *(1+(q`2/q`1) ^2)
by A4;
then (q`1)^2+(q`2)^2=1+(q`2/q`1)^2 by A155,XCMPLX_1:87;
then
A156: (px`1)^2+(px`2)^2-1=(px`2)^2/(px`1)^2 by XCMPLX_1:76;
px`1<>0 by A126,A121,A127,A128,XREAL_1:64;
then ((px`1)^2+(px`2)^2-1)*(px`1)^2=(px`2)^2 by A156,XCMPLX_1:6,87;
then
A157: ((px`1)^2-1)*((px`1)^2+(px`2)^2)=0;
consider p3 being Point of TOP-REAL 2 such that
A158: g.O=p3 and
A159: |.p3.|=1 and
A160: p3`2<=p3`1 and
A161: p3`2<=-p3`1 by A2;
A162: p3<>0.TOP-REAL 2 by A159,TOPRNS_1:23;
A163: (gg.O)=(Sq_Circ").(g.O) by A8,FUNCT_1:12;
then
A164: p3=Sq_Circ.pz by A158,Th43,FUNCT_1:32;
A165: -p3`2>=--p3`1 by A161,XREAL_1:24;
then
A166: Sq_Circ".p3=|[p3`1*sqrt(1+(p3`1/p3`2)^2),p3`2*sqrt(1+(p3`1/p3`2) ^2)
]| by A160,A162,Th30;
then
A167: pz`2 = p3`2*sqrt(1+(p3`1/p3`2)^2) by A163,A158,EUCLID:52;
A168: sqrt(1+(p3`1/p3`2)^2)>0 by Lm1,SQUARE_1:25;
A169: pz`1 = p3`1*sqrt(1+(p3`1/p3`2)^2) by A163,A158,A166,EUCLID:52;
A170: now
assume pz`2=0 & pz`1=0;
then p3`2=0 & p3`1=0 by A167,A169,A168,XCMPLX_1:6;
hence contradiction by A162,EUCLID:53,54;
end;
p3`1<=p3`2 & -p3`2<=p3`1 or p3`1>=p3`2 & p3`1*sqrt(1+(p3`1/p3`2)^2)
<= (-p3`2)*sqrt(1+(p3`1/p3`2)^2) by A160,A165,A168,XREAL_1:64;
then
A171: p3`1<=p3`2 & (-p3`2)*sqrt(1+(p3`1/p3`2)^2) <= p3`1*sqrt(1+(p3`1/p3 `2
)^2) or pz`1>=pz`2 & pz`1<=-pz`2 by A167,A169,A168,XREAL_1:64;
then
p3`1*sqrt(1+(p3`1/p3`2)^2) <= p3`2*sqrt(1+(p3`1/p3`2)^2) & -pz `2<=pz
`1 or pz`1>=pz`2 & pz`1<=-pz`2 by A163,A158,A166,A167,A168,EUCLID:52,XREAL_1:64
;
then
A172: Sq_Circ.pz=|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|
by A167,A169,A170,Th4,JGRAPH_2:3;
A173: sqrt(1+(pz`1/pz`2)^2)>0 by Lm1,SQUARE_1:25;
A174: now
assume
A175: pz`2=1;
then -p3`1>0 by A161,A164,A172,A154,A173,XREAL_1:139;
then --p3`1<-0;
hence contradiction by A160,A164,A172,A173,A175,EUCLID:52;
end;
(|[pz`1/sqrt(1+(pz`1/pz`2)^2),pz`2/sqrt(1+(pz`1/pz`2)^2)]|)`1 = pz`1/
sqrt(1+(pz`1/pz`2)^2) by EUCLID:52;
then (|.p3.|)^2= (pz`2/sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)
^2))^2 by A164,A172,A154,JGRAPH_1:29
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2+(pz`1/sqrt(1+(pz`1/pz`2)^2))^2
by XCMPLX_1:76
.= (pz`2)^2/(sqrt(1+(pz`1/pz`2)^2))^2 +(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2)
)^2 by XCMPLX_1:76
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(sqrt(1+(pz`1/pz`2)^2))^2 by A125,
SQUARE_1:def 2
.= (pz`2)^2/(1+(pz`1/pz`2)^2)+(pz`1)^2/(1+(pz`1/pz`2)^2) by A125,
SQUARE_1:def 2
.= ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2) by XCMPLX_1:62;
then ((pz`2)^2+(pz`1)^2)/(1+(pz`1/pz`2)^2)*(1+(pz`1/pz`2)^2)=1 *(1+(pz`1/
pz `2)^2) by A159;
then ((pz`2)^2+(pz`1)^2)=(1+(pz`1/pz`2)^2) by A125,XCMPLX_1:87;
then
A176: (pz`2)^2+(pz`1)^2-1=(pz`1)^2/(pz`2)^2 by XCMPLX_1:76;
pz`2<>0 by A167,A169,A168,A170,A171,XREAL_1:64;
then ((pz`2)^2+(pz`1)^2-1)*(pz`2)^2=(pz`1)^2 by A176,XCMPLX_1:6,87;
then
A177: ((pz`2)^2-1)*((pz`2)^2+(pz`1)^2)=0;
((pz`2)^2+(pz`1)^2)<>0 by A170,COMPLEX1:1;
then
A178: (pz`2)^2-1=0 by A177,XCMPLX_1:6;
((px`1)^2+(px`2)^2)<>0 by A127,COMPLEX1:1;
then ((px`1)^2-1)=0 by A157,XCMPLX_1:6;
hence thesis by A131,A153,A148,A178,A174,A120,A123,Lm20;
end;
then rng ff meets rng gg by A2,A12,Th42,JGRAPH_1:47;
then
A179: rng ff /\ rng gg <>{};
then y in rng ff by XBOOLE_0:def 4;
then consider x1 being object such that
A180: x1 in dom ff and
A181: y=ff.x1 by FUNCT_1:def 3;
x1 in dom f by A180,FUNCT_1:11;
then
A182: f.x1 in rng f by FUNCT_1:def 3;
y in rng gg by A179,XBOOLE_0:def 4;
then consider x2 being object such that
A183: x2 in dom gg and
A184: y=gg.x2 by FUNCT_1:def 3;
A185: gg.x2=Sq_Circ".(g.x2) by A183,FUNCT_1:12;
x2 in dom g by A183,FUNCT_1:11;
then
A186: g.x2 in rng g by FUNCT_1:def 3;
ff.x1=Sq_Circ".(f.x1) by A180,FUNCT_1:12;
then f.x1=g.x2 by A181,A184,A1,A182,A186,A185,FUNCT_1:def 4;
then rng f /\ rng g <> {} by A182,A186,XBOOLE_0:def 4;
hence thesis;
end;