:: Fan Homeomorphisms in the Plane
:: by Yatsuka Nakamura
::
:: Received January 8, 2002
:: Copyright (c) 2002-2026 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, PRE_TOPC, EUCLID, SUPINF_2, COMPLEX1, XXREAL_0, CARD_1,
XBOOLE_0, FUNCT_1, TOPMETR, SUBSET_1, ORDINAL2, PARTFUN1, RCOMP_1,
REAL_1, TARSKI, STRUCT_0, RELAT_1, ARYTM_3, TOPS_2, ARYTM_1, SQUARE_1,
NAT_1, METRIC_1, XXREAL_2, PCOMPS_1, MCART_1, FUNCT_4, JGRAPH_4, FUNCT_2;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0,
REAL_1, COMPLEX1, RELAT_1, FUNCT_1, RELSET_1, FUNCT_2, DOMAIN_1,
STRUCT_0, PARTFUN1, TOPMETR, RLVECT_1, EUCLID, COMPTS_1, TOPS_2,
METRIC_1, PCOMPS_1, SQUARE_1, TBSP_1, PSCOMP_1, PRE_TOPC, RLTOPSP1,
FUNCT_4, XXREAL_0;
constructors FUNCT_4, REAL_1, SQUARE_1, BINOP_2, COMPLEX1, TOPS_2, COMPTS_1,
TBSP_1, TOPMETR, PSCOMP_1, FUNCSDOM, PCOMPS_1;
registrations XBOOLE_0, NUMBERS, XXREAL_0, XREAL_0, SQUARE_1, MEMBERED,
STRUCT_0, PRE_TOPC, EUCLID, TOPMETR, TOPREAL1, PSCOMP_1, FUNCT_2,
COMPTS_1, TBSP_1, RELSET_1, FUNCT_1, JORDAN2C, ORDINAL1;
requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM;
definitions TARSKI;
equalities SQUARE_1, STRUCT_0, PCOMPS_1, ALGSTR_0;
expansions TARSKI;
theorems TARSKI, RELAT_1, SUBSET_1, FUNCT_1, FUNCT_2, FUNCT_4, TOPS_1, TOPS_2,
PARTFUN1, PRE_TOPC, TOPMETR, JORDAN6, EUCLID, JGRAPH_1, SQUARE_1,
PSCOMP_1, METRIC_1, JGRAPH_2, GOBOARD6, GOBOARD9, TBSP_1, TOPREAL6,
JGRAPH_3, ABSVALUE, JORDAN2C, TSEP_1, XREAL_0, XBOOLE_0, XBOOLE_1,
TOPRNS_1, XCMPLX_0, XCMPLX_1, COMPLEX1, XREAL_1, JORDAN2B, XXREAL_0,
RLVECT_1;
schemes FUNCT_2, DOMAIN_1, JGRAPH_2, BINOP_2, JGRAPH_3;
begin :: Preliminaries
reserve a for Real;
reserve p,q for Point of TOP-REAL 2;
Lm1: p <> 0.TOP-REAL 2 implies |.p.| > 0
proof
assume p <> 0.TOP-REAL 2;
then |.p.| <> 0 by TOPRNS_1:24;
hence thesis;
end;
theorem Th1:
for X being non empty TopStruct, g being Function of X,R^1, B
being Subset of X, a being Real st g is continuous & B = {p where p is
Point of X: g/.p > a } holds B is open
proof
let X be non empty TopStruct,g be Function of X,R^1,B be Subset of X, a be
Real;
assume that
A1: g is continuous and
A2: B={p where p is Point of X:g/.p > a };
{r where r is Real : r>a} c= the carrier of R^1
proof
let x be object;
assume x in {r where r is Real: r>a};
then consider r being Real such that
A3: r=x & r>a;
r in REAL by XREAL_0:def 1;
hence thesis by A3,TOPMETR:17;
end;
then reconsider D={r where r is Real: r>a} as Subset of R^1;
A4: g"D c= B
proof
let x be object;
assume
A5: x in g"D;
then reconsider p=x as Point of X;
g.x in D by A5,FUNCT_1:def 7;
then
A6: ex r being Real st r=g.x & r>a;
g/.p=g.p;
hence thesis by A2,A6;
end;
A7: [#]R^1 <> {} & D is open by JORDAN2B:25;
B c= g"D
proof
let x be object;
assume x in B;
then consider p being Point of X such that
A8: p=x and
A9: g/.p > a by A2;
dom g=the carrier of X & g.x in D by A8,A9,FUNCT_2:def 1;
hence thesis by A8,FUNCT_1:def 7;
end;
then B=g"D by A4,XBOOLE_0:def 10;
hence thesis by A1,A7,TOPS_2:43;
end;
theorem Th2:
for X being non empty TopStruct, g being Function of X,R^1,B
being Subset of X,
a being Real st g is continuous & B={p where p is Point of X
: g/.p < a } holds B is open
proof
let X be non empty TopStruct,g be Function of X,R^1, B be Subset of X,a be
Real;
assume that
A1: g is continuous and
A2: B={p where p is Point of X:g/.p < a };
{r where r is Real: r<a} c= the carrier of R^1
proof
let x be object;
assume x in {r where r is Real: r<a};
then consider r being Real such that
A3: r=x & r<a;
r in REAL by XREAL_0:def 1;
hence thesis by A3,TOPMETR:17;
end;
then reconsider D={r where r is Real: r<a} as Subset of R^1;
A4: g"D c= B
proof
let x be object;
assume
A5: x in g"D;
then reconsider p=x as Point of X;
g.x in D by A5,FUNCT_1:def 7;
then
A6: ex r being Real st r=g.x & r<a;
g/.p=g.p;
hence thesis by A2,A6;
end;
A7: [#]R^1 <> {} & D is open by JORDAN2B:24;
B c= g"D
proof
let x be object;
assume x in B;
then consider p being Point of X such that
A8: p=x and
A9: g/.p < a by A2;
dom g=the carrier of X & g.x in D by A8,A9,FUNCT_2:def 1;
hence thesis by A8,FUNCT_1:def 7;
end;
then B=g"D by A4,XBOOLE_0:def 10;
hence thesis by A1,A7,TOPS_2:43;
end;
theorem Th3:
for f being Function of TOP-REAL 2,TOP-REAL 2 st f is continuous
one-to-one & rng f=[#](TOP-REAL 2) & (for p2 being Point of TOP-REAL 2 ex K
being non empty compact Subset of TOP-REAL 2 st K = f.:K & (ex V2 being Subset
of TOP-REAL 2 st p2 in V2 & V2 is open & V2 c= K & f.p2 in V2)) holds f is
being_homeomorphism
proof
let f be Function of TOP-REAL 2,TOP-REAL 2;
assume that
A1: f is continuous one-to-one and
A2: rng f=[#](TOP-REAL 2) and
A3: for p2 being Point of TOP-REAL 2 ex K being non empty compact Subset
of TOP-REAL 2 st K = f.:K & ex V2 being Subset of TOP-REAL 2 st p2 in V2 & V2
is open & V2 c= K & f.p2 in V2;
reconsider g=(f qua Function)" as Function of TOP-REAL 2,TOP-REAL 2 by A1,A2,
FUNCT_2:25;
A4: dom f=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
for p being Point of TOP-REAL 2, V being Subset of TOP-REAL 2 st g.p in
V & V is open ex W being Subset of TOP-REAL 2 st p in W & W is open & g.:W c= V
proof
let p be Point of TOP-REAL 2, V be Subset of TOP-REAL 2;
assume that
A5: g.p in V and
A6: V is open;
consider K being non empty compact Subset of TOP-REAL 2 such that
A7: K = f.:K and
A8: ex V2 being Subset of TOP-REAL 2 st g.p in V2 & V2 is open & V2 c=
K & f.(g.p) in V2 by A3;
consider V2 being Subset of TOP-REAL 2 such that
A9: g.p in V2 and
A10: V2 is open and
A11: V2 c= K and
A12: f.(g.p) in V2 by A8;
A13: dom (f|K)=dom f /\ K by RELAT_1:61
.=K by A4,XBOOLE_1:28;
A14: g.p in V /\ V2 by A5,A9,XBOOLE_0:def 4;
the carrier of ((TOP-REAL 2)|K)=K by PRE_TOPC:8;
then reconsider R=V /\ V2 /\ K as Subset of (TOP-REAL 2)|K by XBOOLE_1:17;
A15: R=V /\ V2 /\ [#]((TOP-REAL 2)|K) by PRE_TOPC:def 5;
V /\ V2 is open by A6,A10,TOPS_1:11;
then
A16: R is open by A15,TOPS_2:24;
A17: p in V2 by A1,A2,A12,FUNCT_1:35;
then reconsider q=p as Point of ((TOP-REAL 2)|K) by A11,PRE_TOPC:8;
A18: rng (f|K) c= the carrier of TOP-REAL 2;
dom (f|K)=dom f /\ K by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K by FUNCT_2:def 1
.=K by XBOOLE_1:28
.=the carrier of ((TOP-REAL 2)|K) by PRE_TOPC:8;
then reconsider h=f|K as Function of (TOP-REAL 2)|K,TOP-REAL 2 by A18,
FUNCT_2:2;
A19: h is one-to-one by A1,FUNCT_1:52;
A20: K=(f|K).:K by A7,RELAT_1:129
.=rng (f|K) by A13,RELAT_1:113;
then consider f1 being Function of (TOP-REAL 2)|K,(TOP-REAL 2)|K such that
A21: h=f1 and
A22: f1 is being_homeomorphism by A1,A19,JGRAPH_1:46,TOPMETR:7;
A23: rng f1=[#]((TOP-REAL 2)|K) by A22,TOPS_2:def 5;
A24: f1 is onto by A23,FUNCT_2:def 3;
dom ((f1 qua Function)")=rng f1 & rng ((f1 qua Function)")=dom f1 by A19
,A21,FUNCT_1:33;
then reconsider
g1=(f1 qua Function)" as Function of (TOP-REAL 2)|K,(TOP-REAL 2
)|K by A23,FUNCT_2:2;
g1=f1" by A19,A21,A24,TOPS_2:def 4;
then
A25: g1 is continuous by A22,TOPS_2:def 5;
A26: f1.(g.p)=f.(g.p) by A9,A11,A21,FUNCT_1:49
.=p by A1,A2,FUNCT_1:35;
A27: dom f1=dom f /\ K by A21,RELAT_1:61
.=K by A4,XBOOLE_1:28;
rng f1=dom ((f1 qua Function)") by A19,A21,FUNCT_1:33;
then
A28: ((f1 qua Function)").p in rng ((f1 qua Function)") by A11,A17,A20,A21,
FUNCT_1:3;
A29: rng ((f1 qua Function)")=dom f1 by A19,A21,FUNCT_1:33;
f1.((f1 qua Function)".p)=p by A11,A17,A19,A20,A21,FUNCT_1:35;
then ((f1 qua Function)").p=g.p by A8,A19,A21,A26,A27,A29,A28,FUNCT_1:def 4
;
then ((f1 qua Function)").p in R by A9,A11,A14,XBOOLE_0:def 4;
then consider W3 being Subset of (TOP-REAL 2)|K such that
A30: q in W3 and
A31: W3 is open and
A32: ((f1 qua Function)").:W3 c= R by A16,A25,JGRAPH_2:10;
R=V /\ (V2 /\ K) by XBOOLE_1:16;
then
A33: R c= V by XBOOLE_1:17;
consider W5 being Subset of TOP-REAL 2 such that
A34: W5 is open and
A35: W3=W5 /\ [#]((TOP-REAL 2)|K) by A31,TOPS_2:24;
reconsider W4=W5 /\ V2 as Subset of TOP-REAL 2;
p in W5 by A30,A35,XBOOLE_0:def 4;
then
A36: p in W4 by A17,XBOOLE_0:def 4;
A37: dom f1=the carrier of (TOP-REAL 2)|K by FUNCT_2:def 1;
A38: ((f qua Function)").:W3 c= R
proof
let y be object;
assume y in ((f qua Function)").:W3;
then consider x being object such that
A39: x in dom ((f qua Function)") and
A40: x in W3 and
A41: y=((f qua Function)").x by FUNCT_1:def 6;
A42: x in rng f by A1,A39,FUNCT_1:33;
then
A43: y in dom f by A1,A41,FUNCT_1:32;
A44: f.y=x by A1,A41,A42,FUNCT_1:32;
the carrier of (TOP-REAL 2)|K=K by PRE_TOPC:8;
then ex z2 being object st z2 in dom f & z2 in K & f.y=f.z2
by A7,A40,A44,
FUNCT_1:def 6;
then
A45: y in K by A1,A43,FUNCT_1:def 4;
then
A46: y in the carrier of ((TOP-REAL 2)|K) by PRE_TOPC:8;
A47: dom ((f1 qua Function)")=the carrier of (TOP-REAL 2)|K by A19,A21,A23,
FUNCT_1:33;
f1.y=x by A21,A44,A45,FUNCT_1:49;
then y=((f1 qua Function)").x by A19,A21,A37,A46,FUNCT_1:32;
then y in ((f1 qua Function)").:W3 by A40,A47,FUNCT_1:def 6;
hence thesis by A32;
end;
W4=W5 /\ (V2 /\ K) by A11,XBOOLE_1:28
.=W5 /\ K /\ V2 by XBOOLE_1:16
.=W3 /\ V2 by A35,PRE_TOPC:def 5;
then
A48: g.:W4 c= g.:W3 /\ g.:V2 by RELAT_1:121;
g.:W3 /\ g.:V2 c= g.:W3 by XBOOLE_1:17;
then g.:W4 c= g.:W3 by A48;
then
A49: g.:W4 c= R by A38;
W4 is open by A10,A34,TOPS_1:11;
hence thesis by A36,A49,A33,XBOOLE_1:1;
end;
then
A50: g is continuous by JGRAPH_2:10;
f is onto by A2,FUNCT_2:def 3;
then g=f" by A1,TOPS_2:def 4;
hence thesis by A1,A2,A4,A50,TOPS_2:def 5;
end;
theorem Th4:
for X being non empty TopSpace, f1,f2 being Function of X,R^1,a,b
being Real st f1 is continuous & f2 is continuous & b<>0 & (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-a)/b)
& g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1,a,b be Real;
assume that
A1: f1 is continuous & f2 is continuous and
A2: b<>0 and
A3: for q being Point of X holds f2.q<>0;
consider g3 being Function of X,R^1 such that
A4: for p being Point of X,r1,r0 being Real st f1.p=r1 & f2.p=r0
holds g3.p=r1/r0 and
A5: g3 is continuous by A1,A3,JGRAPH_2:27;
consider g1 being Function of X,R^1 such that
A6: for p being Point of X holds g1.p=b & g1 is continuous by JGRAPH_2:20;
consider g2 being Function of X,R^1 such that
A7: for p being Point of X holds g2.p=a & g2 is continuous by JGRAPH_2:20;
consider g4 being Function of X,R^1 such that
A8: for p being Point of X,r1,r0 being Real st g3.p=r1 & g2.p=r0
holds g4.p=r1-r0 and
A9: g4 is continuous by A7,A5,JGRAPH_2:21;
for q being Point of X holds g1.q<>0 by A2,A6;
then consider g5 being Function of X,R^1 such that
A10: for p being Point of X,r1,r0 being Real st g4.p=r1 & g1.p=r0
holds g5.p=r1/r0 and
A11: g5 is continuous by A6,A9,JGRAPH_2:27;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g5.p=(r1/r2-a)/b
proof
let p be Point of X,r1,r2 be Real;
set r8=r1/r2;
A12: g1.p=b by A6;
assume f1.p=r1 & f2.p=r2;
then
A13: g3.p=r8 by A4;
g2.p=a by A7;
then g4.p=r8-a by A8,A13;
hence thesis by A10,A12;
end;
hence thesis by A11;
end;
theorem Th5:
for X being non empty TopSpace, f1,f2 being Function of X,R^1,
a,b being Real
st f1 is continuous & f2 is continuous & b<>0 & (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*((r1/r2-a)/b)) & g is
continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1,a,b be Real;
assume that
A1: f1 is continuous and
A2: f2 is continuous and
A3: b<>0 and
A4: for q being Point of X holds f2.q<>0;
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 & f2.p=r0
holds g3.p=r1/r0 and
A6: g3 is continuous by A1,A2,A4,JGRAPH_2:27;
consider g1 being Function of X,R^1 such that
A7: for p being Point of X holds g1.p=b & g1 is continuous by JGRAPH_2:20;
consider g2 being Function of X,R^1 such that
A8: for p being Point of X holds g2.p=a & g2 is continuous by JGRAPH_2:20;
consider g4 being Function of X,R^1 such that
A9: for p being Point of X,r1,r0 being Real st g3.p=r1 & g2.p=r0
holds g4.p=r1-r0 and
A10: g4 is continuous by A8,A6,JGRAPH_2:21;
for q being Point of X holds g1.q<>0 by A3,A7;
then consider g5 being Function of X,R^1 such that
A11: for p being Point of X,r1,r0 being Real st g4.p=r1 & g1.p=r0
holds g5.p=r1/r0 and
A12: g5 is continuous by A7,A10,JGRAPH_2:27;
consider g6 being Function of X,R^1 such that
A13: for p being Point of X,r1,r0 being Real st f2.p=r1 & g5.p=r0
holds g6.p=r1*r0 and
A14: g6 is continuous by A2,A12,JGRAPH_2:25;
for p being Point of X,r1,r2 being Real
st f1.p=r1 & f2.p=r2 holds g6.p
=r2*((r1/r2-a)/b)
proof
let p be Point of X,r1,r2 be Real;
assume that
A15: f1.p=r1 and
A16: f2.p=r2;
A17: g2.p=a by A8;
set r8=r1/r2;
A18: g1.p=b by A7;
g3.p=r8 by A5,A15,A16;
then g4.p=r8-a by A9,A17;
then g5.p=(r1/r2-a)/b by A11,A18;
hence thesis by A13,A16;
end;
hence thesis by A14;
end;
theorem Th6:
for X being non empty TopSpace, f1 being Function of X,R^1 st f1
is continuous 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=r1^2) & g is continuous
proof
let X be non empty TopSpace, f1 be Function of X,R^1;
assume f1 is continuous;
then consider g1 being Function of X,R^1 such that
A1: for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1
*r1 and
A2: g1 is continuous by JGRAPH_2:22;
for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1^2 by A1;
hence thesis by A2;
end;
theorem Th7:
for X being non empty TopSpace, f1 being Function of X,R^1 st f1
is continuous 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=|.r1.|) & g is continuous
proof
let X be non empty TopSpace, f1 be Function of X,R^1;
assume f1 is continuous;
then consider g1 being Function of X,R^1 such that
A1: for p being Point of X,r1 being Real st f1.p=r1 holds g1.p=r1 ^2 and
A2: g1 is continuous by Th6;
for q being Point of X ex r being Real st g1.q=r & r>=0
proof
let q be Point of X;
reconsider r11=f1.q as Real;
g1.q=r11^2 by A1;
hence thesis by XREAL_1:63;
end;
then consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1 being Real st g1.p=r1 holds g2.p=
sqrt( r1) and
A4: g2 is continuous by A2,JGRAPH_3:5;
for p being Point of X,r1 being Real st f1.p=r1 holds g2.p=|.r1.|
proof
let p be Point of X,r1 be Real;
assume f1.p=r1;
then g1.p=r1^2 by A1;
then g2.p=sqrt(r1^2) by A3
.=|.r1.| by COMPLEX1:72;
hence thesis;
end;
hence thesis by A4;
end;
theorem Th8:
for X being non empty TopSpace, f1 being Function of X,R^1 st f1
is continuous 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=-r1) & g is continuous
proof
let X be non empty TopSpace, f1 be Function of X,R^1;
consider g1 being Function of X,R^1 such that
A1: for p being Point of X holds g1.p=0 and
A2: g1 is continuous by JGRAPH_2:20;
assume f1 is continuous;
then consider g2 being Function of X,R^1 such that
A3: for p being Point of X,r1,r2 being Real st g1.p=r1 & f1.p=r2
holds g2.p=r1-r2 and
A4: g2 is continuous by A2,JGRAPH_2:21;
for p being Point of X,r1 being Real st f1.p=r1 holds g2.p=-r1
proof
let p be Point of X,r1 be Real;
assume
A5: f1.p=r1;
g1.p=0 by A1;
then g2.p=0-r1 by A3,A5;
hence thesis;
end;
hence thesis by A4;
end;
theorem Th9:
for X being non empty TopSpace, f1,f2 being Function of X,R^1,a,
b being Real st f1 is continuous & f2 is continuous & b<>0 & (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-a)/b)^2.|))) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1,a,b be Real;
assume that
A1: f1 is continuous and
A2: f2 is continuous and
A3: b<>0 & for q being Point of X holds f2.q<>0;
consider g1 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 g1.p=(r1/r2-a)/b and
A5: g1 is continuous by A1,A2,A3,Th4;
consider g2 being Function of X,R^1 such that
A6: for p being Point of X,s being Real st g1.p=s holds g2.p=s^2 and
A7: g2 is continuous by A5,Th6;
consider g0 being Function of X,R^1 such that
A8: for p being Point of X holds g0.p=1 and
A9: g0 is continuous by JGRAPH_2:20;
consider g3 being Function of X,R^1 such that
A10: for p being Point of X,s,t being Real st g0.p=s & g2.p=t
holds g3.p=s-t and
A11: g3 is continuous by A7,A9,JGRAPH_2:21;
consider g4 being Function of X,R^1 such that
A12: for p being Point of X,s being Real st g3.p=s holds g4.p=|.s.| and
A13: g4 is continuous by A11,Th7;
for q being Point of X ex r being Real st g4.q=r & r>=0
proof
let q be Point of X;
reconsider s=g3.q as Real;
g4.q=|.s.| by A12;
hence thesis by COMPLEX1:46;
end;
then consider g5 being Function of X,R^1 such that
A14: for p being Point of X,s being Real st g4.p=s holds g5.p=
sqrt(s ) and
A15: g5 is continuous by A13,JGRAPH_3:5;
consider g6 being Function of X,R^1 such that
A16: for p being Point of X,s being Real st g5.p=s holds g6.p=-s and
A17: g6 is continuous by A15,Th8;
consider g7 being Function of X,R^1 such that
A18: for p being Point of X,r1,r0 being Real st f2.p=r1 & g6.p=r0
holds g7.p=r1*r0 and
A19: g7 is continuous by A2,A17,JGRAPH_2:25;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g7.p= r2*(-sqrt(|.1-((r1/r2-a)/b)^2.|))
proof
let p be Point of X,r1,r2 be Real;
assume that
A20: f1.p=r1 and
A21: f2.p=r2;
A22: g0.p=1 by A8;
g1.p=(r1/r2-a)/b by A4,A20,A21;
then g2.p=((r1/r2-a)/b)^2 by A6;
then g3.p=1-((r1/r2-a)/b)^2 by A10,A22;
then g4.p= |.1-((r1/r2-a)/b)^2.| by A12;
then g5.p=sqrt(|.1-((r1/r2-a)/b)^2.|) by A14;
then g6.p= -sqrt(|.1-((r1/r2-a)/b)^2.|) by A16;
hence thesis by A18,A21;
end;
hence thesis by A19;
end;
theorem Th10:
for X being non empty TopSpace, f1,f2 being Function of X,R^1,a,
b being Real st f1 is continuous & f2 is continuous & b<>0 & (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-a)/b)^2.|))) & g is continuous
proof
let X be non empty TopSpace, f1,f2 be Function of X,R^1,a,b be Real;
assume that
A1: f1 is continuous and
A2: f2 is continuous and
A3: b<>0 & for q being Point of X holds f2.q<>0;
consider g1 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 g1.p=(r1/r2-a)/b and
A5: g1 is continuous by A1,A2,A3,Th4;
consider g2 being Function of X,R^1 such that
A6: for p being Point of X,s being Real st g1.p=s holds g2.p=s^2 and
A7: g2 is continuous by A5,Th6;
consider g0 being Function of X,R^1 such that
A8: for p being Point of X holds g0.p=1 and
A9: g0 is continuous by JGRAPH_2:20;
consider g3 being Function of X,R^1 such that
A10: for p being Point of X,s,t being Real st g0.p=s & g2.p=t
holds g3.p=s-t and
A11: g3 is continuous by A7,A9,JGRAPH_2:21;
consider g4 being Function of X,R^1 such that
A12: for p being Point of X,s being Real st g3.p=s holds g4.p=|.s.| and
A13: g4 is continuous by A11,Th7;
for q being Point of X ex r being Real st g4.q=r & r>=0
proof
let q be Point of X;
reconsider s=g3.q as Real;
g4.q=|.s.| by A12;
hence thesis by COMPLEX1:46;
end;
then consider g5 being Function of X,R^1 such that
A14: for p being Point of X,s being Real st g4.p=s holds g5.p=
sqrt s and
A15: g5 is continuous by A13,JGRAPH_3:5;
consider g7 being Function of X,R^1 such that
A16: for p being Point of X,r1,r0 being Real st f2.p=r1 & g5.p=r0
holds g7.p=r1*r0 and
A17: g7 is continuous by A2,A15,JGRAPH_2:25;
for p being Point of X,r1,r2 being Real st f1.p=r1 & f2.p=r2
holds g7.p= r2*( sqrt(|.1-((r1/r2-a)/b)^2.|))
proof
let p be Point of X,r1,r2 be Real;
assume that
A18: f1.p=r1 and
A19: f2.p=r2;
A20: g0.p=1 by A8;
g1.p=(r1/r2-a)/b by A4,A18,A19;
then g2.p=((r1/r2-a)/b)^2 by A6;
then g3.p=1-((r1/r2-a)/b)^2 by A10,A20;
then g4.p= |.1-((r1/r2-a)/b)^2.| by A12;
then g5.p=sqrt(|.1-((r1/r2-a)/b)^2.|) by A14;
hence thesis by A16,A19;
end;
hence thesis by A17;
end;
definition
let n be Nat;
deffunc F(Point of TOP-REAL n)=|.$1.|;
func n NormF -> Function of TOP-REAL n, R^1 means
:Def1:
for q being Point of TOP-REAL n holds it.q=|.q.|;
existence
proof
A1: for x being Element of TOP-REAL n holds F(x) in the carrier of R^1
by TOPMETR:17,XREAL_0:def 1;
thus ex IT being Function of TOP-REAL n, R^1 st for q being Point of
TOP-REAL n holds IT.q = F(q) from FUNCT_2:sch 8(A1);
end;
uniqueness
proof
thus for f,g being Function of TOP-REAL n, R^1 st (for q being Point of
TOP-REAL n holds f.q=F(q)) & (for q being Point of TOP-REAL n holds g.q=F(q))
holds f=g from BINOP_2:sch 1;
end;
end;
theorem
for n being Nat holds dom (n NormF)=the carrier of TOP-REAL
n & dom (n NormF)=REAL n
proof
let n be Nat;
thus dom (n NormF)=the carrier of TOP-REAL n by FUNCT_2:def 1;
hence thesis by EUCLID:22;
end;
theorem Th12:
for n being Nat holds n NormF is continuous
proof
let n be Nat;
for q being Point of TOP-REAL n holds (n NormF).q=|.q.| by Def1;
hence thesis by JORDAN2C:83;
end;
registration
let n be Nat;
cluster n NormF -> continuous;
coherence by Th12;
end;
theorem Th13:
for n being Element of NAT,K0 being Subset of TOP-REAL n, f
being Function of (TOP-REAL n)|K0,R^1 st (for p being Point of (TOP-REAL n)|K0
holds f.p=(n NormF).p) holds f is continuous
proof
let n be Element of NAT,K0 be Subset of TOP-REAL n, f be Function of (
TOP-REAL n)|K0,R^1;
A1: (the carrier of TOP-REAL n)/\K0=K0 by XBOOLE_1:28;
reconsider g=(n NormF) as Function of TOP-REAL n,R^1;
assume for p being Point of (TOP-REAL n)|K0 holds f.p=(n NormF).p;
then
A2: for x being object st x in dom f holds f.x=(n NormF).x;
dom f= the carrier of (TOP-REAL n)|K0 & the carrier of (TOP-REAL n)|K0=
K0 by FUNCT_2:def 1,PRE_TOPC:8;
then dom f=dom (n NormF) /\ K0 by A1,FUNCT_2:def 1;
then f=g|K0 by A2,FUNCT_1:46;
hence thesis by TOPMETR:7;
end;
theorem Th14:
for n being Element of NAT,p being Point of Euclid n,r being
Real, B being Subset of TOP-REAL n st B=cl_Ball(p,r)
holds B is bounded closed
proof
let n be Element of NAT,p be Point of Euclid n,
r be Real, B be Subset of
TOP-REAL n;
assume
A1: B=cl_Ball(p,r);
cl_Ball(p,r) c= Ball(p,r+1)
proof
let x be object;
A2: r<r+1 by XREAL_1:29;
assume
A3: x in cl_Ball(p,r);
then reconsider q=x as Point of Euclid n;
dist(p,q)<=r by A3,METRIC_1:12;
then dist(p,q)<r+1 by A2,XXREAL_0:2;
hence thesis by METRIC_1:11;
end;
then cl_Ball(p,r) is bounded by TBSP_1:14;
hence B is bounded by A1,JORDAN2C:11;
A4: the TopStruct of TOP-REAL n = TopSpaceMetr Euclid n by EUCLID:def 8;
then reconsider BB = B as Subset of TopSpaceMetr Euclid n;
BB is closed by A1,TOPREAL6:57;
hence thesis by A4,PRE_TOPC:31;
end;
theorem Th15:
for p being Point of Euclid 2,r being Real, B being Subset of
TOP-REAL 2 st B=cl_Ball(p,r) holds B is compact
proof
let p be Point of Euclid 2,r be Real, B be Subset of TOP-REAL 2;
assume B=cl_Ball(p,r);
then B is bounded closed by Th14;
hence thesis by TOPREAL6:79;
end;
begin :: Fan Morphism for West
definition
let s be Real, q be Point of TOP-REAL 2;
func FanW(s,q) -> Point of TOP-REAL 2 equals
:Def2:
|.q.|*|[-sqrt(1-((q`2/|.
q.|-s)/(1-s))^2), (q`2/|.q.|-s)/(1-s)]| if q`2/|.q.|>=s & q`1<0, |.q.|*|[-sqrt(
1-((q`2/|.q.|-s)/(1+s))^2), (q`2/|.q.|-s)/(1+s)]| if q`2/|.q.|<s & q`1<0
otherwise q;
correctness;
end;
definition
let s be Real;
func s-FanMorphW -> Function of TOP-REAL 2, TOP-REAL 2 means
:Def3:
for q being Point of TOP-REAL 2 holds it.q=FanW(s,q);
existence
proof
deffunc F(Point of TOP-REAL 2)=FanW(s,$1);
thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point
of TOP-REAL 2 holds IT.q=F(q) from FUNCT_2:sch 4;
end;
uniqueness
proof
deffunc F(Point of TOP-REAL 2)=FanW(s,$1);
thus for f,g being Function of TOP-REAL 2, TOP-REAL 2 st (for q being
Point of TOP-REAL 2 holds f.q=F(q)) & (for q being Point of TOP-REAL 2 holds g.
q=F(q)) holds f=g from BINOP_2:sch 1;
end;
end;
theorem Th16:
for sn being Real holds (q`2/|.q.|>=sn & q`1<0 implies sn
-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|
-sn)/(1-sn))]|) & (q`1>=0 implies sn-FanMorphW.q=q)
proof
let sn be Real;
hereby
assume q`2/|.q.|>=sn & q`1<0;
then
FanW(sn,q)= |.q.|*|[-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn)
/(1-sn)]| by Def2
.= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1-sn))]| by EUCLID:58;
hence
sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1-sn))]| by Def3;
end;
assume
A1: q`1>=0;
sn-FanMorphW.q=FanW(sn,q) by Def3;
hence thesis by A1,Def2;
end;
theorem Th17:
for sn being Real st q`2/|.q.|<=sn & q`1<0 holds sn-FanMorphW.q=
|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|*((q`2/|.q.|-sn)/(1+sn))]|
proof
let sn be Real;
assume that
A1: q`2/|.q.|<=sn and
A2: q`1<0;
per cases by A1,XXREAL_0:1;
suppose
q`2/|.q.|<sn;
then
FanW(sn,q)= |.q.|*|[-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2), (q`2/|.q.|-sn)
/(1+sn)]| by A2,Def2
.= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1+sn))]| by EUCLID:58;
hence thesis by Def3;
end;
suppose
A3: q`2/|.q.|=sn;
then (q`2/|.q.|-sn)/(1-sn)=0;
hence thesis by A2,A3,Th16;
end;
end;
theorem Th18:
for sn being Real st -1<sn & sn<1 holds (q`2/|.q.|>=sn & q`1<=0
& q<>0.TOP-REAL 2 implies sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-
sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1-sn))]|) & (q`2/|.q.|<=sn & q`1<=0 & q<>0.
TOP-REAL 2 implies sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)
), |.q.|*((q`2/|.q.|-sn)/(1+sn))]|)
proof
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
per cases;
suppose
A3: q`2/|.q.|>=sn & q`1<=0 & q<>0.TOP-REAL 2;
per cases;
suppose
A4: q`1<0;
then
FanW(sn,q)= |.q.|*|[-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-
sn)/(1-sn)]| by A3,Def2
.= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|
-sn)/(1-sn))]| by EUCLID:58;
hence thesis by A4,Def3,Th17;
end;
suppose
A5: q`1>=0;
then
A6: sn-FanMorphW.q=q by Th16;
A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A8: 1-sn>0 by A2,XREAL_1:149;
A9: q`1=0 by A3,A5;
|.q.|<>0 by A3,TOPRNS_1:24;
then |.q.|^2>0 by SQUARE_1:12;
then (q`2)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60;
then ((q`2)/|.q.|)^2=1^2 by XCMPLX_1:76;
then
A10: sqrt(((q`2)/|.q.|)^2)=1;
A11: now
assume q`2<0;
then -((q`2)/|.q.|)=1 by A10,SQUARE_1:23;
hence contradiction by A1,A3;
end;
sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22;
then
A12: |.q.|=q`2 by A7,A9,A11,SQUARE_1:22;
then 1=q`2/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60;
then (q`2/|.q.|-sn)/(1-sn)=1 by A8,XCMPLX_1:60;
hence thesis by A2,A6,A9,A12,EUCLID:53,TOPRNS_1:24
,XCMPLX_1:60;
end;
end;
suppose
A13: q`2/|.q.|<=sn & q`1<=0 & q<>0.TOP-REAL 2;
per cases;
suppose
q`1<0;
hence thesis by Th16,Th17;
end;
suppose
A14: q`1>=0;
A15: 1+sn>0 by A1,XREAL_1:148;
A16: |.q.|<>0 by A13,TOPRNS_1:24;
A17: q`1=0 by A13,A14;
|.q.|>0 & 1>q`2/|.q.| by A2,A13,Lm1,XXREAL_0:2;
then 1 *(|.q.|)>q`2/|.q.|*(|.q.|) by XREAL_1:68;
then
A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`2 by A13,JGRAPH_3:1,TOPRNS_1:24
,XCMPLX_1:87;
then
A19: q`2= -(|.q.|) by A17,SQUARE_1:40;
then -1=q`2/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197;
then
A20: (q`2/|.q.|-sn)/(1+sn) =(-(1+sn))/(1+sn) .=-1 by A15,XCMPLX_1:197;
|.q.|=-q`2 by A17,A18,SQUARE_1:40;
then |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1+sn))]| =q by A17,A20,EUCLID:53;
hence thesis by A1,A14,A16,A19,Th16,XCMPLX_1:197;
end;
end;
suppose
q`1>0 or q=0.TOP-REAL 2;
hence thesis;
end;
end;
Lm2: for K being non empty Subset of TOP-REAL 2 holds proj1|K is continuous
Function of (TOP-REAL 2)|K,R^1 & for q being Point of (TOP-REAL 2)|K holds (
proj1|K).q=proj1.q
proof
let K be non empty Subset of TOP-REAL 2;
reconsider g2=proj1|K as Function of (TOP-REAL 2)|K,R^1 by TOPMETR:17;
A1: the carrier of (TOP-REAL 2)|K=K by PRE_TOPC:8;
for q being Point of (TOP-REAL 2)|K holds g2.q=proj1.q
proof
let q be Point of (TOP-REAL 2)|K;
q in the carrier of (TOP-REAL 2)|K & dom proj1=the carrier of TOP-REAL
2 by FUNCT_2:def 1;
then q in dom proj1 /\ K by A1,XBOOLE_0:def 4;
hence thesis by FUNCT_1:48;
end;
hence thesis by JGRAPH_2:29;
end;
Lm3: for K being non empty Subset of TOP-REAL 2 holds proj2|K is continuous
Function of (TOP-REAL 2)|K,R^1 & for q being Point of (TOP-REAL 2)|K holds (
proj2|K).q=proj2.q
proof
let K be non empty Subset of TOP-REAL 2;
reconsider g2=proj2|K as Function of (TOP-REAL 2)|K,R^1 by TOPMETR:17;
A1: the carrier of (TOP-REAL 2)|K=K by PRE_TOPC:8;
for q being Point of (TOP-REAL 2)|K holds g2.q=proj2.q
proof
let q be Point of (TOP-REAL 2)|K;
q in the carrier of (TOP-REAL 2)|K & dom proj2=the carrier of TOP-REAL
2 by FUNCT_2:def 1;
then q in dom proj2 /\ K by A1,XBOOLE_0:def 4;
hence thesis by FUNCT_1:48;
end;
hence thesis by JGRAPH_2:30;
end;
Lm4: dom (2 NormF)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
Lm5: for K being non empty Subset of TOP-REAL 2 holds (2 NormF)|K is
continuous Function of (TOP-REAL 2)|K,R^1 & for q being Point of (TOP-REAL 2)|K
holds ((2 NormF)|K).q=(2 NormF).q
proof
let K1 be non empty Subset of TOP-REAL 2;
A1: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then reconsider g1=(2 NormF)|K1 as Function of (TOP-REAL 2)|K1,R^1 by
FUNCT_2:32;
for q being Point of (TOP-REAL 2)|K1 holds g1.q=(2 NormF).q
proof
let q be Point of (TOP-REAL 2)|K1;
q in the carrier of (TOP-REAL 2)|K1;
then q in dom (2 NormF) /\ K1 by A1,Lm4,XBOOLE_0:def 4;
hence thesis by FUNCT_1:48;
end;
hence thesis by Th13;
end;
Lm6: for K1 be non empty Subset of TOP-REAL 2, g1 be Function of (TOP-REAL 2)|
K1,R^1 st g1=(2 NormF)|K1 & (for q being Point of TOP-REAL 2 st q in the
carrier of (TOP-REAL 2)|K1 holds q<>0.TOP-REAL 2) holds for q being Point of (
TOP-REAL 2)|K1 holds g1.q<>0
proof
let K1 be non empty Subset of TOP-REAL 2, g1 be Function of (TOP-REAL 2)|K1,
R^1;
assume that
A1: g1=(2 NormF)|K1 and
A2: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q<>0.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 reconsider q2=q as Point of TOP-REAL 2;
g1.q=(2 NormF).q by A1,Lm5
.= |.q2.| by Def1;
hence thesis by A2,TOPRNS_1:24;
end;
theorem Th19:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st sn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1-
sn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1<=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=1-sn;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: sn<1 and
A2: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in K1 by A7,A8,A9,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
A10: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
A11: g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
f.r=(|.r.|)* ((r`2/|.r.|-sn)/(1-sn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th20:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<sn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`2/|.p.|
-sn)/(1+sn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`1<=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=1+sn;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: -1<sn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1+sn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
1+sn>0 by A1,XREAL_1:148;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
A8: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in dom g3 by A7,A9;
then x in K1 by A7,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
A10: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
A11: g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
f.r=(|.r.|)* ((r`2/|.r.|-sn)/(1+sn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
dom f=dom g3 by A7,FUNCT_2:def 1;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th21:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st sn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*(-sqrt(1-((p`2/|.p.|-
sn)/(1-sn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`1<=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2) holds f is
continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=1-sn;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: sn<1 and
A2: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f.p=|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th9;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
A9: 1-sn>0 by A1,XREAL_1:149;
assume
A10: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
A11: |.r.|<>0 by A3,A10,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`2) -(|.r.|))*((r`2)+|.r.|) =-(r`1)^2;
(r`1)^2>=0 by XREAL_1:63;
then r`2<= |.r.| by A12,XREAL_1:93;
then r`2/|.r.| <= |.r.|/|.r.| by XREAL_1:72;
then r`2/|.r.|<=1 by A11,XCMPLX_1:60;
then
A13: r`2/|.r.|-sn<=(1-sn) by XREAL_1:9;
reconsider s=x as Point of (TOP-REAL 2)|K1 by A10;
A14: now
assume (1-sn)^2=0;
then 1-sn+sn=0+sn by XCMPLX_1:6;
hence contradiction by A1;
end;
sn-r`2/|.r.|<=0 by A3,A10,XREAL_1:47;
then -(sn- r`2/|.r.|)>=-(1-sn) by A9,XREAL_1:24;
then (1-sn)^2>=0 & (r`2/|.r.|-sn)^2<=(1-sn)^2 by A13,SQUARE_1:49,XREAL_1:63
;
then (r`2/|.r.|-sn)^2/(1-sn)^2<=(1-sn)^2/(1-sn)^2 by XREAL_1:72;
then (r`2/|.r.|-sn)^2/(1-sn)^2<=1 by A14,XCMPLX_1:60;
then ((r`2/|.r.|-sn)/(1-sn))^2<=1 by XCMPLX_1:76;
then 1-((r`2/|.r.|-sn)/(1-sn))^2>=0 by XREAL_1:48;
then |.1-((r`2/|.r.|-sn)/(1-sn))^2.| =1-((r`2/|.r.|-sn)/(1-sn))^2 by
ABSVALUE:def 1;
then
A15: f.r=(|.r.|)*(-sqrt(|.1-((r`2/|.r.|-sn)/(1-sn))^2.|)) by A2,A10;
A16: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
hence thesis by A5,A15,A16;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th22:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<sn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*(-sqrt(1-((p
`2/|.p.|-sn)/(1+sn))^2))) & (for q being Point of TOP-REAL 2 st q in the
carrier of (TOP-REAL 2)|K1 holds q`1<=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2)
holds f is continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=1+sn;
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: -1<sn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1<=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2;
A4: 1+sn>0 by A1,XREAL_1:148;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th9;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
reconsider s=x as Point of (TOP-REAL 2)|K1 by A9;
A10: (1+sn)^2>0 by A4,SQUARE_1:12;
A11: |.r.|<>0 by A3,A9,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`2) -(|.r.|))*((r`2)+|.r.|) =-(r`1)^2;
(r`1)^2>=0 by XREAL_1:63;
then -(|.r.|)<=r`2 by A12,XREAL_1:93;
then r`2/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72;
then r`2/|.r.|>= -1 by A11,XCMPLX_1:197;
then r`2/|.r.|-sn>=-1-sn by XREAL_1:9;
then
A13: r`2/|.r.|-sn>=-(1+sn);
sn-r`2/|.r.|>=0 by A3,A9,XREAL_1:48;
then -(sn-r`2/|.r.|)<=-0;
then (r`2/|.r.|-sn)^2<=(1+sn)^2 by A4,A13,SQUARE_1:49;
then (r`2/|.r.|-sn)^2/(1+sn)^2<=(1+sn)^2/(1+sn)^2 by A4,XREAL_1:72;
then (r`2/|.r.|-sn)^2/(1+sn)^2<=1 by A10,XCMPLX_1:60;
then ((r`2/|.r.|-sn)/(1+sn))^2<=1 by XCMPLX_1:76;
then 1-((r`2/|.r.|-sn)/(1+sn))^2>=0 by XREAL_1:48;
then |.1-((r`2/|.r.|-sn)/(1+sn))^2.| =1-((r`2/|.r.|-sn)/(1+sn))^2 by
ABSVALUE:def 1;
then
A14: f.r=(|.r.|)*(-sqrt(|.1-((r`2/|.r.|-sn)/(1+sn))^2.|)) by A2,A9;
A15: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
hence thesis by A5,A14,A15;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th23:
for sn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphW)|
K0 & B0={q where q is Point of TOP-REAL 2: q`1<=0 & q<>0.TOP-REAL 2} & K0={p: p
`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[-cn,sn]|;
A1: p0`1=-cn by EUCLID:52;
p0`2=sn by EUCLID:52;
then
A2: |.p0.|=sqrt((-cn)^2+sn^2) by A1,JGRAPH_3:1
.=sqrt((cn)^2+sn^2);
assume
A3: -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0={q where q is Point of
TOP-REAL 2: q`1<=0 & q<>0.TOP-REAL 2} & K0={p: p`2/|.p.|>=sn & p`1<=0 & p<>0.
TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A4: 1-sn^2>0 by XREAL_1:50;
then
A5: --cn>0 by SQUARE_1:25;
A6: now
assume p0=0.TOP-REAL 2;
then --cn=-0 by EUCLID:52,JGRAPH_2:3;
hence contradiction by A4,SQUARE_1:25;
end;
cn^2=1-sn^2 by A4,SQUARE_1:def 2;
then p0`2/|.p0.|=sn by A2,EUCLID:52;
then
A7: p0 in K0 by A3,A1,A6,A5;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A8: rng (proj1*((sn-FanMorphW)|K1)) c= the carrier of R^1 by TOPMETR:17;
A9: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2/|.p8.|>= sn & p8`1<=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A10: dom ((sn-FanMorphW)|K1) c= dom (proj2*((sn-FanMorphW)|K1))
proof
let x be object;
assume
A11: x in dom ((sn-FanMorphW)|K1);
then x in dom (sn-FanMorphW) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphW) by XBOOLE_0:def 4;
then
A12: dom proj2 = (the carrier of TOP-REAL 2) & (sn-FanMorphW).x in rng (sn
-FanMorphW) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphW)|K1).x=(sn-FanMorphW).x by A11,FUNCT_1:47;
hence thesis by A11,A12,FUNCT_1:11;
end;
A13: rng (proj2*((sn-FanMorphW)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj2*((sn-FanMorphW)|K1)) c= dom ((sn-FanMorphW)|K1) by RELAT_1:25;
then dom (proj2*((sn-FanMorphW)|K1)) =dom ((sn-FanMorphW)|K1) by A10,
XBOOLE_0:def 10
.=dom (sn-FanMorphW) /\ 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;
then reconsider
g2=proj2*((sn-FanMorphW)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A13,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.|* ((p`2/|.p.|-sn)/(1-sn))
proof
let p be Point of TOP-REAL 2;
A14: dom ((sn-FanMorphW)|K1)=dom (sn-FanMorphW) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A15: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A16: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|>= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A15;
then
A17: (sn-FanMorphW).p =|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|
* ((p`2/|.p.|-sn)/(1-sn))]| by A3,Th18;
((sn-FanMorphW)|K1).p=(sn-FanMorphW).p by A16,A15,FUNCT_1:49;
then g2.p=proj2.(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1-sn))]|) by A16,A14,A15,A17,FUNCT_1:13
.=(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn
)/(1-sn))]|)`2 by PSCOMP_1:def 6
.=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A18: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn));
A19: dom ((sn-FanMorphW)|K1) c= dom (proj1*((sn-FanMorphW)|K1))
proof
let x be object;
assume
A20: x in dom ((sn-FanMorphW)|K1);
then x in dom (sn-FanMorphW) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphW) by XBOOLE_0:def 4;
then
A21: dom proj1 = (the carrier of TOP-REAL 2) & (sn-FanMorphW).x in rng (sn
-FanMorphW) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphW)|K1).x=(sn-FanMorphW).x by A20,FUNCT_1:47;
hence thesis by A20,A21,FUNCT_1:11;
end;
dom (proj1*((sn-FanMorphW)|K1)) c= dom ((sn-FanMorphW)|K1) by RELAT_1:25;
then dom (proj1*((sn-FanMorphW)|K1)) =dom ((sn-FanMorphW)|K1) by A19,
XBOOLE_0:def 10
.=dom (sn-FanMorphW) /\ 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;
then reconsider
g1=proj1*((sn-FanMorphW)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A8,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.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))
proof
let p be Point of TOP-REAL 2;
A22: dom ((sn-FanMorphW)|K1)=dom (sn-FanMorphW) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A23: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A24: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|>= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A23;
then
A25: (sn-FanMorphW).p=|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|*
((p`2/|.p.|-sn)/(1-sn))]| by A3,Th18;
((sn-FanMorphW)|K1).p=(sn-FanMorphW).p by A24,A23,FUNCT_1:49;
then g1.p=proj1.(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1-sn))]|) by A24,A22,A23,A25,FUNCT_1:13
.=(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn
)/(1-sn)) ]|)`1 by PSCOMP_1:def 5
.= |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A26: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1<=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A27: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|>= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A27;
hence thesis;
end;
then
A28: f1 is continuous by A3,A26,Th21;
A29: 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
A30: |[x,y]| in K1 and
A31: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
set p99=|[x,y]|;
A32: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`2/|.p3.| >=sn & p3`1<=0
& p3<>0.TOP-REAL 2 by A3,A30;
A33: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A34: f1.p99=|.p99.|*(-sqrt(1-((p99`2/|.p99.|-sn)/(1-sn))^2)) by A26,A30;
((sn-FanMorphW)|K0).(|[x,y]|)=((sn-FanMorphW)).(|[x,y]|) by A30,FUNCT_1:49
.= |[|.p99.|*(-sqrt(1-((p99`2/|.p99.|-sn)/(1-sn))^2)), |.p99.|* ((p99
`2/|.p99.|-sn)/(1-sn))]| by A3,A32,Th18
.=|[r,s]| by A18,A30,A31,A33,A34;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1<=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A35: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|>= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A35;
hence thesis;
end;
then f2 is continuous by A3,A18,Th19;
hence thesis by A7,A9,A28,A29,JGRAPH_2:35;
end;
theorem Th24:
for sn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphW)|
K0 & B0={q where q is Point of TOP-REAL 2: q`1<=0 & q<>0.TOP-REAL 2} & K0={p: p
`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[-cn,sn]|;
A1: p0`1=-cn by EUCLID:52;
p0`2=sn by EUCLID:52;
then
A2: |.p0.|=sqrt((-cn)^2+sn^2) by A1,JGRAPH_3:1
.=sqrt((cn)^2+sn^2);
assume
A3: -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0={q where q is Point of
TOP-REAL 2: q`1<=0 & q<>0.TOP-REAL 2} & K0={p: p`2/|.p.|<=sn & p`1<=0 & p<>0.
TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A4: 1-sn^2>0 by XREAL_1:50;
then
A5: --cn>0 by SQUARE_1:25;
A6: now
assume p0=0.TOP-REAL 2;
then --cn=-0 by EUCLID:52,JGRAPH_2:3;
hence contradiction by A4,SQUARE_1:25;
end;
cn^2=1-sn^2 by A4,SQUARE_1:def 2;
then p0`2/|.p0.|=sn by A2,EUCLID:52;
then
A7: p0 in K0 by A3,A1,A6,A5;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A8: rng (proj1*((sn-FanMorphW)|K1)) c= the carrier of R^1 by TOPMETR:17;
A9: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2/|.p8.|<= sn & p8`1<=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A10: dom ((sn-FanMorphW)|K1) c= dom (proj2*((sn-FanMorphW)|K1))
proof
let x be object;
assume
A11: x in dom ((sn-FanMorphW)|K1);
then x in dom (sn-FanMorphW) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphW) by XBOOLE_0:def 4;
then
A12: dom proj2 = (the carrier of TOP-REAL 2) & (sn-FanMorphW).x in rng (sn
-FanMorphW) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphW)|K1).x=(sn-FanMorphW).x by A11,FUNCT_1:47;
hence thesis by A11,A12,FUNCT_1:11;
end;
A13: rng (proj2*((sn-FanMorphW)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj2*((sn-FanMorphW)|K1)) c= dom ((sn-FanMorphW)|K1) by RELAT_1:25;
then dom (proj2*((sn-FanMorphW)|K1)) =dom ((sn-FanMorphW)|K1) by A10,
XBOOLE_0:def 10
.=dom (sn-FanMorphW) /\ 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;
then reconsider
g2=proj2*((sn-FanMorphW)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A13,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.|* ((p`2/|.p.|-sn)/(1+sn))
proof
let p be Point of TOP-REAL 2;
A14: dom ((sn-FanMorphW)|K1)=dom (sn-FanMorphW) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A15: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A16: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|<= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A15;
then
A17: (sn-FanMorphW).p =|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|
* ((p`2/|.p.|-sn)/(1+sn))]| by A3,Th18;
((sn-FanMorphW)|K1).p=(sn-FanMorphW).p by A16,A15,FUNCT_1:49;
then g2.p=proj2.(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1+sn))]|) by A16,A14,A15,A17,FUNCT_1:13
.=(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn
)/(1+sn))]|)`2 by PSCOMP_1:def 6
.=|.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A18: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`2/|.p.|-sn)/(1+sn));
A19: dom ((sn-FanMorphW)|K1) c= dom (proj1*((sn-FanMorphW)|K1))
proof
let x be object;
assume
A20: x in dom ((sn-FanMorphW)|K1);
then x in dom (sn-FanMorphW) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphW) by XBOOLE_0:def 4;
then
A21: dom proj1 = (the carrier of TOP-REAL 2) & (sn-FanMorphW).x in rng (sn
-FanMorphW) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphW)|K1).x=(sn-FanMorphW).x by A20,FUNCT_1:47;
hence thesis by A20,A21,FUNCT_1:11;
end;
dom (proj1*((sn-FanMorphW)|K1)) c= dom ((sn-FanMorphW)|K1) by RELAT_1:25;
then dom (proj1*((sn-FanMorphW)|K1)) =dom ((sn-FanMorphW)|K1) by A19,
XBOOLE_0:def 10
.=dom (sn-FanMorphW) /\ 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;
then reconsider
g1=proj1*((sn-FanMorphW)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A8,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.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))
proof
let p be Point of TOP-REAL 2;
A22: dom ((sn-FanMorphW)|K1)=dom (sn-FanMorphW) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A23: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A24: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|<= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A23;
then
A25: (sn-FanMorphW).p=|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|*
((p`2/|.p.|-sn)/(1+sn))]| by A3,Th18;
((sn-FanMorphW)|K1).p=(sn-FanMorphW).p by A24,A23,FUNCT_1:49;
then g1.p=proj1.(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1+sn))]|) by A24,A22,A23,A25,FUNCT_1:13
.=(|[|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn
)/(1+sn)) ]|)`1 by PSCOMP_1:def 5
.= |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A26: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1<=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A27: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|<= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A27;
hence thesis;
end;
then
A28: f1 is continuous by A3,A26,Th22;
A29: 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
A30: |[x,y]| in K1 and
A31: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
set p99=|[x,y]|;
A32: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`2/|.p3.| <=sn & p3`1<=0
& p3<>0.TOP-REAL 2 by A3,A30;
A33: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A34: f1.p99=|.p99.|*(-sqrt(1-((p99`2/|.p99.|-sn)/(1+sn))^2)) by A26,A30;
((sn-FanMorphW)|K0).(|[x,y]|)=((sn-FanMorphW)).(|[x,y]|) by A30,FUNCT_1:49
.= |[|.p99.|*(-sqrt(1-((p99`2/|.p99.|-sn)/(1+sn))^2)), |.p99.|* ((p99
`2/|.p99.|-sn)/(1+sn))]| by A3,A32,Th18
.=|[r,s]| by A18,A30,A31,A33,A34;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1<=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A35: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|<= sn & p3`1<=0 &
p3<>0.TOP-REAL 2 by A3,A35;
hence thesis;
end;
then f2 is continuous by A3,A18,Th20;
hence thesis by A7,A9,A28,A29,JGRAPH_2:35;
end;
Lm7: for sn being Real , K1 being Subset of TOP-REAL 2
st K1={p7 where p7 is
Point of TOP-REAL 2:p7`2>=sn*|.p7.|} holds K1 is closed
proof
set K10=[#]TOP-REAL 2;
reconsider g0=(2 NormF)|K10 as continuous Function of (TOP-REAL 2)|K10,R^1
by Lm5;
reconsider g1=proj2|K10 as continuous Function of (TOP-REAL 2)|K10,R^1 by Lm3
;
let sn be Real, K1 be Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means ($1`2>=sn*|.$1.|);
consider g2 being Function of (TOP-REAL 2)|K10,R^1 such that
A1: for q being Point of (TOP-REAL 2)|K10,r1 being Real st g0.q=
r1 holds g2.q=sn*r1 and
A2: g2 is continuous by JGRAPH_2:23;
consider g3 being Function of (TOP-REAL 2)|K10,R^1 such that
A3: for q being Point of (TOP-REAL 2)|K10,r1,r2 being Real st g2.
q= r1 & g1.q=r2 holds g3.q=r1-r2 and
A4: g3 is continuous by A2,JGRAPH_2:21;
A5: (TOP-REAL 2)|K10=the TopStruct of TOP-REAL 2 by TSEP_1:93;
then reconsider g=g3 as Function of TOP-REAL 2,R^1;
reconsider K2=K1 as Subset of the TopStruct of TOP-REAL 2;
assume K1={p: p`2>=(sn)*(|.p.|)};
then
A6: K1={p7 where p7 is Point of TOP-REAL 2:P[p7]};
A7: K1`={p7 where p7 is Point of TOP-REAL 2:not P[p7]} from JGRAPH_2:sch 2(
A6);
A8: for p being Point of TOP-REAL 2 holds g3.p=sn*|.p.|-p`2
proof
let p be Point of TOP-REAL 2;
g0.p=(2 NormF).p by A5,Lm5
.=|.p.| by Def1;
then
A9: g2.p=sn*|.p.| by A1,A5;
g1.p=proj2.p by A5,Lm3
.=p`2 by PSCOMP_1:def 6;
hence thesis by A3,A5,A9;
end;
A10: K1` c= {p7 where p7 is Point of TOP-REAL 2:g/.p7>0}
proof
let x be object;
assume x in K1`;
then consider p9 being Point of TOP-REAL 2 such that
A11: x=p9 and
A12: p9`2<sn*|.p9.| by A7;
A13: g/.p9=sn*|.p9.|-p9`2 by A8;
sn*|.p9.| - p9`2>0 by A12,XREAL_1:50;
hence thesis by A11,A13;
end;
{p7 where p7 is Point of TOP-REAL 2: g/.p7>0} c= K1`
proof
let x be object;
assume x in {p7 where p7 is Point of TOP-REAL 2: g/.p7>0};
then consider p7 being Point of TOP-REAL 2 such that
A14: p7=x and
A15: g/.p7>0;
g/.p7=sn*|.p7.|-p7`2 by A8;
then sn*|.p7.| - p7`2+p7`2>0+p7`2 by A15,XREAL_1:8;
hence thesis by A7,A14;
end;
then K1`={p7 where p7 is Point of TOP-REAL 2:g/.p7>0} by A10,XBOOLE_0:def 10;
then K2` is open by A4,A5,Th1;
then K1` is open by PRE_TOPC:30;
hence thesis by TOPS_1:3;
end;
Lm8: for sn being Real, K1 being Subset of TOP-REAL 2
st K1={p7 where p7 is
Point of TOP-REAL 2:p7`1>=sn*|.p7.|} holds K1 is closed
proof
set K10=[#](TOP-REAL 2);
reconsider g0=(2 NormF)|K10 as continuous Function of (TOP-REAL 2)|K10,R^1
by Lm5;
reconsider g1=proj1|K10 as continuous Function of (TOP-REAL 2)|K10,R^1 by Lm2
;
let sn be Real, K1 be Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means ($1`1>=sn*|.$1.|);
consider g2 being Function of (TOP-REAL 2)|K10,R^1 such that
A1: for q being Point of (TOP-REAL 2)|K10,r1 being Real st g0.q=
r1 holds g2.q=sn*r1 and
A2: g2 is continuous by JGRAPH_2:23;
consider g3 being Function of (TOP-REAL 2)|K10,R^1 such that
A3: for q being Point of (TOP-REAL 2)|K10,r1,r2 being Real st g2.
q= r1 & g1.q=r2 holds g3.q=r1-r2 and
A4: g3 is continuous by A2,JGRAPH_2:21;
A5: (TOP-REAL 2)|K10=the TopStruct of TOP-REAL 2 by TSEP_1:93;
then reconsider g=g3 as Function of TOP-REAL 2,R^1;
reconsider K2=K1 as Subset of the TopStruct of TOP-REAL 2;
assume K1={p: p`1>=(sn)*(|.p.|)};
then
A6: K1={p7 where p7 is Point of TOP-REAL 2:P[p7]};
A7: K1`={p7 where p7 is Point of TOP-REAL 2:not P[p7]} from JGRAPH_2:sch 2(
A6);
A8: for p being Point of TOP-REAL 2 holds g3.p=sn*|.p.|-p`1
proof
let p be Point of TOP-REAL 2;
g0.p=(2 NormF).p by A5,Lm5
.=|.p.| by Def1;
then
A9: g2.p=sn*|.p.| by A1,A5;
g1.p=proj1.p by A5,Lm2
.=p`1 by PSCOMP_1:def 5;
hence thesis by A3,A5,A9;
end;
A10: K1` c= {p7 where p7 is Point of TOP-REAL 2:g/.p7>0}
proof
let x be object;
assume x in K1`;
then consider p9 being Point of TOP-REAL 2 such that
A11: x=p9 and
A12: p9`1<sn*|.p9.| by A7;
A13: g/.p9=sn*|.p9.|-p9`1 by A8;
sn*|.p9.| - p9`1>0 by A12,XREAL_1:50;
hence thesis by A11,A13;
end;
{p7 where p7 is Point of TOP-REAL 2: g/.p7>0} c= K1`
proof
let x be object;
assume x in {p7 where p7 is Point of TOP-REAL 2: g/.p7>0};
then consider p7 being Point of TOP-REAL 2 such that
A14: p7=x and
A15: g/.p7>0;
g/.p7=sn*|.p7.|-p7`1 by A8;
then sn*|.p7.| - p7`1+p7`1>0+p7`1 by A15,XREAL_1:8;
hence thesis by A7,A14;
end;
then K1`={p7 where p7 is Point of TOP-REAL 2:g/.p7>0} by A10,XBOOLE_0:def 10;
then K2` is open by A4,A5,Th1;
then K1` is open by PRE_TOPC:30;
hence thesis by TOPS_1:3;
end;
theorem Th25:
for sn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`2
>=sn*(|.p.|) & p`1<=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`1<=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means ($1`2>=sn*|.$1.|);
assume
A1: K003={p: p`2>=(sn)*(|.p.|) & p`1<=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm7,JORDAN6:5;
hence thesis by A1,A2,TOPS_1:8;
end;
Lm9: for sn being Real, K1 being Subset of TOP-REAL 2
st K1={p7 where p7 is
Point of TOP-REAL 2:p7`2<=sn*|.p7.|} holds K1 is closed
proof
set K10=[#](TOP-REAL 2);
reconsider g0=(2 NormF)|K10 as continuous Function of (TOP-REAL 2)|K10,R^1
by Lm5;
reconsider g1=proj2|K10 as continuous Function of (TOP-REAL 2)|K10,R^1 by Lm3
;
let sn be Real, K1 be Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means ($1`2<=sn*|.$1.|);
consider g2 being Function of (TOP-REAL 2)|K10,R^1 such that
A1: for q being Point of (TOP-REAL 2)|K10,r1 being Real st g0.q=
r1 holds g2.q=sn*r1 and
A2: g2 is continuous by JGRAPH_2:23;
consider g3 being Function of (TOP-REAL 2)|K10,R^1 such that
A3: for q being Point of (TOP-REAL 2)|K10,r1,r2 being Real st g2.
q= r1 & g1.q=r2 holds g3.q=r1-r2 and
A4: g3 is continuous by A2,JGRAPH_2:21;
A5: (TOP-REAL 2)|K10=the TopStruct of TOP-REAL 2 by TSEP_1:93;
then reconsider g=g3 as Function of TOP-REAL 2,R^1;
reconsider K2=K1 as Subset of the TopStruct of TOP-REAL 2;
assume K1={p: p`2<=(sn)*(|.p.|)};
then
A6: K1={p7 where p7 is Point of TOP-REAL 2:P[p7]};
A7: K1`={p7 where p7 is Point of TOP-REAL 2:not P[p7]} from JGRAPH_2:sch 2(
A6);
A8: for p being Point of TOP-REAL 2 holds g3.p=sn*|.p.|-p`2
proof
let p be Point of TOP-REAL 2;
g0.p=(2 NormF).p by A5,Lm5
.=|.p.| by Def1;
then
A9: g2.p=sn*|.p.| by A1,A5;
g1.p=proj2.p by A5,Lm3
.=p`2 by PSCOMP_1:def 6;
hence thesis by A3,A5,A9;
end;
A10: K1` c= {p7 where p7 is Point of TOP-REAL 2:g/.p7<0}
proof
let x be object;
assume x in K1`;
then consider p9 being Point of TOP-REAL 2 such that
A11: x=p9 and
A12: p9`2>sn*|.p9.| by A7;
A13: g/.p9=sn*|.p9.|-p9`2 by A8;
sn*|.p9.| - p9`2<0 by A12,XREAL_1:49;
hence thesis by A11,A13;
end;
{p7 where p7 is Point of TOP-REAL 2: g/.p7<0} c= K1`
proof
let x be object;
assume x in {p7 where p7 is Point of TOP-REAL 2: g/.p7<0};
then consider p7 being Point of TOP-REAL 2 such that
A14: p7=x and
A15: g/.p7<0;
g/.p7=sn*|.p7.|-p7`2 by A8;
then sn*|.p7.| - p7`2+p7`2<0+p7`2 by A15,XREAL_1:8;
hence thesis by A7,A14;
end;
then K1`={p7 where p7 is Point of TOP-REAL 2:g/.p7<0} by A10,XBOOLE_0:def 10;
then K2` is open by A4,A5,Th2;
then K1` is open by PRE_TOPC:30;
hence thesis by TOPS_1:3;
end;
Lm10: for sn being Real, K1 being Subset of TOP-REAL 2
st K1={p7 where p7 is
Point of TOP-REAL 2:p7`1<=sn*|.p7.|} holds K1 is closed
proof
set K10=[#](TOP-REAL 2);
reconsider g0=(2 NormF)|K10 as continuous Function of (TOP-REAL 2)|K10,R^1
by Lm5;
reconsider g1=proj1|K10 as continuous Function of (TOP-REAL 2)|K10,R^1 by Lm2
;
let sn be Real, K1 be Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means ($1`1<=sn*|.$1.|);
consider g2 being Function of (TOP-REAL 2)|K10,R^1 such that
A1: for q being Point of (TOP-REAL 2)|K10,r1 being Real st g0.q=
r1 holds g2.q=sn*r1 and
A2: g2 is continuous by JGRAPH_2:23;
consider g3 being Function of (TOP-REAL 2)|K10,R^1 such that
A3: for q being Point of (TOP-REAL 2)|K10,r1,r2 being Real st g2.
q= r1 & g1.q=r2 holds g3.q=r1-r2 and
A4: g3 is continuous by A2,JGRAPH_2:21;
A5: (TOP-REAL 2)|K10=the TopStruct of TOP-REAL 2 by TSEP_1:93;
then reconsider g=g3 as Function of TOP-REAL 2,R^1;
reconsider K2=K1 as Subset of the TopStruct of TOP-REAL 2;
assume K1={p: p`1<=(sn)*(|.p.|)};
then
A6: K1={p7 where p7 is Point of TOP-REAL 2:P[p7]};
A7: K1`={p7 where p7 is Point of TOP-REAL 2:not P[p7]} from JGRAPH_2:sch 2(
A6);
A8: for p being Point of TOP-REAL 2 holds g3.p=sn*|.p.|-p`1
proof
let p be Point of TOP-REAL 2;
g0.p=(2 NormF).p by A5,Lm5
.=|.p.| by Def1;
then
A9: g2.p=sn*|.p.| by A1,A5;
g1.p=proj1.p by A5,Lm2
.=p`1 by PSCOMP_1:def 5;
hence thesis by A3,A5,A9;
end;
A10: K1` c= {p7 where p7 is Point of TOP-REAL 2:g/.p7<0}
proof
let x be object;
assume x in K1`;
then consider p9 being Point of TOP-REAL 2 such that
A11: x=p9 and
A12: p9`1>sn*|.p9.| by A7;
A13: g/.p9=sn*|.p9.|-p9`1 by A8;
sn*|.p9.| - p9`1<0 by A12,XREAL_1:49;
hence thesis by A11,A13;
end;
{p7 where p7 is Point of TOP-REAL 2: g/.p7<0} c= K1`
proof
let x be object;
assume x in {p7 where p7 is Point of TOP-REAL 2: g/.p7<0};
then consider p7 being Point of TOP-REAL 2 such that
A14: p7=x and
A15: g/.p7<0;
g/.p7=sn*|.p7.|-p7`1 by A8;
then sn*|.p7.| - p7`1+p7`1<0+p7`1 by A15,XREAL_1:8;
hence thesis by A7,A14;
end;
then K1`={p7 where p7 is Point of TOP-REAL 2:g/.p7<0} by A10,XBOOLE_0:def 10;
then K2` is open by A4,A5,Th2;
then K1` is open by PRE_TOPC:30;
hence thesis by TOPS_1:3;
end;
theorem Th26:
for sn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`2
<=(sn)*(|.p.|) & p`1<=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`1<=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means ($1`2<=sn*|.$1.|);
assume
A1: K003={p: P[p] & Q[p]};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm9,JORDAN6:5;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th27:
for sn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphW)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`1<=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[-cn,sn]|;
A1: p0`1=-cn by EUCLID:52;
p0`2=sn by EUCLID:52;
then
A2: |.p0.|=sqrt((-cn)^2+sn^2) by A1,JGRAPH_3:1
.=sqrt((cn)^2+sn^2);
assume
A3: -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1<=0 & p<>0.TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A4: 1-sn^2>0 by XREAL_1:50;
then
A5: --cn>0 by SQUARE_1:25;
A6: now
assume p0=0.TOP-REAL 2;
then --cn=-0 by EUCLID:52,JGRAPH_2:3;
hence contradiction by A4,SQUARE_1:25;
end;
then p0 in K0 by A3,A1,A5;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
cn^2=1-sn^2 by A4,SQUARE_1:def 2;
then
A7: p0`2/|.p0.|=sn by A2,EUCLID:52;
then
A8: p0 in {p: p`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2} by A1,A6,A5;
not p0 in {0.TOP-REAL 2} by A6,TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A3,XBOOLE_0:def 5;
K1 c= D
proof
let x be object;
assume
A9: x in K1;
then ex p6 being Point of TOP-REAL 2 st p6=x & p6`1<=0 & p6 <>0.TOP-REAL 2
by A3;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A3,A9,XBOOLE_0:def 5;
end;
then D=K1 \/ D by XBOOLE_1:12;
then
A10: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4;
A11: {p: p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2;
hence thesis by A3;
end;
A12: {p: p`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2;
hence thesis by A3;
end;
then reconsider
K00={p: p`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A8,PRE_TOPC:8;
the carrier of (TOP-REAL 2)|D=D by PRE_TOPC:8;
then
A13: rng (f|K00) c=D;
p0 in {p: p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2} by A1,A6,A5,A7;
then reconsider
K11={p: p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A11,PRE_TOPC:8;
the carrier of (TOP-REAL 2)|D=D by PRE_TOPC:8;
then
A14: rng (f|K11) c=D;
the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D;
then
A15: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1
.=K1 by PRE_TOPC:8;
then dom (f|K00)=K00 by A12,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8;
then reconsider
f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A13,FUNCT_2:2
;
dom (f|K11)=K11 by A11,A15,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8;
then reconsider
f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A14,FUNCT_2:2
;
A16: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`2/|.$1.|>=sn & $1`1<=0 & $1<>0.
TOP-REAL 2;
A17: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1
.=K11 by PRE_TOPC:8;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K001={p: p`2/|.p.|>=sn & p`1<=0 & p<>0.TOP-REAL 2} as non empty
Subset of TOP-REAL 2 by A8;
A18: the carrier of (TOP-REAL 2)|K1 = K1 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`2>=(sn)*(|.$1.|) & $1`1<=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K003={p: p`2>=(sn)*(|.p.|) & p`1<=0} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`2/|.$1.|<=sn & $1`1<=0 & $1<>0.
TOP-REAL 2;
A19: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
A20: rng ((sn-FanMorphW)|K001) c= K1
proof
let y be object;
assume y in rng ((sn-FanMorphW)|K001);
then consider x being object such that
A21: x in dom ((sn-FanMorphW)|K001) and
A22: y=((sn-FanMorphW)|K001).x by FUNCT_1:def 3;
x in dom (sn-FanMorphW) by A21,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A23: y=(sn-FanMorphW).q by A21,A22,FUNCT_1:47;
dom ((sn-FanMorphW)|K001)=(dom (sn-FanMorphW))/\ K001 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1
.=K001 by XBOOLE_1:28;
then
A24: ex p2 being Point of TOP-REAL 2 st p2=q & p2`2/|.p2.|>= sn & p2`1<=0 &
p2<>0.TOP-REAL 2 by A21;
then
A25: (q`2/|.q.|-sn)>= 0 by XREAL_1:48;
|.q.|<>0 by A24,TOPRNS_1:24;
then
A26: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1-sn))]|;
A27: q4`2= |.q.|*((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
A28: 1-sn>0 by A3,XREAL_1:149;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2 <= (|.q.|)^2 by JGRAPH_3:1;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A28,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A28,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A28,A25,SQUARE_1:49;
then
A29: 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A30: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by A29,SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0;
then
A31: sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
A32: q4`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
then
A33: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A30,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A27,A33;
then
A34: q4<>0.TOP-REAL 2 by A26,TOPRNS_1:23;
sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1-sn))]| by A3,A24,Th18;
hence thesis by A3,A23,A32,A31,A34;
end;
A35: dom (sn-FanMorphW)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then dom ((sn-FanMorphW)|K001)=K001 by RELAT_1:62
.= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8;
then reconsider
f3=(sn-FanMorphW)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL
2)|K1 by A18,A20,FUNCT_2:2;
A36: K003 is closed by Th25;
defpred P[Point of TOP-REAL 2] means $1`2<=(sn)*(|.$1.|) & $1`1<=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K004={p: p`2<=(sn)*(|.p.|) & p`1<=0} as Subset of TOP-REAL 2;
A37: K004 /\ K1 c= K11
proof
let x be object;
assume
A38: x in K004 /\ K1;
then x in K004 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A39: q1=x and
A40: q1`2<=(sn)*(|.q1.|) and
q1`1<=0;
x in K1 by A38,XBOOLE_0:def 4;
then
A41: ex q2 being Point of TOP-REAL 2 st q2=x & q2`1<=0 & q2 <>0.TOP-REAL 2
by A3;
q1`2/|.q1.|<=(sn)*(|.q1.|)/|.q1.| by A40,XREAL_1:72;
then q1`2/|.q1.|<=(sn) by A39,A41,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A39,A41;
end;
A42: K004 is closed by Th26;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A3,FUNCT_1:51
,GOBOARD9:2;
then
A43: f1 is continuous by A3,A10,Th23,PRE_TOPC:26;
A44: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
p0 in {p: p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2} by A1,A6,A5,A7;
then reconsider
K111={p: p`2/|.p.|<=sn & p`1<=0 & p<>0.TOP-REAL 2} as non empty
Subset of TOP-REAL 2 by A19;
A45: rng ((sn-FanMorphW)|K111) c= K1
proof
let y be object;
assume y in rng ((sn-FanMorphW)|K111);
then consider x being object such that
A46: x in dom ((sn-FanMorphW)|K111) and
A47: y=((sn-FanMorphW)|K111).x by FUNCT_1:def 3;
x in dom (sn-FanMorphW) by A46,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A48: y=(sn-FanMorphW).q by A46,A47,FUNCT_1:47;
dom ((sn-FanMorphW)|K111)=(dom (sn-FanMorphW))/\ K111 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1
.=K111 by XBOOLE_1:28;
then
A49: ex p2 being Point of TOP-REAL 2 st p2=q & p2`2/|.p2.|<= sn & p2`1<=0 &
p2<>0.TOP-REAL 2 by A46;
then
A50: (q`2/|.q.|-sn)<=0 by XREAL_1:47;
|.q.|<>0 by A49,TOPRNS_1:24;
then
A51: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1+sn))]|;
A52: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
A53: 1+sn>0 by A3,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1
,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A51,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
then (-(1+sn))/(1+sn)<=(( q`2/|.q.|-sn))/(1+sn) by A53,XREAL_1:72;
then -1<=(( q`2/|.q.|-sn))/(1+sn) by A53,XCMPLX_1:197;
then
A54: ( (q`2/|.q.|-sn) /(1+sn))^2<=1^2 by A53,A50,SQUARE_1:49;
then
A55: 1-((q`2/|.q.|-sn)/(1+sn))^2>=0 by XREAL_1:48;
1-(-((q`2/|.q.|-sn)/(1+sn)))^2>=0 by A54,XREAL_1:48;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)>=0;
then
A56: sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
A57: q4`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
then
A58: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A55,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A52,A58;
then
A59: q4<>0.TOP-REAL 2 by A51,TOPRNS_1:23;
sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1+sn))]| by A3,A49,Th18;
hence thesis by A3,A48,A57,A56,A59;
end;
dom ((sn-FanMorphW)|K111)=K111 by A35,RELAT_1:62
.= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8;
then reconsider
f4=(sn-FanMorphW)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL
2)|K1 by A16,A45,FUNCT_2:2;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A3,FUNCT_1:51
,GOBOARD9:2;
then
A60: f2 is continuous by A3,A10,Th24,PRE_TOPC:26;
set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11;
A61: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5;
K11 c= K004 /\ K1
proof
let x be object;
assume x in K11;
then consider p such that
A62: p=x and
A63: p`2/|.p.|<=sn and
A64: p`1<=0 and
A65: p<>0.TOP-REAL 2;
p`2/|.p.|*|.p.|<=(sn)*(|.p.|) by A63,XREAL_1:64;
then p`2<=(sn)*(|.p.|) by A65,TOPRNS_1:24,XCMPLX_1:87;
then
A66: x in K004 by A62,A64;
x in K1 by A3,A62,A64,A65;
hence thesis by A66,XBOOLE_0:def 4;
end;
then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A44,A37,XBOOLE_0:def 10;
then
A67: K11 is closed by A42,PRE_TOPC:13;
A68: K003 /\ K1 c= K00
proof
let x be object;
assume
A69: x in K003 /\ K1;
then x in K003 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A70: q1=x and
A71: q1`2>=(sn)*(|.q1.|) and
q1`1<=0;
x in K1 by A69,XBOOLE_0:def 4;
then
A72: ex q2 being Point of TOP-REAL 2 st q2=x & q2`1<=0 & q2 <>0.TOP-REAL 2
by A3;
q1`2/|.q1.|>=(sn)*(|.q1.|)/|.q1.| by A71,XREAL_1:72;
then q1`2/|.q1.|>=(sn) by A70,A72,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A70,A72;
end;
A73: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8;
A74: D<>{};
A75: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5;
A76: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p
proof
let p be object;
assume
A77: p in ([#]T1)/\([#]T2);
then p in K00 by A75,XBOOLE_0:def 4;
hence f1.p=f.p by FUNCT_1:49
.=f2.p by A61,A77,FUNCT_1:49;
end;
K00 c= K003 /\ K1
proof
let x be object;
assume x in K00;
then consider p such that
A78: p=x and
A79: p`2/|.p.|>=sn and
A80: p`1<=0 and
A81: p<>0.TOP-REAL 2;
p`2/|.p.|*|.p.|>=(sn)*(|.p.|) by A79,XREAL_1:64;
then p`2>=(sn)*(|.p.|) by A81,TOPRNS_1:24,XCMPLX_1:87;
then
A82: x in K003 by A78,A80;
x in K1 by A3,A78,A80,A81;
hence thesis by A82,XBOOLE_0:def 4;
end;
then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A44,A68,XBOOLE_0:def 10;
then
A83: K00 is closed by A36,PRE_TOPC:13;
A84: K1 c= K00 \/ K11
proof
let x be object;
assume x in K1;
then consider p such that
A85: p=x & p`1<=0 & p<>0.TOP-REAL 2 by A3;
per cases;
suppose
p`2/|.p.|>=sn;
then x in K00 by A85;
hence thesis by XBOOLE_0:def 3;
end;
suppose
p`2/|.p.|<sn;
then x in K11 by A85;
hence thesis by XBOOLE_0:def 3;
end;
end;
then [#](((TOP-REAL 2)|K1)|K00) \/ [#](((TOP-REAL 2)|K1)|K11) =[#]((
TOP-REAL 2)|K1) by A75,A61,A44,XBOOLE_0:def 10;
then consider h being Function of (TOP-REAL 2)|K1,(TOP-REAL 2)|D such that
A86: h=f1+*f2 and
A87: h is continuous by A75,A61,A83,A67,A43,A60,A76,JGRAPH_2:1;
A88: dom h=the carrier of ((TOP-REAL 2)|K1) by FUNCT_2:def 1;
A89: dom f1=the carrier of ((TOP-REAL 2)|K1)|K00 by FUNCT_2:def 1
.=K00 by PRE_TOPC:8;
A90: for y being object st y in dom h holds h.y=f.y
proof
let y be object;
assume
A91: y in dom h;
now
per cases by A84,A88,A73,A91,XBOOLE_0:def 3;
case
A92: y in K00 & not y in K11;
then y in dom f1 \/ dom f2 by A89,XBOOLE_0:def 3;
hence h.y=f1.y by A17,A86,A92,FUNCT_4:def 1
.=f.y by A92,FUNCT_1:49;
end;
case
A93: y in K11;
then y in dom f1 \/ dom f2 by A17,XBOOLE_0:def 3;
hence h.y=f2.y by A17,A86,A93,FUNCT_4:def 1
.=f.y by A93,FUNCT_1:49;
end;
end;
hence thesis;
end;
K0=the carrier of ((TOP-REAL 2)|K0) by PRE_TOPC:8
.=dom f by A74,FUNCT_2:def 1;
hence thesis by A87,A88,A90,FUNCT_1:2,PRE_TOPC:8;
end;
theorem Th28:
for sn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphW)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`1>=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[cn,-sn]|;
assume
A1: -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1>=0 & p<>0.TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A2: p0`1=cn & 1-sn^2>0 by EUCLID:52,XREAL_1:50;
then p0<>0.TOP-REAL 2 by JGRAPH_2:3,SQUARE_1:25;
then not p0 in {0.TOP-REAL 2} by TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A1,XBOOLE_0:def 5;
p0`1>0 by A2,SQUARE_1:25;
then p0 in K0 by A1,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A3: K1 c= D
proof
let x be object;
assume x in K1;
then consider p2 being Point of TOP-REAL 2 such that
A4: p2=x and
p2`1>=0 and
A5: p2<>0.TOP-REAL 2 by A1;
not p2 in {0.TOP-REAL 2} by A5,TARSKI:def 1;
hence thesis by A1,A4,XBOOLE_0:def 5;
end;
for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D
st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W &
W is open & f.:W c= V
proof
let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D;
assume that
A6: f.p in V and
A7: V is open;
consider V2 being Subset of TOP-REAL 2 such that
A8: V2 is open and
A9: V2 /\ [#]((TOP-REAL 2)|D)=V by A7,TOPS_2:24;
reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1;
A10: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
then
A11: f.p=(sn-FanMorphW).p by A1,FUNCT_1:49;
A12: f.:W2 c= V
proof
let y be object;
assume y in f.:W2;
then consider x being object such that
A13: x in dom f and
A14: x in W2 and
A15: y=f.x by FUNCT_1:def 6;
f is Function of (TOP-REAL 2)|K1, (TOP-REAL 2)|D;
then dom f= K1 by A10,FUNCT_2:def 1;
then consider p4 being Point of TOP-REAL 2 such that
A16: x=p4 and
A17: p4`1>=0 and
p4<>0.TOP-REAL 2 by A1,A13;
A18: p4 in V2 by A14,A16,XBOOLE_0:def 4;
p4 in [#]((TOP-REAL 2)|K1) by A13,A16;
then p4 in D by A3,A10;
then
A19: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
f.p4=(sn-FanMorphW).p4 by A1,A10,A13,A16,FUNCT_1:49
.=p4 by A17,Th16;
hence thesis by A9,A15,A16,A18,A19,XBOOLE_0:def 4;
end;
p in the carrier of (TOP-REAL 2)|K1;
then consider q being Point of TOP-REAL 2 such that
A20: q=p and
A21: q`1>=0 and
q <>0.TOP-REAL 2 by A1,A10;
(sn-FanMorphW).q=q by A21,Th16;
then p in V2 by A6,A9,A11,A20,XBOOLE_0:def 4;
then
A22: p in W2 by XBOOLE_0:def 4;
W2 is open by A8,TOPS_2:24;
hence thesis by A22,A12;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th29:
for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL
2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`1<=0 & p<>0.TOP-REAL 2} holds K0 is
closed
proof
set J0 = NonZero TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`1<=0;
set I1 = {p: P[p] & p<>0.TOP-REAL 2};
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0;
reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3:
sch 2;
assume B0=J0 & K0=I1;
then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:5
,PRE_TOPC:def 5;
hence thesis by PRE_TOPC:13;
end;
theorem Th30:
for sn being Real, 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 -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p
`1<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
defpred P[Point of TOP-REAL 2] means $1`1<=0 & $1<>0.TOP-REAL 2;
let sn be Real,B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
reconsider K1={p: P[p] } as Subset of TOP-REAL 2 from JGRAPH_2:sch 1;
assume
A1: -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1<=0 & p<>0.TOP-REAL 2 };
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1<=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by A1,PRE_TOPC:7;
hence thesis by A1,Th27;
end;
theorem Th31:
for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL
2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`1>=0 & p<>0.TOP-REAL 2} holds K0 is
closed
proof
set J0 = NonZero TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`1>=0;
set I1 = {p: P[p] & p<>0.TOP-REAL 2};
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0;
reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3:
sch 2;
assume B0=J0 & K0=I1;
then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:4
,PRE_TOPC:def 5;
hence thesis by PRE_TOPC:13;
end;
theorem Th32:
for sn being Real, 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 -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p`1>=0 &
p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<sn & sn<1 & f=(sn-FanMorphW)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1>=0 & p<>0.TOP-REAL 2};
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1>=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th28;
end;
theorem Th33:
for sn being Real,p being Point of TOP-REAL 2 holds |.(sn
-FanMorphW).p.|=|.p.|
proof
let sn be Real,p be Point of TOP-REAL 2;
set z=(sn-FanMorphW).p;
reconsider q=p,qz=z as Point of TOP-REAL 2;
per cases;
suppose
A1: q`2/|.q.|>=sn & q`1<0;
then
A2: (sn-FanMorphW).q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|
* ((q`2/|.q.|-sn)/(1-sn))]| by Th16;
then
A3: qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
A4: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by A2,EUCLID:52;
A5: (q`2/|.q.|-sn)>=0 by A1,XREAL_1:48;
A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
|.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then
A8: 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
per cases;
suppose
A9: 1-sn=0;
A10: ((q`2/|.q.|-sn)/(1-sn))=(q`2/|.q.|-sn)*(1-sn)" by XCMPLX_0:def 9
.= (q`2/|.q.|-sn)*0 by A9
.=0;
then -sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)=-1;
then (sn-FanMorphW).q= |[|.q.|*(-1),|.q.|*0]| by A1,A10,Th16
.=|[-(|.q.|),0]|;
then ((sn-FanMorphW).q)`1=-(|.q.|) & ((sn-FanMorphW).q)`2=0 by EUCLID:52;
then |.(sn-FanMorphW).p.|=sqrt((-(|.q.|))^2+0^2) by JGRAPH_3:1
.=sqrt((|.q.|)^2)
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A11: 1-sn<>0;
per cases by A11;
suppose
A12: 1-sn>0;
-(1-sn)<= -( q`2/|.q.|-sn) by A8,XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A12,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A12,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A5,A12,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A13: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
A14: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2 by A3
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A13,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A4,A14;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A15: 1-sn<0;
0+(q`2)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then
A16: 1 > q`2/|.p.| by SQUARE_1:52;
q`2/|.q.|-sn>=0 by A1,XREAL_1:48;
hence thesis by A15,A16,XREAL_1:9;
end;
end;
end;
suppose
A17: q`2/|.q.|<sn & q`1<0;
then |.q.|<>0 by JGRAPH_2:3,TOPRNS_1:24;
then
A18: (|.q.|)^2>0 by SQUARE_1:12;
A19: (q`2/|.q.|-sn)<0 by A17,XREAL_1:49;
A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then
A21: -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
A22: (sn-FanMorphW).q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q
.|* ((q`2/|.q.|-sn)/(1+sn))]| by A17,Th17;
then
A23: qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
A24: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by A22,EUCLID:52;
per cases;
suppose
A25: 1+sn=0;
((q`2/|.q.|-sn)/(1+sn))=(q`2/|.q.|-sn)*(1+sn)" by XCMPLX_0:def 9
.= (q`2/|.q.|-sn)*0 by A25
.=0;
then ((sn-FanMorphW).q)`1=-(|.q.|) & ((sn-FanMorphW).q)`2=0 by A22,
EUCLID:52;
then |.(sn-FanMorphW).p.|=sqrt((-(|.q.|))^2+0^2) by JGRAPH_3:1
.=sqrt((|.q.|)^2)
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A26: 1+sn<>0;
per cases by A26;
suppose
A27: 1+sn>0;
then (-(1+sn))/(1+sn)<=(( q`2/|.q.|-sn))/(1+sn) by A21,XREAL_1:72;
then -1<=(( q`2/|.q.|-sn))/(1+sn) by A27,XCMPLX_1:197;
then ( (q`2/|.q.|-sn) /(1+sn))^2<=1^2 by A19,A27,SQUARE_1:49;
then
A28: 1-(((q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
A29: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2 by A23
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A28,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A24,A29;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A30: 1+sn<0;
0+(q`2)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1 < q`2/|.p.| by SQUARE_1:52;
then
A31: q`2/|.q.|-sn>-1-sn by XREAL_1:9;
-(1+sn)>-0 by A30,XREAL_1:24;
hence thesis by A17,A31,XREAL_1:49;
end;
end;
end;
suppose
q`1>=0;
hence thesis by Th16;
end;
end;
theorem Th34:
for sn being Real,x,K0 being set st -1<sn & sn<1 & x in K0 & K0=
{p: p`1<=0 & p<>0.TOP-REAL 2} holds (sn-FanMorphW).x in K0
proof
let sn be Real,x,K0 be set;
assume
A1: -1<sn & sn<1 & x in K0 & K0={p: p`1<=0 & p<>0.TOP-REAL 2};
then consider p such that
A2: p=x and
A3: p`1<=0 and
A4: p<>0.TOP-REAL 2;
A5: now
assume |.p.|<=0;
then |.p.|=0;
hence contradiction by A4,TOPRNS_1:24;
end;
then
A6: (|.p.|)^2>0 by SQUARE_1:12;
per cases;
suppose
A7: p`2/|.p.|<=sn;
reconsider p9= (sn-FanMorphW).p as Point of TOP-REAL 2;
(sn-FanMorphW).p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p
.|*((p`2/|.p.|-sn)/(1+sn))]| by A1,A3,A4,A7,Th18;
then
A8: p9`1=|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A10: 1+sn>0 by A1,XREAL_1:148;
per cases;
suppose
p`1=0;
hence thesis by A1,A2,Th16;
end;
suppose
p`1<>0;
then 0+(p`2)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`2)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74;
then (p`2)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 < 1 by XCMPLX_1:76;
then -1 < p`2/|.p.| by SQUARE_1:52;
then -1-sn< p`2/|.p.|-sn by XREAL_1:9;
then (-1)*(1+sn)/(1+sn)< (p`2/|.p.|-sn)/(1+sn) by A10,XREAL_1:74;
then
A11: -1< (p`2/|.p.|-sn)/(1+sn) by A10,XCMPLX_1:89;
p`2/|.p.|-sn<=0 by A7,XREAL_1:47;
then 1^2> ((p`2/|.p.|-sn)/(1+sn))^2 by A10,A11,SQUARE_1:50;
then 1-((p`2/|.p.|-sn)/(1+sn))^2>0 by XREAL_1:50;
then --sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>0 by SQUARE_1:25;
then -sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)<0;
then |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))<0 by A5,XREAL_1:132;
hence thesis by A1,A2,A8,JGRAPH_2:3;
end;
end;
suppose
A12: p`2/|.p.|>sn;
reconsider p9= (sn-FanMorphW).p as Point of TOP-REAL 2;
(sn-FanMorphW).p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p
.|*((p`2/|.p.|-sn)/(1-sn))]| by A1,A3,A4,A12,Th18;
then
A13: p9`1=|.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A15: 1-sn>0 by A1,XREAL_1:149;
per cases;
suppose
p`1=0;
hence thesis by A1,A2,Th16;
end;
suppose
p`1<>0;
then 0+(p`2)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`2)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74;
then (p`2)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 < 1 by XCMPLX_1:76;
then p`2/|.p.|<1 by SQUARE_1:52;
then (p`2/|.p.|-sn)<1-sn by XREAL_1:9;
then (p`2/|.p.|-sn)/(1-sn)<(1-sn)/(1-sn) by A15,XREAL_1:74;
then
A16: (p`2/|.p.|-sn)/(1-sn)<1 by A15,XCMPLX_1:60;
-(1-sn)< -0 & p`2/|.p.|-sn>=sn-sn by A12,A15,XREAL_1:9,24;
then (-1)*(1-sn)/(1-sn)< (p`2/|.p.|-sn)/(1-sn) by A15,XREAL_1:74;
then -1< (p`2/|.p.|-sn)/(1-sn) by A15,XCMPLX_1:89;
then 1^2> ((p`2/|.p.|-sn)/(1-sn))^2 by A16,SQUARE_1:50;
then 1-((p`2/|.p.|-sn)/(1-sn))^2>0 by XREAL_1:50;
then --sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>0 by SQUARE_1:25;
then -sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)<0;
then p9`1<0 by A5,A13,XREAL_1:132;
hence thesis by A1,A2,JGRAPH_2:3;
end;
end;
end;
theorem Th35:
for sn being Real,x,K0 being set st -1<sn & sn<1 & x in K0 & K0=
{p: p`1>=0 & p<>0.TOP-REAL 2} holds (sn-FanMorphW).x in K0
proof
let sn be Real,x,K0 be set;
assume
A1: -1<sn & sn<1 & x in K0 & K0={p: p`1>=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`1>=0 & p<>0.TOP-REAL 2;
hence thesis by A1,Th16;
end;
scheme
InclSub { D() -> non empty Subset of TOP-REAL 2, P[set] } : { p : P[p] & p<>
0.TOP-REAL 2 } c= the carrier of (TOP-REAL 2)|D()
provided
A1: D() = NonZero TOP-REAL 2
proof
let x be object;
assume x in {p: P[p] & p<>0.TOP-REAL 2};
then
A2: ex p st x=p &( P[p])& p<>0.TOP-REAL 2;
A3: D()` = {0.TOP-REAL 2} by A1,JGRAPH_3:20;
now
assume not x in D();
then x in (the carrier of TOP-REAL 2) \ D() by A2,XBOOLE_0:def 5;
then x in D()` by SUBSET_1:def 4;
hence contradiction by A3,A2,TARSKI:def 1;
end;
hence thesis by PRE_TOPC:8;
end;
theorem Th36:
for sn being Real, D being non empty Subset of TOP-REAL 2 st -1<
sn & sn<1 & D`={0.TOP-REAL 2} holds ex h being Function of (TOP-REAL 2)|D,(
TOP-REAL 2)|D st h=(sn-FanMorphW)|D & h is continuous
proof
(|[0,1]|)`1=0 & (|[0,1]|)`2=1 by EUCLID:52;
then
A1: |[0,1]| in {p where p is Point of TOP-REAL 2: p`1<=0 & p<>0.TOP-REAL 2}
by JGRAPH_2:3;
set Y1=|[0,1]|;
defpred P[Point of TOP-REAL 2] means $1`1<=0;
reconsider B0= {0.TOP-REAL 2} as Subset of TOP-REAL 2;
let sn be Real,D be non empty Subset of TOP-REAL 2;
assume that
A2: -1<sn & sn<1 and
A3: D`={0.TOP-REAL 2};
A4: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
A5: D =(B0)` by A3
.=(NonZero TOP-REAL 2) by SUBSET_1:def 4;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A5);
then reconsider
K0={p:p`1<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A1;
A6: K0 =the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8;
A7: the carrier of ((TOP-REAL 2)|D) =D by PRE_TOPC:8;
A8: rng ((sn-FanMorphW)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
let y be object;
assume y in rng ((sn-FanMorphW)|K0);
then consider x being object such that
A9: x in dom ((sn-FanMorphW)|K0) and
A10: y=((sn-FanMorphW)|K0).x by FUNCT_1:def 3;
x in (dom (sn-FanMorphW)) /\ K0 by A9,RELAT_1:61;
then
A11: x in K0 by XBOOLE_0:def 4;
K0 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A11;
(sn-FanMorphW).p=y by A10,A11,FUNCT_1:49;
then y in K0 by A2,A11,Th34;
hence thesis by PRE_TOPC:8;
end;
A12: 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`1<=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
Y1`1=0 & Y1`2=1 by EUCLID:52;
then
A13: Y1 in {p where p is Point of TOP-REAL 2: p`1>=0 & p<>0.TOP-REAL 2} by
JGRAPH_2:3;
A14: the carrier of ((TOP-REAL 2)|D) = (NonZero TOP-REAL 2) by A5,PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`1>=0;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A5);
then reconsider
K1={p: p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A13;
A15: K0 is closed & K1 is closed by A5,Th29,Th31;
dom ((sn-FanMorphW)|K0)= dom ((sn-FanMorphW)) /\ K0 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1
.=K0 by A12,XBOOLE_1:28;
then reconsider f=(sn-FanMorphW)|K0 as Function of ((TOP-REAL 2)|D)|K0, ((
TOP-REAL 2)|D) by A6,A8,FUNCT_2:2,XBOOLE_1:1;
A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8;
A17: rng ((sn-FanMorphW)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1
proof
let y be object;
assume y in rng ((sn-FanMorphW)|K1);
then consider x being object such that
A18: x in dom ((sn-FanMorphW)|K1) and
A19: y=((sn-FanMorphW)|K1).x by FUNCT_1:def 3;
x in (dom (sn-FanMorphW)) /\ K1 by A18,RELAT_1:61;
then
A20: x in K1 by XBOOLE_0:def 4;
K1 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A20;
(sn-FanMorphW).p=y by A19,A20,FUNCT_1:49;
then y in K1 by A2,A20,Th35;
hence thesis by PRE_TOPC:8;
end;
A21: 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>=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
dom ((sn-FanMorphW)|K1)= dom ((sn-FanMorphW)) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1
.=K1 by A21,XBOOLE_1:28;
then reconsider g=(sn-FanMorphW)|K1 as Function of ((TOP-REAL 2)|D)|K1, ((
TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1;
A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
A23: D c= K0 \/ K1
proof
let x be object;
assume
A24: x in D;
then reconsider px=x as Point of TOP-REAL 2;
not x in {0.TOP-REAL 2} by A5,A24,XBOOLE_0:def 5;
then px`1<=0 & px<>0.TOP-REAL 2 or px`1>=0 & px<>0.TOP-REAL 2 by
TARSKI:def 1;
then x in K0 or x in K1;
hence thesis by XBOOLE_0:def 3;
end;
A25: dom f=K0 by A6,FUNCT_2:def 1;
A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
A27: 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
A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) );
then x in K0 by A26,XBOOLE_0:def 4;
then f.x=(sn-FanMorphW).x by FUNCT_1:49;
hence thesis by A22,A28,FUNCT_1:49;
end;
D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
then
A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#]((
TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10;
A30: f is continuous & g is continuous by A2,A5,Th30,Th32;
then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that
A31: h= f+*g and
h is continuous by A26,A22,A29,A15,A27,JGRAPH_2:1;
A32: dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1;
A33: dom g=K1 by A16,FUNCT_2:def 1;
K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
then
A34: f tolerates g by A27,A25,A33,PARTFUN1:def 4;
A35: for x being object st x in dom h holds h.x=((sn-FanMorphW)|D).x
proof
let x be object;
assume
A36: x in dom h;
then reconsider p=x as Point of TOP-REAL 2 by A14,XBOOLE_0:def 5;
A37: x in D`` by A32,A36,PRE_TOPC:8;
not x in {0.TOP-REAL 2} by A14,A36,XBOOLE_0:def 5;
then
A38: x <>0.TOP-REAL 2 by TARSKI:def 1;
per cases;
suppose
A39: x in K0;
A40: (sn-FanMorphW)|D.p=(sn-FanMorphW).p by A37,FUNCT_1:49
.=f.p by A39,FUNCT_1:49;
h.p=(g+*f).p by A31,A34,FUNCT_4:34
.=f.p by A25,A39,FUNCT_4:13;
hence thesis by A40;
end;
suppose
not x in K0;
then not p`1<=0 by A38;
then
A41: x in K1 by A38;
(sn-FanMorphW)|D.p=(sn-FanMorphW).p by A37,FUNCT_1:49
.=g.p by A41,FUNCT_1:49;
hence thesis by A31,A33,A41,FUNCT_4:13;
end;
end;
dom (sn-FanMorphW)=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1;
then dom ((sn-FanMorphW)|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|D) by A4,XBOOLE_1:28;
then f+*g=(sn-FanMorphW)|D by A31,A32,A35,FUNCT_1:2;
hence thesis by A26,A22,A29,A30,A15,A27,JGRAPH_2:1;
end;
Lm11: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
theorem Th37:
for sn being Real st -1<sn & sn<1 holds ex h being Function of (
TOP-REAL 2),(TOP-REAL 2) st h=(sn-FanMorphW) & h is continuous
proof
reconsider D=NonZero TOP-REAL 2 as non empty Subset of TOP-REAL 2 by
JGRAPH_2:9;
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
reconsider f=(sn-FanMorphW) as Function of (TOP-REAL 2),(TOP-REAL 2);
A3: f.(0.TOP-REAL 2)=0.TOP-REAL 2 by Th16,JGRAPH_2:3;
A4: 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;
A5: [#]((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 A5,XBOOLE_0:def 5;
then
A6: p<>0.TOP-REAL 2 by TARSKI:def 1;
per cases;
suppose
A7: q`2/|.q.|>=sn & q`1<=0;
set q9= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q
.|-sn)/(1-sn))]|;
A8: q9`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
A9: q9`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
now
assume
A10: q9=0.TOP-REAL 2;
A11: |.q.|<>0^2 by A6,TOPRNS_1:24;
then -sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)=-sqrt(1-0) by A8,A10,JGRAPH_2:3
,XCMPLX_1:6
.=-1;
hence contradiction by A9,A10,A11,JGRAPH_2:3,XCMPLX_1:6;
end;
hence thesis by A1,A2,A3,A6,A7,Th18;
end;
suppose
A12: q`2/|.q.|<sn & q`1<=0;
set q9=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|
-sn)/(1+sn))]|;
A13: q9`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
A14: q9`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
now
assume
A15: q9=0.TOP-REAL 2;
A16: |.q.|<>0^2 by A6,TOPRNS_1:24;
then -sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)=-sqrt(1-0) by A13,A15,
JGRAPH_2:3,XCMPLX_1:6
.=-1;
hence contradiction by A14,A15,A16,JGRAPH_2:3,XCMPLX_1:6;
end;
hence thesis by A1,A2,A3,A6,A12,Th18;
end;
suppose
q`1>0;
then f.p=p by Th16;
hence thesis by A6,Th16,JGRAPH_2:3;
end;
end;
A17: 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 Lm11;
assume that
A18: f.(0.TOP-REAL 2) in V and
A19: V is open;
VV is open by A19,Lm11,PRE_TOPC:30;
then consider r being Real such that
A20: r>0 and
A21: Ball(u0,r) c= V by A3,A18,TOPMETR:15;
reconsider r as Real;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2;
A22: W1 is open by GOBOARD6:3;
A23: f.:W1 c= W1
proof
let z be object;
assume z in f.:W1;
then consider y being object such that
A24: y in dom f and
A25: y in W1 and
A26: z=f.y by FUNCT_1:def 6;
z in rng f by A24,A26,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 A24;
reconsider qy=q as Point of Euclid 2 by EUCLID:67;
dist(u0,qy)<r by A25,METRIC_1:11;
then
A27: |.(0.TOP-REAL 2) - q.|<r by JGRAPH_1:28;
per cases by JGRAPH_2:3;
suppose
q`1>=0;
hence thesis by A25,A26,Th16;
end;
suppose
A28: q<>0.TOP-REAL 2 & q`2/|.q.|>=sn & q`1<=0;
then
A29: (q`2/|.q.|-sn)>= 0 by XREAL_1:48;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A30: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A31: 1-sn>0 by A2,XREAL_1:149;
|.q.|<>0 by A28,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`2)^2/(|.q.|)^2 <= 1 by A30,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A31,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A31,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A31,A29,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A32: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
A33: (sn-FanMorphW).q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn) )^2)),
|.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A28,Th18;
then
A34: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by A26,EUCLID:52;
qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by A26,A33,EUCLID:52;
then
A35: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A32,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A34,A35;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A36: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A27,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A36,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
suppose
A37: q<>0.TOP-REAL 2 & q`2/|.q.|<sn & q`1<=0;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A38: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A39: 1+sn>0 by A1,XREAL_1:148;
|.q.|<>0 by A37,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`2)^2/(|.q.|)^2 <= 1 by A38,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then --1>=-q`2/|.q.| by XREAL_1:24;
then 1+sn>=-q`2/|.q.|+sn by XREAL_1:7;
then
A40: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A39,XREAL_1:185;
(sn-q`2/|.q.|)>=0 by A37,XREAL_1:48;
then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A39;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A40,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A41: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
A42: (sn-FanMorphW).q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn) )^2)),
|.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A37,Th18;
then
A43: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by A26,EUCLID:52;
qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by A26,A42,EUCLID:52;
then
A44: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A41,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A43,A44;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A45: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A27,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A45,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
end;
u0 in W1 by A20,GOBOARD6:1;
hence thesis by A21,A22,A23,XBOOLE_1:1;
end;
A46: D`= {0.TOP-REAL 2} by JGRAPH_3:20;
then
ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=(sn-FanMorphW
)|D & h is continuous by A1,A2,Th36;
hence thesis by A3,A46,A4,A17,JGRAPH_3:3;
end;
theorem Th38:
for sn being Real st -1<sn & sn<1 holds sn-FanMorphW is one-to-one
proof
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
for x1,x2 being object st x1 in dom (sn-FanMorphW) & x2 in dom (sn
-FanMorphW) & (sn-FanMorphW).x1=(sn-FanMorphW).x2 holds x1=x2
proof
let x1,x2 be object;
assume that
A3: x1 in dom (sn-FanMorphW) and
A4: x2 in dom (sn-FanMorphW) and
A5: (sn-FanMorphW).x1=(sn-FanMorphW).x2;
reconsider p2=x2 as Point of TOP-REAL 2 by A4;
reconsider p1=x1 as Point of TOP-REAL 2 by A3;
set q=p1,p=p2;
A6: 1-sn>0 by A2,XREAL_1:149;
now
per cases by JGRAPH_2:3;
case
A7: q`1>=0;
then
A8: (sn-FanMorphW).q=q by Th16;
now
per cases by JGRAPH_2:3;
case
p`1>=0;
hence thesis by A5,A8,Th16;
end;
case
A9: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1<=0;
set p4= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1-sn))]|;
A10: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
0<=(p`1)^2 by XREAL_1:63;
then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then
A11: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A10,XREAL_1:72;
A12: |.p.|>0 by A9,Lm1;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`2)^2/(|.p.|)^2 <= 1 by A11,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`2/|.p.| by SQUARE_1:51;
then 1-sn>=p`2/|.p.|-sn by XREAL_1:9;
then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72;
then
A13: -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197;
A14: (p`2/|.p.|-sn)>=0 by A9,XREAL_1:48;
A15: sn -FanMorphW.p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)
), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A9,Th18;
(p`2/|.p.|-sn)>= 0 by A9,XREAL_1:48;
then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A13,SQUARE_1:49;
then
A16: 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`2/|.p.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`2/|.p.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(p`2/|.p.|-sn)^2/(1-sn)^2)>=0;
then sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
then p4`1= |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) & q`1=0 by A5
,A7,A8,A15,EUCLID:52;
then
A17: (-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6;
1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by A16,XCMPLX_1:187;
then 1-((p`2/|.p.|-sn)/(1-sn))^2=0 by A17,SQUARE_1:24;
then 1= (p`2/|.p.|-sn)/(1-sn) by A6,A14,SQUARE_1:18,22;
then 1 *(1-sn)=(p`2/|.p.|-sn) by A6,XCMPLX_1:87;
then 1 *|.p.|=p`2 by A12,XCMPLX_1:87;
then p`1=0 by A10,XCMPLX_1:6;
hence thesis by A5,A8,Th16;
end;
case
A18: p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1<=0;
then
A19: sn -FanMorphW.p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)
), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,Th18;
set p4= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1+sn))]|;
A20: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
0<=(p`1)^2 by XREAL_1:63;
then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then
A21: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A20,XREAL_1:72;
A22: 1+sn>0 by A1,XREAL_1:148;
A23: (p`2/|.p.|-sn)<=0 by A18,XREAL_1:47;
then
A24: -1<=(-( p`2/|.p.|-sn))/(1+sn) by A22;
A25: |.p.|>0 by A18,Lm1;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`2)^2/(|.p.|)^2 <= 1 by A21,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then (-((p`2)/|.p.|))^2 <= 1;
then 1>= -p`2/|.p.| by SQUARE_1:51;
then (1+sn)>= -p`2/|.p.|+sn by XREAL_1:7;
then (-(p`2/|.p.|-sn))/(1+sn)<=1 by A22,XREAL_1:185;
then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A24,SQUARE_1:49;
then
A26: 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`2/|.p.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`2/|.p.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((p`2/|.p.|-sn))^2/(1+sn)^2)>=0;
then sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
then p4`1= |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) & q`1=0 by A5
,A7,A8,A19,EUCLID:52;
then
A27: (-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))=0 by A5,A8,A19,A25,XCMPLX_1:6;
1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by A26,XCMPLX_1:187;
then 1-((p`2/|.p.|-sn)/(1+sn))^2=0 by A27,SQUARE_1:24;
then 1=sqrt((-((p`2/|.p.|-sn)/(1+sn)))^2);
then 1= -((p`2/|.p.|-sn)/(1+sn)) by A22,A23,SQUARE_1:22;
then 1= ((-(p`2/|.p.|-sn))/(1+sn)) by XCMPLX_1:187;
then 1 *(1+sn)=-(p`2/|.p.|-sn) by A22,XCMPLX_1:87;
then 1+sn-sn=-p`2/|.p.|;
then 1=(-p`2)/|.p.| by XCMPLX_1:187;
then 1 *|.p.|=-p`2 by A25,XCMPLX_1:87;
then (p`2)^2-(p`2)^2 =(p`1)^2 by A20,XCMPLX_1:26;
then p`1=0 by XCMPLX_1:6;
hence thesis by A5,A8,Th16;
end;
end;
hence thesis;
end;
case
A28: q`2/|.q.|>=sn & q`1<=0 & q<>0.TOP-REAL 2;
then |.q.|>0 by Lm1;
then
A29: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.
q.|-sn)/(1-sn))]|;
A30: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
A31: sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn)) ^2)),
|.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A28,Th18;
A32: q4`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
now
per cases by JGRAPH_2:3;
case
A33: p`1>=0;
A34: (q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48;
A35: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then
A36: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A35,XREAL_1:72;
A37: (sn-FanMorphW).p=p by A33,Th16;
A38: (q`2/|.q.|-sn)>=0 by A28,XREAL_1:48;
A39: 1-sn>0 by A2,XREAL_1:149;
A40: |.q.|>0 by A28,Lm1;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`2)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A39,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A39,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A39,A34,SQUARE_1:49;
then
A41: 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0;
then sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
then p`1=0 by A5,A31,A33,A37,EUCLID:52;
then
A42: (-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))=0 by A5,A31,A32,A37,A40,
XCMPLX_1:6;
1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by A41,XCMPLX_1:187;
then 1-((q`2/|.q.|-sn)/(1-sn))^2=0 by A42,SQUARE_1:24;
then 1= (q`2/|.q.|-sn)/(1-sn) by A39,A38,SQUARE_1:18,22;
then 1 *(1-sn)=(q`2/|.q.|-sn) by A39,XCMPLX_1:87;
then 1 *|.q.|=q`2 by A40,XCMPLX_1:87;
then q`1=0 by A35,XCMPLX_1:6;
hence thesis by A5,A37,Th16;
end;
case
A43: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1<=0;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XREAL_1:72;
then
A44: -1<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XCMPLX_1:197;
(q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A6,A44,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A45: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
q4`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
then
A46: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A45,SQUARE_1:def 2;
A47: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A47,A46;
then sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A48: |.q4.|=|.q.| by SQUARE_1:22;
0<=(p`1)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A49: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A50: |.p.|>0 by A43,Lm1;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`2)^2/(|.p.|)^2 <= 1 by A49,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`2/|.p.| by SQUARE_1:51;
then 1-sn>=p`2/|.p.|-sn by XREAL_1:9;
then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72;
then
A51: -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197;
set p4 = |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1-sn))]|;
A52: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52;
(p`2/|.p.|-sn)>= 0 by A43,XREAL_1:48;
then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A51,SQUARE_1:49;
then 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A53: 1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
p4`1= |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
then
A54: (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))^2
.= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1-sn))^2) by A53,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A52,A54;
then sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A55: |.p4.|=|.p.| by SQUARE_1:22;
A56: sn-FanMorphW.p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/ (1-sn))^2
)), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A43,Th18;
then ((p`2/|.p.|-sn)/(1-sn)) =|.q.|* ((q`2/|.q.|-sn)/(1-sn))/|.p
.| by A5,A31,A30,A52,A50,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1-sn)=(q`2/|.q.|-sn)/(1-sn) by A5,A31,A56,A48
,A50,A55,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1-sn)*(1-sn)=q`2/|.q.|-sn by A6,XCMPLX_1:87;
then p`2/|.p.|-sn=q`2/|.q.|-sn by A6,XCMPLX_1:87;
then p`2/|.p.|*|.p.|=q`2 by A5,A31,A56,A48,A50,A55,XCMPLX_1:87;
then
A57: p`2=q`2 by A50,XCMPLX_1:87;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then (-p`1)^2=(q`1)^2 by A5,A31,A56,A48,A55,A57;
then -p`1=sqrt((-q`1)^2) by A43,SQUARE_1:22;
then
A58: --p`1=--q`1 by A28,SQUARE_1:22;
p=|[p`1,p`2]| by EUCLID:53;
hence thesis by A57,A58,EUCLID:53;
end;
case
A59: p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1<=0;
then p`2/|.p.|-sn<0 by XREAL_1:49;
then
A60: ((p`2/|.p.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148;
set p4= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p
`2/|.p.|-sn)/(1+sn))]|;
A61: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) & q`2/|.q.|-sn>=0 by A28,
EUCLID:52,XREAL_1:48;
A62: 1-sn>0 by A2,XREAL_1:149;
sn-FanMorphW.p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/ (1+sn))^2
)), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A59,Th18;
hence thesis by A5,A31,A30,A59,A60,A61,A62,Lm1,XREAL_1:132;
end;
end;
hence thesis;
end;
case
A63: q`2/|.q.|<sn & q`1<=0 & q<>0.TOP-REAL 2;
then
A64: |.q.|>0 by Lm1;
then
A65: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.
q.|-sn)/(1+sn))]|;
A66: q4`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
A67: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
A68: sn-FanMorphW.q= |[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn)) ^2)),
|.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A63,Th18;
per cases by JGRAPH_2:3;
suppose
A69: p`1>=0;
A70: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A71: 1+sn>0 by A1,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A70,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A65,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then (-((q`2)/|.q.|))^2 <= 1;
then 1>= -q`2/|.q.| by SQUARE_1:51;
then (1+sn)>= -q`2/|.q.|+sn by XREAL_1:7;
then
A72: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A71,XREAL_1:185;
A73: (q`2/|.q.|-sn)<=0 by A63,XREAL_1:47;
then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A71;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A72,SQUARE_1:49;
then
A74: 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A75: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
A76: (sn-FanMorphW).p=p by A69,Th16;
sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by A74,SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((q`2/|.q.|-sn))^2/(1+sn)^2)>=0;
then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
then p`1=0 by A5,A68,A69,A76,EUCLID:52;
then (-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))=0 by A5,A68,A66,A64,A76,
XCMPLX_1:6;
then 1-((q`2/|.q.|-sn)/(1+sn))^2=0 by A75,SQUARE_1:24;
then 1=sqrt((-((q`2/|.q.|-sn)/(1+sn)))^2);
then 1= -((q`2/|.q.|-sn)/(1+sn)) by A71,A73,SQUARE_1:22;
then 1= ((-(q`2/|.q.|-sn))/(1+sn)) by XCMPLX_1:187;
then 1 *(1+sn)=-(q`2/|.q.|-sn) by A71,XCMPLX_1:87;
then 1+sn-sn=-q`2/|.q.|;
then 1=(-q`2)/|.q.| by XCMPLX_1:187;
then 1 *|.q.|=-q`2 by A64,XCMPLX_1:87;
then (q`2)^2-(q`2)^2 =(q`1)^2 by A70,XCMPLX_1:26;
then q`1=0 by XCMPLX_1:6;
hence thesis by A5,A76,Th16;
end;
suppose
A77: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1<=0;
set p4= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/
|.p.|-sn)/(1-sn))]|;
A78: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) & 1-sn>0 by A2,EUCLID:52
,XREAL_1:149;
q`2/|.q.|-sn<0 by A63,XREAL_1:49;
then
A79: ((q`2/|.q.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148;
sn-FanMorphW.p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1- sn))^2))
, |.p.|* ((p`2/ |.p.|-sn)/(1-sn))]| & p`2/|.p.|-sn>=0 by A1,A2,A77,Th18,
XREAL_1:48;
hence thesis by A5,A63,A68,A67,A79,A78,Lm1,XREAL_1:132;
end;
suppose
A80: p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1<=0;
0<=(p`1)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A81: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A82: 1+sn>0 by A1,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A65,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
then -(-1-sn)>= -(q`2/|.q.|-sn) by XREAL_1:24;
then
A83: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A82,XREAL_1:185;
(q`2/|.q.|-sn)<=0 by A63,XREAL_1:47;
then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A82;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A83,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A84: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
q4`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
then
A85: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A84,SQUARE_1:def 2;
A86: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
set p4= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/
|.p.|-sn)/(1+sn))]|;
A87: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A86,A85;
then sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A88: |.q4.|=|.q.| by SQUARE_1:22;
(p`2/|.p.|-sn)<=0 by A80,XREAL_1:47;
then
A89: -1<=(-( p`2/|.p.|-sn))/(1+sn) by A82;
A90: |.p.|>0 by A80,Lm1;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`2)^2/(|.p.|)^2 <= 1 by A81,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then -1<=p`2/|.p.| by SQUARE_1:51;
then -1-sn<=p`2/|.p.|-sn by XREAL_1:9;
then -(-1-sn)>= -(p`2/|.p.|-sn) by XREAL_1:24;
then (-(p`2/|.p.|-sn))/(1+sn)<=1 by A82,XREAL_1:185;
then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A89,SQUARE_1:49;
then 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A91: 1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
p4`1= |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
then
A92: (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))^2
.= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1+sn))^2) by A91,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A87,A92;
then sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A93: |.p4.|=|.p.| by SQUARE_1:22;
A94: sn-FanMorphW.p= |[ |.p.|*(-sqrt(1-((p`2/|.p.|-sn)/(1+ sn))^2))
, |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A80,Th18;
then ((p`2/|.p.|-sn)/(1+sn)) =|.q.|* ((q`2/|.q.|-sn)/(1+sn))/|.p.|
by A5,A68,A67,A87,A90,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1+sn)=(q`2/|.q.|-sn)/(1+sn) by A5,A68,A94,A88
,A90,A93,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1+sn)*(1+sn)=q`2/|.q.|-sn by A82,XCMPLX_1:87;
then p`2/|.p.|-sn=q`2/|.q.|-sn by A82,XCMPLX_1:87;
then p`2/|.p.|*|.p.|=q`2 by A5,A68,A94,A88,A90,A93,XCMPLX_1:87;
then
A95: p`2=q`2 by A90,XCMPLX_1:87;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then (-p`1)^2=(q`1)^2 by A5,A68,A94,A88,A93,A95;
then -p`1=sqrt((-q`1)^2) by A80,SQUARE_1:22;
then
A96: --p`1=--q`1 by A63,SQUARE_1:22;
p=|[p`1,p`2]| by EUCLID:53;
hence thesis by A95,A96,EUCLID:53;
end;
end;
end;
hence thesis;
end;
hence thesis by FUNCT_1:def 4;
end;
theorem Th39:
for sn being Real st -1<sn & sn<1 holds sn-FanMorphW is Function
of TOP-REAL 2,TOP-REAL 2 & rng (sn-FanMorphW) = the carrier of TOP-REAL 2
proof
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
thus sn-FanMorphW is Function of TOP-REAL 2,TOP-REAL 2;
for f being Function of TOP-REAL 2,TOP-REAL 2 st f=(sn-FanMorphW) holds
rng (sn-FanMorphW)=the carrier of TOP-REAL 2
proof
let f be Function of TOP-REAL 2,TOP-REAL 2;
assume
A3: f=sn-FanMorphW;
A4: 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 by JGRAPH_2:3;
case
q`1>=0;
then y=(sn-FanMorphW).q by Th16;
hence ex x being set st x in dom (sn-FanMorphW) & y=(sn-FanMorphW).x
by A3,A4;
end;
case
A5: q`2/|.q.|>=0 & q`1<=0 & q<>0.TOP-REAL 2;
A6: --(1+sn)>0 by A1,XREAL_1:148;
A7: 1-sn>=0 by A2,XREAL_1:149;
then q`2/|.q.|*(1-sn)>=0 by A5;
then -(1+sn)<= q`2/|.q.|*(1-sn) by A6;
then
A8: -1-sn+sn<= q`2/|.q.|*(1-sn)+sn by XREAL_1:7;
set px=|[ -(|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2), |.q.|*(q`2/|.q.|
*(1-sn)+sn)]|;
A9: px`2 = |.q.|*(q`2/|.q.|*(1-sn)+sn) by EUCLID:52;
|.q.|<>0 by A5,TOPRNS_1:24;
then
A10: |.q.|^2>0 by SQUARE_1:12;
A11: |.q.|>0 by A5,Lm1;
A12: dom (sn-FanMorphW)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A13: 1-sn>0 by A2,XREAL_1:149;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`2/|.q.|<=1 by SQUARE_1:51;
then q`2/|.q.|*(1-sn) <=1 *(1-sn) by A13,XREAL_1:64;
then q`2/|.q.|*(1-sn)+sn-sn <=1-sn;
then (q`2/|.q.|*(1-sn)+sn) <=1 by XREAL_1:9;
then 1^2>=(q`2/|.q.|*(1-sn)+sn)^2 by A8,SQUARE_1:49;
then
A14: 1-(q`2/|.q.|*(1-sn)+sn)^2>=0 by XREAL_1:48;
then
A15: sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2)>=0 by SQUARE_1:def 2;
A16: px`1 = -(|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2) by EUCLID:52;
then
|.px.|^2=(-(|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2))^2 +(|.q.|*(
q`2/|.q.|*(1-sn)+sn))^2 by A9,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2))^2 +(|.q.|)^2*((q`2
/|.q.|*(1-sn)+sn))^2;
then
A17: |.px.|^2=(|.q.|)^2*(1-(q`2/|.q.|*(1-sn)+sn)^2) +(|.q.|)^2*((q`2
/|.q.|*(1-sn)+sn))^2 by A14,SQUARE_1:def 2
.= (|.q.|)^2;
then
A18: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A19: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24;
(q`2/|.q.|*(1-sn)+sn)>=0+sn by A5,A7,XREAL_1:7;
then px`2/|.px.| >=sn by A5,A9,A18,TOPRNS_1:24,XCMPLX_1:89;
then
A20: (sn-FanMorphW).px =|[ |.px.|*(-sqrt(1-((px`2/|.px.|-sn )/(1-sn)
)^2)), |.px.|* ((px`2/|.px.|-sn)/(1-sn))]| by A1,A2,A16,A15,A19,Th18;
A21: |.px.|*(-sqrt((q`1/|.q.|)^2))=|.px.|*(--(q`1/|.q.|)) by A5,
SQUARE_1:23
.=q`1 by A11,A18,XCMPLX_1:87;
A22: |.px.|* ((px`2/|.px.|-sn)/(1-sn)) =|.q.|* (( ((q`2/|.q.|*(1-sn)
+sn))-sn)/(1-sn)) by A5,A9,A18,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`2/|.q.|) by A13,XCMPLX_1:89
.= q`2 by A5,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*(-sqrt(1-((px`2/|.px.|-sn)/(1-sn))^2)) = |.px.|*(-sqrt(1
-(q`2/|.px.|)^2)) by A5,A18,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*(-sqrt(1-(q`2)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*(-sqrt( (|.px.|)^2/(|.px.|)^2-(q`2)^2/(|.px.|)^2)) by A10
,A17,XCMPLX_1:60
.= |.px.|*(-sqrt( ((|.px.|)^2-(q`2)^2)/(|.px.|)^2)) by XCMPLX_1:120
.= |.px.|*(-sqrt( ((q`1)^2+(q`2)^2-(q`2)^2)/(|.px.|)^2)) by A17,
JGRAPH_3:1
.= |.px.|*(-sqrt((q`1/|.q.|)^2)) by A18,XCMPLX_1:76;
hence ex x being set st x in dom (sn-FanMorphW) & y=(sn-FanMorphW).x
by A20,A22,A21,A12,EUCLID:53;
end;
case
A23: q`2/|.q.|<0 & q`1<=0 & q<>0.TOP-REAL 2;
A24: 1+sn>=0 by A1,XREAL_1:148;
then q`2/|.q.|*(1+sn)<=0 by A23;
then 1-sn>= q`2/|.q.|*(1+sn) by A2,XREAL_1:149;
then
A25: 1-sn+sn>= q`2/|.q.|*(1+sn)+sn by XREAL_1:7;
set px=|[ -(|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2), |.q.|*(q`2/|.q.|
*(1+sn)+sn)]|;
A26: px`2 = |.q.|*(q`2/|.q.|*(1+sn)+sn) by EUCLID:52;
|.q.|<>0 by A23,TOPRNS_1:24;
then
A27: |.q.|^2>0 by SQUARE_1:12;
A28: |.q.|>0 by A23,Lm1;
A29: dom (sn-FanMorphW)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A30: 1+sn>0 by A1,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A27,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`2/|.q.|>=-1 by SQUARE_1:51;
then q`2/|.q.|*(1+sn) >=(-1)*(1+sn) by A30,XREAL_1:64;
then q`2/|.q.|*(1+sn)+sn-sn >=-1-sn;
then (q`2/|.q.|*(1+sn)+sn) >=-1 by XREAL_1:9;
then 1^2>=(q`2/|.q.|*(1+sn)+sn)^2 by A25,SQUARE_1:49;
then
A31: 1-(q`2/|.q.|*(1+sn)+sn)^2>=0 by XREAL_1:48;
then
A32: sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2)>=0 by SQUARE_1:def 2;
A33: px`1 = -(|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2) by EUCLID:52;
then
|.px.|^2=(-(|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2))^2 +(|.q.|*(
q`2/|.q.|*(1+sn)+sn))^2 by A26,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2))^2 +(|.q.|)^2*((q`2
/|.q.|*(1+sn)+sn))^2;
then
A34: |.px.|^2=(|.q.|)^2*(1-(q`2/|.q.|*(1+sn)+sn)^2) +(|.q.|)^2*((q`2
/|.q.|*(1+sn)+sn))^2 by A31,SQUARE_1:def 2
.= (|.q.|)^2;
then
A35: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A36: px<>0.TOP-REAL 2 by A23,TOPRNS_1:23,24;
(q`2/|.q.|*(1+sn)+sn)<=0+sn by A23,A24,XREAL_1:7;
then px`2/|.px.| <=sn by A23,A26,A35,TOPRNS_1:24,XCMPLX_1:89;
then
A37: (sn-FanMorphW).px =|[ |.px.|*(-sqrt(1-((px`2/|.px.|-sn )/(1+sn)
)^2)), |.px.|* ((px`2/|.px.|-sn)/(1+sn))]| by A1,A2,A33,A32,A36,Th18;
A38: |.px.|*(-sqrt((q`1/|.q.|)^2))=|.px.|*(--(q`1/|.q.|)) by A23,
SQUARE_1:23
.=q`1 by A28,A35,XCMPLX_1:87;
A39: |.px.|* ((px`2/|.px.|-sn)/(1+sn)) =|.q.|* (( ((q`2/|.q.|*(1+sn)
+sn))-sn)/(1+sn)) by A23,A26,A35,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`2/|.q.|) by A30,XCMPLX_1:89
.= q`2 by A23,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*(-sqrt(1-((px`2/|.px.|-sn)/(1+sn))^2)) = |.px.|*(-sqrt(1
-(q`2/|.px.|)^2)) by A23,A35,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*(-sqrt(1-(q`2)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*(-sqrt( (|.px.|)^2/(|.px.|)^2-(q`2)^2/(|.px.|)^2)) by A27
,A34,XCMPLX_1:60
.= |.px.|*(-sqrt( ((|.px.|)^2-(q`2)^2)/(|.px.|)^2)) by XCMPLX_1:120
.= |.px.|*(-sqrt( ((q`1)^2+(q`2)^2-(q`2)^2)/(|.px.|)^2)) by A34,
JGRAPH_3:1
.= |.px.|*(-sqrt((q`1/|.q.|)^2)) by A35,XCMPLX_1:76;
hence ex x being set st x in dom (sn-FanMorphW) & y=(sn-FanMorphW).x
by A37,A39,A38,A29,EUCLID:53;
end;
end;
hence thesis by A3,FUNCT_1:def 3;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
hence thesis;
end;
Lm12: for q4,q,p2 be Point of TOP-REAL 2, O,u,uq be Point of Euclid 2 st u in
cl_Ball(O,|.p2.|+1) & q = uq & q4 = u & O=0.TOP-REAL 2 & |.q4.|=|.q.| holds q
in cl_Ball(O,|.p2.|+1)
proof
let q4,q,p2 be Point of TOP-REAL 2, O,u,uq be Point of Euclid 2;
assume
A1: u in cl_Ball(O,|.p2.|+1);
assume that
A2: q = uq and
A3: q4 = u and
A4: O=0.TOP-REAL 2;
assume
A5: |.q4.|=|.q.|;
now
assume not q in cl_Ball(O,|.p2.|+1);
then not dist(O,uq)<=|.p2.|+1 by A2,METRIC_1:12;
then |.0.TOP-REAL 2 - q.|> |.p2.|+1 by A2,A4,JGRAPH_1:28;
then |. -q.|> |.p2.|+1 by RLVECT_1:4;
then |.q.|> |.p2.|+1 by TOPRNS_1:26;
then |. -q4.|> |.p2.|+1 by A5,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q4.|> |.p2.|+1 by RLVECT_1:4;
then dist(O,u)> |.p2.|+1 by A3,A4,JGRAPH_1:28;
hence contradiction by A1,METRIC_1:12;
end;
hence q in cl_Ball(O,|.p2.|+1);
end;
theorem Th40:
for sn being Real,p2 being Point of TOP-REAL 2 st -1<sn & sn<1
ex K being non empty compact Subset of TOP-REAL 2 st K = (sn-FanMorphW).:K & ex
V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is open & V2 c= K & (sn
-FanMorphW).p2 in V2
proof
reconsider O=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67;
let sn be Real,p2 be Point of TOP-REAL 2;
A1: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider V0=Ball(O,|.p2.|+1) as Subset of TOP-REAL 2;
O in V0 & V0 c= cl_Ball(O,|.p2.|+1) by GOBOARD6:1,METRIC_1:14;
then reconsider
K0=cl_Ball(O,|.p2.|+1) as non empty compact Subset of TOP-REAL 2
by A1,Th15;
set q3= (sn-FanMorphW).p2;
reconsider VV0 = V0 as Subset of TopSpaceMetr Euclid 2;
reconsider u2=p2 as Point of Euclid 2 by EUCLID:67;
reconsider u3=q3 as Point of Euclid 2 by EUCLID:67;
A2: (sn-FanMorphW).:K0 c= K0
proof
let y be object;
assume y in (sn-FanMorphW).:K0;
then consider x being object such that
A3: x in dom (sn-FanMorphW) and
A4: x in K0 and
A5: y=(sn-FanMorphW).x by FUNCT_1:def 6;
reconsider q=x as Point of TOP-REAL 2 by A3;
reconsider uq=q as Point of Euclid 2 by EUCLID:67;
dist(O,uq)<= |.p2.|+1 by A4,METRIC_1:12;
then |.0.TOP-REAL 2 - q.|<= |.p2.|+1 by JGRAPH_1:28;
then |. -q.|<= |.p2.|+1 by RLVECT_1:4;
then
A6: |.q.|<= |.p2.|+1 by TOPRNS_1:26;
A7: y in rng (sn-FanMorphW) by A3,A5,FUNCT_1:def 3;
then reconsider u=y as Point of Euclid 2 by EUCLID:67;
reconsider q4=y as Point of TOP-REAL 2 by A7;
|.q4.|=|.q.| by A5,Th33;
then |. -q4.|<= |.p2.|+1 by A6,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q4.|<= |.p2.|+1 by RLVECT_1:4;
then dist(O,u)<= |.p2.|+1 by JGRAPH_1:28;
hence thesis by METRIC_1:12;
end;
VV0 is open by TOPMETR:14;
then
A8: V0 is open by Lm11,PRE_TOPC:30;
A9: |. p2.|<|.p2.|+1 by XREAL_1:29;
then |. -p2.|<|.p2.|+1 by TOPRNS_1:26;
then |.0.TOP-REAL 2 - p2.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u2)<|.p2.|+1 by JGRAPH_1:28;
then
A10: p2 in V0 by METRIC_1:11;
|.q3.|=|.p2.| by Th33;
then |. -q3.|<|.p2.|+1 by A9,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q3.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u3)< |.p2.|+1 by JGRAPH_1:28;
then
A11: (sn-FanMorphW).p2 in V0 by METRIC_1:11;
assume
A12: -1<sn & sn<1;
K0 c= (sn-FanMorphW).:K0
proof
let y be object;
assume
A13: y in K0;
then reconsider y as Point of Euclid 2;
reconsider q4=y as Point of TOP-REAL 2 by A13;
the carrier of TOP-REAL 2 c= rng (sn-FanMorphW) by A12,Th39;
then q4 in rng (sn-FanMorphW);
then consider x being object such that
A14: x in dom (sn-FanMorphW) and
A15: y=(sn-FanMorphW).x by FUNCT_1:def 3;
reconsider x as Point of Euclid 2 by A14,Lm11;
reconsider q=x as Point of TOP-REAL 2 by A14;
|.q4.|=|.q.| by A15,Th33;
then q in K0 by A13,Lm12;
hence thesis by A14,A15,FUNCT_1:def 6;
end;
then K0= (sn-FanMorphW).:K0 by A2,XBOOLE_0:def 10;
hence thesis by A10,A8,A11,METRIC_1:14;
end;
theorem
for sn being Real st -1<sn & sn<1 ex f being Function of TOP-REAL 2,
TOP-REAL 2 st f=(sn-FanMorphW) & f is being_homeomorphism
proof
let sn be Real;
reconsider f=(sn-FanMorphW) as Function of TOP-REAL 2,TOP-REAL 2;
assume
A1: -1<sn & sn<1;
then
A2: for p2 being Point of TOP-REAL 2 ex K being non empty compact Subset of
TOP-REAL 2 st K = f.:K & ex V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is
open & V2 c= K & f.p2 in V2 by Th40;
rng (sn-FanMorphW) = the carrier of TOP-REAL 2 & ex h being Function of
( TOP-REAL 2),(TOP-REAL 2) st h=(sn -FanMorphW) & h is continuous by A1,Th37
,Th39;
then f is being_homeomorphism by A1,A2,Th3,Th38;
hence thesis;
end;
Lm13: now
let q be Point of TOP-REAL 2, sn,t be Real;
assume ((-(t/|.q.|-sn))/(1-sn))^2<1^2;
then 1-((-(t/|.q.|-sn))/(1-sn))^2>0 by XREAL_1:50;
then sqrt(1-((-(t/|.q.|-sn))/(1-sn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(t/|.q.|-sn))^2/(1-sn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(t/|.q.|-sn)^2/(1-sn)^2)> 0;
then sqrt(1-((t/|.q.|-sn)/(1-sn))^2)> 0 by XCMPLX_1:76;
hence -sqrt(1-((t/|.q.|-sn)/(1-sn))^2)< -0 by XREAL_1:24;
end;
theorem Th42:
for sn being Real,q being Point of TOP-REAL 2 st sn<1 & q`1<0 &
q`2/|.q.|>=sn holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphW).q holds
p`1<0 & p`2>=0
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: sn<1 and
A2: q`1<0 and
A3: q`2/|.q.|>=sn;
A4: 1-sn>0 by A1,XREAL_1:149;
let p be Point of TOP-REAL 2;
set qz=p;
assume p=(sn-FanMorphW).q;
then
A5: p=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/
(1-sn))]| by A2,A3,Th16;
then
A6: qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
A7: (q`2/|.q.|-sn)>= 0 by A3,XREAL_1:48;
A8: |.q.|>0 by A2,Lm1,JGRAPH_2:3;
then
A9: (|.q.|)^2>0 by SQUARE_1:12;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A9,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A9,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then 1>q`2/|.q.| by SQUARE_1:52;
then 1-sn>q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)< -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<(-( q`2/|.q.|-sn))/(1-sn) by A4,XREAL_1:74;
then -1<(-( q`2/|.q.|-sn))/(1-sn) by A4,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<1^2 by A4,A7,SQUARE_1:50;
hence thesis by A5,A8,A4,A6,A7,Lm13,EUCLID:52,XREAL_1:132;
end;
theorem Th43:
for sn being Real,q being Point of TOP-REAL 2 st -1<sn & q`1<0 &
q`2/|.q.|<sn holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphW).q holds
p`1<0 & p`2<0
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: q`1<0 and
A3: q`2/|.q.|<sn;
A4: 1+sn>0 by A1,XREAL_1:148;
A5: (q`2/|.q.|-sn)< 0 by A3,XREAL_1:49;
then -( q`2/|.q.|-sn)>0 by XREAL_1:58;
then (-(1+sn))/(1+sn)<(-( q`2/|.q.|-sn))/(1+sn) by A4,XREAL_1:74;
then
A6: -1<(-( q`2/|.q.|-sn))/(1+sn) by A4,XCMPLX_1:197;
|.q.|>0 by A2,Lm1,JGRAPH_2:3;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1<q`2/|.q.| by SQUARE_1:52;
then -1-sn<q`2/|.q.|-sn by XREAL_1:9;
then --(1+sn)> -(q`2/|.q.|-sn) by XREAL_1:24;
then (-(q`2/|.q.|-sn))/(1+sn)<1 by A4,XREAL_1:191;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<1^2 by A6,SQUARE_1:50;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>0 by XREAL_1:50;
then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)> 0;
then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)> 0 by XCMPLX_1:76;
then
A8: -sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)< -0 by XREAL_1:24;
let p be Point of TOP-REAL 2;
set qz=p;
assume p=(sn-FanMorphW).q;
then
p=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn)/
(1+sn))]| by A2,A3,Th17;
then
A9: qz`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) & qz`2= |.q.|* ((q`2/
|.q.|- sn)/(1+sn)) by EUCLID:52;
((q`2/|.q.|-sn)/(1+sn))<0 by A1,A5,XREAL_1:141,148;
hence thesis by A2,A9,A8,Lm1,JGRAPH_2:3,XREAL_1:132;
end;
theorem Th44:
for sn being Real,q1,q2 being Point of TOP-REAL 2 st sn<1 & q1`1
<0 & q1`2/|.q1.|>=sn & q2`1<0 & q2`2/|.q2.|>=sn & q1`2/|.q1.|<q2`2/|.q2.| holds
for p1,p2 being Point of TOP-REAL 2 st p1=(sn-FanMorphW).q1 & p2=(sn-FanMorphW)
.q2 holds p1`2/|.p1.|<p2`2/|.p2.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: sn<1 and
A2: q1`1<0 and
A3: q1`2/|.q1.|>=sn and
A4: q2`1<0 and
A5: q2`2/|.q2.|>=sn and
A6: q1`2/|.q1.|<q2`2/|.q2.|;
A7: q1`2/|.q1.|-sn< q2`2/|.q2.|-sn & 1-sn>0 by A1,A6,XREAL_1:9,149;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(sn-FanMorphW).q1 and
A9: p2=(sn-FanMorphW).q2;
A10: |.p2.|=|.q2.| by A9,Th33;
p2=|[ |.q2.|*(-sqrt(1-((q2`2/|.q2.|-sn)/(1-sn))^2)), |.q2.|* ((q2`2/|.
q2.|-sn)/(1-sn))]| by A4,A5,A9,Th16;
then
A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1-sn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`2/|.p2.|= ((q2`2/|.q2.|-sn)/(1-sn)) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|*(-sqrt(1-((q1`2/|.q1.|-sn)/(1-sn))^2)), |.q1.|* ((q1`2/|.q1
.|-sn)/(1-sn))]| by A2,A3,A8,Th16;
then
A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1-sn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th33;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`2/|.p1.|= ((q1`2/|.q1.|-sn)/(1-sn)) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem Th45:
for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1<sn & q1
`1<0 & q1`2/|.q1.|<sn & q2`1<0 & q2`2/|.q2.|<sn & q1`2/|.q1.|<q2`2/|.q2.| holds
for p1,p2 being Point of TOP-REAL 2 st p1=(sn-FanMorphW).q1 & p2=(sn-FanMorphW)
.q2 holds p1`2/|.p1.|<p2`2/|.p2.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: q1`1<0 and
A3: q1`2/|.q1.|<sn and
A4: q2`1<0 and
A5: q2`2/|.q2.|<sn and
A6: q1`2/|.q1.|<q2`2/|.q2.|;
A7: q1`2/|.q1.|-sn< q2`2/|.q2.|-sn & 1+sn>0 by A1,A6,XREAL_1:9,148;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(sn-FanMorphW).q1 and
A9: p2=(sn-FanMorphW).q2;
A10: |.p2.|=|.q2.| by A9,Th33;
p2=|[ |.q2.|*(-sqrt(1-((q2`2/|.q2.|-sn)/(1+sn))^2)), |.q2.|* ((q2`2/|.
q2.|-sn)/(1+sn))]| by A4,A5,A9,Th17;
then
A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1+sn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`2/|.p2.|= ((q2`2/|.q2.|-sn)/(1+sn)) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|*(-sqrt(1-((q1`2/|.q1.|-sn)/(1+sn))^2)), |.q1.|* ((q1`2/|.q1
.|-sn)/(1+sn))]| by A2,A3,A8,Th17;
then
A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1+sn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th33;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`2/|.p1.|= ((q1`2/|.q1.|-sn)/(1+sn)) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem
for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1<sn & sn<1 & q1
`1 < 0 & q2`1<0 & q1`2/|.q1.|<q2`2/|.q2.| holds for p1,p2 being Point of
TOP-REAL 2 st p1=(sn-FanMorphW).q1 & p2=(sn-FanMorphW).q2 holds p1`2/|.p1.|<p2
`2/|.p2.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: sn<1 and
A3: q1`1<0 and
A4: q2`1<0 and
A5: q1`2/|.q1.|<q2`2/|.q2.|;
let p1,p2 be Point of TOP-REAL 2;
assume that
A6: p1=(sn-FanMorphW).q1 and
A7: p2=(sn-FanMorphW).q2;
now
per cases;
case
q1`2/|.q1.|>=sn & q2`2/|.q2.|>=sn;
hence thesis by A2,A3,A4,A5,A6,A7,Th44;
end;
case
q1`2/|.q1.|>=sn & q2`2/|.q2.|<sn;
hence thesis by A5,XXREAL_0:2;
end;
case
A8: q1`2/|.q1.|<sn & q2`2/|.q2.|>=sn;
then p2`2>=0 by A2,A4,A7,Th42;
then
A9: p2`2/|.p2.|>=0;
p1`2<0 by A1,A3,A6,A8,Th43;
hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141;
end;
case
q1`2/|.q1.|<sn & q2`2/|.q2.|<sn;
hence thesis by A1,A3,A4,A5,A6,A7,Th45;
end;
end;
hence thesis;
end;
theorem
for sn being Real,q being Point of TOP-REAL 2 st q`1<0 & q`2/|.q.|=sn
holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphW).q holds p`1<0 & p`2=0
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: q`1<0 and
A2: q`2/|.q.|=sn;
let p be Point of TOP-REAL 2;
A3: |.q.|>0 by A1,Lm1,JGRAPH_2:3;
assume p=(sn-FanMorphW).q;
then
A4: p=|[ |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/
(1-sn))]| by A1,A2,Th16;
then p`1= |.q.|*(-sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
hence thesis by A2,A4,A3,EUCLID:52,XREAL_1:132;
end;
theorem
for sn being Real holds 0.TOP-REAL 2=(sn-FanMorphW).(0.TOP-REAL
2) by Th16,JGRAPH_2:3;
begin :: Fan Morphism for North
definition
let s be Real, q be Point of TOP-REAL 2;
func FanN(s,q) -> Point of TOP-REAL 2 equals
:Def4:
|.q.|*|[(q`1/|.q.|-s)/(1
-s), sqrt(1-((q`1/|.q.|-s)/(1-s))^2)]| if q`1/|.q.|>=s & q`2>0, |.q.|*|[(q`1/|.
q.|-s)/(1+s), sqrt(1-((q`1/|.q.|-s)/(1+s))^2)]| if q`1/|.q.|<s & q`2>0
otherwise q;
correctness;
end;
definition
let c be Real;
func c-FanMorphN -> Function of TOP-REAL 2, TOP-REAL 2 means
:Def5:
for q being Point of TOP-REAL 2 holds it.q=FanN(c,q);
existence
proof
deffunc F(Point of TOP-REAL 2)=FanN(c,$1);
thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point
of TOP-REAL 2 holds IT.q=F(q) from FUNCT_2:sch 4;
end;
uniqueness
proof
deffunc F(Point of TOP-REAL 2)=FanN(c,$1);
thus for a,b being Function of TOP-REAL 2, TOP-REAL 2 st (for q being
Point of TOP-REAL 2 holds a.q=F(q)) & (for q being Point of TOP-REAL 2 holds b.
q=F(q)) holds a = b from BINOP_2:sch 1;
end;
end;
theorem Th49:
for cn being Real holds (q`1/|.q.|>=cn & q`2>0 implies cn
-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q`1/|.q.|-cn)/
(1-cn))^2))]|)& (q`2<=0 implies cn-FanMorphN.q=q)
proof
let cn be Real;
hereby
assume q`1/|.q.|>=cn & q`2>0;
then
FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), sqrt(1-((q`1/|.q.|-cn)/(1-
cn))^2)]| by Def4
.= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q`1/|.q.|-cn)/(1
-cn))^2))]| by EUCLID:58;
hence cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(sqrt(1-((q
`1/|.q.|-cn)/(1-cn))^2))]| by Def5;
end;
assume
A1: q`2<=0;
cn-FanMorphN.q=FanN(cn,q) by Def5;
hence thesis by A1,Def4;
end;
theorem Th50:
for cn being Real holds (q`1/|.q.|<=cn & q`2>0 implies cn
-FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/
(1+cn))^2))]|)
proof
let cn be Real;
assume that
A1: q`1/|.q.|<=cn and
A2: q`2>0;
per cases by A1,XXREAL_0:1;
suppose
q`1/|.q.|<cn;
then
FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1+cn), sqrt(1-((q`1/|.q.|-cn)/(1+
cn))^2)]| by A2,Def4
.= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(
1+cn))^2))]| by EUCLID:58;
hence thesis by Def5;
end;
suppose
A3: q`1/|.q.|=cn;
then (q`1/|.q.|-cn)/(1-cn)=0;
hence thesis by A2,A3,Th49;
end;
end;
theorem Th51:
for cn being Real st -1<cn & cn<1 holds (q`1/|.q.|>=cn & q`2>=0
& q<>0.TOP-REAL 2 implies cn-FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1-cn)), |.q
.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]|) & (q`1/|.q.|<=cn & q`2>=0 & q<>0.
TOP-REAL 2 implies cn-FanMorphN.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*(
sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]|)
proof
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
per cases;
suppose
A3: q`1/|.q.|>=cn & q`2>=0 & q<>0.TOP-REAL 2;
per cases;
suppose
A4: q`2>0;
then
FanN(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), sqrt(1-((q`1/|.q.|-cn)/(1
-cn))^2)]| by A3,Def4
.= |[|.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/
(1-cn))^2))]| by EUCLID:58;
hence thesis by A4,Def5,Th50;
end;
suppose
A5: q`2<=0;
then
A6: cn-FanMorphN.q=q by Th49;
A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A8: 1-cn>0 by A2,XREAL_1:149;
A9: q`2=0 by A3,A5;
|.q.|<>0 by A3,TOPRNS_1:24;
then |.q.|^2>0 by SQUARE_1:12;
then (q`1)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60;
then ((q`1)/|.q.|)^2=1^2 by XCMPLX_1:76;
then
A10: sqrt(((q`1)/|.q.|)^2)=1;
A11: now
assume q`1<0;
then -((q`1)/|.q.|)=1 by A10,SQUARE_1:23;
hence contradiction by A1,A3;
end;
sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22;
then
A12: |.q.|=q`1 by A7,A9,A11,SQUARE_1:22;
then 1=q`1/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60;
then (q`1/|.q.|-cn)/(1-cn)=1 by A8,XCMPLX_1:60;
hence thesis by A2,A6,A9,A12,EUCLID:53,TOPRNS_1:24
,XCMPLX_1:60;
end;
end;
suppose
A13: q`1/|.q.|<=cn & q`2>=0 & q<>0.TOP-REAL 2;
per cases;
suppose
q`2>0;
hence thesis by Th49,Th50;
end;
suppose
A14: q`2<=0;
A15: 1+cn>0 by A1,XREAL_1:148;
A16: |.q.|<>0 by A13,TOPRNS_1:24;
A17: q`2=0 by A13,A14;
|.q.|>0 & 1>q`1/|.q.| by A2,A13,Lm1,XXREAL_0:2;
then 1 *(|.q.|)>q`1/|.q.|*(|.q.|) by XREAL_1:68;
then
A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`1 by A13,JGRAPH_3:1,TOPRNS_1:24
,XCMPLX_1:87;
then
A19: q`1= -(|.q.|) by A17,SQUARE_1:40;
then -1=q`1/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197;
then
A20: (q`1/|.q.|-cn)/(1+cn) =(-(1+cn))/(1+cn) .=-1 by A15,XCMPLX_1:197;
|.q.|=-q`1 by A17,A18,SQUARE_1:40;
then
|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(
1+cn))^2))]| =q by A17,A20,EUCLID:53;
hence thesis by A1,A14,A16,A19,Th49,XCMPLX_1:197;
end;
end;
suppose
q`2<0 or q=0.TOP-REAL 2;
hence thesis;
end;
end;
theorem Th52:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1-
cn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2>=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=(1-cn);
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: cn<1 and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2>=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in K1 by A7,A8,A9,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1-cn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th53:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<cn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`1/|.p.|
-cn)/(1+cn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`2>=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=(1+cn);
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: -1<cn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2>=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
1+cn>0 by A1,XREAL_1:148;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
A8: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in dom g3 by A7,A9;
then x in K1 by A7,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1+cn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
dom f=dom g3 by A7,FUNCT_2:def 1;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th54:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*( sqrt(1-((p`1/|.p.|-
cn)/(1-cn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`2>=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2) holds f is
continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=(1-cn);
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: cn<1 and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2>=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th10;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
A9: 1-cn>0 by A1,XREAL_1:149;
assume
A10: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
A11: |.r.|<>0 by A3,A10,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2;
(r`2)^2>=0 by XREAL_1:63;
then r`1<= |.r.| by A12,XREAL_1:93;
then r`1/|.r.| <= |.r.|/|.r.| by XREAL_1:72;
then r`1/|.r.|<=1 by A11,XCMPLX_1:60;
then
A13: r`1/|.r.|-cn<=(1-cn) by XREAL_1:9;
reconsider s=x as Point of (TOP-REAL 2)|K1 by A10;
A14: now
assume (1-cn)^2=0;
then 1-cn+cn=0+cn by XCMPLX_1:6;
hence contradiction by A1;
end;
cn-r`1/|.r.|<=0 by A3,A10,XREAL_1:47;
then -(cn- r`1/|.r.|)>=-(1-cn) by A9,XREAL_1:24;
then (1-cn)^2>=0 & (r`1/|.r.|-cn)^2<=(1-cn)^2 by A13,SQUARE_1:49,XREAL_1:63
;
then (r`1/|.r.|-cn)^2/(1-cn)^2<=(1-cn)^2/(1-cn)^2 by XREAL_1:72;
then (r`1/|.r.|-cn)^2/(1-cn)^2<=1 by A14,XCMPLX_1:60;
then ((r`1/|.r.|-cn)/(1-cn))^2<=1 by XCMPLX_1:76;
then 1-((r`1/|.r.|-cn)/(1-cn))^2>=0 by XREAL_1:48;
then |.1-((r`1/|.r.|-cn)/(1-cn))^2.| =1-((r`1/|.r.|-cn)/(1-cn))^2 by
ABSVALUE:def 1;
then
A15: f.r=(|.r.|)*( sqrt(|.1-((r`1/|.r.|-cn)/(1-cn))^2.|)) by A2,A10;
A16: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
hence thesis by A5,A15,A16;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th55:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<cn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*( sqrt(1-((p
`1/|.p.|-cn)/(1+cn))^2))) & (for q being Point of TOP-REAL 2 st q in the
carrier of (TOP-REAL 2)|K1 holds q`2>=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2)
holds f is continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=(1+cn);
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: -1<cn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2>=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2;
A4: 1+cn>0 by A1,XREAL_1:148;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th10;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of TOP-REAL 2;
reconsider s=x as Point of (TOP-REAL 2)|K1 by A9;
A10: (1+cn)^2>0 by A4,SQUARE_1:12;
A11: |.r.|<>0 by A3,A9,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2;
(r`2)^2>=0 by XREAL_1:63;
then -(|.r.|)<=r`1 by A12,XREAL_1:93;
then r`1/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72;
then r`1/|.r.|>= -1 by A11,XCMPLX_1:197;
then r`1/|.r.|-cn>=-1-cn by XREAL_1:9;
then
A13: r`1/|.r.|-cn>=-(1+cn);
cn-r`1/|.r.|>=0 by A3,A9,XREAL_1:48;
then -(cn-r`1/|.r.|)<=-0;
then (r`1/|.r.|-cn)^2<=(1+cn)^2 by A4,A13,SQUARE_1:49;
then (r`1/|.r.|-cn)^2/(1+cn)^2<=(1+cn)^2/(1+cn)^2 by A4,XREAL_1:72;
then (r`1/|.r.|-cn)^2/(1+cn)^2<=1 by A10,XCMPLX_1:60;
then ((r`1/|.r.|-cn)/(1+cn))^2<=1 by XCMPLX_1:76;
then 1-((r`1/|.r.|-cn)/(1+cn))^2>=0 by XREAL_1:48;
then |.1-((r`1/|.r.|-cn)/(1+cn))^2.| =1-((r`1/|.r.|-cn)/(1+cn))^2 by
ABSVALUE:def 1;
then
A14: f.r=(|.r.|)*( sqrt(|.1-((r`1/|.r.|-cn)/(1+cn))^2.|)) by A2,A9;
A15: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
hence thesis by A5,A14,A15;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th56:
for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphN)|
K0 & B0={q where q is Point of TOP-REAL 2: q`2>=0 & q<>0.TOP-REAL 2} & K0={p: p
`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
p0`1=cn by EUCLID:52;
then
A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1;
assume
A3: -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0={q where q is Point of
TOP-REAL 2: q`2>=0 & q<>0.TOP-REAL 2} & K0={p: p`1/|.p.|>=cn & p`2>=0 & p<>0.
TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A4: 1-cn^2>0 by XREAL_1:50;
then sn^2=1-cn^2 by SQUARE_1:def 2;
then
A5: p0`1/|.p0.|=cn by A2,EUCLID:52;
p0`2>0 by A1,A4,SQUARE_1:25;
then
A6: p0 in K0 by A3,A5,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A7: rng (proj2*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17;
A8: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|>= cn & p8`2>=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A9: dom ((cn-FanMorphN)|K1) c= dom (proj1*((cn-FanMorphN)|K1))
proof
let x be object;
assume
A10: x in dom ((cn-FanMorphN)|K1);
then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphN) by XBOOLE_0:def 4;
then
A11: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn
-FanMorphN) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A10,FUNCT_1:47;
hence thesis by A10,A11,FUNCT_1:11;
end;
A12: rng (proj1*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj1*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25;
then dom (proj1*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A9,
XBOOLE_0:def 10
.=dom (cn-FanMorphN) /\ 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;
then reconsider
g2=proj1*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A12,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.|* ((p`1/|.p.|-cn)/(1-cn))
proof
let p be Point of TOP-REAL 2;
A13: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A15: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A14;
then
A16: (cn-FanMorphN).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-(
(p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th51;
((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A15,A14,FUNCT_1:49;
then g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/
|.p.|-cn)/(1-cn))^2))]|) by A15,A13,A14,A16,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(
1-cn))^2))]|)`1 by PSCOMP_1:def 5
.=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn));
A18: dom ((cn-FanMorphN)|K1) c= dom (proj2*((cn-FanMorphN)|K1))
proof
let x be object;
assume
A19: x in dom ((cn-FanMorphN)|K1);
then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphN) by XBOOLE_0:def 4;
then
A20: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn
-FanMorphN) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A19,FUNCT_1:47;
hence thesis by A19,A20,FUNCT_1:11;
end;
dom (proj2*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25;
then dom (proj2*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A18,
XBOOLE_0:def 10
.=dom (cn-FanMorphN) /\ 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;
then reconsider
g1=proj2*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A7,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.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))
proof
let p be Point of TOP-REAL 2;
A21: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A23: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A22;
then
A24: (cn-FanMorphN).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((
p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th51;
((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A23,A22,FUNCT_1:49;
then g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/
|.p.|-cn)/(1-cn))^2))]|) by A23,A21,A22,A24,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(
1-cn))^2))]|)`2 by PSCOMP_1:def 6
.= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2>=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A26;
hence thesis;
end;
then
A27: f1 is continuous by A3,A25,Th54;
A28: 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
A29: |[x,y]| in K1 and
A30: s=f2.(|[x,y]|) & r=f1.(|[x,y]|);
set p99=|[x,y]|;
A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| >=cn & p3`2>=0
& p3<>0.TOP-REAL 2 by A3,A29;
A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A33: f1.p99=|.p99.|*( sqrt(1-((p99`1/|.p99.|-cn)/(1-cn))^2)) by A25,A29;
((cn-FanMorphN)|K0).(|[x,y]|)=((cn-FanMorphN)).(|[x,y]|) by A29,FUNCT_1:49
.= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1-cn)), |.p99.|*( sqrt(1-((p99`1/
|.p99.|-cn)/(1-cn))^2))]| by A3,A31,Th51
.=|[s,r]| by A17,A29,A30,A32,A33;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2>=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A34;
hence thesis;
end;
then f2 is continuous by A3,A17,Th52;
hence thesis by A6,A8,A27,A28,JGRAPH_2:35;
end;
theorem Th57:
for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphN)|
K0 & B0={q where q is Point of TOP-REAL 2: q`2>=0 & q<>0.TOP-REAL 2} & K0={p: p
`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
p0`1=cn by EUCLID:52;
then
A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1;
assume
A3: -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0={q where q is Point of
TOP-REAL 2: q`2>=0 & q<>0.TOP-REAL 2} & K0={p: p`1/|.p.|<=cn & p`2>=0 & p<>0.
TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A4: 1-cn^2>0 by XREAL_1:50;
then sn^2=1-cn^2 by SQUARE_1:def 2;
then
A5: p0`1/|.p0.|=cn by A2,EUCLID:52;
p0`2>0 by A1,A4,SQUARE_1:25;
then
A6: p0 in K0 by A3,A5,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A7: rng (proj2*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17;
A8: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|<= cn & p8`2>=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A9: dom ((cn-FanMorphN)|K1) c= dom (proj1*((cn-FanMorphN)|K1))
proof
let x be object;
assume
A10: x in dom ((cn-FanMorphN)|K1);
then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphN) by XBOOLE_0:def 4;
then
A11: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn
-FanMorphN) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A10,FUNCT_1:47;
hence thesis by A10,A11,FUNCT_1:11;
end;
A12: rng (proj1*((cn-FanMorphN)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj1*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25;
then dom (proj1*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A9,
XBOOLE_0:def 10
.=dom (cn-FanMorphN) /\ 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;
then reconsider
g2=proj1*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A12,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.|* ((p`1/|.p.|-cn)/(1+cn))
proof
let p be Point of TOP-REAL 2;
A13: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A15: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A14;
then
A16: (cn-FanMorphN).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-(
(p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th51;
((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A15,A14,FUNCT_1:49;
then g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/
|.p.|-cn)/(1+cn))^2))]|) by A15,A13,A14,A16,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(
1+cn))^2))]|)`1 by PSCOMP_1:def 5
.=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn));
A18: dom ((cn-FanMorphN)|K1) c= dom (proj2*((cn-FanMorphN)|K1))
proof
let x be object;
assume
A19: x in dom ((cn-FanMorphN)|K1);
then x in dom (cn-FanMorphN) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphN) by XBOOLE_0:def 4;
then
A20: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphN).x in rng (cn
-FanMorphN) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphN)|K1).x=(cn-FanMorphN).x by A19,FUNCT_1:47;
hence thesis by A19,A20,FUNCT_1:11;
end;
dom (proj2*((cn-FanMorphN)|K1)) c= dom ((cn-FanMorphN)|K1) by RELAT_1:25;
then dom (proj2*((cn-FanMorphN)|K1)) =dom ((cn-FanMorphN)|K1) by A18,
XBOOLE_0:def 10
.=dom (cn-FanMorphN) /\ 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;
then reconsider
g1=proj2*((cn-FanMorphN)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A7,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.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))
proof
let p be Point of TOP-REAL 2;
A21: dom ((cn-FanMorphN)|K1)=dom (cn-FanMorphN) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A23: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A22;
then
A24: (cn-FanMorphN).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((
p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th51;
((cn-FanMorphN)|K1).p=(cn-FanMorphN).p by A23,A22,FUNCT_1:49;
then g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/
|.p.|-cn)/(1+cn))^2))]|) by A23,A21,A22,A24,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(
1+cn))^2))]|)`2 by PSCOMP_1:def 6
.= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2>=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A26;
hence thesis;
end;
then
A27: f1 is continuous by A3,A25,Th55;
A28: 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
A29: |[x,y]| in K1 and
A30: s=f2.(|[x,y]|) & r=f1.(|[x,y]|);
set p99=|[x,y]|;
A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| <=cn & p3`2>=0
& p3<>0.TOP-REAL 2 by A3,A29;
A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A33: f1.p99=|.p99.|*( sqrt(1-((p99`1/|.p99.|-cn)/(1+cn))^2)) by A25,A29;
((cn-FanMorphN)|K0).(|[x,y]|)=((cn-FanMorphN)).(|[x,y]|) by A29,FUNCT_1:49
.= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1+cn)), |.p99.|*( sqrt(1-((p99`1/
|.p99.|-cn)/(1+cn))^2))]| by A3,A31,Th51
.=|[s,r]| by A17,A29,A30,A32,A33;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2>=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2>=0 &
p3<>0.TOP-REAL 2 by A3,A34;
hence thesis;
end;
then f2 is continuous by A3,A17,Th53;
hence thesis by A6,A8,A27,A28,JGRAPH_2:35;
end;
theorem Th58:
for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`1
>=(cn)*(|.p.|) & p`2>=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`2>=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
assume
A1: K003={p: p`1>=(sn)*(|.p.|) & p`2>=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`1>=sn*|.$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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm8,JORDAN6:7;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th59:
for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`1
<=(cn)*(|.p.|) & p`2>=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`2>=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
assume
A1: K003={p: p`1<=(sn)*(|.p.|) & p`2>=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`1<=sn*|.$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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm10,JORDAN6:7;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th60:
for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphN)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`2>=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
assume
A2: -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2>=0 & p<>0.TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A3: 1-cn^2>0 by XREAL_1:50;
then
A4: p0`2>0 by A1,SQUARE_1:25;
then p0 in K0 by A2,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
p0<>0.TOP-REAL 2 by A1,A3,JGRAPH_2:3,SQUARE_1:25;
then not p0 in {0.TOP-REAL 2} by TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A2,XBOOLE_0:def 5;
A5: the carrier of (TOP-REAL 2)|K1 = K1 by PRE_TOPC:8;
p0`1=cn by EUCLID:52;
then
A6: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1;
A7: D<>{};
sn^2=1-cn^2 by A3,SQUARE_1:def 2;
then
A8: p0`1/|.p0.|=cn by A6,EUCLID:52;
then
A9: p0 in {p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} by A4,JGRAPH_2:3;
A10: {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2;
hence thesis by A2;
end;
A11: {p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2;
hence thesis by A2;
end;
then reconsider
K00={p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A9,PRE_TOPC:8;
the carrier of (TOP-REAL 2)|D =D by PRE_TOPC:8;
then
A12: rng (f|K00) c=D;
p0 in {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} by A4,A8,JGRAPH_2:3;
then reconsider
K11={p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A10,PRE_TOPC:8;
the carrier of (TOP-REAL 2)|D =D by PRE_TOPC:8;
then
A13: rng (f|K11) c=D;
the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D;
then
A14: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1
.=K1 by PRE_TOPC:8;
then dom (f|K00)=K00 by A11,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8;
then reconsider
f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A12,FUNCT_2:2
;
dom (f|K11)=K11 by A10,A14,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8;
then reconsider
f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A13,FUNCT_2:2
;
defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|>=cn & $1`2>=0 & $1<>0.
TOP-REAL 2;
A15: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1
.=K11 by PRE_TOPC:8;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K001={p: p`1/|.p.|>=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty
Subset of (TOP-REAL 2) by A9;
A16: the carrier of (TOP-REAL 2)|K1 =K1 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`1>=(cn)*(|.$1.|) & $1`2>=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K003={p: p`1>=(cn)*(|.p.|) & p`2>=0} as Subset of (TOP-REAL 2);
defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|<=cn & $1`2>=0 & $1<>0.
TOP-REAL 2;
A17: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
A18: rng ((cn-FanMorphN)|K001) c= K1
proof
let y be object;
assume y in rng ((cn-FanMorphN)|K001);
then consider x being object such that
A19: x in dom ((cn-FanMorphN)|K001) and
A20: y=((cn-FanMorphN)|K001).x by FUNCT_1:def 3;
x in dom (cn-FanMorphN) by A19,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A21: y=(cn-FanMorphN).q by A19,A20,FUNCT_1:47;
dom ((cn-FanMorphN)|K001)=(dom (cn-FanMorphN))/\ K001 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1
.=K001 by XBOOLE_1:28;
then
A22: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|>= cn & p2`2>=0 &
p2<>0.TOP-REAL 2 by A19;
then
A23: (q`1/|.q.|-cn)>= 0 by XREAL_1:48;
|.q.|<>0 by A22,TOPRNS_1:24;
then
A24: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/
(1-cn))^2))]|;
A25: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
A26: 1-cn>0 by A2,XREAL_1:149;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2 <= (|.q.|)^2 by JGRAPH_3:1;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A24,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A26,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A26,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A26,A23,SQUARE_1:49;
then
A27: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A28: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by A27,SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0;
then
A29: sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
A30: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
then
A31: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A28,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A25,A31;
then
A32: q4<>0.TOP-REAL 2 by A24,TOPRNS_1:23;
cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q
`1/|.q.|-cn)/(1-cn))^2))]| by A2,A22,Th51;
hence thesis by A2,A21,A30,A29,A32;
end;
A33: dom (cn-FanMorphN)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then dom ((cn-FanMorphN)|K001)=K001 by RELAT_1:62
.= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8;
then reconsider
f3=(cn-FanMorphN)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL
2)|K1 by A5,A18,FUNCT_2:2;
A34: K003 is closed by Th58;
K1 c= D
proof
let x be object;
assume
A35: x in K1;
then ex p6 being Point of TOP-REAL 2 st p6=x & p6`2>=0 & p6 <>0.TOP-REAL 2
by A2;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A2,A35,XBOOLE_0:def 5;
end;
then D=K1 \/ D by XBOOLE_1:12;
then
A36: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4;
p0 in {p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} by A4,A8,JGRAPH_2:3;
then reconsider
K111={p: p`1/|.p.|<=cn & p`2>=0 & p<>0.TOP-REAL 2} as non empty
Subset of TOP-REAL 2 by A17;
A37: rng ((cn-FanMorphN)|K111) c= K1
proof
let y be object;
assume y in rng ((cn-FanMorphN)|K111);
then consider x being object such that
A38: x in dom ((cn-FanMorphN)|K111) and
A39: y=((cn-FanMorphN)|K111).x by FUNCT_1:def 3;
x in dom (cn-FanMorphN) by A38,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A40: y=(cn-FanMorphN).q by A38,A39,FUNCT_1:47;
dom ((cn-FanMorphN)|K111)=(dom (cn-FanMorphN))/\ K111 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1
.=K111 by XBOOLE_1:28;
then
A41: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|<= cn & p2`2>=0 &
p2<>0.TOP-REAL 2 by A38;
then
A42: (q`1/|.q.|-cn)<=0 by XREAL_1:47;
|.q.|<>0 by A41,TOPRNS_1:24;
then
A43: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/
(1+cn))^2))]|;
A44: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
A45: 1+cn>0 by A2,XREAL_1:148;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1
,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A43,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A45,XREAL_1:72;
then -1<=(( q`1/|.q.|-cn))/(1+cn) by A45,XCMPLX_1:197;
then
A46: ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A45,A42,SQUARE_1:49;
then
A47: 1-((q`1/|.q.|-cn)/(1+cn))^2>=0 by XREAL_1:48;
1-(-((q`1/|.q.|-cn)/(1+cn)))^2>=0 by A46,XREAL_1:48;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)>=0;
then
A48: sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
A49: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
then
A50: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A47,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A44,A50;
then
A51: q4<>0.TOP-REAL 2 by A43,TOPRNS_1:23;
cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q
`1/|.q.|-cn)/(1+cn))^2))]| by A2,A41,Th51;
hence thesis by A2,A40,A49,A48,A51;
end;
dom ((cn-FanMorphN)|K111)=K111 by A33,RELAT_1:62
.= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8;
then reconsider
f4=(cn-FanMorphN)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL
2)|K1 by A16,A37,FUNCT_2:2;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A2,FUNCT_1:51
,GOBOARD9:2;
then
A52: f2 is continuous by A2,A36,Th57,PRE_TOPC:26;
A53: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8;
set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11;
A54: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5;
defpred P[Point of TOP-REAL 2] means $1`1<=cn*(|.$1.|) & $1`2>=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K004={p: p`1<=(cn)*(|.p.|) & p`2>=0} as Subset of (TOP-REAL 2);
A55: K004 /\ K1 c= K11
proof
let x be object;
assume
A56: x in K004 /\ K1;
then x in K004 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A57: q1=x and
A58: q1`1<=(cn)*(|.q1.|) and
q1`2>=0;
x in K1 by A56,XBOOLE_0:def 4;
then
A59: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2>=0 & q2 <>0.TOP-REAL 2
by A2;
q1`1/|.q1.|<=(cn)*(|.q1.|)/|.q1.| by A58,XREAL_1:72;
then q1`1/|.q1.|<=(cn) by A57,A59,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A57,A59;
end;
A60: K004 is closed by Th59;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A2,FUNCT_1:51
,GOBOARD9:2;
then
A61: f1 is continuous by A2,A36,Th56,PRE_TOPC:26;
A62: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
K11 c= K004 /\ K1
proof
let x be object;
assume x in K11;
then consider p such that
A63: p=x and
A64: p`1/|.p.|<=cn and
A65: p`2>=0 and
A66: p<>0.TOP-REAL 2;
p`1/|.p.|*|.p.|<=(cn)*(|.p.|) by A64,XREAL_1:64;
then p`1<=(cn)*(|.p.|) by A66,TOPRNS_1:24,XCMPLX_1:87;
then
A67: x in K004 by A63,A65;
x in K1 by A2,A63,A65,A66;
hence thesis by A67,XBOOLE_0:def 4;
end;
then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A62,A55,XBOOLE_0:def 10;
then
A68: K11 is closed by A60,PRE_TOPC:13;
A69: K003 /\ K1 c= K00
proof
let x be object;
assume
A70: x in K003 /\ K1;
then x in K003 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A71: q1=x and
A72: q1`1>=(cn)*(|.q1.|) and
q1`2>=0;
x in K1 by A70,XBOOLE_0:def 4;
then
A73: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2>=0 & q2 <>0.TOP-REAL 2
by A2;
q1`1/|.q1.|>=(cn)*(|.q1.|)/|.q1.| by A72,XREAL_1:72;
then q1`1/|.q1.|>=(cn) by A71,A73,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A71,A73;
end;
K00 c= K003 /\ K1
proof
let x be object;
assume x in K00;
then consider p such that
A74: p=x and
A75: p`1/|.p.|>=cn and
A76: p`2>=0 and
A77: p<>0.TOP-REAL 2;
p`1/|.p.|*|.p.|>=(cn)*(|.p.|) by A75,XREAL_1:64;
then p`1>=(cn)*(|.p.|) by A77,TOPRNS_1:24,XCMPLX_1:87;
then
A78: x in K003 by A74,A76;
x in K1 by A2,A74,A76,A77;
hence thesis by A78,XBOOLE_0:def 4;
end;
then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A62,A69,XBOOLE_0:def 10;
then
A79: K00 is closed by A34,PRE_TOPC:13;
A80: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5;
A81: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p
proof
let p be object;
assume
A82: p in ([#]T1)/\([#]T2);
then p in K00 by A80,XBOOLE_0:def 4;
hence f1.p=f.p by FUNCT_1:49
.=f2.p by A54,A82,FUNCT_1:49;
end;
A83: K1 c= K00 \/ K11
proof
let x be object;
assume x in K1;
then consider p such that
A84: p=x & p`2>=0 & p<>0.TOP-REAL 2 by A2;
per cases;
suppose
p`1/|.p.|>=cn;
then x in K00 by A84;
hence thesis by XBOOLE_0:def 3;
end;
suppose
p`1/|.p.|<cn;
then x in K11 by A84;
hence thesis by XBOOLE_0:def 3;
end;
end;
then [#](((TOP-REAL 2)|K1)|K00) \/ [#](((TOP-REAL 2)|K1)|K11) =[#]((
TOP-REAL 2)|K1) by A80,A54,A62,XBOOLE_0:def 10;
then consider h being Function of (TOP-REAL 2)|K1,(TOP-REAL 2)|D such that
A85: h=f1+*f2 and
A86: h is continuous by A80,A54,A79,A68,A61,A52,A81,JGRAPH_2:1;
A87: dom h=the carrier of ((TOP-REAL 2)|K1) by FUNCT_2:def 1;
A88: dom f1=the carrier of ((TOP-REAL 2)|K1)|K00 by FUNCT_2:def 1
.=K00 by PRE_TOPC:8;
A89: for y being object st y in dom h holds h.y=f.y
proof
let y be object;
assume
A90: y in dom h;
per cases by A83,A87,A53,A90,XBOOLE_0:def 3;
suppose
A91: y in K00 & not y in K11;
then y in dom f1 \/ dom f2 by A88,XBOOLE_0:def 3;
hence h.y=f1.y by A15,A85,A91,FUNCT_4:def 1
.=f.y by A91,FUNCT_1:49;
end;
suppose
A92: y in K11;
then y in dom f1 \/ dom f2 by A15,XBOOLE_0:def 3;
hence h.y=f2.y by A15,A85,A92,FUNCT_4:def 1
.=f.y by A92,FUNCT_1:49;
end;
end;
K0=the carrier of ((TOP-REAL 2)|K0) by PRE_TOPC:8
.=dom f by A7,FUNCT_2:def 1;
hence thesis by A86,A87,A89,FUNCT_1:2,PRE_TOPC:8;
end;
theorem Th61:
for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphN)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`2<=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=sqrt(1-cn^2);
set p0=|[cn,-sn]|;
assume
A1: -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2<=0 & p<>0.TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then 1-cn^2>0 by XREAL_1:50;
then p0`2=-sn & --sn>0 by EUCLID:52,SQUARE_1:25;
then
A2: p0`2<0;
then p0 in K0 by A1,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
not p0 in {0.TOP-REAL 2} by A2,JGRAPH_2:3,TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A1,XBOOLE_0:def 5;
A3: K1 c= D
proof
let x be object;
assume x in K1;
then consider p2 being Point of TOP-REAL 2 such that
A4: p2=x and
p2`2<=0 and
A5: p2<>0.TOP-REAL 2 by A1;
not p2 in {0.TOP-REAL 2} by A5,TARSKI:def 1;
hence thesis by A1,A4,XBOOLE_0:def 5;
end;
for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D
st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W &
W is open & f.:W c= V
proof
let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D;
assume that
A6: f.p in V and
A7: V is open;
consider V2 being Subset of TOP-REAL 2 such that
A8: V2 is open and
A9: V2 /\ [#]((TOP-REAL 2)|D)=V by A7,TOPS_2:24;
reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1;
A10: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
then
A11: f.p=(cn-FanMorphN).p by A1,FUNCT_1:49;
A12: f.:W2 c= V
proof
let y be object;
assume y in f.:W2;
then consider x being object such that
A13: x in dom f and
A14: x in W2 and
A15: y=f.x by FUNCT_1:def 6;
f is Function of (TOP-REAL 2)|K1, (TOP-REAL 2)|D;
then dom f= K1 by A10,FUNCT_2:def 1;
then consider p4 being Point of TOP-REAL 2 such that
A16: x=p4 and
A17: p4`2<=0 and
p4<>0.TOP-REAL 2 by A1,A13;
A18: p4 in V2 by A14,A16,XBOOLE_0:def 4;
p4 in [#]((TOP-REAL 2)|K1) by A13,A16;
then p4 in D by A3,A10;
then
A19: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
f.p4=(cn-FanMorphN).p4 by A1,A10,A13,A16,FUNCT_1:49
.=p4 by A17,Th49;
hence thesis by A9,A15,A16,A18,A19,XBOOLE_0:def 4;
end;
p in the carrier of (TOP-REAL 2)|K1;
then consider q being Point of TOP-REAL 2 such that
A20: q=p and
A21: q`2<=0 and
q <>0.TOP-REAL 2 by A1,A10;
(cn-FanMorphN).q=q by A21,Th49;
then p in V2 by A6,A9,A11,A20,XBOOLE_0:def 4;
then
A22: p in W2 by XBOOLE_0:def 4;
W2 is open by A8,TOPS_2:24;
hence thesis by A22,A12;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th62:
for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL
2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`2>=0 & p<>0.TOP-REAL 2} holds K0 is
closed
proof
set J0 = NonZero TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`2>=0;
set I1 = {p: P[p] & p<>0.TOP-REAL 2};
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0;
reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3:
sch 2;
assume B0=J0 & K0=I1;
then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:7
,PRE_TOPC:def 5;
hence thesis by PRE_TOPC:13;
end;
theorem Th63:
for B0 being Subset of TOP-REAL 2, K0 being Subset of (TOP-REAL
2)|B0 st B0=NonZero TOP-REAL 2 & K0={p: p`2<=0 & p<>0.TOP-REAL 2} holds K0 is
closed
proof
set J0 = NonZero TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`2<=0;
set I1 = {p: P[p] & p<>0.TOP-REAL 2};
let B0 be Subset of TOP-REAL 2,K0 be Subset of (TOP-REAL 2)|B0;
reconsider K1={p7 where p7 is Point of TOP-REAL 2:P[p7]} as Subset of
TOP-REAL 2 from JGRAPH_2:sch 1;
A1: I1 = {p7 where p7 is Point of TOP-REAL 2 : P[p7]} /\ J0 from JGRAPH_3:
sch 2;
assume B0=J0 & K0=I1;
then K1 is closed & K0=K1 /\ [#]((TOP-REAL 2)|B0) by A1,JORDAN6:8
,PRE_TOPC:def 5;
hence thesis by PRE_TOPC:13;
end;
theorem Th64:
for cn being Real, 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 -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p`2>=0 &
p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2>=0 & p<>0.TOP-REAL 2 };
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2>=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th60;
end;
theorem Th65:
for cn being Real, 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 -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p`2<=0 &
p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,
f be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<cn & cn<1 & f=(cn-FanMorphN)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2<=0 & p<>0.TOP-REAL 2};
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2<=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th61;
end;
theorem Th66:
for cn being Real,p being Point of TOP-REAL 2 holds |.(cn
-FanMorphN).p.|=|.p.|
proof
let cn be Real,p be Point of TOP-REAL 2;
set f=cn-FanMorphN;
set z=f.p;
reconsider q=p as Point of TOP-REAL 2;
reconsider qz=z as Point of TOP-REAL 2;
per cases;
suppose
A1: q`1/|.q.|>=cn & q`2>0;
then
A2: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((
q`1/|.q.|-cn)/(1-cn))^2))]| by Th49;
then
A3: qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
A4: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A2,EUCLID:52;
A5: (q`1/|.q.|-cn)>=0 by A1,XREAL_1:48;
A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
|.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then
A8: 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
per cases;
suppose
A9: 1-cn=0;
A10: ((q`1/|.q.|-cn)/(1-cn))=(q`1/|.q.|-cn)*(1-cn)" by XCMPLX_0:def 9
.= (q`1/|.q.|-cn)*0 by A9
.=0;
then 1-((q`1/|.q.|-cn)/(1-cn))^2=1;
then (cn-FanMorphN).q= |[ |.q.|*0,|.q.|*1]| by A1,A10,Th49,SQUARE_1:18
.=|[0,(|.q.|)]|;
then ((cn-FanMorphN).q)`2=(|.q.|) & ((cn-FanMorphN).q)`1=0 by EUCLID:52;
then |.(cn-FanMorphN).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A11: 1-cn<>0;
per cases by A11;
suppose
A12: 1-cn>0;
-(1-cn)<= -( q`1/|.q.|-cn) by A8,XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A5,A12,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A13: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
A14: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 by A3
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A13,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A4,A14;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A15: 1-cn<0;
0+(q`1)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then
A16: 1 > q`1/|.p.| by SQUARE_1:52;
q`1/|.q.|-cn>=0 by A1,XREAL_1:48;
hence thesis by A15,A16,XREAL_1:9;
end;
end;
end;
suppose
A17: q`1/|.q.|<cn & q`2>0;
then |.q.|<>0 by JGRAPH_2:3,TOPRNS_1:24;
then
A18: (|.q.|)^2>0 by SQUARE_1:12;
A19: (q`1/|.q.|-cn)<0 by A17,XREAL_1:49;
A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then
A21: -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
A22: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-(
(q`1/|.q.|-cn)/(1+cn))^2))]| by A17,Th50;
then
A23: qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
A24: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A22,EUCLID:52;
per cases;
suppose
A25: 1+cn=0;
((q`1/|.q.|-cn)/(1+cn))=(q`1/|.q.|-cn)*(1+cn)" by XCMPLX_0:def 9
.= (q`1/|.q.|-cn)*0 by A25
.=0;
then ((cn-FanMorphN).q)`2=(|.q.|) & ((cn-FanMorphN).q)`1=0 by A22,
EUCLID:52;
then |.(cn-FanMorphN).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A26: 1+cn<>0;
per cases by A26;
suppose
A27: 1+cn>0;
then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A21,XREAL_1:72;
then -1<=(( q`1/|.q.|-cn))/(1+cn) by A27,XCMPLX_1:197;
then ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A19,A27,SQUARE_1:49;
then
A28: 1-(((q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
A29: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 by A23
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A28,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A24,A29;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A30: 1+cn<0;
0+(q`1)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1 < q`1/|.p.| by SQUARE_1:52;
then
A31: q`1/|.q.|-cn>-1-cn by XREAL_1:9;
-(1+cn)>-0 by A30,XREAL_1:24;
hence thesis by A17,A31,XREAL_1:49;
end;
end;
end;
suppose
q`2<=0;
hence thesis by Th49;
end;
end;
theorem Th67:
for cn being Real,x,K0 being set st -1<cn & cn<1 & x in K0 & K0=
{p: p`2>=0 & p<>0.TOP-REAL 2} holds (cn-FanMorphN).x in K0
proof
let cn be Real,x,K0 be set;
assume
A1: -1<cn & cn<1 & x in K0 & K0={p: p`2>=0 & p<>0.TOP-REAL 2};
then consider p such that
A2: p=x and
A3: p`2>=0 and
A4: p<>0.TOP-REAL 2;
A5: now
assume |.p.|<=0;
then |.p.|=0;
hence contradiction by A4,TOPRNS_1:24;
end;
then
A6: (|.p.|)^2>0 by SQUARE_1:12;
per cases;
suppose
A7: p`1/|.p.|<=cn;
reconsider p9= (cn-FanMorphN).p as Point of TOP-REAL 2;
(cn-FanMorphN).p= |[ |.p.|*((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((
p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A3,A4,A7,Th51;
then
A8: p9`2=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A10: 1+cn>0 by A1,XREAL_1:148;
per cases;
suppose
p`2=0;
hence thesis by A1,A2,Th49;
end;
suppose
p`2<>0;
then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74;
then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76;
then -1 < p`1/|.p.| by SQUARE_1:52;
then -1-cn< p`1/|.p.|-cn by XREAL_1:9;
then (-1)*(1+cn)/(1+cn)< (p`1/|.p.|-cn)/(1+cn) by A10,XREAL_1:74;
then
A11: -1< (p`1/|.p.|-cn)/(1+cn) by A10,XCMPLX_1:89;
p`1/|.p.|-cn<=0 by A7,XREAL_1:47;
then 1^2> ((p`1/|.p.|-cn)/(1+cn))^2 by A10,A11,SQUARE_1:50;
then 1-((p`1/|.p.|-cn)/(1+cn))^2>0 by XREAL_1:50;
then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>0 by SQUARE_1:25;
then |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))>0 by A5,XREAL_1:129;
hence thesis by A1,A2,A8,JGRAPH_2:3;
end;
end;
suppose
A12: p`1/|.p.|>cn;
reconsider p9= (cn-FanMorphN).p as Point of TOP-REAL 2;
(cn-FanMorphN).p= |[ |.p.|*((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((
p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A3,A4,A12,Th51;
then
A13: p9`2=|.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A15: 1-cn>0 by A1,XREAL_1:149;
per cases;
suppose
p`2=0;
hence thesis by A1,A2,Th49;
end;
suppose
p`2<>0;
then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74;
then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76;
then p`1/|.p.|<1 by SQUARE_1:52;
then (p`1/|.p.|-cn)<1-cn by XREAL_1:9;
then (p`1/|.p.|-cn)/(1-cn)<(1-cn)/(1-cn) by A15,XREAL_1:74;
then
A16: (p`1/|.p.|-cn)/(1-cn)<1 by A15,XCMPLX_1:60;
-(1-cn)< -0 & p`1/|.p.|-cn>=cn-cn by A12,A15,XREAL_1:9,24;
then (-1)*(1-cn)/(1-cn)< (p`1/|.p.|-cn)/(1-cn) by A15,XREAL_1:74;
then -1< (p`1/|.p.|-cn)/(1-cn) by A15,XCMPLX_1:89;
then 1^2> ((p`1/|.p.|-cn)/(1-cn))^2 by A16,SQUARE_1:50;
then 1-((p`1/|.p.|-cn)/(1-cn))^2>0 by XREAL_1:50;
then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>0 by SQUARE_1:25;
then p9`2>0 by A5,A13,XREAL_1:129;
hence thesis by A1,A2,JGRAPH_2:3;
end;
end;
end;
theorem Th68:
for cn being Real,x,K0 being set st -1<cn & cn<1 & x in K0 & K0=
{p: p`2<=0 & p<>0.TOP-REAL 2} holds (cn-FanMorphN).x in K0
proof
let cn be Real,x,K0 be set;
assume
A1: -1<cn & cn<1 & x in K0 & K0={p: p`2<=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`2<=0 & p<>0.TOP-REAL 2;
hence thesis by A1,Th49;
end;
theorem Th69:
for cn being Real, D being non empty Subset of TOP-REAL 2 st -1<
cn & cn<1 & D`={0.TOP-REAL 2} holds ex h being Function of (TOP-REAL 2)|D,(
TOP-REAL 2)|D st h=(cn-FanMorphN)|D & h is continuous
proof
(|[0,1]|)`2=1 by EUCLID:52;
then
A1: |[0,1]| in {p where p is Point of TOP-REAL 2: p`2>=0 & p<>0.TOP-REAL 2}
by JGRAPH_2:3;
set Y1=|[0,-1]|;
reconsider B0= {0.TOP-REAL 2} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`2>=0;
let cn be Real,D be non empty Subset of TOP-REAL 2;
assume that
A2: -1<cn & cn<1 and
A3: D`={0.TOP-REAL 2};
A4: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
A5: D =(B0)` by A3
.=(NonZero TOP-REAL 2) by SUBSET_1:def 4;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A5);
then reconsider
K0={p:p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A1;
A6: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8;
A7: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
A8: rng ((cn-FanMorphN)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
let y be object;
assume y in rng ((cn-FanMorphN)|K0);
then consider x being object such that
A9: x in dom ((cn-FanMorphN)|K0) and
A10: y=((cn-FanMorphN)|K0).x by FUNCT_1:def 3;
x in (dom (cn-FanMorphN)) /\ K0 by A9,RELAT_1:61;
then
A11: x in K0 by XBOOLE_0:def 4;
K0 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A11;
(cn-FanMorphN).p=y by A10,A11,FUNCT_1:49;
then y in K0 by A2,A11,Th67;
hence thesis by PRE_TOPC:8;
end;
A12: 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>=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
Y1`2=-1 & 0.TOP-REAL 2 <> Y1 by EUCLID:52,JGRAPH_2:3;
then
A13: Y1 in {p where p is Point of TOP-REAL 2: p`2<=0 & p<>0.TOP-REAL 2};
A14: the carrier of ((TOP-REAL 2)|D) = (NonZero TOP-REAL 2) by A5,PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`2<=0;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A5);
then reconsider
K1={p: p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A13;
A15: K0 is closed & K1 is closed by A5,Th62,Th63;
dom ((cn-FanMorphN)|K0)= dom ((cn-FanMorphN)) /\ K0 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1
.=K0 by A12,XBOOLE_1:28;
then reconsider f=(cn-FanMorphN)|K0 as Function of ((TOP-REAL 2)|D)|K0, ((
TOP-REAL 2)|D) by A6,A8,FUNCT_2:2,XBOOLE_1:1;
A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8;
A17: rng ((cn-FanMorphN)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1
proof
let y be object;
assume y in rng ((cn-FanMorphN)|K1);
then consider x being object such that
A18: x in dom ((cn-FanMorphN)|K1) and
A19: y=((cn-FanMorphN)|K1).x by FUNCT_1:def 3;
x in (dom (cn-FanMorphN)) /\ K1 by A18,RELAT_1:61;
then
A20: x in K1 by XBOOLE_0:def 4;
K1 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A20;
(cn-FanMorphN).p=y by A19,A20,FUNCT_1:49;
then y in K1 by A2,A20,Th68;
hence thesis by PRE_TOPC:8;
end;
A21: 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`2<=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
dom ((cn-FanMorphN)|K1)= dom ((cn-FanMorphN)) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1
.=K1 by A21,XBOOLE_1:28;
then reconsider g=(cn-FanMorphN)|K1 as Function of ((TOP-REAL 2)|D)|K1, ((
TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1;
A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
A23: D c= K0 \/ K1
proof
let x be object;
assume
A24: x in D;
then reconsider px=x as Point of TOP-REAL 2;
not x in {0.TOP-REAL 2} by A5,A24,XBOOLE_0:def 5;
then px`2>=0 & px<>0.TOP-REAL 2 or px`2<=0 & px<>0.TOP-REAL 2 by
TARSKI:def 1;
then x in K0 or x in K1;
hence thesis by XBOOLE_0:def 3;
end;
A25: dom f=K0 by A6,FUNCT_2:def 1;
A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
A27: 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
A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) );
then x in K0 by A26,XBOOLE_0:def 4;
then f.x=(cn-FanMorphN).x by FUNCT_1:49;
hence thesis by A22,A28,FUNCT_1:49;
end;
D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
then
A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#]((
TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10;
A30: f is continuous & g is continuous by A2,A5,Th64,Th65;
then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that
A31: h= f+*g and
h is continuous by A26,A22,A29,A15,A27,JGRAPH_2:1;
A32: dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1;
A33: dom g=K1 by A16,FUNCT_2:def 1;
K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
then
A34: f tolerates g by A27,A25,A33,PARTFUN1:def 4;
A35: for x being object st x in dom h holds h.x=((cn-FanMorphN)|D).x
proof
let x be object;
assume
A36: x in dom h;
then reconsider p=x as Point of TOP-REAL 2 by A14,XBOOLE_0:def 5;
not x in {0.TOP-REAL 2} by A14,A36,XBOOLE_0:def 5;
then
A37: x <>0.TOP-REAL 2 by TARSKI:def 1;
A38: x in D`` by A32,A36,PRE_TOPC:8;
now
per cases;
case
A39: x in K0;
A40: (cn-FanMorphN)|D.p=(cn-FanMorphN).p by A38,FUNCT_1:49
.=f.p by A39,FUNCT_1:49;
h.p=(g+*f).p by A31,A34,FUNCT_4:34
.=f.p by A25,A39,FUNCT_4:13;
hence thesis by A40;
end;
case
not x in K0;
then not p`2>=0 by A37;
then
A41: x in K1 by A37;
(cn-FanMorphN)|D.p=(cn-FanMorphN).p by A38,FUNCT_1:49
.=g.p by A41,FUNCT_1:49;
hence thesis by A31,A33,A41,FUNCT_4:13;
end;
end;
hence thesis;
end;
dom (cn-FanMorphN)=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1;
then dom ((cn-FanMorphN)|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|D) by A4,XBOOLE_1:28;
then f+*g=(cn-FanMorphN)|D by A31,A32,A35,FUNCT_1:2;
hence thesis by A26,A22,A29,A30,A15,A27,JGRAPH_2:1;
end;
theorem Th70:
for cn being Real st -1<cn & cn<1 holds ex h being Function of (
TOP-REAL 2),(TOP-REAL 2) st h=(cn-FanMorphN) & h is continuous
proof
reconsider D=NonZero TOP-REAL 2 as non empty Subset of TOP-REAL 2 by
JGRAPH_2:9;
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
reconsider f=(cn-FanMorphN) as Function of (TOP-REAL 2),(TOP-REAL 2);
A3: f.(0.TOP-REAL 2)=0.TOP-REAL 2 by Th49,JGRAPH_2:3;
A4: 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;
A5: [#]((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 A5,XBOOLE_0:def 5;
then
A6: not p=0.TOP-REAL 2 by TARSKI:def 1;
now
per cases;
case
A7: q`1/|.q.|>=cn & q`2>=0;
set q9= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-
cn)/(1-cn))^2))]|;
A8: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
now
assume
A9: q9=0.TOP-REAL 2;
A10: |.q.|<>0^2 by A6,TOPRNS_1:24;
then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)= sqrt(1-0) by A8,A9,JGRAPH_2:3
,XCMPLX_1:6
.=1;
hence contradiction by A9,A10,EUCLID:52,JGRAPH_2:3;
end;
hence thesis by A1,A2,A3,A6,A7,Th51;
end;
case
A11: q`1/|.q.|<cn & q`2>=0;
set q9=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-
cn)/(1+cn))^2))]|;
A12: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
now
assume
A13: q9=0.TOP-REAL 2;
A14: |.q.|<>0^2 by A6,TOPRNS_1:24;
then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)= sqrt(1-0) by A12,A13,
JGRAPH_2:3,XCMPLX_1:6
.=1;
hence contradiction by A13,A14,EUCLID:52,JGRAPH_2:3;
end;
hence thesis by A1,A2,A3,A6,A11,Th51;
end;
case
q`2<0;
then f.p=p by Th49;
hence thesis by A6,Th49,JGRAPH_2:3;
end;
end;
hence thesis;
end;
A15: 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 Lm11;
assume that
A16: f.(0.TOP-REAL 2) in V and
A17: V is open;
VV is open by A17,Lm11,PRE_TOPC:30;
then consider r being Real such that
A18: r>0 and
A19: Ball(u0,r) c= V by A3,A16,TOPMETR:15;
reconsider r as Real;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2;
A20: W1 is open by GOBOARD6:3;
A21: f.:W1 c= W1
proof
let z be object;
assume z in f.:W1;
then consider y being object such that
A22: y in dom f and
A23: y in W1 and
A24: z=f.y by FUNCT_1:def 6;
z in rng f by A22,A24,FUNCT_1:def 3;
then reconsider qz=z as Point of TOP-REAL 2;
reconsider q=y as Point of TOP-REAL 2 by A22;
reconsider qy=q as Point of Euclid 2 by EUCLID:67;
reconsider pz=qz as Point of Euclid 2 by EUCLID:67;
dist(u0,qy)<r by A23,METRIC_1:11;
then
A25: |.(0.TOP-REAL 2) - q.|<r by JGRAPH_1:28;
now
per cases by JGRAPH_2:3;
case
q`2<=0;
hence thesis by A23,A24,Th49;
end;
case
A26: q<>0.TOP-REAL 2 & q`1/|.q.|>=cn & q`2>=0;
then
A27: (q`1/|.q.|-cn)>= 0 by XREAL_1:48;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A28: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A29: 1-cn>0 by A2,XREAL_1:149;
|.q.|<>0 by A26,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`1)^2/(|.q.|)^2 <= 1 by A28,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A29,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A29,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A29,A27,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A30: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
A31: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)) , |.q.|*(
sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A26,Th51;
then
A32: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A24,EUCLID:52;
qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by A24,A31,EUCLID:52
;
then
A33: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A30,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A32,A33;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A34: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A25,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A34,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
case
A35: q<>0.TOP-REAL 2 & q`1/|.q.|<cn & q`2>=0;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A36: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A37: 1+cn>0 by A1,XREAL_1:148;
|.q.|<>0 by A35,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`1)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then --1>=-q`1/|.q.| by XREAL_1:24;
then 1+cn>=-q`1/|.q.|+cn by XREAL_1:7;
then
A38: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A37,XREAL_1:185;
(cn-q`1/|.q.|)>=0 by A35,XREAL_1:48;
then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A37;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A38,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A39: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
A40: (cn-FanMorphN).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*(
sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A35,Th51;
then
A41: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A24,EUCLID:52;
qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by A24,A40,EUCLID:52
;
then
A42: (qz`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A39,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A41,A42;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A43: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A25,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A43,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
end;
hence thesis;
end;
u0 in W1 by A18,GOBOARD6:1;
hence thesis by A19,A20,A21,XBOOLE_1:1;
end;
A44: D`= {0.TOP-REAL 2} by JGRAPH_3:20;
then
ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=(cn-FanMorphN
)|D & h is continuous by A1,A2,Th69;
hence thesis by A3,A44,A4,A15,JGRAPH_3:3;
end;
theorem Th71:
for cn being Real st -1<cn & cn<1 holds cn-FanMorphN is one-to-one
proof
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
for x1,x2 being object st x1 in dom (cn-FanMorphN) & x2 in dom (cn
-FanMorphN) & (cn-FanMorphN).x1=(cn-FanMorphN).x2 holds x1=x2
proof
let x1,x2 be object;
assume that
A3: x1 in dom (cn-FanMorphN) and
A4: x2 in dom (cn-FanMorphN) and
A5: (cn-FanMorphN).x1=(cn-FanMorphN).x2;
reconsider p2=x2 as Point of TOP-REAL 2 by A4;
reconsider p1=x1 as Point of TOP-REAL 2 by A3;
set q=p1,p=p2;
A6: 1-cn>0 by A2,XREAL_1:149;
now
per cases by JGRAPH_2:3;
case
A7: q`2<=0;
then
A8: (cn-FanMorphN).q=q by Th49;
now
per cases by JGRAPH_2:3;
case
p`2<=0;
hence thesis by A5,A8,Th49;
end;
case
A9: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.
p.|-cn)/(1-cn))^2))]|;
A10: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
0<=(p`2)^2 by XREAL_1:63;
then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then
A11: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A10,XREAL_1:72;
A12: |.p.|>0 by A9,Lm1;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`1)^2/(|.p.|)^2 <= 1 by A11,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`1/|.p.| by SQUARE_1:51;
then 1-cn>=p`1/|.p.|-cn by XREAL_1:9;
then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72;
then
A13: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197;
A14: (p`1/|.p.|-cn)>=0 by A9,XREAL_1:48;
A15: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*(
sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A9,Th51;
(p`1/|.p.|-cn)>= 0 by A9,XREAL_1:48;
then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A13,SQUARE_1:49;
then
A16: 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`1/|.p.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`1/|.p.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(p`1/|.p.|-cn)^2/(1-cn)^2)>=0;
then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
then p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) & q`2=0 by A5
,A7,A8,A15,EUCLID:52;
then
A17: ( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6;
1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by A16,XCMPLX_1:187;
then 1-((p`1/|.p.|-cn)/(1-cn))^2=0 by A17,SQUARE_1:24;
then 1= (p`1/|.p.|-cn)/(1-cn) by A6,A14,SQUARE_1:18,22;
then 1 *(1-cn)=(p`1/|.p.|-cn) by A6,XCMPLX_1:87;
then 1 *|.p.|=p`1 by A12,XCMPLX_1:87;
then p`2=0 by A10,XCMPLX_1:6;
hence thesis by A5,A8,Th49;
end;
case
A18: p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2>=0;
then
A19: |.p.|<>0 by TOPRNS_1:24;
then
A20: (|.p.|)^2>0 by SQUARE_1:12;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.
p.|-cn)/(1+cn))^2))]|;
A21: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A22: 1+cn>0 by A1,XREAL_1:148;
A23: (p`1/|.p.|-cn)<=0 by A18,XREAL_1:47;
then
A24: -1<=(-( p`1/|.p.|-cn))/(1+cn) by A22;
A25: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*(
sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A18,Th51;
0<=(p`2)^2 by XREAL_1:63;
then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A21,XREAL_1:72;
then (p`1)^2/(|.p.|)^2 <= 1 by A20,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then (-((p`1)/|.p.|))^2 <= 1;
then 1>= -p`1/|.p.| by SQUARE_1:51;
then (1+cn)>= -p`1/|.p.|+cn by XREAL_1:7;
then (-(p`1/|.p.|-cn))/(1+cn)<=1 by A22,XREAL_1:185;
then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A24,SQUARE_1:49;
then
A26: 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`1/|.p.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`1/|.p.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((p`1/|.p.|-cn))^2/(1+cn)^2)>=0;
then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
then p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) & q`2=0 by A5
,A7,A8,A25,EUCLID:52;
then
A27: ( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))=0 by A5,A8,A25,A19,XCMPLX_1:6;
1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by A26,XCMPLX_1:187;
then 1-((p`1/|.p.|-cn)/(1+cn))^2=0 by A27,SQUARE_1:24;
then 1=sqrt((-((p`1/|.p.|-cn)/(1+cn)))^2);
then 1= -((p`1/|.p.|-cn)/(1+cn)) by A22,A23,SQUARE_1:22;
then 1= ((-(p`1/|.p.|-cn))/(1+cn)) by XCMPLX_1:187;
then 1 *(1+cn)=-(p`1/|.p.|-cn) by A22,XCMPLX_1:87;
then 1+cn-cn=-p`1/|.p.|;
then 1=(-p`1)/|.p.| by XCMPLX_1:187;
then 1 *|.p.|=-p`1 by A18,TOPRNS_1:24,XCMPLX_1:87;
then (p`1)^2-(p`1)^2 =(p`2)^2 by A21,XCMPLX_1:26;
then p`2=0 by XCMPLX_1:6;
hence thesis by A5,A8,Th49;
end;
end;
hence thesis;
end;
case
A28: q`1/|.q.|>=cn & q`2>=0 & q<>0.TOP-REAL 2;
then |.q.|<>0 by TOPRNS_1:24;
then
A29: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-
cn)/(1-cn))^2))]|;
A30: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
A31: cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1
-((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A28,Th51;
A32: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
now
per cases by JGRAPH_2:3;
case
A33: p`2<=0;
then
A34: (cn-FanMorphN).p=p by Th49;
A35: |.q.|<>0 by A28,TOPRNS_1:24;
then
A36: (|.q.|)^2>0 by SQUARE_1:12;
A37: (q`1/|.q.|-cn)>= 0 by A28,XREAL_1:48;
A38: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A39: (q`1/|.q.|-cn)>=0 by A28,XREAL_1:48;
A40: 1-cn>0 by A2,XREAL_1:149;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A38,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A40,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A40,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A40,A37,SQUARE_1:49;
then
A41: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0;
then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
then p`2=0 by A5,A31,A33,A34,EUCLID:52;
then
A42: ( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))=0 by A5,A31,A32,A34,A35,
XCMPLX_1:6;
1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by A41,XCMPLX_1:187;
then 1-((q`1/|.q.|-cn)/(1-cn))^2=0 by A42,SQUARE_1:24;
then 1= (q`1/|.q.|-cn)/(1-cn) by A40,A39,SQUARE_1:18,22;
then 1 *(1-cn)=(q`1/|.q.|-cn) by A40,XCMPLX_1:87;
then 1 *|.q.|=q`1 by A28,TOPRNS_1:24,XCMPLX_1:87;
then q`2=0 by A38,XCMPLX_1:6;
hence thesis by A5,A34,Th49;
end;
case
A43: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XREAL_1:72;
then
A44: -1<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XCMPLX_1:197;
(q`1/|.q.|-cn)>= 0 by A28,XREAL_1:48;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A6,A44,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A45: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
then
A46: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A45,SQUARE_1:def 2;
A47: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A47,A46;
then
A48: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A49: |.q4.|=|.q.| by SQUARE_1:22;
0<=(p`2)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A50: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
|.p.|<>0 by A43,TOPRNS_1:24;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`1)^2/(|.p.|)^2 <= 1 by A50,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`1/|.p.| by SQUARE_1:51;
then 1-cn>=p`1/|.p.|-cn by XREAL_1:9;
then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72;
then
A51: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197;
(p`1/|.p.|-cn)>= 0 by A43,XREAL_1:48;
then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A51,SQUARE_1:49;
then 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A52: 1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.
p.|-cn)/(1-cn))^2))]|;
A53: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52;
p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
then
A54: (p4`2)^2= (|.p.|)^2*( sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))^2
.= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1-cn))^2) by A52,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A53,A54;
then
A55: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A56: |.p4.|=|.p.| by SQUARE_1:22;
A57: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*(
sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A43,Th51;
then ((p`1/|.p.|-cn)/(1-cn)) =|.q.|* ((q`1/|.q.|-cn)/(1-cn))/|.p
.| by A5,A31,A30,A43,A53,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1-cn)=(q`1/|.q.|-cn)/(1-cn) by A5,A31,A43,A57
,A48,A55,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1-cn)*(1-cn)=q`1/|.q.|-cn by A6,XCMPLX_1:87;
then p`1/|.p.|-cn=q`1/|.q.|-cn by A6,XCMPLX_1:87;
then p`1/|.p.|*|.p.|=q`1 by A5,A31,A43,A57,A49,A56,TOPRNS_1:24
,XCMPLX_1:87;
then
A58: p`1=q`1 by A43,TOPRNS_1:24,XCMPLX_1:87;
A59: p=|[p`1,p`2]| by EUCLID:53;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then p`2=sqrt((q`2)^2) by A5,A31,A43,A57,A49,A56,A58,SQUARE_1:22;
then p`2=q`2 by A28,SQUARE_1:22;
hence thesis by A58,A59,EUCLID:53;
end;
case
A60: p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2>=0;
then p`1/|.p.|-cn<0 by XREAL_1:49;
then
A61: ((p`1/|.p.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.
p.|-cn)/(1+cn))^2))]|;
A62: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) & q`1/|.q.|-cn>=0 by A28,
EUCLID:52,XREAL_1:48;
A63: 1-cn>0 by A2,XREAL_1:149;
cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*(
sqrt(1-((p`1/|.p.| -cn)/(1+cn))^2))]| & |.p.|<>0 by A1,A2,A60,Th51,TOPRNS_1:24;
hence thesis by A5,A31,A30,A61,A62,A63,XREAL_1:132;
end;
end;
hence thesis;
end;
case
A64: q`1/|.q.|<cn & q`2>=0 & q<>0.TOP-REAL 2;
then
A65: |.q.|<>0 by TOPRNS_1:24;
then
A66: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-
cn)/(1+cn))^2))]|;
A67: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
A68: cn-FanMorphN.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(
1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A64,Th51;
A69: q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
now
per cases by JGRAPH_2:3;
case
A70: p`2<=0;
A71: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A72: 1+cn>0 by A1,XREAL_1:148;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A71,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then (-((q`1)/|.q.|))^2 <= 1;
then 1>= -q`1/|.q.| by SQUARE_1:51;
then (1+cn)>= -q`1/|.q.|+cn by XREAL_1:7;
then
A73: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A72,XREAL_1:185;
A74: (q`1/|.q.|-cn)<=0 by A64,XREAL_1:47;
then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A72;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A73,SQUARE_1:49;
then
A75: 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A76: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
A77: (cn-FanMorphN).p=p by A70,Th49;
sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by A75,SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((q`1/|.q.|-cn))^2/(1+cn)^2)>=0;
then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
then p`2=0 by A5,A68,A70,A77,EUCLID:52;
then ( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))=0 by A5,A68,A69,A65,A77,
XCMPLX_1:6;
then 1-((q`1/|.q.|-cn)/(1+cn))^2=0 by A76,SQUARE_1:24;
then 1=sqrt((-((q`1/|.q.|-cn)/(1+cn)))^2);
then 1= -((q`1/|.q.|-cn)/(1+cn)) by A72,A74,SQUARE_1:22;
then 1= ((-(q`1/|.q.|-cn))/(1+cn)) by XCMPLX_1:187;
then 1 *(1+cn)=-(q`1/|.q.|-cn) by A72,XCMPLX_1:87;
then 1+cn-cn=-q`1/|.q.|;
then 1=(-q`1)/|.q.| by XCMPLX_1:187;
then 1 *|.q.|=-q`1 by A64,TOPRNS_1:24,XCMPLX_1:87;
then (q`1)^2-(q`1)^2 =(q`2)^2 by A71,XCMPLX_1:26;
then q`2=0 by XCMPLX_1:6;
hence thesis by A5,A77,Th49;
end;
case
A78: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2>=0;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( sqrt(1-((p`1/|.
p.|-cn)/(1-cn))^2))]|;
A79: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) & |.q.|<>0 by A64,EUCLID:52
,TOPRNS_1:24;
q`1/|.q.|-cn<0 by A64,XREAL_1:49;
then
A80: ((q`1/|.q.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148;
A81: 1-cn>0 by A2,XREAL_1:149;
cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*(
sqrt(1-((p`1/|.p.| -cn)/(1-cn))^2))]| & p`1/|.p.|-cn>=0 by A1,A2,A78,Th51,
XREAL_1:48;
hence thesis by A5,A68,A67,A80,A79,A81,XREAL_1:132;
end;
case
A82: p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2>=0;
0<=(p`2)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A83: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A84: 1+cn>0 by A1,XREAL_1:148;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
then -(-1-cn)>= -(q`1/|.q.|-cn) by XREAL_1:24;
then
A85: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A84,XREAL_1:185;
(q`1/|.q.|-cn)<=0 by A64,XREAL_1:47;
then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A84;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A85,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A86: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
q4`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
then
A87: (q4`2)^2= (|.q.|)^2*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A86,SQUARE_1:def 2;
A88: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( sqrt(1-((p`1/|.
p.|-cn)/(1+cn))^2))]|;
A89: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52;
|.p.|<>0 by A82,TOPRNS_1:24;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`1)^2/(|.p.|)^2 <= 1 by A83,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then -1<=p`1/|.p.| by SQUARE_1:51;
then -1-cn<=p`1/|.p.|-cn by XREAL_1:9;
then -(-1-cn)>= -(p`1/|.p.|-cn) by XREAL_1:24;
then
A90: (-(p`1/|.p.|-cn))/(1+cn)<=1 by A84,XREAL_1:185;
(p`1/|.p.|-cn)<=0 by A82,XREAL_1:47;
then -1<=(-( p`1/|.p.|-cn))/(1+cn) by A84;
then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A90,SQUARE_1:49;
then 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A91: 1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
p4`2= |.p.|*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
then
A92: (p4`2)^2= (|.p.|)^2*( sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))^2
.= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1+cn))^2) by A91,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A89,A92;
then
A93: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A94: |.p4.|=|.p.| by SQUARE_1:22;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A88,A87;
then
A95: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A96: |.q4.|=|.q.| by SQUARE_1:22;
A97: cn-FanMorphN.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*(
sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A82,Th51;
then ((p`1/|.p.|-cn)/(1+cn))=|.q.|* ((q`1/|.q.|-cn)/(1+cn))/|.p.|
by A5,A68,A67,A82,A89,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1+cn)=(q`1/|.q.|-cn)/(1+cn) by A5,A68,A82,A97
,A95,A93,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1+cn)*(1+cn)=q`1/|.q.|-cn by A84,XCMPLX_1:87;
then p`1/|.p.|-cn=q`1/|.q.|-cn by A84,XCMPLX_1:87;
then p`1/|.p.|*|.p.|=q`1 by A5,A68,A82,A97,A96,A94,TOPRNS_1:24
,XCMPLX_1:87;
then
A98: p`1=q`1 by A82,TOPRNS_1:24,XCMPLX_1:87;
A99: p=|[p`1,p`2]| by EUCLID:53;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then p`2=sqrt((q`2)^2) by A5,A68,A82,A97,A96,A94,A98,SQUARE_1:22;
then p`2=q`2 by A64,SQUARE_1:22;
hence thesis by A98,A99,EUCLID:53;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
hence thesis by FUNCT_1:def 4;
end;
theorem Th72:
for cn being Real st -1<cn & cn<1 holds (cn-FanMorphN) is
Function of TOP-REAL 2,TOP-REAL 2 & rng (cn-FanMorphN) = the carrier of
TOP-REAL 2
proof
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
thus (cn-FanMorphN) is Function of TOP-REAL 2,TOP-REAL 2;
for f being Function of TOP-REAL 2,TOP-REAL 2 st f=(cn-FanMorphN) holds
rng (cn-FanMorphN)=the carrier of TOP-REAL 2
proof
let f be Function of TOP-REAL 2,TOP-REAL 2;
assume
A3: f=(cn-FanMorphN);
A4: 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 by JGRAPH_2:3;
case
q`2<=0;
then y=(cn-FanMorphN).q by Th49;
hence ex x being set st x in dom (cn-FanMorphN) & y=(cn-FanMorphN).x
by A3,A4;
end;
case
A5: q`1/|.q.|>=0 & q`2>=0 & q<>0.TOP-REAL 2;
--(1+cn)>0 by A1,XREAL_1:148;
then
A6: -(-1-cn)>0;
A7: 1-cn>=0 by A2,XREAL_1:149;
then q`1/|.q.|*(1-cn)>=0 by A5;
then -1-cn<= q`1/|.q.|*(1-cn) by A6;
then
A8: -1-cn+cn<= q`1/|.q.|*(1-cn)+cn by XREAL_1:7;
set px=|[ |.q.|*(q`1/|.q.|*(1-cn)+cn), (|.q.|)*sqrt(1-(q`1/|.q.|*(1-
cn)+cn)^2)]|;
A9: px`1 = |.q.|*(q`1/|.q.|*(1-cn)+cn) by EUCLID:52;
|.q.|<>0 by A5,TOPRNS_1:24;
then
A10: |.q.|^2>0 by SQUARE_1:12;
A11: dom (cn-FanMorphN)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A12: 1-cn>0 by A2,XREAL_1:149;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`1/|.q.|<=1 by SQUARE_1:51;
then q`1/|.q.|*(1-cn) <=1 *(1-cn) by A12,XREAL_1:64;
then q`1/|.q.|*(1-cn)+cn-cn <=1-cn;
then (q`1/|.q.|*(1-cn)+cn) <=1 by XREAL_1:9;
then 1^2>=(q`1/|.q.|*(1-cn)+cn)^2 by A8,SQUARE_1:49;
then
A13: 1-(q`1/|.q.|*(1-cn)+cn)^2>=0 by XREAL_1:48;
then
A14: sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2)>=0 by SQUARE_1:def 2;
A15: px`2 = (|.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2) by EUCLID:52;
then
|.px.|^2=((|.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|*(q
`1/|.q.|*(1-cn)+cn))^2 by A9,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|)^2*((q`1
/|.q.|*(1-cn)+cn))^2;
then
A16: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1-cn)+cn)^2) +(|.q.|)^2*((q`1
/|.q.|*(1-cn)+cn))^2 by A13,SQUARE_1:def 2
.= (|.q.|)^2;
then
A17: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A18: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24;
(q`1/|.q.|*(1-cn)+cn)>=0+cn by A5,A7,XREAL_1:7;
then px`1/|.px.| >=cn by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89;
then
A19: (cn-FanMorphN).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1-cn) ), |.px
.|*( sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2))]| by A1,A2,A15,A14,A18,Th51;
A20: |.px.|*( sqrt((q`2/|.q.|)^2))=|.q.|*(q`2/|.q.|) by A5,A17,SQUARE_1:22
.=q`2 by A5,TOPRNS_1:24,XCMPLX_1:87;
A21: |.px.|* ((px`1/|.px.|-cn)/(1-cn)) =|.q.|* (( ((q`1/|.q.|*(1-cn)
+cn))-cn)/(1-cn)) by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`1/|.q.|) by A12,XCMPLX_1:89
.= q`1 by A5,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*( sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2)) = |.px.|*( sqrt(1
-(q`1/|.px.|)^2)) by A5,A17,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*( sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*( sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by A10
,A16,XCMPLX_1:60
.= |.px.|*( sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by XCMPLX_1:120
.= |.px.|*( sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A16,
JGRAPH_3:1
.= |.px.|*( sqrt((q`2/|.q.|)^2)) by A17,XCMPLX_1:76;
hence ex x being set st x in dom (cn-FanMorphN) & y=(cn-FanMorphN).x
by A19,A21,A20,A11,EUCLID:53;
end;
case
A22: q`1/|.q.|<0 & q`2>=0 & q<>0.TOP-REAL 2;
A23: 1+cn>=0 by A1,XREAL_1:148;
(1-cn)>0 by A2,XREAL_1:149;
then
A24: 1-cn+cn>= q`1/|.q.|*(1+cn)+cn by A22,A23,XREAL_1:7;
A25: 1+cn>0 by A1,XREAL_1:148;
|.q.|<>0 by A22,TOPRNS_1:24;
then
A26: |.q.|^2>0 by SQUARE_1:12;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`1/|.q.|>=-1 by SQUARE_1:51;
then q`1/|.q.|*(1+cn) >=(-1)*(1+cn) by A25,XREAL_1:64;
then q`1/|.q.|*(1+cn)+cn-cn >=-1-cn;
then (q`1/|.q.|*(1+cn)+cn) >=-1 by XREAL_1:9;
then 1^2>=(q`1/|.q.|*(1+cn)+cn)^2 by A24,SQUARE_1:49;
then
A27: 1-(q`1/|.q.|*(1+cn)+cn)^2>=0 by XREAL_1:48;
then
A28: sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2)>=0 by SQUARE_1:def 2;
A29: dom (cn-FanMorphN)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set px=|[ |.q.|*(q`1/|.q.|*(1+cn)+cn), (|.q.|)*sqrt(1-(q`1/|.q.|*(1+
cn)+cn)^2)]|;
A30: px`1 = |.q.|*(q`1/|.q.|*(1+cn)+cn) by EUCLID:52;
A31: px`2 = (|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2) by EUCLID:52;
then
|.px.|^2=((|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|*(q
`1/|.q.|*(1+cn)+cn))^2 by A30,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|)^2*((q`1
/|.q.|*(1+cn)+cn))^2;
then
A32: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1+cn)+cn)^2) +(|.q.|)^2*((q`1
/|.q.|*(1+cn)+cn))^2 by A27,SQUARE_1:def 2
.= (|.q.|)^2;
then
A33: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A34: px<>0.TOP-REAL 2 by A22,TOPRNS_1:23,24;
(q`1/|.q.|*(1+cn)+cn)<=0+cn by A22,A23,XREAL_1:7;
then px`1/|.px.| <=cn by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89;
then
A35: (cn-FanMorphN).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1+cn) ), |.px
.|*( sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2))]| by A1,A2,A31,A28,A34,Th51;
A36: |.px.|*( sqrt((q`2/|.q.|)^2))=|.q.|*(q`2/|.q.|) by A22,A33,
SQUARE_1:22
.=q`2 by A22,TOPRNS_1:24,XCMPLX_1:87;
A37: |.px.|* ((px`1/|.px.|-cn)/(1+cn)) =|.q.|* (( ((q`1/|.q.|*(1+cn)
+cn))-cn)/(1+cn)) by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`1/|.q.|) by A25,XCMPLX_1:89
.= q`1 by A22,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*( sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2)) = |.px.|*( sqrt(1
-(q`1/|.px.|)^2)) by A22,A33,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*( sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*( sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by A26
,A32,XCMPLX_1:60
.= |.px.|*( sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by XCMPLX_1:120
.= |.px.|*( sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A32,
JGRAPH_3:1
.= |.px.|*( sqrt((q`2/|.q.|)^2)) by A33,XCMPLX_1:76;
hence ex x being set st x in dom (cn-FanMorphN) & y=(cn-FanMorphN).x
by A35,A37,A36,A29,EUCLID:53;
end;
end;
hence thesis by A3,FUNCT_1:def 3;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
hence thesis;
end;
theorem Th73:
for cn being Real,p2 being Point of TOP-REAL 2 st -1<cn & cn<1
ex K being non empty compact Subset of TOP-REAL 2 st K = (cn-FanMorphN).:K & ex
V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is open & V2 c= K & (cn
-FanMorphN).p2 in V2
proof
reconsider O=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67;
let cn be Real,p2 be Point of TOP-REAL 2;
A1: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider V0=Ball(O,|.p2.|+1) as Subset of TOP-REAL 2;
O in V0 & V0 c= cl_Ball(O,|.p2.|+1) by GOBOARD6:1,METRIC_1:14;
then reconsider
K0=cl_Ball(O,|.p2.|+1) as non empty compact Subset of TOP-REAL 2
by A1,Th15;
set q3= (cn-FanMorphN).p2;
reconsider VV0 = V0 as Subset of TopSpaceMetr Euclid 2;
reconsider u2=p2 as Point of Euclid 2 by EUCLID:67;
reconsider u3=q3 as Point of Euclid 2 by EUCLID:67;
A2: (cn-FanMorphN).:K0 c= K0
proof
let y be object;
assume y in (cn-FanMorphN).:K0;
then consider x being object such that
A3: x in dom (cn-FanMorphN) and
A4: x in K0 and
A5: y=(cn-FanMorphN).x by FUNCT_1:def 6;
reconsider q=x as Point of TOP-REAL 2 by A3;
reconsider uq=q as Point of Euclid 2 by EUCLID:67;
dist(O,uq)<= |.p2.|+1 by A4,METRIC_1:12;
then |.0.TOP-REAL 2 - q.|<= |.p2.|+1 by JGRAPH_1:28;
then |. -q.|<= |.p2.|+1 by RLVECT_1:4;
then
A6: |.q.|<= |.p2.|+1 by TOPRNS_1:26;
A7: y in rng (cn-FanMorphN) by A3,A5,FUNCT_1:def 3;
then reconsider u=y as Point of Euclid 2 by EUCLID:67;
reconsider q4=y as Point of TOP-REAL 2 by A7;
|.q4.|=|.q.| by A5,Th66;
then |. -q4.|<= |.p2.|+1 by A6,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q4.|<= |.p2.|+1 by RLVECT_1:4;
then dist(O,u)<= |.p2.|+1 by JGRAPH_1:28;
hence thesis by METRIC_1:12;
end;
VV0 is open by TOPMETR:14;
then
A8: V0 is open by Lm11,PRE_TOPC:30;
A9: |. p2.|<|.p2.|+1 by XREAL_1:29;
then |. -p2.|<|.p2.|+1 by TOPRNS_1:26;
then |.0.TOP-REAL 2 - p2.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u2)<|.p2.|+1 by JGRAPH_1:28;
then
A10: p2 in V0 by METRIC_1:11;
|.q3.|=|.p2.| by Th66;
then |. -q3.|<|.p2.|+1 by A9,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q3.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u3)< |.p2.|+1 by JGRAPH_1:28;
then
A11: (cn-FanMorphN).p2 in V0 by METRIC_1:11;
assume
A12: -1<cn & cn<1;
K0 c= (cn-FanMorphN).:K0
proof
let y be object;
assume
A13: y in K0;
then reconsider q4=y as Point of TOP-REAL 2;
reconsider y as Point of Euclid 2 by A13;
the carrier of TOP-REAL 2 c= rng (cn-FanMorphN) by A12,Th72;
then q4 in rng (cn-FanMorphN);
then consider x being object such that
A14: x in dom (cn-FanMorphN) and
A15: y=(cn-FanMorphN).x by FUNCT_1:def 3;
reconsider x as Point of Euclid 2 by A14,Lm11;
reconsider q=x as Point of TOP-REAL 2 by A14;
|.q4.|=|.q.| by A15,Th66;
then q in K0 by A13,Lm12;
hence thesis by A14,A15,FUNCT_1:def 6;
end;
then K0= (cn-FanMorphN).:K0 by A2,XBOOLE_0:def 10;
hence thesis by A10,A8,A11,METRIC_1:14;
end;
theorem
for cn being Real st -1<cn & cn<1 ex f being Function of TOP-REAL 2,
TOP-REAL 2 st f=cn-FanMorphN & f is being_homeomorphism
proof
let cn be Real;
reconsider f=(cn-FanMorphN) as Function of TOP-REAL 2,TOP-REAL 2;
assume
A1: -1<cn & cn<1;
then
A2: for p2 being Point of TOP-REAL 2 ex K being non empty compact Subset of
TOP-REAL 2 st K = f.:K & ex V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is
open & V2 c= K & f.p2 in V2 by Th73;
rng (cn-FanMorphN) = the carrier of TOP-REAL 2 & ex h being Function of
( TOP-REAL 2),(TOP-REAL 2) st h=( cn-FanMorphN) & h is continuous by A1,Th70
,Th72;
then f is being_homeomorphism by A1,A2,Th3,Th71;
hence thesis;
end;
theorem Th75:
for cn being Real,q being Point of TOP-REAL 2 st cn<1 & q`2>0 &
q`1/|.q.|>=cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds
p`2>0 & p`1>=0
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: cn<1 and
A2: q`2>0 and
A3: q`1/|.q.|>=cn;
A4: (q`1/|.q.|-cn)>= 0 by A3,XREAL_1:48;
let p be Point of TOP-REAL 2;
set qz=p;
A5: 1-cn>0 by A1,XREAL_1:149;
A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then 1>q`1/|.q.| by SQUARE_1:52;
then 1-cn>q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)< -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<(-( q`1/|.q.|-cn))/(1-cn) by A5,XREAL_1:74;
then -1<(-( q`1/|.q.|-cn))/(1-cn) by A5,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<1^2 by A5,A4,SQUARE_1:50;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>0 by XREAL_1:50;
then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)> 0;
then
A8: sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)> 0 by XCMPLX_1:76;
assume p=(cn-FanMorphN).q;
then
A9: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-
cn))^2))]| by A2,A3,Th49;
then qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
hence thesis by A9,A6,A5,A4,A8,EUCLID:52,XREAL_1:129;
end;
theorem Th76:
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & q`2>0 &
q`1/|.q.|<cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds
p`2>0 & p`1<0
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: q`2>0 and
A3: q`1/|.q.|<cn;
A4: 1+cn>0 by A1,XREAL_1:148;
let p be Point of TOP-REAL 2;
set qz=p;
assume p=(cn-FanMorphN).q;
then
p=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+
cn))^2))]| by A2,A3,Th50;
then
A5: qz`2= |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) & qz`1= |.q.|* ((q`1/
|.q.|- cn)/(1+cn)) by EUCLID:52;
A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
A8: (q`1/|.q.|-cn)< 0 by A3,XREAL_1:49;
then -( q`1/|.q.|-cn)>0 by XREAL_1:58;
then (-(1+cn))/(1+cn)<(-( q`1/|.q.|-cn))/(1+cn) by A4,XREAL_1:74;
then
A9: -1<(-( q`1/|.q.|-cn))/(1+cn) by A4,XCMPLX_1:197;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1<q`1/|.q.| by SQUARE_1:52;
then -1-cn<q`1/|.q.|-cn by XREAL_1:9;
then --(1+cn)> -(q`1/|.q.|-cn) by XREAL_1:24;
then (-(q`1/|.q.|-cn))/(1+cn)<1 by A4,XREAL_1:191;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<1^2 by A9,SQUARE_1:50;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>0 by XREAL_1:50;
then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)> 0;
then
A10: sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)> 0 by XCMPLX_1:76;
((q`1/|.q.|-cn)/(1+cn))<0 by A1,A8,XREAL_1:141,148;
hence thesis by A6,A5,A10,XREAL_1:129,132;
end;
theorem Th77:
for cn being Real,q1,q2 being Point of TOP-REAL 2 st cn<1 & q1`2
>0 & q1`1/|.q1.|>=cn & q2`2>0 & q2`1/|.q2.|>=cn & q1`1/|.q1.|<q2`1/|.q2.| holds
for p1,p2 being Point of TOP-REAL 2 st p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN)
.q2 holds p1`1/|.p1.|<p2`1/|.p2.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: cn<1 and
A2: q1`2>0 and
A3: q1`1/|.q1.|>=cn and
A4: q2`2>0 and
A5: q2`1/|.q2.|>=cn and
A6: q1`1/|.q1.|<q2`1/|.q2.|;
A7: q1`1/|.q1.|-cn< q2`1/|.q2.|-cn & 1-cn>0 by A1,A6,XREAL_1:9,149;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(cn-FanMorphN).q1 and
A9: p2=(cn-FanMorphN).q2;
A10: |.p2.|=|.q2.| by A9,Th66;
p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)), |.q2.|*( sqrt(1-((q2`1/|.q2.|-
cn)/(1-cn))^2))]| by A4,A5,A9,Th49;
then
A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`1/|.p2.|= ((q2`1/|.q2.|-cn)/(1-cn)) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)), |.q1.|*( sqrt(1-((q1`1/|.q1.|-
cn)/(1-cn))^2))]| by A2,A3,A8,Th49;
then
A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th66;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`1/|.p1.|= ((q1`1/|.q1.|-cn)/(1-cn)) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem Th78:
for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & q1
`2>0 & q1`1/|.q1.|<cn & q2`2>0 & q2`1/|.q2.|<cn & q1`1/|.q1.|<q2`1/|.q2.| holds
for p1,p2 being Point of TOP-REAL 2 st p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN)
.q2 holds p1`1/|.p1.|<p2`1/|.p2.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: q1`2>0 and
A3: q1`1/|.q1.|<cn and
A4: q2`2>0 and
A5: q2`1/|.q2.|<cn and
A6: q1`1/|.q1.|<q2`1/|.q2.|;
A7: q1`1/|.q1.|-cn< q2`1/|.q2.|-cn & 1+cn>0 by A1,A6,XREAL_1:9,148;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(cn-FanMorphN).q1 and
A9: p2=(cn-FanMorphN).q2;
A10: |.p2.|=|.q2.| by A9,Th66;
p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)), |.q2.|*( sqrt(1-((q2`1/|.q2.|-
cn)/(1+cn))^2))]| by A4,A5,A9,Th50;
then
A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`1/|.p2.|= ((q2`1/|.q2.|-cn)/(1+cn)) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)), |.q1.|*( sqrt(1-((q1`1/|.q1.|-
cn)/(1+cn))^2))]| by A2,A3,A8,Th50;
then
A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th66;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`1/|.p1.|= ((q1`1/|.q1.|-cn)/(1+cn)) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem
for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & cn<1 & q1
`2 > 0 & q2`2>0 & q1`1/|.q1.|<q2`1/|.q2.| holds for p1,p2 being Point of
TOP-REAL 2 st p1=(cn-FanMorphN).q1 & p2=(cn-FanMorphN).q2 holds p1`1/|.p1.|<p2
`1/|.p2.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q1`2>0 and
A4: q2`2>0 and
A5: q1`1/|.q1.|<q2`1/|.q2.|;
let p1,p2 be Point of TOP-REAL 2;
assume that
A6: p1=(cn-FanMorphN).q1 and
A7: p2=(cn-FanMorphN).q2;
per cases;
suppose
q1`1/|.q1.|>=cn & q2`1/|.q2.|>=cn;
hence thesis by A2,A3,A4,A5,A6,A7,Th77;
end;
suppose
q1`1/|.q1.|>=cn & q2`1/|.q2.|<cn;
hence thesis by A5,XXREAL_0:2;
end;
suppose
A8: q1`1/|.q1.|<cn & q2`1/|.q2.|>=cn;
then p2`1>=0 by A2,A4,A7,Th75;
then
A9: p2`1/|.p2.|>=0;
p1`1<0 by A1,A3,A6,A8,Th76;
hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141;
end;
suppose
q1`1/|.q1.|<cn & q2`1/|.q2.|<cn;
hence thesis by A1,A3,A4,A5,A6,A7,Th78;
end;
end;
theorem
for cn being Real,q being Point of TOP-REAL 2 st q`2>0 & q`1/|.q.|=cn
holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphN).q holds p`2>0 & p`1=0
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: q`2>0 and
A2: q`1/|.q.|=cn;
A3: |.q.|<>0 & sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>0 by A1,A2,JGRAPH_2:3
,TOPRNS_1:24;
let p be Point of TOP-REAL 2;
assume p=(cn-FanMorphN).q;
then p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( sqrt(1-((q`1/|.q.|-cn)/(1-
cn))^2))]| by A1,A2,Th49;
hence thesis by A2,A3,EUCLID:52;
end;
theorem
for cn being Real holds 0.TOP-REAL 2=(cn-FanMorphN).(0.TOP-REAL
2) by Th49,JGRAPH_2:3;
begin :: Fan Morphism for East
definition
let s be Real, q be Point of TOP-REAL 2;
func FanE(s,q) -> Point of TOP-REAL 2 equals
:Def6:
|.q.|*|[sqrt(1-((q`2/|.q
.|-s)/(1-s))^2), (q`2/|.q.|-s)/(1-s)]| if q`2/|.q.|>=s & q`1>0, |.q.|*|[sqrt(1-
((q`2/|.q.|-s)/(1+s))^2), (q`2/|.q.|-s)/(1+s)]| if q`2/|.q.|<s & q`1>0
otherwise q;
correctness;
end;
definition
let s be Real;
func s-FanMorphE -> Function of TOP-REAL 2, TOP-REAL 2 means
:Def7:
for q being Point of TOP-REAL 2 holds it.q=FanE(s,q);
existence
proof
deffunc F(Point of TOP-REAL 2)=FanE(s,$1);
thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point
of TOP-REAL 2 holds IT.q=F(q)from FUNCT_2:sch 4;
end;
uniqueness
proof
deffunc F(Point of TOP-REAL 2)=FanE(s,$1);
thus for a,b being Function of TOP-REAL 2, TOP-REAL 2 st (for q being
Point of TOP-REAL 2 holds a.q=F(q)) & (for q being Point of TOP-REAL 2 holds b.
q=F(q)) holds a=b from BINOP_2:sch 1;
end;
end;
theorem Th82:
for sn being Real holds (q`2/|.q.|>=sn & q`1>0 implies sn
-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1-sn))]|)& (q`1<=0 implies sn-FanMorphE.q=q)
proof
let sn be Real;
hereby
assume q`2/|.q.|>=sn & q`1>0;
then
FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn)/
(1-sn)]| by Def6
.= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn
)/(1-sn))]| by EUCLID:58;
hence sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1-sn))]| by Def7;
end;
assume
A1: q`1<=0;
sn-FanMorphE.q=FanE(sn,q) by Def7;
hence thesis by A1,Def6;
end;
theorem Th83:
for sn being Real holds (q`2/|.q.|<=sn & q`1>0 implies sn
-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|*((q`2/|.q.|-
sn)/(1+sn))]|)
proof
let sn be Real;
assume that
A1: q`2/|.q.|<=sn and
A2: q`1>0;
now
per cases by A1,XXREAL_0:1;
case
q`2/|.q.|<sn;
then
FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2), (q`2/|.q.|-sn
)/(1+sn)]| by A2,Def6
.= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1+sn))]| by EUCLID:58;
hence thesis by Def7;
end;
case
A3: q`2/|.q.|=sn;
then (q`2/|.q.|-sn)/(1-sn)=0;
hence thesis by A2,A3,Th82;
end;
end;
hence thesis;
end;
theorem Th84:
for sn being Real st -1<sn & sn<1 holds (q`2/|.q.|>=sn & q`1>=0
& q<>0.TOP-REAL 2 implies sn-FanMorphE.q = |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-
sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(1-sn))]|) & (q`2/|.q.|<=sn & q`1>=0 & q<>0.
TOP-REAL 2 implies sn-FanMorphE.q = |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)
), |.q.|*((q`2/|.q.|-sn)/(1+sn))]|)
proof
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
per cases;
suppose
A3: q`2/|.q.|>=sn & q`1>=0 & q<>0.TOP-REAL 2;
per cases;
suppose
A4: q`1>0;
then
FanE(sn,q)= |.q.|*|[sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2), (q`2/|.q.|-sn
)/(1-sn)]| by A3,Def6
.= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1-sn))]| by EUCLID:58;
hence thesis by A4,Def7,Th83;
end;
suppose
A5: q`1<=0;
then
A6: sn-FanMorphE.q=q by Th82;
A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A8: 1-sn>0 by A2,XREAL_1:149;
A9: q`1=0 by A3,A5;
|.q.|<>0 by A3,TOPRNS_1:24;
then |.q.|^2>0 by SQUARE_1:12;
then (q`2)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60;
then ((q`2)/|.q.|)^2=1^2 by XCMPLX_1:76;
then
A10: sqrt(((q`2)/|.q.|)^2)=1;
A11: now
assume q`2<0;
then -((q`2)/|.q.|)=1 by A10,SQUARE_1:23;
hence contradiction by A1,A3;
end;
sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22;
then
A12: |.q.|=q`2 by A7,A9,A11,SQUARE_1:22;
then 1=q`2/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60;
then (q`2/|.q.|-sn)/(1-sn)=1 by A8,XCMPLX_1:60;
hence thesis by A2,A6,A9,A12,EUCLID:53,TOPRNS_1:24
,XCMPLX_1:60;
end;
end;
suppose
A13: q`2/|.q.|<=sn & q`1>=0 & q<>0.TOP-REAL 2;
per cases;
suppose
q`1>0;
hence thesis by Th82,Th83;
end;
suppose
A14: q`1<=0;
then
A15: q`1=0 by A13;
A16: 1+sn>0 by A1,XREAL_1:148;
A17: |.q.|<>0 by A13,TOPRNS_1:24;
1>q`2/|.q.| by A2,A13,XXREAL_0:2;
then 1 *(|.q.|)>q`2/|.q.|*(|.q.|) by A17,XREAL_1:68;
then
A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`2 by A13,JGRAPH_3:1,TOPRNS_1:24
,XCMPLX_1:87;
then
A19: |.q.|=-q`2 by A15,SQUARE_1:40;
A20: q`2= -(|.q.|) by A15,A18,SQUARE_1:40;
then -1=q`2/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197;
then (q`2/|.q.|-sn)/(1+sn) =(-(1+sn))/(1+sn) .=-1 by A16,XCMPLX_1:197;
then
|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn
)/(1+sn))]| =q by A15,A19,EUCLID:53;
hence thesis by A1,A14,A17,A20,Th82,XCMPLX_1:197;
end;
end;
suppose
q`1<0 or q=0.TOP-REAL 2;
hence thesis;
end;
end;
theorem Th85:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st sn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1-
sn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1>=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=(1-sn);
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: sn<1 and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1>=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in K1 by A7,A8,A9,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A10: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
A11: g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
f.r=(|.r.|)* ((r`2/|.r.|-sn)/(1-sn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th86:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<sn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`2/|.p.|
-sn)/(1+sn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`1>=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=(1+sn);
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: -1<sn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`2/|.p.|-sn)/(1+sn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1>=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
1+sn>0 by A1,XREAL_1:148;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
A8: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in dom g3 by A7,A9;
then x in K1 by A7,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A10: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
A11: g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
f.r=(|.r.|)* ((r`2/|.r.|-sn)/(1+sn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
dom f=dom g3 by A7,FUNCT_2:def 1;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th87:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st sn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*(sqrt(1-((p`2/|.p.|-
sn)/(1-sn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`1>=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2) holds f is
continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=(1-sn);
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: sn<1 and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1>=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th10;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
A9: 1-sn>0 by A1,XREAL_1:149;
assume
A10: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A11: |.r.|<>0 by A3,A10,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`2) -(|.r.|))*((r`2)+|.r.|) =-(r`1)^2;
(r`1)^2>=0 by XREAL_1:63;
then r`2<= |.r.| by A12,XREAL_1:93;
then r`2/|.r.| <= |.r.|/|.r.| by XREAL_1:72;
then r`2/|.r.|<=1 by A11,XCMPLX_1:60;
then
A13: r`2/|.r.|-sn<=(1-sn) by XREAL_1:9;
reconsider s=x as Point of (TOP-REAL 2)|K1 by A10;
A14: now
assume (1-sn)^2=0;
then 1-sn+sn=0+sn by XCMPLX_1:6;
hence contradiction by A1;
end;
sn-r`2/|.r.|<=0 by A3,A10,XREAL_1:47;
then -(sn- r`2/|.r.|)>=-(1-sn) by A9,XREAL_1:24;
then (1-sn)^2>=0 & (r`2/|.r.|-sn)^2<=(1-sn)^2 by A13,SQUARE_1:49,XREAL_1:63
;
then (r`2/|.r.|-sn)^2/(1-sn)^2<=(1-sn)^2/(1-sn)^2 by XREAL_1:72;
then (r`2/|.r.|-sn)^2/(1-sn)^2<=1 by A14,XCMPLX_1:60;
then ((r`2/|.r.|-sn)/(1-sn))^2<=1 by XCMPLX_1:76;
then 1-((r`2/|.r.|-sn)/(1-sn))^2>=0 by XREAL_1:48;
then |.1-((r`2/|.r.|-sn)/(1-sn))^2.| =1-((r`2/|.r.|-sn)/(1-sn))^2 by
ABSVALUE:def 1;
then
A15: f.r=(|.r.|)*(sqrt(|.1-((r`2/|.r.|-sn)/(1-sn))^2.|)) by A2,A10;
A16: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
hence thesis by A5,A15,A16;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th88:
for sn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<sn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*(sqrt(1-((p
`2/|.p.|-sn)/(1+sn))^2))) & (for q being Point of TOP-REAL 2 st q in the
carrier of (TOP-REAL 2)|K1 holds q`1>=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2)
holds f is continuous
proof
let sn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=sn, b=(1+sn);
reconsider g2=proj2|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm3;
assume that
A1: -1<sn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`1>=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2;
A4: 1+sn>0 by A1,XREAL_1:148;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th10;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
reconsider s=x as Point of (TOP-REAL 2)|K1 by A9;
A10: (1+sn)^2>0 by A4,SQUARE_1:12;
A11: |.r.|<>0 by A3,A9,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`2) -(|.r.|))*((r`2)+|.r.|) =-(r`1)^2;
(r`1)^2>=0 by XREAL_1:63;
then -(|.r.|)<=r`2 by A12,XREAL_1:93;
then r`2/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72;
then r`2/|.r.|>= -1 by A11,XCMPLX_1:197;
then r`2/|.r.|-sn>=-1-sn by XREAL_1:9;
then
A13: r`2/|.r.|-sn>=-(1+sn);
sn-r`2/|.r.|>=0 by A3,A9,XREAL_1:48;
then -(sn-r`2/|.r.|)<=-0;
then (r`2/|.r.|-sn)^2<=(1+sn)^2 by A4,A13,SQUARE_1:49;
then (r`2/|.r.|-sn)^2/(1+sn)^2<=(1+sn)^2/(1+sn)^2 by A4,XREAL_1:72;
then (r`2/|.r.|-sn)^2/(1+sn)^2<=1 by A10,XCMPLX_1:60;
then ((r`2/|.r.|-sn)/(1+sn))^2<=1 by XCMPLX_1:76;
then 1-((r`2/|.r.|-sn)/(1+sn))^2>=0 by XREAL_1:48;
then |.1-((r`2/|.r.|-sn)/(1+sn))^2.| =1-((r`2/|.r.|-sn)/(1+sn))^2 by
ABSVALUE:def 1;
then
A14: f.r=(|.r.|)*(sqrt(|.1-((r`2/|.r.|-sn)/(1+sn))^2.|)) by A2,A9;
A15: proj2.r=r`2 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 6;
g2.s=proj2.s & g1.s=(2 NormF).s by Lm3,Lm5;
hence thesis by A5,A14,A15;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th89:
for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphE)|
K0 & B0={q where q is Point of TOP-REAL 2: q`1>=0 & q<>0.TOP-REAL 2} & K0={p: p
`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[cn,sn]|;
A1: p0`1=cn by EUCLID:52;
p0`2=sn by EUCLID:52;
then
A2: |.p0.|=sqrt((cn)^2+sn^2) by A1,JGRAPH_3:1;
assume
A3: -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0={q where q is Point of
TOP-REAL 2: q`1>=0 & q<>0.TOP-REAL 2} & K0={p: p`2/|.p.|>=sn & p`1>=0 & p<>0.
TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A4: 1-sn^2>0 by XREAL_1:50;
then
A5: --cn>0 by SQUARE_1:25;
cn^2=1-sn^2 by A4,SQUARE_1:def 2;
then p0`2/|.p0.|=sn by A2,EUCLID:52;
then
A6: p0 in K0 by A3,A1,A5,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A7: rng (proj1*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17;
A8: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2/|.p8.|>= sn & p8`1>=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A9: dom ((sn-FanMorphE)|K1) c= dom (proj2*((sn-FanMorphE)|K1))
proof
let x be object;
assume
A10: x in dom ((sn-FanMorphE)|K1);
then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphE) by XBOOLE_0:def 4;
then
A11: dom proj2 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn
-FanMorphE) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A10,FUNCT_1:47;
hence thesis by A10,A11,FUNCT_1:11;
end;
A12: rng (proj2*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj2*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25;
then dom (proj2*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A9,
XBOOLE_0:def 10
.=dom (sn-FanMorphE) /\ 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;
then reconsider
g2=proj2*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A12,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.|* ((p`2/|.p.|-sn)/(1-sn))
proof
let p be Point of TOP-REAL 2;
A13: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A15: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|>= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A14;
then
A16: (sn-FanMorphE).p =|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|*
((p`2/|.p.|-sn)/(1-sn))]| by A3,Th84;
((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A15,A14,FUNCT_1:49;
then
g2.p=proj2.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1-sn))]|) by A15,A13,A14,A16,FUNCT_1:13
.=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn)
/(1-sn))]|)`2 by PSCOMP_1:def 6
.=|.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`2/|.p.|-sn)/(1-sn));
A18: dom ((sn-FanMorphE)|K1) c= dom (proj1*((sn-FanMorphE)|K1))
proof
let x be object;
assume
A19: x in dom ((sn-FanMorphE)|K1);
then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphE) by XBOOLE_0:def 4;
then
A20: dom proj1 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn
-FanMorphE) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A19,FUNCT_1:47;
hence thesis by A19,A20,FUNCT_1:11;
end;
dom (proj1*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25;
then dom (proj1*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A18,
XBOOLE_0:def 10
.=dom (sn-FanMorphE) /\ 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;
then reconsider
g1=proj1*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A7,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.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))
proof
let p be Point of TOP-REAL 2;
A21: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A23: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|>= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A22;
then
A24: (sn-FanMorphE).p=|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|*
((p`2/|.p.|-sn)/(1-sn))]| by A3,Th84;
((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A23,A22,FUNCT_1:49;
then
g1.p=proj1.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1-sn))]|) by A23,A21,A22,A24,FUNCT_1:13
.=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2/|.p.|-sn)
/(1-sn)) ]|)`1 by PSCOMP_1:def 5
.= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1>=0 & q`2/|.q.|>=sn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|>= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A26;
hence thesis;
end;
then
A27: f1 is continuous by A3,A25,Th87;
A28: 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
A29: |[x,y]| in K1 and
A30: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
set p99=|[x,y]|;
A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`2/|.p3.| >=sn & p3`1>=0
& p3<>0.TOP-REAL 2 by A3,A29;
A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A33: f1.p99=|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1-sn))^2)) by A25,A29;
((sn-FanMorphE)|K0).(|[x,y]|)=((sn-FanMorphE)).(|[x,y]|) by A29,FUNCT_1:49
.= |[|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1-sn))^2)), |.p99.|* ((p99`2
/|.p99.|-sn)/(1-sn))]| by A3,A31,Th84
.=|[r,s]| by A17,A29,A30,A32,A33;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1>=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|>= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A34;
hence thesis;
end;
then f2 is continuous by A3,A17,Th85;
hence thesis by A6,A8,A27,A28,JGRAPH_2:35;
end;
theorem Th90:
for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphE)|
K0 & B0={q where q is Point of TOP-REAL 2: q`1>=0 & q<>0.TOP-REAL 2} & K0={p: p
`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[cn,sn]|;
A1: p0`1=cn by EUCLID:52;
p0`2=sn by EUCLID:52;
then
A2: |.p0.|=sqrt((cn)^2+sn^2) by A1,JGRAPH_3:1;
assume
A3: -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0={q where q is Point of
TOP-REAL 2: q`1>=0 & q<>0.TOP-REAL 2} & K0={p: p`2/|.p.|<=sn & p`1>=0 & p<>0.
TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A4: 1-sn^2>0 by XREAL_1:50;
then
A5: --cn>0 by SQUARE_1:25;
cn^2=1-sn^2 by A4,SQUARE_1:def 2;
then p0`2/|.p0.|=sn by A2,EUCLID:52;
then
A6: p0 in K0 by A3,A1,A5,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A7: rng (proj1*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17;
A8: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2/|.p8.|<= sn & p8`1>=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A9: dom ((sn-FanMorphE)|K1) c= dom (proj2*((sn-FanMorphE)|K1))
proof
let x be object;
assume
A10: x in dom ((sn-FanMorphE)|K1);
then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphE) by XBOOLE_0:def 4;
then
A11: dom proj2 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn
-FanMorphE) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A10,FUNCT_1:47;
hence thesis by A10,A11,FUNCT_1:11;
end;
A12: rng (proj2*((sn-FanMorphE)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj2*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25;
then dom (proj2*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A9,
XBOOLE_0:def 10
.=dom (sn-FanMorphE) /\ 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;
then reconsider
g2=proj2*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A12,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.|* ((p`2/|.p.|-sn)/(1+sn))
proof
let p be Point of TOP-REAL 2;
A13: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A14: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A15: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|<= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A14;
then
A16: (sn-FanMorphE).p =|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|*
((p`2/|.p.|-sn)/(1+sn))]| by A3,Th84;
((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A15,A14,FUNCT_1:49;
then
g2.p=proj2.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1+sn))]|) by A15,A13,A14,A16,FUNCT_1:13
.=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn)
/(1+sn))]|)`2 by PSCOMP_1:def 6
.=|.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A17: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`2/|.p.|-sn)/(1+sn));
A18: dom ((sn-FanMorphE)|K1) c= dom (proj1*((sn-FanMorphE)|K1))
proof
let x be object;
assume
A19: x in dom ((sn-FanMorphE)|K1);
then x in dom (sn-FanMorphE) /\ K1 by RELAT_1:61;
then x in dom (sn-FanMorphE) by XBOOLE_0:def 4;
then
A20: dom proj1 = (the carrier of TOP-REAL 2) & (sn-FanMorphE).x in rng (sn
-FanMorphE) by FUNCT_1:3,FUNCT_2:def 1;
((sn-FanMorphE)|K1).x=(sn-FanMorphE).x by A19,FUNCT_1:47;
hence thesis by A19,A20,FUNCT_1:11;
end;
dom (proj1*((sn-FanMorphE)|K1)) c= dom ((sn-FanMorphE)|K1) by RELAT_1:25;
then dom (proj1*((sn-FanMorphE)|K1)) =dom ((sn-FanMorphE)|K1) by A18,
XBOOLE_0:def 10
.=dom (sn-FanMorphE) /\ 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;
then reconsider
g1=proj1*((sn-FanMorphE)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A7,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.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))
proof
let p be Point of TOP-REAL 2;
A21: dom ((sn-FanMorphE)|K1)=dom (sn-FanMorphE) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A22: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A23: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`2/|.p3.|<= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A22;
then
A24: (sn-FanMorphE).p=|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|*
((p`2/|.p.|-sn)/(1+sn))]| by A3,Th84;
((sn-FanMorphE)|K1).p=(sn-FanMorphE).p by A23,A22,FUNCT_1:49;
then
g1.p=proj1.(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1+sn))]|) by A23,A21,A22,A24,FUNCT_1:13
.=(|[|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2/|.p.|-sn)
/(1+sn)) ]|)`1 by PSCOMP_1:def 5
.= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A25: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1>=0 & q`2/|.q.|<=sn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A26: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|<= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A26;
hence thesis;
end;
then
A27: f1 is continuous by A3,A25,Th88;
A28: 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
A29: |[x,y]| in K1 and
A30: r=f1.(|[x,y]|) & s=f2.(|[x,y]|);
set p99=|[x,y]|;
A31: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`2/|.p3.| <=sn & p3`1>=0
& p3<>0.TOP-REAL 2 by A3,A29;
A32: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A33: f1.p99=|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1+sn))^2)) by A25,A29;
((sn-FanMorphE)|K0).(|[x,y]|)=((sn-FanMorphE)).(|[x,y]|) by A29,FUNCT_1:49
.= |[|.p99.|*(sqrt(1-((p99`2/|.p99.|-sn)/(1+sn))^2)), |.p99.|* ((p99`2
/|.p99.|-sn)/(1+sn))]| by A3,A31,Th84
.=|[r,s]| by A17,A29,A30,A32,A33;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`1>=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A34: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`2/|.p3.|<= sn & p3`1>=0 &
p3<>0.TOP-REAL 2 by A3,A34;
hence thesis;
end;
then f2 is continuous by A3,A17,Th86;
hence thesis by A6,A8,A27,A28,JGRAPH_2:35;
end;
theorem Th91:
for sn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`2
>=(sn)*(|.p.|) & p`1>=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`1>=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
assume
A1: K003={p: p`2>=(sn)*(|.p.|) & p`1>=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`2>=sn*|.$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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm7,JORDAN6:4;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th92:
for sn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p`2
<=(sn)*(|.p.|) & p`1>=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`1>=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
assume
A1: K003={p: p`2<=(sn)*(|.p.|) & p`1>=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`2<=sn*|.$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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm9,JORDAN6:4;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th93:
for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphE)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`1>=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[cn,sn]|;
A1: p0`1=cn by EUCLID:52;
defpred P[Point of TOP-REAL 2] means $1`2/|.$1.|>=sn & $1`1>=0 & $1<>0.
TOP-REAL 2;
p0`2=sn by EUCLID:52;
then
A2: |.p0.|=sqrt((cn)^2+sn^2) by A1,JGRAPH_3:1;
assume
A3: -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0= NonZero TOP-REAL 2 & K0={p:
p`1>=0 & p<>0.TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then
A4: 1-sn^2>0 by XREAL_1:50;
then
A5: p0`1>0 by A1,SQUARE_1:25;
then p0 in K0 by A3,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
cn^2=1-sn^2 by A4,SQUARE_1:def 2;
then
A6: p0`2/|.p0.|=sn by A2,EUCLID:52;
then
A7: p0 in {p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} by A5,JGRAPH_2:3;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K001={p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty
Subset of (TOP-REAL 2) by A7;
A8: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`2>=(sn)*(|.$1.|) & $1`1>=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K003={p: p`2>=(sn)*(|.p.|) & p`1>=0} as Subset of (TOP-REAL 2);
defpred P[Point of TOP-REAL 2] means $1`2/|.$1.|<=sn & $1`1>=0 & $1<>0.
TOP-REAL 2;
A9: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
A10: --cn>0 by A4,SQUARE_1:25;
then p0 in {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} by A1,A6,JGRAPH_2:3;
then reconsider
K111={p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty
Subset of (TOP-REAL 2) by A9;
A11: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
A12: rng ((sn-FanMorphE)|K001) c= K1
proof
let y be object;
assume y in rng ((sn-FanMorphE)|K001);
then consider x being object such that
A13: x in dom ((sn-FanMorphE)|K001) and
A14: y=((sn-FanMorphE)|K001).x by FUNCT_1:def 3;
x in dom (sn-FanMorphE) by A13,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A15: y=(sn-FanMorphE).q by A13,A14,FUNCT_1:47;
dom ((sn-FanMorphE)|K001)=(dom (sn-FanMorphE))/\ K001 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1
.=K001 by XBOOLE_1:28;
then
A16: ex p2 being Point of TOP-REAL 2 st p2=q & p2`2/|.p2.|>= sn & p2`1>=0 &
p2<>0.TOP-REAL 2 by A13;
then
A17: (q`2/|.q.|-sn)>= 0 by XREAL_1:48;
|.q.|<>0 by A16,TOPRNS_1:24;
then
A18: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1-sn))]|;
A19: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
A20: 1-sn>0 by A3,XREAL_1:149;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2 <= (|.q.|)^2 by JGRAPH_3:1;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A20,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A20,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A20,A17,SQUARE_1:49;
then
A21: 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A22: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by A21,SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0;
then
A23: sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
A24: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
then
A25: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A22,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A19,A25;
then
A26: q4<>0.TOP-REAL 2 by A18,TOPRNS_1:23;
sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1-sn))]| by A3,A16,Th84;
hence thesis by A3,A15,A24,A23,A26;
end;
A27: {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2;
hence thesis by A3;
end;
p0<>0.TOP-REAL 2 by A1,A4,JGRAPH_2:3,SQUARE_1:25;
then not p0 in {0.TOP-REAL 2} by TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A3,XBOOLE_0:def 5;
K1 c= D
proof
let x be object;
assume
A28: x in K1;
then ex p6 being Point of TOP-REAL 2 st p6=x & p6`1>=0 & p6 <>0.TOP-REAL 2
by A3;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A3,A28,XBOOLE_0:def 5;
end;
then D=K1 \/ D by XBOOLE_1:12;
then
A29: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4;
A30: {p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2;
hence thesis by A3;
end;
then reconsider
K00={p: p`2/|.p.|>=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A7,PRE_TOPC:8;
A31: K003 is closed by Th91;
A32: rng ((sn-FanMorphE)|K111) c= K1
proof
let y be object;
assume y in rng ((sn-FanMorphE)|K111);
then consider x being object such that
A33: x in dom ((sn-FanMorphE)|K111) and
A34: y=((sn-FanMorphE)|K111).x by FUNCT_1:def 3;
x in dom (sn-FanMorphE) by A33,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A35: y=(sn-FanMorphE).q by A33,A34,FUNCT_1:47;
dom ((sn-FanMorphE)|K111)=(dom (sn-FanMorphE))/\ K111 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1
.=K111 by XBOOLE_1:28;
then
A36: ex p2 being Point of TOP-REAL 2 st p2=q & p2`2/|.p2.|<= sn & p2`1>=0 &
p2<>0.TOP-REAL 2 by A33;
then
A37: (q`2/|.q.|-sn)<=0 by XREAL_1:47;
|.q.|<>0 by A36,TOPRNS_1:24;
then
A38: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-
sn)/(1+sn))]|;
A39: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
A40: 1+sn>0 by A3,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1
,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A38,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
then (-(1+sn))/(1+sn)<=(( q`2/|.q.|-sn))/(1+sn) by A40,XREAL_1:72;
then -1<=(( q`2/|.q.|-sn))/(1+sn) by A40,XCMPLX_1:197;
then
A41: ( (q`2/|.q.|-sn) /(1+sn))^2<=1^2 by A40,A37,SQUARE_1:49;
then
A42: 1-((q`2/|.q.|-sn)/(1+sn))^2>=0 by XREAL_1:48;
1-(-((q`2/|.q.|-sn)/(1+sn)))^2>=0 by A41,XREAL_1:48;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)>=0;
then
A43: sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
A44: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
then
A45: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A42,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A39,A45;
then
A46: q4<>0.TOP-REAL 2 by A38,TOPRNS_1:23;
sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1+sn))]| by A3,A36,Th84;
hence thesis by A3,A35,A44,A43,A46;
end;
the carrier of (TOP-REAL 2)|D=D by PRE_TOPC:8;
then
A47: rng (f|K00) c=D;
the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D;
then
A48: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1
.=K1 by PRE_TOPC:8;
then dom (f|K00)=K00 by A30,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8;
then reconsider
f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A47,FUNCT_2:2
;
A49: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8;
p0 in {p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} by A1,A10,A6,JGRAPH_2:3;
then reconsider
K11={p: p`2/|.p.|<=sn & p`1>=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A27,PRE_TOPC:8;
A50: the carrier of (TOP-REAL 2)|K1 =K1 by PRE_TOPC:8;
A51: dom (sn-FanMorphE)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then dom ((sn-FanMorphE)|K001)=K001 by RELAT_1:62
.= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8;
then reconsider
f3=(sn-FanMorphE)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL
2)|K1 by A8,A12,FUNCT_2:2;
A52: D<>{};
dom ((sn-FanMorphE)|K111)=K111 by A51,RELAT_1:62
.= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8;
then reconsider
f4=(sn-FanMorphE)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL
2)|K1 by A50,A32,FUNCT_2:2;
the carrier of (TOP-REAL 2)|D =D by PRE_TOPC:8;
then
A53: rng (f|K11) c=D;
dom (f|K11)=K11 by A27,A48,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8;
then reconsider
f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A53,FUNCT_2:2
;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A3,FUNCT_1:51
,GOBOARD9:2;
then
A54: f2 is continuous by A3,A29,Th90,PRE_TOPC:26;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A3,FUNCT_1:51
,GOBOARD9:2;
then
A55: f1 is continuous by A3,A29,Th89,PRE_TOPC:26;
A56: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1
.=K11 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`2<=(sn)*(|.$1.|) & $1`1>=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K004={p: p`2<=(sn)*(|.p.|) & p`1>=0} as Subset of (TOP-REAL 2);
A57: K004 /\ K1 c= K11
proof
let x be object;
assume
A58: x in K004 /\ K1;
then x in K004 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A59: q1=x and
A60: q1`2<=(sn)*(|.q1.|) and
q1`1>=0;
x in K1 by A58,XBOOLE_0:def 4;
then
A61: ex q2 being Point of TOP-REAL 2 st q2=x & q2`1>=0 & q2 <>0.TOP-REAL 2
by A3;
q1`2/|.q1.|<=(sn)*(|.q1.|)/|.q1.| by A60,XREAL_1:72;
then q1`2/|.q1.|<=(sn) by A59,A61,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A59,A61;
end;
A62: K004 is closed by Th92;
K11 c= K004 /\ K1
proof
let x be object;
assume x in K11;
then consider p such that
A63: p=x and
A64: p`2/|.p.|<=sn and
A65: p`1>=0 and
A66: p<>0.TOP-REAL 2;
p`2/|.p.|*|.p.|<=(sn)*(|.p.|) by A64,XREAL_1:64;
then p`2<=(sn)*(|.p.|) by A66,TOPRNS_1:24,XCMPLX_1:87;
then
A67: x in K004 by A63,A65;
x in K1 by A3,A63,A65,A66;
hence thesis by A67,XBOOLE_0:def 4;
end;
then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A11,A57,XBOOLE_0:def 10;
then
A68: K11 is closed by A62,PRE_TOPC:13;
A69: K003 /\ K1 c= K00
proof
let x be object;
assume
A70: x in K003 /\ K1;
then x in K003 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A71: q1=x and
A72: q1`2>=(sn)*(|.q1.|) and
q1`1>=0;
x in K1 by A70,XBOOLE_0:def 4;
then
A73: ex q2 being Point of TOP-REAL 2 st q2=x & q2`1>=0 & q2 <>0.TOP-REAL 2
by A3;
q1`2/|.q1.|>=(sn)*(|.q1.|)/|.q1.| by A72,XREAL_1:72;
then q1`2/|.q1.|>=(sn) by A71,A73,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A71,A73;
end;
K00 c= K003 /\ K1
proof
let x be object;
assume x in K00;
then consider p such that
A74: p=x and
A75: p`2/|.p.|>=sn and
A76: p`1>=0 and
A77: p<>0.TOP-REAL 2;
p`2/|.p.|*|.p.|>=(sn)*(|.p.|) by A75,XREAL_1:64;
then p`2>=(sn)*(|.p.|) by A77,TOPRNS_1:24,XCMPLX_1:87;
then
A78: x in K003 by A74,A76;
x in K1 by A3,A74,A76,A77;
hence thesis by A78,XBOOLE_0:def 4;
end;
then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A11,A69,XBOOLE_0:def 10;
then
A79: K00 is closed by A31,PRE_TOPC:13;
set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11;
A80: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5;
A81: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5;
A82: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p
proof
let p be object;
assume
A83: p in ([#]T1)/\([#]T2);
then p in K00 by A81,XBOOLE_0:def 4;
hence f1.p=f.p by FUNCT_1:49
.=f2.p by A80,A83,FUNCT_1:49;
end;
A84: K1 c= K00 \/ K11
proof
let x be object;
assume x in K1;
then consider p such that
A85: p=x & p`1>=0 & p<>0.TOP-REAL 2 by A3;
per cases;
suppose
p`2/|.p.|>=sn;
then x in K00 by A85;
hence thesis by XBOOLE_0:def 3;
end;
suppose
p`2/|.p.|<sn;
then x in K11 by A85;
hence thesis by XBOOLE_0:def 3;
end;
end;
then [#](((TOP-REAL 2)|K1)|K00) \/ [#](((TOP-REAL 2)|K1)|K11) =[#]((
TOP-REAL 2)|K1) by A81,A80,A11,XBOOLE_0:def 10;
then consider h being Function of (TOP-REAL 2)|K1,(TOP-REAL 2)|D such that
A86: h=f1+*f2 and
A87: h is continuous by A81,A80,A79,A68,A55,A54,A82,JGRAPH_2:1;
A88: dom h=the carrier of ((TOP-REAL 2)|K1) by FUNCT_2:def 1;
A89: dom f1=the carrier of ((TOP-REAL 2)|K1)|K00 by FUNCT_2:def 1
.=K00 by PRE_TOPC:8;
A90: for y being object st y in dom h holds h.y=f.y
proof
let y be object;
assume
A91: y in dom h;
now
per cases by A84,A88,A49,A91,XBOOLE_0:def 3;
suppose
A92: y in K00 & not y in K11;
then y in dom f1 \/ dom f2 by A89,XBOOLE_0:def 3;
hence h.y=f1.y by A56,A86,A92,FUNCT_4:def 1
.=f.y by A92,FUNCT_1:49;
end;
suppose
A93: y in K11;
then y in dom f1 \/ dom f2 by A56,XBOOLE_0:def 3;
hence h.y=f2.y by A56,A86,A93,FUNCT_4:def 1
.=f.y by A93,FUNCT_1:49;
end;
end;
hence thesis;
end;
K0=the carrier of ((TOP-REAL 2)|K0) by PRE_TOPC:8
.=dom f by A52,FUNCT_2:def 1;
hence thesis by A87,A88,A90,FUNCT_1:2,PRE_TOPC:8;
end;
theorem Th94:
for sn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<sn & sn<1 & f=(sn-FanMorphE)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`1<=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let sn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set cn=sqrt(1-sn^2);
set p0=|[-cn,-sn]|;
assume
A1: -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1<=0 & p<>0.TOP-REAL 2};
then sn^2<1^2 by SQUARE_1:50;
then 1-sn^2>0 by XREAL_1:50;
then --cn>0 by SQUARE_1:25;
then
A2: p0`1=-cn & -cn<0 by EUCLID:52;
then p0 in K0 by A1,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
not p0 in {0.TOP-REAL 2} by A2,JGRAPH_2:3,TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A1,XBOOLE_0:def 5;
A3: K1 c= D
proof
let x be object;
assume x in K1;
then consider p2 being Point of TOP-REAL 2 such that
A4: p2=x and
p2`1<=0 and
A5: p2<>0.TOP-REAL 2 by A1;
not p2 in {0.TOP-REAL 2} by A5,TARSKI:def 1;
hence thesis by A1,A4,XBOOLE_0:def 5;
end;
for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D
st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W &
W is open & f.:W c= V
proof
let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D;
assume that
A6: f.p in V and
A7: V is open;
consider V2 being Subset of TOP-REAL 2 such that
A8: V2 is open and
A9: V2 /\ [#]((TOP-REAL 2)|D)=V by A7,TOPS_2:24;
reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1;
A10: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
then
A11: f.p=(sn-FanMorphE).p by A1,FUNCT_1:49;
A12: f.:W2 c= V
proof
let y be object;
assume y in f.:W2;
then consider x being object such that
A13: x in dom f and
A14: x in W2 and
A15: y=f.x by FUNCT_1:def 6;
f is Function of (TOP-REAL 2)|K1, (TOP-REAL 2)|D;
then dom f= K1 by A10,FUNCT_2:def 1;
then consider p4 being Point of TOP-REAL 2 such that
A16: x=p4 and
A17: p4`1<=0 and
p4<>0.TOP-REAL 2 by A1,A13;
A18: p4 in V2 by A14,A16,XBOOLE_0:def 4;
p4 in [#]((TOP-REAL 2)|K1) by A13,A16;
then p4 in D by A3,A10;
then
A19: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
f.p4=(sn-FanMorphE).p4 by A1,A10,A13,A16,FUNCT_1:49
.=p4 by A17,Th82;
hence thesis by A9,A15,A16,A18,A19,XBOOLE_0:def 4;
end;
p in the carrier of (TOP-REAL 2)|K1;
then consider q being Point of TOP-REAL 2 such that
A20: q=p and
A21: q`1<=0 and
q <>0.TOP-REAL 2 by A1,A10;
(sn-FanMorphE).q=q by A21,Th82;
then p in V2 by A6,A9,A11,A20,XBOOLE_0:def 4;
then
A22: p in W2 by XBOOLE_0:def 4;
W2 is open by A8,TOPS_2:24;
hence thesis by A22,A12;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th95:
for sn being Real, 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 -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p
`1>=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1>=0 & p<>0.TOP-REAL 2 };
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1>=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th93;
end;
theorem Th96:
for sn being Real, 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 -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p
`1<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let sn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<sn & sn<1 & f=(sn-FanMorphE)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`1<=0 & p<>0.TOP-REAL 2};
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1<=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th94;
end;
theorem Th97:
for sn being Real,p being Point of TOP-REAL 2 holds |.(sn
-FanMorphE).p.|=|.p.|
proof
let sn be Real,p be Point of TOP-REAL 2;
set f=sn-FanMorphE;
set z=f.p;
reconsider q=p as Point of TOP-REAL 2;
reconsider qz=z as Point of TOP-REAL 2;
per cases;
suppose
A1: q`2/|.q.|>=sn & q`1>0;
then
A2: (sn-FanMorphE).q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|*
((q`2/|.q.|-sn)/(1-sn))]| by Th82;
then
A3: qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
A4: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by A2,EUCLID:52;
A5: (q`2/|.q.|-sn)>=0 by A1,XREAL_1:48;
A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
|.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then
A8: 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
per cases;
suppose
A9: 1-sn=0;
A10: ((q`2/|.q.|-sn)/(1-sn))=(q`2/|.q.|-sn)*(1-sn)" by XCMPLX_0:def 9
.= (q`2/|.q.|-sn)*0 by A9
.=0;
then 1-((q`2/|.q.|-sn)/(1-sn))^2=1;
then (sn-FanMorphE).q= |[|.q.|*1,|.q.|*0]| by A1,A10,Th82,SQUARE_1:18
.=|[(|.q.|),0]|;
then ((sn-FanMorphE).q)`1=(|.q.|) & ((sn-FanMorphE).q)`2=0 by EUCLID:52;
then |.(sn-FanMorphE).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A11: 1-sn<>0;
per cases by A11;
suppose
A12: 1-sn>0;
-(1-sn)<= -( q`2/|.q.|-sn) by A8,XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A12,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A12,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A5,A12,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A13: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
A14: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2 by A3
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A13,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A4,A14;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A15: 1-sn<0;
0+(q`2)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then
A16: 1 > q`2/|.p.| by SQUARE_1:52;
q`2/|.q.|-sn>=0 by A1,XREAL_1:48;
hence thesis by A15,A16,XREAL_1:9;
end;
end;
end;
suppose
A17: q`2/|.q.|<sn & q`1>0;
then |.q.|<>0 by JGRAPH_2:3,TOPRNS_1:24;
then
A18: (|.q.|)^2>0 by SQUARE_1:12;
A19: (q`2/|.q.|-sn)<0 by A17,XREAL_1:49;
A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then
A21: -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
A22: (sn-FanMorphE).q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|
* ((q`2/|.q.|-sn)/(1+sn))]| by A17,Th83;
then
A23: qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
A24: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by A22,EUCLID:52;
per cases;
suppose
A25: 1+sn=0;
((q`2/|.q.|-sn)/(1+sn))=(q`2/|.q.|-sn)*(1+sn)" by XCMPLX_0:def 9
.= (q`2/|.q.|-sn)*0 by A25
.=0;
then ((sn-FanMorphE).q)`1=(|.q.|) & ((sn-FanMorphE).q)`2=0 by A22,
EUCLID:52;
then |.(sn-FanMorphE).p.|=sqrt(((|.q.|))^2+0^2) by JGRAPH_3:1
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A26: 1+sn<>0;
per cases by A26;
suppose
A27: 1+sn>0;
then (-(1+sn))/(1+sn)<=(( q`2/|.q.|-sn))/(1+sn) by A21,XREAL_1:72;
then -1<=(( q`2/|.q.|-sn))/(1+sn) by A27,XCMPLX_1:197;
then ( (q`2/|.q.|-sn) /(1+sn))^2<=1^2 by A19,A27,SQUARE_1:49;
then
A28: 1-(((q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
A29: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2 by A23
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A28,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A24,A29;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A30: 1+sn<0;
0+(q`2)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1 < q`2/|.p.| by SQUARE_1:52;
then
A31: q`2/|.q.|-sn>-1-sn by XREAL_1:9;
-(1+sn)>-0 by A30,XREAL_1:24;
hence thesis by A17,A31,XREAL_1:49;
end;
end;
end;
suppose
q`1<=0;
hence thesis by Th82;
end;
end;
theorem Th98:
for sn being Real,x,K0 being set st -1<sn & sn<1 & x in K0 & K0
={p: p`1>=0 & p<>0.TOP-REAL 2} holds (sn-FanMorphE).x in K0
proof
let sn be Real,x,K0 be set;
assume
A1: -1<sn & sn<1 & x in K0 & K0={p: p`1>=0 & p<>0.TOP-REAL 2};
then consider p such that
A2: p=x and
A3: p`1>=0 and
A4: p<>0.TOP-REAL 2;
A5: now
assume |.p.|<=0;
then |.p.|=0;
hence contradiction by A4,TOPRNS_1:24;
end;
then
A6: (|.p.|)^2>0 by SQUARE_1:12;
per cases;
suppose
A7: p`2/|.p.|<=sn;
reconsider p9= (sn-FanMorphE).p as Point of TOP-REAL 2;
(sn-FanMorphE).p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|
*((p`2/|.p.|-sn)/(1+sn))]| by A1,A3,A4,A7,Th84;
then
A8: p9`1=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A10: 1+sn>0 by A1,XREAL_1:148;
per cases;
suppose
p`1=0;
hence thesis by A1,A2,Th82;
end;
suppose
p`1<>0;
then 0+(p`2)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`2)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74;
then (p`2)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 < 1 by XCMPLX_1:76;
then -1 < p`2/|.p.| by SQUARE_1:52;
then -1-sn< p`2/|.p.|-sn by XREAL_1:9;
then (-1)*(1+sn)/(1+sn)< (p`2/|.p.|-sn)/(1+sn) by A10,XREAL_1:74;
then
A11: -1< (p`2/|.p.|-sn)/(1+sn) by A10,XCMPLX_1:89;
p`2/|.p.|-sn<=0 by A7,XREAL_1:47;
then 1^2> ((p`2/|.p.|-sn)/(1+sn))^2 by A10,A11,SQUARE_1:50;
then 1-((p`2/|.p.|-sn)/(1+sn))^2>0 by XREAL_1:50;
then sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>0 by SQUARE_1:25;
then |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))>0 by A5,XREAL_1:129;
hence thesis by A1,A2,A8,JGRAPH_2:3;
end;
end;
suppose
A12: p`2/|.p.|>sn;
reconsider p9= (sn-FanMorphE).p as Point of TOP-REAL 2;
(sn-FanMorphE).p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|
*((p`2/|.p.|-sn)/(1-sn))]| by A1,A3,A4,A12,Th84;
then
A13: p9`1=|.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A15: 1-sn>0 by A1,XREAL_1:149;
per cases;
suppose
p`1=0;
hence thesis by A1,A2,Th82;
end;
suppose
p`1<>0;
then 0+(p`2)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`2)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74;
then (p`2)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 < 1 by XCMPLX_1:76;
then p`2/|.p.|<1 by SQUARE_1:52;
then (p`2/|.p.|-sn)<1-sn by XREAL_1:9;
then (p`2/|.p.|-sn)/(1-sn)<(1-sn)/(1-sn) by A15,XREAL_1:74;
then
A16: (p`2/|.p.|-sn)/(1-sn)<1 by A15,XCMPLX_1:60;
-(1-sn)< -0 & p`2/|.p.|-sn>=sn-sn by A12,A15,XREAL_1:9,24;
then (-1)*(1-sn)/(1-sn)< (p`2/|.p.|-sn)/(1-sn) by A15,XREAL_1:74;
then -1< (p`2/|.p.|-sn)/(1-sn) by A15,XCMPLX_1:89;
then 1^2> ((p`2/|.p.|-sn)/(1-sn))^2 by A16,SQUARE_1:50;
then 1-((p`2/|.p.|-sn)/(1-sn))^2>0 by XREAL_1:50;
then sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>0 by SQUARE_1:25;
then p9`1>0 by A5,A13,XREAL_1:129;
hence thesis by A1,A2,JGRAPH_2:3;
end;
end;
end;
theorem Th99:
for sn being Real,x,K0 being set st -1<sn & sn<1 & x in K0 & K0
={p: p`1<=0 & p<>0.TOP-REAL 2} holds (sn-FanMorphE).x in K0
proof
let sn be Real,x,K0 be set;
assume
A1: -1<sn & sn<1 & x in K0 & K0={p: p`1<=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`1<=0 & p<>0.TOP-REAL 2;
hence thesis by A1,Th82;
end;
theorem Th100:
for sn being Real, D being non empty Subset of TOP-REAL 2 st -1
<sn & sn<1 & D`={0.TOP-REAL 2} holds ex h being Function of (TOP-REAL 2)|D,(
TOP-REAL 2)|D st h=(sn-FanMorphE)|D & h is continuous
proof
(|[0,1]|)`1=0 & (|[0,1]|)`2=1 by EUCLID:52;
then
A1: |[0,1]| in {p where p is Point of TOP-REAL 2: p`1>=0 & p<>0.TOP-REAL 2}
by JGRAPH_2:3;
set Y1=|[0,1]|;
defpred P[Point of TOP-REAL 2] means $1`1>=0;
reconsider B0= {0.TOP-REAL 2} as Subset of TOP-REAL 2;
let sn be Real,D be non empty Subset of TOP-REAL 2;
assume that
A2: -1<sn & sn<1 and
A3: D`={0.TOP-REAL 2};
A4: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
A5: D =(B0)` by A3
.=(NonZero TOP-REAL 2) by SUBSET_1:def 4;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A5);
then reconsider
K0={p:p`1>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A1;
A6: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8;
A7: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
A8: rng ((sn-FanMorphE)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
let y be object;
assume y in rng ((sn-FanMorphE)|K0);
then consider x being object such that
A9: x in dom ((sn-FanMorphE)|K0) and
A10: y=((sn-FanMorphE)|K0).x by FUNCT_1:def 3;
x in (dom (sn-FanMorphE)) /\ K0 by A9,RELAT_1:61;
then
A11: x in K0 by XBOOLE_0:def 4;
K0 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A11;
(sn-FanMorphE).p=y by A10,A11,FUNCT_1:49;
then y in K0 by A2,A11,Th98;
hence thesis by PRE_TOPC:8;
end;
A12: 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`1>=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
Y1`1=0 & Y1`2=1 by EUCLID:52;
then
A13: Y1 in {p where p is Point of TOP-REAL 2: p`1<=0 & p<>0.TOP-REAL 2} by
JGRAPH_2:3;
A14: the carrier of ((TOP-REAL 2)|D)= (NonZero TOP-REAL 2) by A5,PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`1<=0;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A5);
then reconsider
K1={p: p`1<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A13;
A15: K0 is closed & K1 is closed by A5,Th29,Th31;
dom ((sn-FanMorphE)|K0)= dom ((sn-FanMorphE)) /\ K0 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1
.=K0 by A12,XBOOLE_1:28;
then reconsider f=(sn-FanMorphE)|K0 as Function of ((TOP-REAL 2)|D)|K0, ((
TOP-REAL 2)|D) by A6,A8,FUNCT_2:2,XBOOLE_1:1;
A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8;
A17: rng ((sn-FanMorphE)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1
proof
let y be object;
assume y in rng ((sn-FanMorphE)|K1);
then consider x being object such that
A18: x in dom ((sn-FanMorphE)|K1) and
A19: y=((sn-FanMorphE)|K1).x by FUNCT_1:def 3;
x in (dom (sn-FanMorphE)) /\ K1 by A18,RELAT_1:61;
then
A20: x in K1 by XBOOLE_0:def 4;
K1 c= the carrier of TOP-REAL 2 by A7,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A20;
(sn-FanMorphE).p=y by A19,A20,FUNCT_1:49;
then y in K1 by A2,A20,Th99;
hence thesis by PRE_TOPC:8;
end;
A21: 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<=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
dom ((sn-FanMorphE)|K1)= dom ((sn-FanMorphE)) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1
.=K1 by A21,XBOOLE_1:28;
then reconsider g=(sn-FanMorphE)|K1 as Function of ((TOP-REAL 2)|D)|K1, ((
TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1;
A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
A23: D c= K0 \/ K1
proof
let x be object;
assume
A24: x in D;
then reconsider px=x as Point of TOP-REAL 2;
not x in {0.TOP-REAL 2} by A5,A24,XBOOLE_0:def 5;
then px`1>=0 & px<>0.TOP-REAL 2 or px`1<=0 & px<>0.TOP-REAL 2 by
TARSKI:def 1;
then x in K0 or x in K1;
hence thesis by XBOOLE_0:def 3;
end;
A25: dom f=K0 by A6,FUNCT_2:def 1;
A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
A27: 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
A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) );
then x in K0 by A26,XBOOLE_0:def 4;
then f.x=(sn-FanMorphE).x by FUNCT_1:49;
hence thesis by A22,A28,FUNCT_1:49;
end;
D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
then
A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#]((
TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10;
A30: f is continuous & g is continuous by A2,A5,Th95,Th96;
then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that
A31: h= f+*g and
h is continuous by A26,A22,A29,A15,A27,JGRAPH_2:1;
A32: dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1;
A33: dom g=K1 by A16,FUNCT_2:def 1;
K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
then
A34: f tolerates g by A27,A25,A33,PARTFUN1:def 4;
A35: for x being object st x in dom h holds h.x=((sn-FanMorphE)|D).x
proof
let x be object;
assume
A36: x in dom h;
then reconsider p=x as Point of TOP-REAL 2 by A14,XBOOLE_0:def 5;
not x in {0.TOP-REAL 2} by A14,A36,XBOOLE_0:def 5;
then
A37: x <>0.TOP-REAL 2 by TARSKI:def 1;
A38: x in D`` by A32,A36,PRE_TOPC:8;
now
per cases;
case
A39: x in K0;
A40: (sn-FanMorphE)|D.p=(sn-FanMorphE).p by A38,FUNCT_1:49
.=f.p by A39,FUNCT_1:49;
h.p=(g+*f).p by A31,A34,FUNCT_4:34
.=f.p by A25,A39,FUNCT_4:13;
hence thesis by A40;
end;
case
not x in K0;
then not p`1>=0 by A37;
then
A41: x in K1 by A37;
(sn-FanMorphE)|D.p=(sn-FanMorphE).p by A38,FUNCT_1:49
.=g.p by A41,FUNCT_1:49;
hence thesis by A31,A33,A41,FUNCT_4:13;
end;
end;
hence thesis;
end;
dom (sn-FanMorphE)=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1;
then dom ((sn-FanMorphE)|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|D) by A4,XBOOLE_1:28;
then f+*g=(sn-FanMorphE)|D by A31,A32,A35,FUNCT_1:2;
hence thesis by A26,A22,A29,A30,A15,A27,JGRAPH_2:1;
end;
theorem Th101:
for sn being Real st -1<sn & sn<1 holds ex h being Function of
(TOP-REAL 2),(TOP-REAL 2) st h=(sn-FanMorphE) & h is continuous
proof
reconsider D=NonZero TOP-REAL 2 as non empty Subset of TOP-REAL 2 by
JGRAPH_2:9;
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
reconsider f=(sn-FanMorphE) as Function of (TOP-REAL 2),(TOP-REAL 2);
A3: f.(0.TOP-REAL 2)=0.TOP-REAL 2 by Th82,JGRAPH_2:3;
A4: 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;
A5: [#]((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 A5,XBOOLE_0:def 5;
then
A6: not p=0.TOP-REAL 2 by TARSKI:def 1;
now
per cases;
case
A7: q`2/|.q.|>=sn & q`1>=0;
set q9= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q
.|-sn)/(1-sn))]|;
A8: q9`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
now
assume
A9: q9=0.TOP-REAL 2;
A10: |.q.|<>0 by A6,TOPRNS_1:24;
then sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)=sqrt(1-0^2) by A8,A9,
JGRAPH_2:3,XCMPLX_1:6
.=1;
hence contradiction by A9,A10,EUCLID:52,JGRAPH_2:3;
end;
hence thesis by A1,A2,A3,A6,A7,Th84;
end;
case
A11: q`2/|.q.|<sn & q`1>=0;
set q9=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q
.|-sn)/(1+sn))]|;
A12: q9`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
now
assume
A13: q9=0.TOP-REAL 2;
A14: |.q.|<>0 by A6,TOPRNS_1:24;
then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)=sqrt(1-0^2) by A12,A13,
JGRAPH_2:3,XCMPLX_1:6
.=1;
hence contradiction by A13,A14,EUCLID:52,JGRAPH_2:3;
end;
hence thesis by A1,A2,A3,A6,A11,Th84;
end;
case
q`1<0;
then f.p=p by Th82;
hence thesis by A6,Th82,JGRAPH_2:3;
end;
end;
hence thesis;
end;
A15: 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 Lm11;
assume that
A16: f.(0.TOP-REAL 2) in V and
A17: V is open;
VV is open by A17,Lm11,PRE_TOPC:30;
then consider r being Real such that
A18: r>0 and
A19: Ball(u0,r) c= V by A3,A16,TOPMETR:15;
reconsider r as Real;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2;
A20: W1 is open by GOBOARD6:3;
A21: f.:W1 c= W1
proof
let z be object;
assume z in f.:W1;
then consider y being object such that
A22: y in dom f and
A23: y in W1 and
A24: z=f.y by FUNCT_1:def 6;
z in rng f by A22,A24,FUNCT_1:def 3;
then reconsider qz=z as Point of TOP-REAL 2;
reconsider q=y as Point of TOP-REAL 2 by A22;
reconsider qy=q as Point of Euclid 2 by EUCLID:67;
reconsider pz=qz as Point of Euclid 2 by EUCLID:67;
dist(u0,qy)<r by A23,METRIC_1:11;
then
A25: |.(0.TOP-REAL 2) - q.|<r by JGRAPH_1:28;
now
per cases by JGRAPH_2:3;
case
q`1<=0;
hence thesis by A23,A24,Th82;
end;
case
A26: q<>0.TOP-REAL 2 & q`2/|.q.|>=sn & q`1>=0;
then
A27: (q`2/|.q.|-sn)>= 0 by XREAL_1:48;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A28: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A29: 1-sn>0 by A2,XREAL_1:149;
|.q.|<>0 by A26,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`2)^2/(|.q.|)^2 <= 1 by A28,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A29,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A29,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A29,A27,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A30: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
A31: (sn-FanMorphE).q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1- sn))^2))
, |.q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A26,Th84;
then
A32: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by A24,EUCLID:52;
qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by A24,A31,EUCLID:52;
then
A33: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A30,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A32,A33;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A34: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A25,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A34,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
case
A35: q<>0.TOP-REAL 2 & q`2/|.q.|<sn & q`1>=0;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A36: (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A37: 1+sn>0 by A1,XREAL_1:148;
|.q.|<>0 by A35,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`2)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then --1>=-q`2/|.q.| by XREAL_1:24;
then 1+sn>=-q`2/|.q.|+sn by XREAL_1:7;
then
A38: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A37,XREAL_1:185;
(sn-q`2/|.q.|)>=0 by A35,XREAL_1:48;
then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A37;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A38,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A39: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
A40: (sn-FanMorphE).q= |[|.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+ sn))^2) )
, |.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A35,Th84;
then
A41: qz`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by A24,EUCLID:52;
qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by A24,A40,EUCLID:52;
then
A42: (qz`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A39,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A41,A42;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A43: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A25,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A43,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
end;
hence thesis;
end;
u0 in W1 by A18,GOBOARD6:1;
hence thesis by A19,A20,A21,XBOOLE_1:1;
end;
A44: D`= {0.TOP-REAL 2} by JGRAPH_3:20;
then
ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=(sn-FanMorphE
)|D & h is continuous by A1,A2,Th100;
hence thesis by A3,A44,A4,A15,JGRAPH_3:3;
end;
theorem Th102:
for sn being Real st -1<sn & sn<1 holds sn-FanMorphE is one-to-one
proof
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
for x1,x2 being object st x1 in dom (sn-FanMorphE) & x2 in dom (sn
-FanMorphE) & (sn-FanMorphE).x1=(sn-FanMorphE).x2 holds x1=x2
proof
let x1,x2 be object;
assume that
A3: x1 in dom (sn-FanMorphE) and
A4: x2 in dom (sn-FanMorphE) and
A5: (sn-FanMorphE).x1=(sn-FanMorphE).x2;
reconsider p2=x2 as Point of TOP-REAL 2 by A4;
reconsider p1=x1 as Point of TOP-REAL 2 by A3;
set q=p1,p=p2;
A6: 1-sn>0 by A2,XREAL_1:149;
now
per cases by JGRAPH_2:3;
case
A7: q`1<=0;
then
A8: (sn-FanMorphE).q=q by Th82;
now
per cases by JGRAPH_2:3;
case
p`1<=0;
hence thesis by A5,A8,Th82;
end;
case
A9: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0;
then
A10: (p`2/|.p.|-sn)>=0 by XREAL_1:48;
set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1-sn))]|;
A11: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A12: |.p.|<>0 by A9,TOPRNS_1:24;
then
A13: (|.p.|)^2>0 by SQUARE_1:12;
0<=(p`1)^2 by XREAL_1:63;
then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A11,XREAL_1:72;
then (p`2)^2/(|.p.|)^2 <= 1 by A13,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`2/|.p.| by SQUARE_1:51;
then 1-sn>=p`2/|.p.|-sn by XREAL_1:9;
then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72;
then
A14: -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197;
A15: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))
, |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A9,Th84;
(p`2/|.p.|-sn)>= 0 by A9,XREAL_1:48;
then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A14,SQUARE_1:49;
then
A16: 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`2/|.p.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`2/|.p.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(p`2/|.p.|-sn)^2/(1-sn)^2)>=0;
then sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
then
p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) & q`1=0 by A5,A7,A8
,A15,EUCLID:52;
then
A17: (sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))=0 by A5,A8,A15,A12,XCMPLX_1:6;
1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by A16,XCMPLX_1:187;
then 1-((p`2/|.p.|-sn)/(1-sn))^2=0 by A17,SQUARE_1:24;
then 1= (p`2/|.p.|-sn)/(1-sn) by A6,A10,SQUARE_1:18,22;
then 1 *(1-sn)=(p`2/|.p.|-sn) by A6,XCMPLX_1:87;
then 1 *|.p.|=p`2 by A9,TOPRNS_1:24,XCMPLX_1:87;
then p`1=0 by A11,XCMPLX_1:6;
hence thesis by A5,A8,Th82;
end;
case
A18: p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1>=0;
then
A19: |.p.|<>0 by TOPRNS_1:24;
then
A20: (|.p.|)^2>0 by SQUARE_1:12;
set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1+sn))]|;
A21: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A22: 1+sn>0 by A1,XREAL_1:148;
A23: (p`2/|.p.|-sn)<=0 by A18,XREAL_1:47;
then
A24: -1<=(-( p`2/|.p.|-sn))/(1+sn) by A22;
A25: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))
, |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A18,Th84;
0<=(p`1)^2 by XREAL_1:63;
then 0+(p`2)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A21,XREAL_1:72;
then (p`2)^2/(|.p.|)^2 <= 1 by A20,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then (-((p`2)/|.p.|))^2 <= 1;
then 1>= -p`2/|.p.| by SQUARE_1:51;
then (1+sn)>= -p`2/|.p.|+sn by XREAL_1:7;
then (-(p`2/|.p.|-sn))/(1+sn)<=1 by A22,XREAL_1:185;
then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A24,SQUARE_1:49;
then
A26: 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`2/|.p.|-sn))/(1+sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`2/|.p.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((p`2/|.p.|-sn))^2/(1+sn)^2)>=0;
then sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
then
p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) & q`1=0 by A5,A7,A8
,A25,EUCLID:52;
then
A27: (sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))=0 by A5,A8,A25,A19,XCMPLX_1:6;
1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by A26,XCMPLX_1:187;
then 1-((p`2/|.p.|-sn)/(1+sn))^2=0 by A27,SQUARE_1:24;
then 1=sqrt((-((p`2/|.p.|-sn)/(1+sn)))^2);
then 1= -((p`2/|.p.|-sn)/(1+sn)) by A22,A23,SQUARE_1:22;
then 1= ((-(p`2/|.p.|-sn))/(1+sn)) by XCMPLX_1:187;
then 1 *(1+sn)=-(p`2/|.p.|-sn) by A22,XCMPLX_1:87;
then 1+sn-sn=-p`2/|.p.|;
then 1=(-p`2)/|.p.| by XCMPLX_1:187;
then 1 *|.p.|=-p`2 by A18,TOPRNS_1:24,XCMPLX_1:87;
then (p`2)^2-(p`2)^2 =(p`1)^2 by A21,XCMPLX_1:26;
then p`1=0 by XCMPLX_1:6;
hence thesis by A5,A8,Th82;
end;
end;
hence thesis;
end;
case
A28: q`2/|.q.|>=sn & q`1>=0 & q<>0.TOP-REAL 2;
then |.q.|<>0 by TOPRNS_1:24;
then
A29: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q
.|-sn)/(1-sn))]|;
A30: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
A31: sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2 )), |.
q.|* ((q`2/|.q.|-sn)/(1-sn))]| by A1,A2,A28,Th84;
A32: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
now
per cases by JGRAPH_2:3;
case
A33: p`1<=0;
then
A34: (sn-FanMorphE).p=p by Th82;
A35: |.q.|<>0 by A28,TOPRNS_1:24;
then
A36: (|.q.|)^2>0 by SQUARE_1:12;
A37: (q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48;
A38: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A39: (q`2/|.q.|-sn)>=0 by A28,XREAL_1:48;
A40: 1-sn>0 by A2,XREAL_1:149;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A38,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A36,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A40,XREAL_1:72;
then -1<=(-( q`2/|.q.|-sn))/(1-sn) by A40,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A40,A37,SQUARE_1:49;
then
A41: 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)>=0;
then sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)>=0 by XCMPLX_1:76;
then p`1=0 by A5,A31,A33,A34,EUCLID:52;
then
A42: (sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))=0 by A5,A31,A32,A34,A35,
XCMPLX_1:6;
1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by A41,XCMPLX_1:187;
then 1-((q`2/|.q.|-sn)/(1-sn))^2=0 by A42,SQUARE_1:24;
then 1= (q`2/|.q.|-sn)/(1-sn) by A40,A39,SQUARE_1:18,22;
then 1 *(1-sn)=(q`2/|.q.|-sn) by A40,XCMPLX_1:87;
then 1 *|.q.|=q`2 by A28,TOPRNS_1:24,XCMPLX_1:87;
then q`1=0 by A38,XCMPLX_1:6;
hence thesis by A5,A34,Th82;
end;
case
A43: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A29,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`2/|.q.| by SQUARE_1:51;
then 1-sn>=q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)<= -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XREAL_1:72;
then
A44: -1<=(-( q`2/|.q.|-sn))/(1-sn) by A6,XCMPLX_1:197;
(q`2/|.q.|-sn)>= 0 by A28,XREAL_1:48;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<=1^2 by A6,A44,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A45: 1-(-((q`2/|.q.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
then
A46: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1-sn))^2) by A45,SQUARE_1:def 2;
A47: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1-sn)) by EUCLID:52;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A47,A46;
then
A48: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A49: |.q4.|=|.q.| by SQUARE_1:22;
0<=(p`1)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A50: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
|.p.|<>0 by A43,TOPRNS_1:24;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`2)^2/(|.p.|)^2 <= 1 by A50,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`2/|.p.| by SQUARE_1:51;
then 1-sn>=p`2/|.p.|-sn by XREAL_1:9;
then -(1-sn)<= -( p`2/|.p.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XREAL_1:72;
then
A51: -1<=(-( p`2/|.p.|-sn))/(1-sn) by A6,XCMPLX_1:197;
(p`2/|.p.|-sn)>= 0 by A43,XREAL_1:48;
then ((-(p`2/|.p.|-sn))/(1-sn))^2<=1^2 by A6,A51,SQUARE_1:49;
then 1-((-(p`2/|.p.|-sn))/(1-sn))^2>=0 by XREAL_1:48;
then
A52: 1-(-((p`2/|.p.|-sn))/(1-sn))^2>=0 by XCMPLX_1:187;
set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1-sn))]|;
A53: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) by EUCLID:52;
p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)) by EUCLID:52;
then
A54: (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2))^2
.= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1-sn))^2) by A52,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A53,A54;
then
A55: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A56: |.p4.|=|.p.| by SQUARE_1:22;
A57: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)
), |.p.|* ((p`2/|.p.|-sn)/(1-sn))]| by A1,A2,A43,Th84;
then ((p`2/|.p.|-sn)/(1-sn)) =|.q.|* ((q`2/|.q.|-sn)/(1-sn))/|.p
.| by A5,A31,A30,A43,A53,TOPRNS_1:24,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1-sn)=(q`2/|.q.|-sn)/(1-sn) by A5,A31,A43,A57
,A48,A55,TOPRNS_1:24,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1-sn)*(1-sn)=q`2/|.q.|-sn by A6,XCMPLX_1:87;
then p`2/|.p.|-sn=q`2/|.q.|-sn by A6,XCMPLX_1:87;
then p`2/|.p.|*|.p.|=q`2 by A5,A31,A43,A57,A49,A56,TOPRNS_1:24
,XCMPLX_1:87;
then
A58: p`2=q`2 by A43,TOPRNS_1:24,XCMPLX_1:87;
A59: p=|[p`1,p`2]| by EUCLID:53;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then p`1=sqrt((q`1)^2) by A5,A31,A43,A57,A49,A56,A58,SQUARE_1:22;
then p`1=q`1 by A28,SQUARE_1:22;
hence thesis by A58,A59,EUCLID:53;
end;
case
A60: p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1>=0;
then p`2/|.p.|-sn<0 by XREAL_1:49;
then
A61: ((p`2/|.p.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148;
set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1+sn))]|;
A62: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) & q`2/|.q.|-sn>=0 by A28,
EUCLID:52,XREAL_1:48;
A63: 1-sn>0 by A2,XREAL_1:149;
sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)
), |.p.|* ((p`2/ |.p.|-sn)/(1+sn))]| & |.p.|<>0 by A1,A2,A60,Th84,TOPRNS_1:24;
hence thesis by A5,A31,A30,A61,A62,A63,XREAL_1:132;
end;
end;
hence thesis;
end;
case
A64: q`2/|.q.|<sn & q`1>=0 & q<>0.TOP-REAL 2;
then
A65: |.q.|<>0 by TOPRNS_1:24;
then
A66: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q
.|-sn)/(1+sn))]|;
A67: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
A68: sn-FanMorphE.q= |[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2 )),
|.q.|* ((q`2/|.q.|-sn)/(1+sn))]| by A1,A2,A64,Th84;
A69: q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
now
per cases by JGRAPH_2:3;
case
A70: p`1<=0;
A71: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A72: 1+sn>0 by A1,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then 0+(q`2)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A71,XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then (-((q`2)/|.q.|))^2 <= 1;
then 1>= -q`2/|.q.| by SQUARE_1:51;
then (1+sn)>= -q`2/|.q.|+sn by XREAL_1:7;
then
A73: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A72,XREAL_1:185;
A74: (q`2/|.q.|-sn)<=0 by A64,XREAL_1:47;
then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A72;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A73,SQUARE_1:49;
then
A75: 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A76: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
A77: (sn-FanMorphE).p=p by A70,Th82;
sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>=0 by A75,SQUARE_1:def 2;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((q`2/|.q.|-sn))^2/(1+sn)^2)>=0;
then sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)>=0 by XCMPLX_1:76;
then p`1=0 by A5,A68,A70,A77,EUCLID:52;
then (sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))=0 by A5,A68,A69,A65,A77,
XCMPLX_1:6;
then 1-((q`2/|.q.|-sn)/(1+sn))^2=0 by A76,SQUARE_1:24;
then 1=sqrt((-((q`2/|.q.|-sn)/(1+sn)))^2);
then 1= -((q`2/|.q.|-sn)/(1+sn)) by A72,A74,SQUARE_1:22;
then 1= ((-(q`2/|.q.|-sn))/(1+sn)) by XCMPLX_1:187;
then 1 *(1+sn)=-(q`2/|.q.|-sn) by A72,XCMPLX_1:87;
then 1+sn-sn=-q`2/|.q.|;
then 1=(-q`2)/|.q.| by XCMPLX_1:187;
then 1 *|.q.|=-q`2 by A64,TOPRNS_1:24,XCMPLX_1:87;
then (q`2)^2-(q`2)^2 =(q`1)^2 by A71,XCMPLX_1:26;
then q`1=0 by XCMPLX_1:6;
hence thesis by A5,A77,Th82;
end;
case
A78: p<>0.TOP-REAL 2 & p`2/|.p.|>=sn & p`1>=0;
set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1-sn))]|;
A79: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1-sn)) & |.q.|<>0 by A64,EUCLID:52
,TOPRNS_1:24;
q`2/|.q.|-sn<0 by A64,XREAL_1:49;
then
A80: ((q`2/|.q.|-sn)/(1+sn))<0 by A1,XREAL_1:141,148;
A81: 1-sn>0 by A2,XREAL_1:149;
sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1-sn))^2)
), |.p.|* ((p`2/ |.p.|-sn)/(1-sn))]| & p`2/|.p.|-sn>=0 by A1,A2,A78,Th84,
XREAL_1:48;
hence thesis by A5,A68,A67,A80,A79,A81,XREAL_1:132;
end;
case
A82: p<>0.TOP-REAL 2 & p`2/|.p.|<sn & p`1>=0;
0<=(p`1)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`2)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A83: (p`2)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A84: 1+sn>0 by A1,XREAL_1:148;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A66,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`2/|.q.| by SQUARE_1:51;
then -1-sn<=q`2/|.q.|-sn by XREAL_1:9;
then -(-1-sn)>= -(q`2/|.q.|-sn) by XREAL_1:24;
then
A85: (-(q`2/|.q.|-sn))/(1+sn)<=1 by A84,XREAL_1:185;
(q`2/|.q.|-sn)<=0 by A64,XREAL_1:47;
then -1<=(-( q`2/|.q.|-sn))/(1+sn) by A84;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<=1^2 by A85,SQUARE_1:49;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A86: 1-(-((q`2/|.q.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
q4`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) by EUCLID:52;
then
A87: (q4`1)^2= (|.q.|)^2*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2))^2
.= (|.q.|)^2*(1-((q`2/|.q.|-sn)/(1+sn))^2) by A86,SQUARE_1:def 2;
A88: q4`2= |.q.|* ((q`2/|.q.|-sn)/(1+sn)) by EUCLID:52;
set p4= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)), |.p.|* ((p`2
/|.p.|-sn)/(1+sn))]|;
A89: p4`2= |.p.|* ((p`2/|.p.|-sn)/(1+sn)) by EUCLID:52;
|.p.|<>0 by A82,TOPRNS_1:24;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`2)^2/(|.p.|)^2 <= 1 by A83,XCMPLX_1:60;
then ((p`2)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then -1<=p`2/|.p.| by SQUARE_1:51;
then -1-sn<=p`2/|.p.|-sn by XREAL_1:9;
then -(-1-sn)>= -(p`2/|.p.|-sn) by XREAL_1:24;
then
A90: (-(p`2/|.p.|-sn))/(1+sn)<=1 by A84,XREAL_1:185;
(p`2/|.p.|-sn)<=0 by A82,XREAL_1:47;
then -1<=(-( p`2/|.p.|-sn))/(1+sn) by A84;
then ((-(p`2/|.p.|-sn))/(1+sn))^2<=1^2 by A90,SQUARE_1:49;
then 1-((-(p`2/|.p.|-sn))/(1+sn))^2>=0 by XREAL_1:48;
then
A91: 1-(-((p`2/|.p.|-sn))/(1+sn))^2>=0 by XCMPLX_1:187;
p4`1= |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)) by EUCLID:52;
then
A92: (p4`1)^2= (|.p.|)^2*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2))^2
.= (|.p.|)^2*(1-((p`2/|.p.|-sn)/(1+sn))^2) by A91,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A89,A92;
then
A93: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A94: |.p4.|=|.p.| by SQUARE_1:22;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A88,A87;
then
A95: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A96: |.q4.|=|.q.| by SQUARE_1:22;
A97: sn -FanMorphE.p= |[ |.p.|*(sqrt(1-((p`2/|.p.|-sn)/(1+sn))^2)
), |.p.|* ((p`2/|.p.|-sn)/(1+sn))]| by A1,A2,A82,Th84;
then ((p`2/|.p.|-sn)/(1+sn)) =|.q.|* ((q`2/|.q.|-sn)/(1+sn))/|.p
.| by A5,A68,A67,A82,A89,TOPRNS_1:24,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1+sn)=(q`2/|.q.|-sn)/(1+sn) by A5,A68,A82,A97
,A95,A93,TOPRNS_1:24,XCMPLX_1:89;
then (p`2/|.p.|-sn)/(1+sn)*(1+sn)=q`2/|.q.|-sn by A84,XCMPLX_1:87;
then p`2/|.p.|-sn=q`2/|.q.|-sn by A84,XCMPLX_1:87;
then p`2/|.p.|*|.p.|=q`2 by A5,A68,A82,A97,A96,A94,TOPRNS_1:24
,XCMPLX_1:87;
then
A98: p`2=q`2 by A82,TOPRNS_1:24,XCMPLX_1:87;
A99: p=|[p`1,p`2]| by EUCLID:53;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then p`1=sqrt((q`1)^2) by A5,A68,A82,A97,A96,A94,A98,SQUARE_1:22;
then p`1=q`1 by A64,SQUARE_1:22;
hence thesis by A98,A99,EUCLID:53;
end;
end;
hence thesis;
end;
end;
hence thesis;
end;
hence thesis by FUNCT_1:def 4;
end;
theorem Th103:
for sn being Real st -1<sn & sn<1 holds (sn-FanMorphE) is
Function of TOP-REAL 2,TOP-REAL 2 & rng (sn-FanMorphE) = the carrier of
TOP-REAL 2
proof
let sn be Real;
assume that
A1: -1<sn and
A2: sn<1;
thus (sn-FanMorphE) is Function of TOP-REAL 2,TOP-REAL 2;
for f being Function of TOP-REAL 2,TOP-REAL 2 st f=(sn-FanMorphE) holds
rng (sn-FanMorphE)=the carrier of TOP-REAL 2
proof
let f be Function of TOP-REAL 2,TOP-REAL 2;
assume
A3: f=(sn-FanMorphE);
A4: 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 by JGRAPH_2:3;
suppose
q`1<=0;
then y=(sn-FanMorphE).q by Th82;
hence ex x being set st x in dom (sn-FanMorphE) & y=(sn-FanMorphE).x
by A3,A4;
end;
suppose
A5: q`2/|.q.|>=0 & q`1>=0 & q<>0.TOP-REAL 2;
--(1+sn)>0 by A1,XREAL_1:148;
then
A6: -(-1-sn)>0;
A7: 1-sn>=0 by A2,XREAL_1:149;
then q`2/|.q.|*(1-sn)>=0 by A5;
then -1-sn<= q`2/|.q.|*(1-sn) by A6;
then
A8: -1-sn+sn<= q`2/|.q.|*(1-sn)+sn by XREAL_1:7;
set px=|[ (|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2), |.q.|*(q`2/|.q.|*
(1-sn)+sn)]|;
A9: px`2 = |.q.|*(q`2/|.q.|*(1-sn)+sn) by EUCLID:52;
|.q.|<>0 by A5,TOPRNS_1:24;
then
A10: |.q.|^2>0 by SQUARE_1:12;
A11: dom (sn-FanMorphE)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A12: 1-sn>0 by A2,XREAL_1:149;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`2/|.q.|<=1 by SQUARE_1:51;
then q`2/|.q.|*(1-sn) <= 1 *(1-sn) by A12,XREAL_1:64;
then q`2/|.q.|*(1-sn)+sn-sn <=1-sn;
then (q`2/|.q.|*(1-sn)+sn) <=1 by XREAL_1:9;
then 1^2>=(q`2/|.q.|*(1-sn)+sn)^2 by A8,SQUARE_1:49;
then
A13: 1-(q`2/|.q.|*(1-sn)+sn)^2>=0 by XREAL_1:48;
then
A14: sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2)>=0 by SQUARE_1:def 2;
A15: px`1 = (|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2) by EUCLID:52;
then
|.px.|^2=((|.q.|)*sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2))^2 +(|.q.|*(q
`2/|.q.|*(1-sn)+sn))^2 by A9,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`2/|.q.|*(1-sn)+sn)^2))^2 +(|.q.|)^2*((q`2
/|.q.|*(1-sn)+sn))^2;
then
A16: |.px.|^2=(|.q.|)^2*(1-(q`2/|.q.|*(1-sn)+sn)^2) +(|.q.|)^2*((q`2
/|.q.|*(1-sn)+sn))^2 by A13,SQUARE_1:def 2
.= (|.q.|)^2;
then
A17: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A18: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24;
(q`2/|.q.|*(1-sn)+sn)>=0+sn by A5,A7,XREAL_1:7;
then px`2/|.px.| >=sn by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89;
then
A19: (sn-FanMorphE).px =|[ |.px.|*(sqrt(1-((px`2/|.px.|-sn) /(1-sn))
^2)), |.px.|* ((px`2/|.px.|-sn)/(1-sn))]| by A1,A2,A15,A14,A18,Th84;
A20: |.px.|*(sqrt((q`1/|.q.|)^2))=|.q.|*(q`1/|.q.|) by A5,A17,SQUARE_1:22
.=q`1 by A5,TOPRNS_1:24,XCMPLX_1:87;
A21: |.px.|* ((px`2/|.px.|-sn)/(1-sn)) =|.q.|* (( ((q`2/|.q.|*(1-sn)
+sn))-sn)/(1-sn)) by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`2/|.q.|) by A12,XCMPLX_1:89
.= q`2 by A5,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*(sqrt(1-((px`2/|.px.|-sn)/(1-sn))^2)) = |.px.|*(sqrt(1-(
q`2/|.px.|)^2)) by A5,A17,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*(sqrt(1-(q`2)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*(sqrt( (|.px.|)^2/(|.px.|)^2-(q`2)^2/(|.px.|)^2)) by A10
,A16,XCMPLX_1:60
.= |.px.|*(sqrt( ((|.px.|)^2-(q`2)^2)/(|.px.|)^2)) by XCMPLX_1:120
.= |.px.|*(sqrt( ((q`1)^2+(q`2)^2-(q`2)^2)/(|.px.|)^2)) by A16,
JGRAPH_3:1
.= |.px.|*(sqrt((q`1/|.q.|)^2)) by A17,XCMPLX_1:76;
hence ex x being set st x in dom (sn-FanMorphE) & y=(sn-FanMorphE).x
by A19,A21,A20,A11,EUCLID:53;
end;
suppose
A22: q`2/|.q.|<0 & q`1>=0 & q<>0.TOP-REAL 2;
A23: 1+sn>=0 by A1,XREAL_1:148;
(1-sn)>0 by A2,XREAL_1:149;
then
A24: 1-sn+sn>= q`2/|.q.|*(1+sn)+sn by A22,A23,XREAL_1:7;
A25: 1+sn>0 by A1,XREAL_1:148;
|.q.|<>0 by A22,TOPRNS_1:24;
then
A26: |.q.|^2>0 by SQUARE_1:12;
0<=(q`1)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`2)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`2)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`2/|.q.|>=-1 by SQUARE_1:51;
then q`2/|.q.|*(1+sn) >=(-1)*(1+sn) by A25,XREAL_1:64;
then q`2/|.q.|*(1+sn)+sn-sn >=-1-sn;
then (q`2/|.q.|*(1+sn)+sn) >=-1 by XREAL_1:9;
then 1^2>=(q`2/|.q.|*(1+sn)+sn)^2 by A24,SQUARE_1:49;
then
A27: 1-(q`2/|.q.|*(1+sn)+sn)^2>=0 by XREAL_1:48;
then
A28: sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2)>=0 by SQUARE_1:def 2;
A29: dom (sn-FanMorphE)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set px=|[ (|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2), |.q.|*(q`2/|.q.|*
(1+sn)+sn)]|;
A30: px`2 = |.q.|*(q`2/|.q.|*(1+sn)+sn) by EUCLID:52;
A31: px`1 = (|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2) by EUCLID:52;
then
|.px.|^2=((|.q.|)*sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2))^2 +(|.q.|*(q
`2/|.q.|*(1+sn)+sn))^2 by A30,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`2/|.q.|*(1+sn)+sn)^2))^2 +(|.q.|)^2*((q`2
/|.q.|*(1+sn)+sn))^2;
then
A32: |.px.|^2=(|.q.|)^2*(1-(q`2/|.q.|*(1+sn)+sn)^2) +(|.q.|)^2*((q`2
/|.q.|*(1+sn)+sn))^2 by A27,SQUARE_1:def 2
.= (|.q.|)^2;
then
A33: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A34: px<>0.TOP-REAL 2 by A22,TOPRNS_1:23,24;
(q`2/|.q.|*(1+sn)+sn)<=0+sn by A22,A23,XREAL_1:7;
then px`2/|.px.| <=sn by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89;
then
A35: (sn-FanMorphE).px =|[ |.px.|*(sqrt(1-((px`2/|.px.|-sn) /(1+sn))
^2)), |.px.|* ((px`2/|.px.|-sn)/(1+sn))]| by A1,A2,A31,A28,A34,Th84;
A36: |.px.|*(sqrt((q`1/|.q.|)^2)) =|.q.|*(q`1/|.q.|) by A22,A33,
SQUARE_1:22
.=q`1 by A22,TOPRNS_1:24,XCMPLX_1:87;
A37: |.px.|* ((px`2/|.px.|-sn)/(1+sn)) =|.q.|* (( ((q`2/|.q.|*(1+sn)
+sn))-sn)/(1+sn)) by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`2/|.q.|) by A25,XCMPLX_1:89
.= q`2 by A22,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*(sqrt(1-((px`2/|.px.|-sn)/(1+sn))^2)) = |.px.|*(sqrt(1-(
q`2/|.px.|)^2)) by A22,A33,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*(sqrt(1-(q`2)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*(sqrt( (|.px.|)^2/(|.px.|)^2-(q`2)^2/(|.px.|)^2)) by A26
,A32,XCMPLX_1:60
.= |.px.|*(sqrt( ((|.px.|)^2-(q`2)^2)/(|.px.|)^2)) by XCMPLX_1:120
.= |.px.|*(sqrt( ((q`1)^2+(q`2)^2-(q`2)^2)/(|.px.|)^2)) by A32,
JGRAPH_3:1
.= |.px.|*(sqrt((q`1/|.q.|)^2)) by A33,XCMPLX_1:76;
hence ex x being set st x in dom (sn-FanMorphE) & y=(sn-FanMorphE).x
by A35,A37,A36,A29,EUCLID:53;
end;
end;
hence thesis by A3,FUNCT_1:def 3;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
hence thesis;
end;
theorem Th104:
for sn being Real,p2 being Point of TOP-REAL 2 st -1<sn & sn<1
ex K being non empty compact Subset of TOP-REAL 2 st K = (sn-FanMorphE).:K & ex
V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is open & V2 c= K & (sn
-FanMorphE).p2 in V2
proof
reconsider O=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67;
let sn be Real,p2 be Point of TOP-REAL 2;
A1: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider V0=Ball(O,|.p2.|+1) as Subset of TOP-REAL 2;
O in V0 & V0 c= cl_Ball(O,|.p2.|+1) by GOBOARD6:1,METRIC_1:14;
then reconsider
K0=cl_Ball(O,|.p2.|+1) as non empty compact Subset of TOP-REAL 2
by A1,Th15;
set q3= (sn-FanMorphE).p2;
reconsider VV0 = V0 as Subset of TopSpaceMetr Euclid 2;
reconsider u2=p2 as Point of Euclid 2 by EUCLID:67;
reconsider u3=q3 as Point of Euclid 2 by EUCLID:67;
A2: (sn-FanMorphE).:K0 c= K0
proof
let y be object;
assume y in (sn-FanMorphE).:K0;
then consider x being object such that
A3: x in dom (sn-FanMorphE) and
A4: x in K0 and
A5: y=(sn-FanMorphE).x by FUNCT_1:def 6;
reconsider q=x as Point of TOP-REAL 2 by A3;
reconsider uq=q as Point of Euclid 2 by EUCLID:67;
dist(O,uq)<= |.p2.|+1 by A4,METRIC_1:12;
then |.0.TOP-REAL 2 - q.|<= |.p2.|+1 by JGRAPH_1:28;
then |. -q.|<= |.p2.|+1 by RLVECT_1:4;
then
A6: |.q.|<= |.p2.|+1 by TOPRNS_1:26;
A7: y in rng (sn-FanMorphE) by A3,A5,FUNCT_1:def 3;
then reconsider u=y as Point of Euclid 2 by EUCLID:67;
reconsider q4=y as Point of TOP-REAL 2 by A7;
|.q4.|=|.q.| by A5,Th97;
then |. -q4.|<= |.p2.|+1 by A6,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q4.|<= |.p2.|+1 by RLVECT_1:4;
then dist(O,u)<= |.p2.|+1 by JGRAPH_1:28;
hence thesis by METRIC_1:12;
end;
VV0 is open by TOPMETR:14;
then
A8: V0 is open by Lm11,PRE_TOPC:30;
A9: |. p2.|<|.p2.|+1 by XREAL_1:29;
then |. -p2.|<|.p2.|+1 by TOPRNS_1:26;
then |.0.TOP-REAL 2 - p2.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u2)<|.p2.|+1 by JGRAPH_1:28;
then
A10: p2 in V0 by METRIC_1:11;
|.q3.|=|.p2.| by Th97;
then |. -q3.|<|.p2.|+1 by A9,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q3.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u3)< |.p2.|+1 by JGRAPH_1:28;
then
A11: (sn-FanMorphE).p2 in V0 by METRIC_1:11;
assume
A12: -1<sn & sn<1;
K0 c= (sn-FanMorphE).:K0
proof
let y be object;
assume
A13: y in K0;
then reconsider q4=y as Point of TOP-REAL 2;
reconsider y as Point of Euclid 2 by A13;
the carrier of TOP-REAL 2 c= rng (sn-FanMorphE) by A12,Th103;
then q4 in rng (sn-FanMorphE);
then consider x being object such that
A14: x in dom (sn-FanMorphE) and
A15: y=(sn-FanMorphE).x by FUNCT_1:def 3;
reconsider x as Point of Euclid 2 by A14,Lm11;
reconsider q=x as Point of TOP-REAL 2 by A14;
|.q4.|=|.q.| by A15,Th97;
then q in K0 by A13,Lm12;
hence thesis by A14,A15,FUNCT_1:def 6;
end;
then K0= (sn-FanMorphE).:K0 by A2,XBOOLE_0:def 10;
hence thesis by A10,A8,A11,METRIC_1:14;
end;
theorem
for sn being Real st -1<sn & sn<1 ex f being Function of TOP-REAL 2,
TOP-REAL 2 st f=(sn-FanMorphE) & f is being_homeomorphism
proof
let sn be Real;
reconsider f=(sn-FanMorphE) as Function of TOP-REAL 2,TOP-REAL 2;
assume
A1: -1<sn & sn<1;
then
A2: for p2 being Point of TOP-REAL 2 ex K being non empty compact Subset of
TOP-REAL 2 st K = f.:K & ex V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is
open & V2 c= K & f.p2 in V2 by Th104;
rng (sn-FanMorphE) = the carrier of TOP-REAL 2 & ex h being Function of
( TOP-REAL 2),(TOP-REAL 2) st h=( sn-FanMorphE) & h is continuous by A1,Th101
,Th103;
then f is being_homeomorphism by A1,A2,Th3,Th102;
hence thesis;
end;
theorem Th106:
for sn being Real,q being Point of TOP-REAL 2 st sn<1 & q`1>0 &
q`2/|.q.|>=sn holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds
p`1>0 & p`2>=0
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: sn<1 and
A2: q`1>0 and
A3: q`2/|.q.|>=sn;
A4: (q`2/|.q.|-sn)>= 0 by A3,XREAL_1:48;
let p be Point of TOP-REAL 2;
set qz=p;
A5: 1-sn>0 by A1,XREAL_1:149;
A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then 1>q`2/|.q.| by SQUARE_1:52;
then 1-sn>q`2/|.q.|-sn by XREAL_1:9;
then -(1-sn)< -( q`2/|.q.|-sn) by XREAL_1:24;
then (-(1-sn))/(1-sn)<(-( q`2/|.q.|-sn))/(1-sn) by A5,XREAL_1:74;
then -1<(-( q`2/|.q.|-sn))/(1-sn) by A5,XCMPLX_1:197;
then ((-(q`2/|.q.|-sn))/(1-sn))^2<1^2 by A5,A4,SQUARE_1:50;
then 1-((-(q`2/|.q.|-sn))/(1-sn))^2>0 by XREAL_1:50;
then sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1-sn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1-sn)^2)> 0;
then
A8: sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)> 0 by XCMPLX_1:76;
assume p=(sn-FanMorphE).q;
then
A9: p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(
1-sn))]| by A2,A3,Th82;
then qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)) by EUCLID:52;
hence thesis by A9,A6,A5,A4,A8,EUCLID:52,XREAL_1:129;
end;
theorem Th107:
for sn being Real,q being Point of TOP-REAL 2 st -1<sn & q`1>0
& q`2/|.q.|<sn holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q
holds p`1>0 & p`2<0
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: q`1>0 and
A3: q`2/|.q.|<sn;
A4: 1+sn>0 by A1,XREAL_1:148;
let p be Point of TOP-REAL 2;
set qz=p;
assume p=(sn-FanMorphE).q;
then
p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(
1+sn))]| by A2,A3,Th83;
then
A5: qz`1= |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)) & qz`2= |.q.|* ((q`2/|.
q.|- sn)/(1+sn)) by EUCLID:52;
A6: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
A8: (q`2/|.q.|-sn)< 0 by A3,XREAL_1:49;
then -( q`2/|.q.|-sn)>0 by XREAL_1:58;
then (-(1+sn))/(1+sn)<(-( q`2/|.q.|-sn))/(1+sn) by A4,XREAL_1:74;
then
A9: -1<(-( q`2/|.q.|-sn))/(1+sn) by A4,XCMPLX_1:197;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`2)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`2)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,XREAL_1:74;
then (q`2)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`2)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1<q`2/|.q.| by SQUARE_1:52;
then -1-sn<q`2/|.q.|-sn by XREAL_1:9;
then --(1+sn)> -(q`2/|.q.|-sn) by XREAL_1:24;
then (-(q`2/|.q.|-sn))/(1+sn)<1 by A4,XREAL_1:191;
then ((-(q`2/|.q.|-sn))/(1+sn))^2<1^2 by A9,SQUARE_1:50;
then 1-((-(q`2/|.q.|-sn))/(1+sn))^2>0 by XREAL_1:50;
then sqrt(1-((-(q`2/|.q.|-sn))/(1+sn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(q`2/|.q.|-sn))^2/(1+sn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(q`2/|.q.|-sn)^2/(1+sn)^2)> 0;
then
A10: sqrt(1-((q`2/|.q.|-sn)/(1+sn))^2)> 0 by XCMPLX_1:76;
((q`2/|.q.|-sn)/(1+sn))<0 by A1,A8,XREAL_1:141,148;
hence thesis by A6,A5,A10,XREAL_1:129,132;
end;
theorem Th108:
for sn being Real,q1,q2 being Point of TOP-REAL 2 st sn<1 & q1
`1>0 & q1`2/|.q1.|>=sn & q2`1>0 & q2`2/|.q2.|>=sn & q1`2/|.q1.|<q2`2/|.q2.|
holds for p1,p2 being Point of TOP-REAL 2 st p1=(sn-FanMorphE).q1 & p2=(sn
-FanMorphE).q2 holds p1`2/|.p1.|<p2`2/|.p2.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: sn<1 and
A2: q1`1>0 and
A3: q1`2/|.q1.|>=sn and
A4: q2`1>0 and
A5: q2`2/|.q2.|>=sn and
A6: q1`2/|.q1.|<q2`2/|.q2.|;
A7: q1`2/|.q1.|-sn< q2`2/|.q2.|-sn & 1-sn>0 by A1,A6,XREAL_1:9,149;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(sn-FanMorphE).q1 and
A9: p2=(sn-FanMorphE).q2;
A10: |.p2.|=|.q2.| by A9,Th97;
p2=|[ |.q2.|*(sqrt(1-((q2`2/|.q2.|-sn)/(1-sn))^2)), |.q2.|* ((q2`2/|.q2
.|-sn)/(1-sn))]| by A4,A5,A9,Th82;
then
A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1-sn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`2/|.p2.|= ((q2`2/|.q2.|-sn)/(1-sn)) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|*(sqrt(1-((q1`2/|.q1.|-sn)/(1-sn))^2)), |.q1.|* ((q1`2/|.q1
.|-sn)/(1-sn))]| by A2,A3,A8,Th82;
then
A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1-sn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th97;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`2/|.p1.|= ((q1`2/|.q1.|-sn)/(1-sn)) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem Th109:
for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1<sn & q1
`1>0 & q1`2/|.q1.|<sn & q2`1>0 & q2`2/|.q2.|<sn & q1`2/|.q1.|<q2`2/|.q2.| holds
for p1,p2 being Point of TOP-REAL 2 st p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE)
.q2 holds p1`2/|.p1.|<p2`2/|.p2.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: q1`1>0 and
A3: q1`2/|.q1.|<sn and
A4: q2`1>0 and
A5: q2`2/|.q2.|<sn and
A6: q1`2/|.q1.|<q2`2/|.q2.|;
A7: q1`2/|.q1.|-sn< q2`2/|.q2.|-sn & 1+sn>0 by A1,A6,XREAL_1:9,148;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(sn-FanMorphE).q1 and
A9: p2=(sn-FanMorphE).q2;
A10: |.p2.|=|.q2.| by A9,Th97;
p2=|[ |.q2.|*(sqrt(1-((q2`2/|.q2.|-sn)/(1+sn))^2)), |.q2.|* ((q2`2/|.q2
.|-sn)/(1+sn))]| by A4,A5,A9,Th83;
then
A11: p2`2= |.q2.|* ((q2`2/|.q2.|-sn)/(1+sn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`2/|.p2.|= (q2`2/|.q2.|-sn)/(1+sn) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|*(sqrt(1-((q1`2/|.q1.|-sn)/(1+sn))^2)), |.q1.|* ((q1`2/|.q1
.|-sn)/(1+sn))]| by A2,A3,A8,Th83;
then
A13: p1`2= |.q1.|* ((q1`2/|.q1.|-sn)/(1+sn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th97;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`2/|.p1.|= (q1`2/|.q1.|-sn)/(1+sn) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem
for sn being Real,q1,q2 being Point of TOP-REAL 2 st -1<sn & sn<1 & q1
`1>0 & q2`1>0 & q1`2/|.q1.|<q2`2/|.q2.| holds for p1,p2 being Point of TOP-REAL
2 st p1=(sn-FanMorphE).q1 & p2=(sn-FanMorphE).q2 holds p1`2/|.p1.|<p2`2/|.p2.|
proof
let sn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<sn and
A2: sn<1 and
A3: q1`1>0 and
A4: q2`1>0 and
A5: q1`2/|.q1.|<q2`2/|.q2.|;
let p1,p2 be Point of TOP-REAL 2;
assume that
A6: p1=(sn-FanMorphE).q1 and
A7: p2=(sn-FanMorphE).q2;
per cases;
suppose
q1`2/|.q1.|>=sn & q2`2/|.q2.|>=sn;
hence thesis by A2,A3,A4,A5,A6,A7,Th108;
end;
suppose
q1`2/|.q1.|>=sn & q2`2/|.q2.|<sn;
hence thesis by A5,XXREAL_0:2;
end;
suppose
A8: q1`2/|.q1.|<sn & q2`2/|.q2.|>=sn;
then p2`2>=0 by A2,A4,A7,Th106;
then
A9: p2`2/|.p2.|>=0;
p1`2<0 by A1,A3,A6,A8,Th107;
hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141;
end;
suppose
q1`2/|.q1.|<sn & q2`2/|.q2.|<sn;
hence thesis by A1,A3,A4,A5,A6,A7,Th109;
end;
end;
theorem
for sn being Real,q being Point of TOP-REAL 2 st q`1>0 & q`2/|.q.|=sn
holds for p being Point of TOP-REAL 2 st p=(sn-FanMorphE).q holds p`1>0 & p`2=0
proof
let sn be Real,q be Point of TOP-REAL 2;
assume that
A1: q`1>0 and
A2: q`2/|.q.|=sn;
A3: |.q.|<>0 & sqrt(1-((-(q`2/|.q.|-sn))/(1-sn))^2)>0 by A1,A2,JGRAPH_2:3
,TOPRNS_1:24;
let p be Point of TOP-REAL 2;
assume p=(sn-FanMorphE).q;
then p=|[ |.q.|*(sqrt(1-((q`2/|.q.|-sn)/(1-sn))^2)), |.q.|* ((q`2/|.q.|-sn)/(
1-sn))]| by A1,A2,Th82;
hence thesis by A2,A3,EUCLID:52;
end;
theorem
for sn being Real holds 0.TOP-REAL 2=(sn-FanMorphE).(0.TOP-REAL
2) by Th82,JGRAPH_2:3;
begin :: Fan Morphism for South
definition
let s be Real, q be Point of TOP-REAL 2;
func FanS(s,q) -> Point of TOP-REAL 2 equals
:Def8:
|.q.|*|[(q`1/|.q.|-s)/(1
-s), -sqrt(1-((q`1/|.q.|-s)/(1-s))^2)]| if q`1/|.q.|>=s & q`2<0, |.q.|*|[(q`1/
|.q.|-s)/(1+s), -sqrt(1-((q`1/|.q.|-s)/(1+s))^2)]| if q`1/|.q.|<s & q`2<0
otherwise q;
correctness;
end;
definition
let c be Real;
func c-FanMorphS -> Function of TOP-REAL 2, TOP-REAL 2 means
:Def9:
for q being Point of TOP-REAL 2 holds it.q=FanS(c,q);
existence
proof
deffunc F(Point of TOP-REAL 2)=FanS(c,$1);
thus ex IT being Function of TOP-REAL 2, TOP-REAL 2 st for q being Point
of TOP-REAL 2 holds IT.q=F(q)from FUNCT_2:sch 4;
end;
uniqueness
proof
deffunc F(Point of TOP-REAL 2)=FanS(c,$1);
thus for a,b being Function of TOP-REAL 2, TOP-REAL 2 st (for q being
Point of TOP-REAL 2 holds a.q=F(q)) & (for q being Point of TOP-REAL 2 holds b.
q=F(q)) holds a=b from BINOP_2:sch 1;
end;
end;
theorem Th113:
for cn being Real holds (q`1/|.q.|>=cn & q`2<0 implies
cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q`1/|.q.|-
cn)/(1-cn))^2))]|)& (q`2>=0 implies cn-FanMorphS.q=q)
proof
let cn be Real;
hereby
assume q`1/|.q.|>=cn & q`2<0;
then
FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), -sqrt(1-((q`1/|.q.|-cn)/(1-
cn))^2)]| by Def8
.= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q`1/|.q.|-cn)/(
1-cn))^2))]| by EUCLID:58;
hence
cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*(-sqrt(1-((q
`1/|.q.|-cn)/(1-cn))^2))]| by Def9;
end;
assume
A1: q`2>=0;
cn-FanMorphS.q=FanS(cn,q) by Def9;
hence thesis by A1,Def8;
end;
theorem Th114:
for cn being Real holds (q`1/|.q.|<=cn & q`2<0 implies cn
-FanMorphS.q= |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)
/(1+cn))^2))]|)
proof
let cn be Real;
assume that
A1: q`1/|.q.|<=cn and
A2: q`2<0;
per cases by A1,XXREAL_0:1;
suppose
q`1/|.q.|<cn;
then
FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1+cn), -sqrt(1-((q`1/|.q.|-cn)/(1+
cn))^2)]| by A2,Def8
.= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/
(1+cn))^2))]| by EUCLID:58;
hence thesis by Def9;
end;
suppose
A3: q`1/|.q.|=cn;
then (q`1/|.q.|-cn)/(1-cn)=0;
hence thesis by A2,A3,Th113;
end;
end;
theorem Th115:
for cn being Real st -1<cn & cn<1 holds (q`1/|.q.|>=cn & q`2<=0
& q<>0.TOP-REAL 2 implies cn-FanMorphS.q = |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)),
|.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]|) & (q`1/|.q.|<=cn & q`2<=0 & q<>
0.TOP-REAL 2 implies cn-FanMorphS.q = |[ |.q.|*((q`1/|.q.|-cn)/(1+cn)), |.q.|*(
-sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]|)
proof
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
per cases;
suppose
A3: q`1/|.q.|>=cn & q`2<=0 & q<>0.TOP-REAL 2;
per cases;
suppose
A4: q`2<0;
then
FanS(cn,q)= |.q.|*|[(q`1/|.q.|-cn)/(1-cn), -sqrt(1-((q`1/|.q.|-cn)/(
1-cn))^2)]| by A3,Def8
.= |[|.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)
/(1-cn))^2))]| by EUCLID:58;
hence thesis by A4,Def9,Th114;
end;
suppose
A5: q`2>=0;
then
A6: cn-FanMorphS.q=q by Th113;
A7: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A8: 1-cn>0 by A2,XREAL_1:149;
A9: q`2=0 by A3,A5;
|.q.|<>0 by A3,TOPRNS_1:24;
then |.q.|^2>0 by SQUARE_1:12;
then (q`1)^2/|.q.|^2=1^2 by A7,A9,XCMPLX_1:60;
then ((q`1)/|.q.|)^2=1^2 by XCMPLX_1:76;
then
A10: sqrt(((q`1)/|.q.|)^2)=1;
A11: now
assume q`1<0;
then -((q`1)/|.q.|)=1 by A10,SQUARE_1:23;
hence contradiction by A1,A3;
end;
sqrt((|.q.|)^2)=|.q.| by SQUARE_1:22;
then
A12: |.q.|=q`1 by A7,A9,A11,SQUARE_1:22;
then 1=q`1/|.q.| by A3,TOPRNS_1:24,XCMPLX_1:60;
then (q`1/|.q.|-cn)/(1-cn)=1 by A8,XCMPLX_1:60;
hence thesis by A2,A6,A9,A12,EUCLID:53,TOPRNS_1:24
,XCMPLX_1:60;
end;
end;
suppose
A13: q`1/|.q.|<=cn & q`2<=0 & q<>0.TOP-REAL 2;
per cases;
suppose
q`2<0;
hence thesis by Th113,Th114;
end;
suppose
A14: q`2>=0;
then
A15: q`2=0 by A13;
A16: 1+cn>0 by A1,XREAL_1:148;
A17: |.q.|<>0 by A13,TOPRNS_1:24;
1>q`1/|.q.| by A2,A13,XXREAL_0:2;
then 1 *(|.q.|)>q`1/|.q.|*(|.q.|) by A17,XREAL_1:68;
then
A18: (|.q.|)^2 =(q`1)^2+(q`2)^2 & (|.q.|)>q`1 by A13,JGRAPH_3:1,TOPRNS_1:24
,XCMPLX_1:87;
then
A19: |.q.|=-q`1 by A15,SQUARE_1:40;
A20: q`1= -(|.q.|) by A15,A18,SQUARE_1:40;
then -1=q`1/|.q.| by A13,TOPRNS_1:24,XCMPLX_1:197;
then (q`1/|.q.|-cn)/(1+cn) =(-(1+cn))/(1+cn) .=-1 by A16,XCMPLX_1:197;
then
|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*(-sqrt(1-((q`1/|.q.|-cn)/(
1+cn))^2))]| =q by A15,A19,EUCLID:53;
hence thesis by A1,A14,A17,A20,Th113,XCMPLX_1:197;
end;
end;
suppose
q`2>0 or q=0.TOP-REAL 2;
hence thesis;
end;
end;
theorem Th116:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1-
cn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=(1-cn);
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: cn<1 and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in K1 by A7,A8,A9,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1-cn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th117:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<cn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|* ((p`1/|.p.|
-cn)/(1+cn))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`2<=0 & q<>0.TOP-REAL 2) holds f is continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=1+cn;
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: -1<cn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<=0 & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
1+cn>0 by A1,XREAL_1:148;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: for q being Point of (TOP-REAL 2)|K1,r1,r2 being Real st g2.q=r1 &
g1.q =r2 holds g3.q=r2*((r1/r2-a)/b) and
A6: g3 is continuous by A4,Th5;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
A8: for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then reconsider s=x as Point of (TOP-REAL 2)|K1;
x in dom g3 by A7,A9;
then x in K1 by A7,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A10: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
A11: g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
f.r=(|.r.|)* ((r`1/|.r.|-cn)/(1+cn)) by A2,A9;
hence thesis by A5,A11,A10;
end;
dom f=dom g3 by A7,FUNCT_2:def 1;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th118:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st cn<1 & (for p being Point of (TOP-REAL
2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*( -sqrt(1-((p`1/|.p.|
-cn)/(1-cn))^2))) & (for q being Point of TOP-REAL 2 st q in the carrier of (
TOP-REAL 2)|K1 holds q`2<=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2) holds f is
continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=1-cn;
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: cn<1 and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then
A4: for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
b>0 by A1,XREAL_1:149;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th9;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
A9: 1-cn>0 by A1,XREAL_1:149;
assume
A10: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
A11: |.r.|<>0 by A3,A10,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2;
(r`2)^2>=0 by XREAL_1:63;
then r`1<= |.r.| by A12,XREAL_1:93;
then r`1/|.r.| <= |.r.|/|.r.| by XREAL_1:72;
then r`1/|.r.|<=1 by A11,XCMPLX_1:60;
then
A13: r`1/|.r.|-cn<=(1-cn) by XREAL_1:9;
reconsider s=x as Point of (TOP-REAL 2)|K1 by A10;
A14: now
assume (1-cn)^2=0;
then 1-cn+cn=0+cn by XCMPLX_1:6;
hence contradiction by A1;
end;
cn-r`1/|.r.|<=0 by A3,A10,XREAL_1:47;
then -(cn- r`1/|.r.|)>=-(1-cn) by A9,XREAL_1:24;
then (1-cn)^2>=0 & (r`1/|.r.|-cn)^2<=(1-cn)^2 by A13,SQUARE_1:49,XREAL_1:63
;
then (r`1/|.r.|-cn)^2/(1-cn)^2<=(1-cn)^2/(1-cn)^2 by XREAL_1:72;
then (r`1/|.r.|-cn)^2/(1-cn)^2<=1 by A14,XCMPLX_1:60;
then ((r`1/|.r.|-cn)/(1-cn))^2<=1 by XCMPLX_1:76;
then 1-((r`1/|.r.|-cn)/(1-cn))^2>=0 by XREAL_1:48;
then |.1-((r`1/|.r.|-cn)/(1-cn))^2.| =1-((r`1/|.r.|-cn)/(1-cn))^2 by
ABSVALUE:def 1;
then
A15: f.r=(|.r.|)*(-sqrt(|.1-((r`1/|.r.|-cn)/(1-cn))^2.|)) by A2,A10;
A16: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
hence thesis by A5,A15,A16;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th119:
for cn being Real,K1 being non empty Subset of TOP-REAL 2, f
being Function of (TOP-REAL 2)|K1,R^1 st -1<cn & (for p being Point of (
TOP-REAL 2) st p in the carrier of (TOP-REAL 2)|K1 holds f.p=|.p.|*( -sqrt(1-((
p`1/|.p.|-cn)/(1+cn))^2))) & (for q being Point of TOP-REAL 2 st q in the
carrier of (TOP-REAL 2)|K1 holds q`2<=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2)
holds f is continuous
proof
let cn be Real,K1 be non empty Subset of TOP-REAL 2,
f be Function of (
TOP-REAL 2)|K1,R^1;
reconsider g1=(2 NormF)|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by
Lm5;
set a=cn, b=1+cn;
reconsider g2=proj1|K1 as continuous Function of (TOP-REAL 2)|K1,R^1 by Lm2;
assume that
A1: -1<cn and
A2: for p being Point of (TOP-REAL 2) st p in the carrier of (TOP-REAL 2
)|K1 holds f.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) and
A3: for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)
|K1 holds q`2<=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2;
A4: 1+cn>0 by A1,XREAL_1:148;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q<>0.TOP-REAL 2 by A3;
then for q being Point of (TOP-REAL 2)|K1 holds g1.q<>0 by Lm6;
then consider g3 being Function of (TOP-REAL 2)|K1,R^1 such that
A5: 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-a)/b)^2.|)) and
A6: g3 is continuous by A4,Th9;
A7: dom g3=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1;
then
A8: dom f=dom g3 by FUNCT_2:def 1;
for x being object st x in dom f holds f.x=g3.x
proof
let x be object;
assume
A9: x in dom f;
then x in K1 by A7,A8,PRE_TOPC:8;
then reconsider r=x as Point of (TOP-REAL 2);
reconsider s=x as Point of (TOP-REAL 2)|K1 by A9;
A10: (1+cn)^2>0 by A4,SQUARE_1:12;
A11: |.r.|<>0 by A3,A9,TOPRNS_1:24;
|.r.|^2=(r`1)^2+(r`2)^2 by JGRAPH_3:1;
then
A12: ((r`1) -(|.r.|))*((r`1)+|.r.|) =-(r`2)^2;
(r`2)^2>=0 by XREAL_1:63;
then -(|.r.|)<=r`1 by A12,XREAL_1:93;
then r`1/|.r.| >= (-(|.r.|))/|.r.| by XREAL_1:72;
then r`1/|.r.|>= -1 by A11,XCMPLX_1:197;
then r`1/|.r.|-cn>=-1-cn by XREAL_1:9;
then
A13: r`1/|.r.|-cn>=-(1+cn);
cn-r`1/|.r.|>=0 by A3,A9,XREAL_1:48;
then -(cn-r`1/|.r.|)<=-0;
then (r`1/|.r.|-cn)^2<=(1+cn)^2 by A4,A13,SQUARE_1:49;
then (r`1/|.r.|-cn)^2/(1+cn)^2<=(1+cn)^2/(1+cn)^2 by A4,XREAL_1:72;
then (r`1/|.r.|-cn)^2/(1+cn)^2<=1 by A10,XCMPLX_1:60;
then ((r`1/|.r.|-cn)/(1+cn))^2<=1 by XCMPLX_1:76;
then 1-((r`1/|.r.|-cn)/(1+cn))^2>=0 by XREAL_1:48;
then |.1-((r`1/|.r.|-cn)/(1+cn))^2.| =1-((r`1/|.r.|-cn)/(1+cn))^2 by
ABSVALUE:def 1;
then
A14: f.r=(|.r.|)*( -sqrt(|.1-((r`1/|.r.|-cn)/(1+cn))^2.|)) by A2,A9;
A15: proj1.r=r`1 & (2 NormF).r=|.r.| by Def1,PSCOMP_1:def 5;
g2.s=proj1.s & g1.s=(2 NormF).s by Lm2,Lm5;
hence thesis by A5,A14,A15;
end;
hence thesis by A6,A8,FUNCT_1:2;
end;
theorem Th120:
for cn being Real, K0,B0 being Subset of TOP-REAL 2,f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphS)|
K0 & B0={q where q is Point of TOP-REAL 2: q`2<=0 & q<>0.TOP-REAL 2} & K0={p: p
`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=-sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
p0`1=cn by EUCLID:52;
then
A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1;
assume
A3: -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0={q where q is Point of
TOP-REAL 2: q`2<=0 & q<>0.TOP-REAL 2} & K0={p: p`1/|.p.|>=cn & p`2<=0 & p<>0.
TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A4: 1-cn^2>0 by XREAL_1:50;
then
A5: -sn>0 by SQUARE_1:25;
A6: now
assume p0=0.TOP-REAL 2;
then --sn=-0 by EUCLID:52,JGRAPH_2:3;
hence contradiction by A4,SQUARE_1:25;
end;
(-sn)^2=1-cn^2 by A4,SQUARE_1:def 2;
then p0`1/|.p0.|=cn by A2,EUCLID:52;
then
A7: p0 in K0 by A3,A1,A6,A5;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A8: rng (proj2*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17;
A9: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|>= cn & p8`2<=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A10: dom ((cn-FanMorphS)|K1) c= dom (proj1*((cn-FanMorphS)|K1))
proof
let x be object;
assume
A11: x in dom ((cn-FanMorphS)|K1);
then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphS) by XBOOLE_0:def 4;
then
A12: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn
-FanMorphS) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A11,FUNCT_1:47;
hence thesis by A11,A12,FUNCT_1:11;
end;
A13: rng (proj1*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj1*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25;
then dom (proj1*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A10,
XBOOLE_0:def 10
.=dom (cn-FanMorphS) /\ 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;
then reconsider
g2=proj1*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A13,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.|* ((p`1/|.p.|-cn)/(1-cn))
proof
let p be Point of TOP-REAL 2;
A14: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A15: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A16: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A15;
then
A17: (cn-FanMorphS).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-
((p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th115;
((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A16,A15,FUNCT_1:49;
then
g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/
|.p.|-cn)/(1-cn))^2))]|) by A16,A14,A15,A17,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/
(1-cn))^2))]|)`1 by PSCOMP_1:def 5
.=|.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A18: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1-cn));
A19: dom ((cn-FanMorphS)|K1) c= dom (proj2*((cn-FanMorphS)|K1))
proof
let x be object;
assume
A20: x in dom ((cn-FanMorphS)|K1);
then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphS) by XBOOLE_0:def 4;
then
A21: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn
-FanMorphS) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A20,FUNCT_1:47;
hence thesis by A20,A21,FUNCT_1:11;
end;
dom (proj2*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25;
then dom (proj2*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A19,
XBOOLE_0:def 10
.=dom (cn-FanMorphS) /\ 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;
then reconsider
g1=proj2*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A8,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.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))
proof
let p be Point of TOP-REAL 2;
A22: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A23: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A24: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|>= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A23;
then
A25: (cn-FanMorphS).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-(
(p`1/|.p.|-cn)/(1-cn))^2))]| by A3,Th115;
((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A24,A23,FUNCT_1:49;
then
g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/
|.p.|-cn)/(1-cn))^2))]|) by A24,A22,A23,A25,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/
(1-cn))^2))]|)`2 by PSCOMP_1:def 6
.= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A26: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<=0 & q`1/|.q.|>=cn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A27: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A27;
hence thesis;
end;
then
A28: f1 is continuous by A3,A26,Th118;
A29: 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
A30: |[x,y]| in K1 and
A31: s=f2.(|[x,y]|) & r=f1.(|[x,y]|);
set p99=|[x,y]|;
A32: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| >=cn & p3`2<=0
& p3<>0.TOP-REAL 2 by A3,A30;
A33: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A34: f1.p99=|.p99.|*( -sqrt(1-((p99`1/|.p99.|-cn)/(1-cn))^2)) by A26,A30;
((cn-FanMorphS)|K0).(|[x,y]|)=((cn-FanMorphS)).(|[x,y]|) by A30,FUNCT_1:49
.= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1-cn)), |.p99.|*( -sqrt(1-((p99`1/
|.p99.|-cn)/(1-cn))^2))]| by A3,A32,Th115
.=|[s,r]| by A18,A30,A31,A33,A34;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A35: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|>= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A35;
hence thesis;
end;
then f2 is continuous by A3,A18,Th116;
hence thesis by A7,A9,A28,A29,JGRAPH_2:35;
end;
theorem Th121:
for cn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphS)|
K0 & B0={q where q is Point of TOP-REAL 2: q`2<=0 & q<>0.TOP-REAL 2} & K0={p: p
`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=-sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
p0`1=cn by EUCLID:52;
then
A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1;
assume
A3: -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0={q where q is Point of
TOP-REAL 2: q`2<=0 & q<>0.TOP-REAL 2} & K0={p: p`1/|.p.|<=cn & p`2<=0 & p<>0.
TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A4: 1-cn^2>0 by XREAL_1:50;
then
A5: -sn>0 by SQUARE_1:25;
A6: now
assume p0=0.TOP-REAL 2;
then --sn=-0 by EUCLID:52,JGRAPH_2:3;
hence contradiction by A4,SQUARE_1:25;
end;
(-sn)^2=1-cn^2 by A4,SQUARE_1:def 2;
then p0`1/|.p0.|=cn by A2,EUCLID:52;
then
A7: p0 in K0 by A3,A1,A6,A5;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
A8: rng (proj2*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17;
A9: K0 c= B0
proof
let x be object;
assume x in K0;
then
ex p8 being Point of TOP-REAL 2 st x=p8 & p8`1/|.p8.|<= cn & p8`2<=0 &
p8<>0.TOP-REAL 2 by A3;
hence thesis by A3;
end;
A10: dom ((cn-FanMorphS)|K1) c= dom (proj1*((cn-FanMorphS)|K1))
proof
let x be object;
assume
A11: x in dom ((cn-FanMorphS)|K1);
then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphS) by XBOOLE_0:def 4;
then
A12: dom proj1 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn
-FanMorphS) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A11,FUNCT_1:47;
hence thesis by A11,A12,FUNCT_1:11;
end;
A13: rng (proj1*((cn-FanMorphS)|K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj1*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25;
then dom (proj1*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A10,
XBOOLE_0:def 10
.=dom (cn-FanMorphS) /\ 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;
then reconsider
g2=proj1*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A13,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.|* ((p`1/|.p.|-cn)/(1+cn))
proof
let p be Point of TOP-REAL 2;
A14: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A15: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A16: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A15;
then
A17: (cn-FanMorphS).p =|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-
((p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th115;
((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A16,A15,FUNCT_1:49;
then
g2.p=proj1.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/
|.p.|-cn)/(1+cn))^2))]|) by A16,A14,A15,A17,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/
(1+cn))^2))]|)`1 by PSCOMP_1:def 5
.=|.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52;
hence thesis;
end;
then consider f2 being Function of (TOP-REAL 2)|K1,R^1 such that
A18: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f2.p=|.p.|* ((p`1/|.p.|-cn)/(1+cn));
A19: dom ((cn-FanMorphS)|K1) c= dom (proj2*((cn-FanMorphS)|K1))
proof
let x be object;
assume
A20: x in dom ((cn-FanMorphS)|K1);
then x in dom (cn-FanMorphS) /\ K1 by RELAT_1:61;
then x in dom (cn-FanMorphS) by XBOOLE_0:def 4;
then
A21: dom proj2 = (the carrier of TOP-REAL 2) & (cn-FanMorphS).x in rng (cn
-FanMorphS) by FUNCT_1:3,FUNCT_2:def 1;
((cn-FanMorphS)|K1).x=(cn-FanMorphS).x by A20,FUNCT_1:47;
hence thesis by A20,A21,FUNCT_1:11;
end;
dom (proj2*((cn-FanMorphS)|K1)) c= dom ((cn-FanMorphS)|K1) by RELAT_1:25;
then dom (proj2*((cn-FanMorphS)|K1)) =dom ((cn-FanMorphS)|K1) by A19,
XBOOLE_0:def 10
.=dom (cn-FanMorphS) /\ 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;
then reconsider
g1=proj2*((cn-FanMorphS)|K1) as Function of (TOP-REAL 2)|K1,R^1
by A8,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.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))
proof
let p be Point of TOP-REAL 2;
A22: dom ((cn-FanMorphS)|K1)=dom (cn-FanMorphS) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2))/\ K1 by FUNCT_2:def 1
.=K1 by XBOOLE_1:28;
A23: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume
A24: p in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st p=p3 & p3`1/|.p3.|<= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A23;
then
A25: (cn-FanMorphS).p=|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-(
(p`1/|.p.|-cn)/(1+cn))^2))]| by A3,Th115;
((cn-FanMorphS)|K1).p=(cn-FanMorphS).p by A24,A23,FUNCT_1:49;
then
g1.p=proj2.(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/
|.p.|-cn)/(1+cn))^2))]|) by A24,A22,A23,A25,FUNCT_1:13
.=(|[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/
(1+cn))^2))]|)`2 by PSCOMP_1:def 6
.= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
hence thesis;
end;
then consider f1 being Function of (TOP-REAL 2)|K1,R^1 such that
A26: for p being Point of TOP-REAL 2 st p in the carrier of (TOP-REAL 2)
|K1 holds f1.p=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2));
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<=0 & q`1/|.q.|<=cn & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A27: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A27;
hence thesis;
end;
then
A28: f1 is continuous by A3,A26,Th119;
A29: 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
A30: |[x,y]| in K1 and
A31: s=f2.(|[x,y]|) & r=f1.(|[x,y]|);
set p99=|[x,y]|;
A32: ex p3 being Point of TOP-REAL 2 st p99=p3 & p3`1/|.p3.| <=cn & p3`2<=0
& p3<>0.TOP-REAL 2 by A3,A30;
A33: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then
A34: f1.p99=|.p99.|*( -sqrt(1-((p99`1/|.p99.|-cn)/(1+cn))^2)) by A26,A30;
((cn-FanMorphS)|K0).(|[x,y]|)=((cn-FanMorphS)).(|[x,y]|) by A30,FUNCT_1:49
.= |[ |.p99.|* ((p99`1/|.p99.|-cn)/(1+cn)), |.p99.|*( -sqrt(1-((p99`1/
|.p99.|-cn)/(1+cn))^2))]| by A3,A32,Th115
.=|[s,r]| by A18,A30,A31,A33,A34;
hence thesis by A3;
end;
for q being Point of TOP-REAL 2 st q in the carrier of (TOP-REAL 2)|K1
holds q`2<=0 & q<>0.TOP-REAL 2
proof
let q be Point of TOP-REAL 2;
A35: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
assume q in the carrier of (TOP-REAL 2)|K1;
then
ex p3 being Point of TOP-REAL 2 st q=p3 & p3`1/|.p3.|<= cn & p3`2<=0 &
p3<>0.TOP-REAL 2 by A3,A35;
hence thesis;
end;
then f2 is continuous by A3,A18,Th117;
hence thesis by A7,A9,A28,A29,JGRAPH_2:35;
end;
theorem Th122:
for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p
`1>=(cn)*(|.p.|) & p`2<=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`2<=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
assume
A1: K003={p: p`1>=(sn)*(|.p.|) & p`2<=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`1>=sn*|.$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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm8,JORDAN6:8;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th123:
for cn being Real, K03 being Subset of TOP-REAL 2 st K03={p: p
`1<=(cn)*(|.p.|) & p`2<=0} holds K03 is closed
proof
defpred Q[Point of TOP-REAL 2] means $1`2<=0;
let sn be Real, K003 be Subset of TOP-REAL 2;
assume
A1: K003={p: p`1<=(sn)*(|.p.|) & p`2<=0};
reconsider KX={p where p is Point of TOP-REAL 2: Q[p]} as Subset of TOP-REAL
2 from JGRAPH_2:sch 1;
defpred P[Point of TOP-REAL 2] means ($1`1<=sn*|.$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;
A2: {p : P[p] & Q[p]} = {p7 where p7 is Point of TOP-REAL 2:P[p7]} /\ {p1
where p1 is Point of TOP-REAL 2: Q[p1]} from DOMAIN_1:sch 10;
K1 is closed & KX is closed by Lm10,JORDAN6:8;
hence thesis by A1,A2,TOPS_1:8;
end;
theorem Th124:
for cn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphS)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`2<=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=-sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
p0`1=cn by EUCLID:52;
then
A2: |.p0.|=sqrt((sn)^2+cn^2) by A1,JGRAPH_3:1;
assume
A3: -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2<=0 & p<>0.TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A4: 1-cn^2>0 by XREAL_1:50;
then
A5: -sn>0 by SQUARE_1:25;
A6: now
assume p0=0.TOP-REAL 2;
then --sn=-0 by EUCLID:52,JGRAPH_2:3;
hence contradiction by A4,SQUARE_1:25;
end;
then p0 in K0 by A3,A1,A5;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
(-sn)^2=1-cn^2 by A4,SQUARE_1:def 2;
then
A7: p0`1/|.p0.|=cn by A2,EUCLID:52;
then
A8: p0 in {p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} by A1,A6,A5;
not p0 in {0.TOP-REAL 2} by A6,TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A3,XBOOLE_0:def 5;
K1 c= D
proof
let x be object;
assume
A9: x in K1;
then ex p6 being Point of TOP-REAL 2 st p6=x & p6`2<=0 & p6 <>0.TOP-REAL 2
by A3;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A3,A9,XBOOLE_0:def 5;
end;
then D=K1 \/ D by XBOOLE_1:12;
then
A10: (TOP-REAL 2)|K1 is SubSpace of (TOP-REAL 2)|D by TOPMETR:4;
A11: {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2;
hence thesis by A3;
end;
A12: {p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} c= K1
proof
let x be object;
assume x in {p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2;
hence thesis by A3;
end;
then reconsider
K00={p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A8,PRE_TOPC:8;
the carrier of (TOP-REAL 2)|D = D by PRE_TOPC:8;
then
A13: rng (f|K00) c=D;
p0 in {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} by A1,A6,A5,A7;
then reconsider
K11={p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty
Subset of ((TOP-REAL 2)|K1) by A11,PRE_TOPC:8;
the carrier of (TOP-REAL 2)|D = D by PRE_TOPC:8;
then
A14: rng (f|K11) c=D;
the carrier of (TOP-REAL 2)|B0=the carrier of (TOP-REAL 2)|D;
then
A15: dom f=the carrier of (TOP-REAL 2)|K1 by FUNCT_2:def 1
.=K1 by PRE_TOPC:8;
then dom (f|K00)=K00 by A12,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K00 by PRE_TOPC:8;
then reconsider
f1=f|K00 as Function of ((TOP-REAL 2)|K1)|K00,(TOP-REAL 2)|D by A13,FUNCT_2:2
;
dom (f|K11)=K11 by A11,A15,RELAT_1:62
.= the carrier of ((TOP-REAL 2)|K1)|K11 by PRE_TOPC:8;
then reconsider
f2=f|K11 as Function of ((TOP-REAL 2)|K1)|K11,(TOP-REAL 2)|D by A14,FUNCT_2:2
;
A16: the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|>=cn & $1`2<=0 & $1<>0.
TOP-REAL 2;
A17: dom f2=the carrier of ((TOP-REAL 2)|K1)|K11 by FUNCT_2:def 1
.=K11 by PRE_TOPC:8;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K001={p: p`1/|.p.|>=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty
Subset of TOP-REAL 2 by A8;
A18: the carrier of (TOP-REAL 2)|K1 =K1 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`1>=cn*(|.$1.|) & $1`2<=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K003={p: p`1>=(cn)*(|.p.|) & p`2<=0} as Subset of TOP-REAL 2;
defpred P[Point of TOP-REAL 2] means $1`1/|.$1.|<=cn & $1`2<=0 & $1<>0.
TOP-REAL 2;
A19: {p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
A20: rng ((cn-FanMorphS)|K001) c= K1
proof
let y be object;
assume y in rng ((cn-FanMorphS)|K001);
then consider x being object such that
A21: x in dom ((cn-FanMorphS)|K001) and
A22: y=((cn-FanMorphS)|K001).x by FUNCT_1:def 3;
x in dom (cn-FanMorphS) by A21,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A23: y=(cn-FanMorphS).q by A21,A22,FUNCT_1:47;
dom ((cn-FanMorphS)|K001)=(dom (cn-FanMorphS))/\ K001 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K001 by FUNCT_2:def 1
.=K001 by XBOOLE_1:28;
then
A24: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|>= cn & p2`2<=0 &
p2<>0.TOP-REAL 2 by A21;
then
A25: (q`1/|.q.|-cn)>= 0 by XREAL_1:48;
|.q.|<>0 by A24,TOPRNS_1:24;
then
A26: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)
/(1-cn))^2))]|;
A27: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
A28: 1-cn>0 by A3,XREAL_1:149;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2 <= (|.q.|)^2 by JGRAPH_3:1;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A28,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A28,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A28,A25,SQUARE_1:49;
then
A29: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A30: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by A29,SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0;
then
A31: sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
A32: q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
then
A33: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A30,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A27,A33;
then
A34: q4<>0.TOP-REAL 2 by A26,TOPRNS_1:23;
cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((
q`1/|.q.|-cn)/(1-cn))^2))]| by A3,A24,Th115;
hence thesis by A3,A23,A32,A31,A34;
end;
A35: dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
then dom ((cn-FanMorphS)|K001)=K001 by RELAT_1:62
.= the carrier of (TOP-REAL 2)|K001 by PRE_TOPC:8;
then reconsider
f3=(cn-FanMorphS)|K001 as Function of (TOP-REAL 2)|K001,(TOP-REAL
2)|K1 by A18,A20,FUNCT_2:2;
A36: K003 is closed by Th122;
defpred P[Point of TOP-REAL 2] means $1`1<=(cn)*(|.$1.|) & $1`2<=0;
{p: P[p]} is Subset of TOP-REAL 2 from DOMAIN_1:sch 7;
then reconsider
K004={p: p`1<=(cn)*(|.p.|) & p`2<=0} as Subset of (TOP-REAL 2);
A37: K004 /\ K1 c= K11
proof
let x be object;
assume
A38: x in K004 /\ K1;
then x in K004 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A39: q1=x and
A40: q1`1<=(cn)*(|.q1.|) and
q1`2<=0;
x in K1 by A38,XBOOLE_0:def 4;
then
A41: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2<=0 & q2 <>0.TOP-REAL 2
by A3;
q1`1/|.q1.|<=(cn)*(|.q1.|)/|.q1.| by A40,XREAL_1:72;
then q1`1/|.q1.|<=(cn) by A39,A41,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A39,A41;
end;
A42: K004 is closed by Th123;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K00=(TOP-REAL 2)|K001 & f1= f3 by A3,FUNCT_1:51
,GOBOARD9:2;
then
A43: f1 is continuous by A3,A10,Th120,PRE_TOPC:26;
A44: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
p0 in {p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} by A1,A6,A5,A7;
then reconsider
K111={p: p`1/|.p.|<=cn & p`2<=0 & p<>0.TOP-REAL 2} as non empty
Subset of TOP-REAL 2 by A19;
A45: rng ((cn-FanMorphS)|K111) c= K1
proof
let y be object;
assume y in rng ((cn-FanMorphS)|K111);
then consider x being object such that
A46: x in dom ((cn-FanMorphS)|K111) and
A47: y=((cn-FanMorphS)|K111).x by FUNCT_1:def 3;
x in dom (cn-FanMorphS) by A46,RELAT_1:57;
then reconsider q=x as Point of TOP-REAL 2;
A48: y=(cn-FanMorphS).q by A46,A47,FUNCT_1:47;
dom ((cn-FanMorphS)|K111)=(dom (cn-FanMorphS))/\ K111 by RELAT_1:61
.=(the carrier of TOP-REAL 2)/\ K111 by FUNCT_2:def 1
.=K111 by XBOOLE_1:28;
then
A49: ex p2 being Point of TOP-REAL 2 st p2=q & p2`1/|.p2.|<= cn & p2`2<=0 &
p2<>0.TOP-REAL 2 by A46;
then
A50: (q`1/|.q.|-cn)<=0 by XREAL_1:47;
|.q.|<>0 by A49,TOPRNS_1:24;
then
A51: (|.q.|)^2>0^2 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)
/(1+cn))^2))]|;
A52: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
A53: 1+cn>0 by A3,XREAL_1:148;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by JGRAPH_3:1
,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A51,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A53,XREAL_1:72;
then -1<=(( q`1/|.q.|-cn))/(1+cn) by A53,XCMPLX_1:197;
then
A54: ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A53,A50,SQUARE_1:49;
then
A55: 1-((q`1/|.q.|-cn)/(1+cn))^2>=0 by XREAL_1:48;
1-(-((q`1/|.q.|-cn)/(1+cn)))^2>=0 by A54,XREAL_1:48;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)>=0;
then
A56: sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
A57: q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
then
A58: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A55,SQUARE_1:def 2;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A52,A58;
then
A59: q4<>0.TOP-REAL 2 by A51,TOPRNS_1:23;
cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((
q`1/|.q.|-cn)/(1+cn))^2))]| by A3,A49,Th115;
hence thesis by A3,A48,A57,A56,A59;
end;
dom ((cn-FanMorphS)|K111)=K111 by A35,RELAT_1:62
.= the carrier of (TOP-REAL 2)|K111 by PRE_TOPC:8;
then reconsider
f4=(cn-FanMorphS)|K111 as Function of (TOP-REAL 2)|K111,(TOP-REAL
2)|K1 by A16,A45,FUNCT_2:2;
the carrier of (TOP-REAL 2)|K1=K1 by PRE_TOPC:8;
then ((TOP-REAL 2)|K1)|K11=(TOP-REAL 2)|K111 & f2= f4 by A3,FUNCT_1:51
,GOBOARD9:2;
then
A60: f2 is continuous by A3,A10,Th121,PRE_TOPC:26;
set T1= ((TOP-REAL 2)|K1)|K00,T2=((TOP-REAL 2)|K1)|K11;
A61: [#](((TOP-REAL 2)|K1)|K11)=K11 by PRE_TOPC:def 5;
K11 c= K004 /\ K1
proof
let x be object;
assume x in K11;
then consider p such that
A62: p=x and
A63: p`1/|.p.|<=cn and
A64: p`2<=0 and
A65: p<>0.TOP-REAL 2;
p`1/|.p.|*|.p.|<=(cn)*(|.p.|) by A63,XREAL_1:64;
then p`1<=(cn)*(|.p.|) by A65,TOPRNS_1:24,XCMPLX_1:87;
then
A66: x in K004 by A62,A64;
x in K1 by A3,A62,A64,A65;
hence thesis by A66,XBOOLE_0:def 4;
end;
then K11=K004 /\ [#]((TOP-REAL 2)|K1) by A44,A37,XBOOLE_0:def 10;
then
A67: K11 is closed by A42,PRE_TOPC:13;
A68: K003 /\ K1 c= K00
proof
let x be object;
assume
A69: x in K003 /\ K1;
then x in K003 by XBOOLE_0:def 4;
then consider q1 being Point of TOP-REAL 2 such that
A70: q1=x and
A71: q1`1>=(cn)*(|.q1.|) and
q1`2<=0;
x in K1 by A69,XBOOLE_0:def 4;
then
A72: ex q2 being Point of TOP-REAL 2 st q2=x & q2`2<=0 & q2 <>0.TOP-REAL 2
by A3;
q1`1/|.q1.|>=(cn)*(|.q1.|)/|.q1.| by A71,XREAL_1:72;
then q1`1/|.q1.|>=(cn) by A70,A72,TOPRNS_1:24,XCMPLX_1:89;
hence thesis by A70,A72;
end;
A73: the carrier of ((TOP-REAL 2)|K1)=K0 by PRE_TOPC:8;
A74: D<>{};
A75: [#](((TOP-REAL 2)|K1)|K00)=K00 by PRE_TOPC:def 5;
A76: for p being object st p in ([#]T1)/\([#]T2) holds f1.p = f2.p
proof
let p be object;
assume
A77: p in ([#]T1)/\([#]T2);
then p in K00 by A75,XBOOLE_0:def 4;
hence f1.p=f.p by FUNCT_1:49
.=f2.p by A61,A77,FUNCT_1:49;
end;
K00 c= K003 /\ K1
proof
let x be object;
assume x in K00;
then consider p such that
A78: p=x and
A79: p`1/|.p.|>=cn and
A80: p`2<=0 and
A81: p<>0.TOP-REAL 2;
p`1/|.p.|*|.p.|>=(cn)*(|.p.|) by A79,XREAL_1:64;
then p`1>=(cn)*(|.p.|) by A81,TOPRNS_1:24,XCMPLX_1:87;
then
A82: x in K003 by A78,A80;
x in K1 by A3,A78,A80,A81;
hence thesis by A82,XBOOLE_0:def 4;
end;
then K00=K003 /\ [#]((TOP-REAL 2)|K1) by A44,A68,XBOOLE_0:def 10;
then
A83: K00 is closed by A36,PRE_TOPC:13;
A84: K1 c= K00 \/ K11
proof
let x be object;
assume x in K1;
then consider p such that
A85: p=x & p`2<=0 & p<>0.TOP-REAL 2 by A3;
per cases;
suppose
p`1/|.p.|>=cn;
then x in K00 by A85;
hence thesis by XBOOLE_0:def 3;
end;
suppose
p`1/|.p.|<cn;
then x in K11 by A85;
hence thesis by XBOOLE_0:def 3;
end;
end;
then [#](((TOP-REAL 2)|K1)|K00) \/ [#](((TOP-REAL 2)|K1)|K11) =[#]((
TOP-REAL 2)|K1) by A75,A61,A44,XBOOLE_0:def 10;
then consider h being Function of (TOP-REAL 2)|K1,(TOP-REAL 2)|D such that
A86: h=f1+*f2 and
A87: h is continuous by A75,A61,A83,A67,A43,A60,A76,JGRAPH_2:1;
A88: dom h=the carrier of ((TOP-REAL 2)|K1) by FUNCT_2:def 1;
A89: dom f1=the carrier of ((TOP-REAL 2)|K1)|K00 by FUNCT_2:def 1
.=K00 by PRE_TOPC:8;
A90: for y being object st y in dom h holds h.y=f.y
proof
let y be object;
assume
A91: y in dom h;
per cases by A84,A88,A73,A91,XBOOLE_0:def 3;
suppose
A92: y in K00 & not y in K11;
then y in dom f1 \/ dom f2 by A89,XBOOLE_0:def 3;
hence h.y=f1.y by A17,A86,A92,FUNCT_4:def 1
.=f.y by A92,FUNCT_1:49;
end;
suppose
A93: y in K11;
then y in dom f1 \/ dom f2 by A17,XBOOLE_0:def 3;
hence h.y=f2.y by A17,A86,A93,FUNCT_4:def 1
.=f.y by A93,FUNCT_1:49;
end;
end;
K0=the carrier of ((TOP-REAL 2)|K0) by PRE_TOPC:8
.=dom f by A74,FUNCT_2:def 1;
hence thesis by A87,A88,A90,FUNCT_1:2,PRE_TOPC:8;
end;
theorem Th125:
for cn being Real, K0,B0 being Subset of TOP-REAL 2, f being
Function of (TOP-REAL 2)|K0,(TOP-REAL 2)|B0 st -1<cn & cn<1 & f=(cn-FanMorphS)|
K0 & B0=NonZero TOP-REAL 2 & K0={p: p`2>=0 & p<>0.TOP-REAL 2} holds f is
continuous
proof
let cn be Real,K0,B0 be Subset of TOP-REAL 2,
f be Function of (TOP-REAL 2)|
K0,(TOP-REAL 2)|B0;
set sn=sqrt(1-cn^2);
set p0=|[cn,sn]|;
A1: p0`2=sn by EUCLID:52;
assume
A2: -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2>=0 & p<>0.TOP-REAL 2};
then cn^2<1^2 by SQUARE_1:50;
then
A3: 1-cn^2>0 by XREAL_1:50;
then sn>0 by SQUARE_1:25;
then p0 in K0 by A2,A1,JGRAPH_2:3;
then reconsider K1=K0 as non empty Subset of TOP-REAL 2;
p0`2>0 by A1,A3,SQUARE_1:25;
then not p0 in {0.TOP-REAL 2} by JGRAPH_2:3,TARSKI:def 1;
then reconsider D=B0 as non empty Subset of TOP-REAL 2 by A2,XBOOLE_0:def 5;
A4: K1 c= D
proof
let x be object;
assume x in K1;
then consider p2 being Point of TOP-REAL 2 such that
A5: p2=x and
p2`2>=0 and
A6: p2<>0.TOP-REAL 2 by A2;
not p2 in {0.TOP-REAL 2} by A6,TARSKI:def 1;
hence thesis by A2,A5,XBOOLE_0:def 5;
end;
for p being Point of (TOP-REAL 2)|K1,V being Subset of (TOP-REAL 2)|D
st f.p in V & V is open holds ex W being Subset of (TOP-REAL 2)|K1 st p in W &
W is open & f.:W c= V
proof
let p be Point of (TOP-REAL 2)|K1,V be Subset of (TOP-REAL 2)|D;
assume that
A7: f.p in V and
A8: V is open;
consider V2 being Subset of TOP-REAL 2 such that
A9: V2 is open and
A10: V2 /\ [#]((TOP-REAL 2)|D)=V by A8,TOPS_2:24;
reconsider W2=V2 /\ [#]((TOP-REAL 2)|K1) as Subset of (TOP-REAL 2)| K1;
A11: [#]((TOP-REAL 2)|K1)=K1 by PRE_TOPC:def 5;
then
A12: f.p=(cn-FanMorphS).p by A2,FUNCT_1:49;
A13: f.:W2 c= V
proof
let y be object;
assume y in f.:W2;
then consider x being object such that
A14: x in dom f and
A15: x in W2 and
A16: y=f.x by FUNCT_1:def 6;
f is Function of (TOP-REAL 2)|K1,(TOP-REAL 2)|D;
then dom f= K1 by A11,FUNCT_2:def 1;
then consider p4 being Point of TOP-REAL 2 such that
A17: x=p4 and
A18: p4`2>=0 and
p4<>0.TOP-REAL 2 by A2,A14;
A19: p4 in V2 by A15,A17,XBOOLE_0:def 4;
p4 in [#]((TOP-REAL 2)|K1) by A14,A17;
then p4 in D by A4,A11;
then
A20: p4 in [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
f.p4=(cn-FanMorphS).p4 by A2,A11,A14,A17,FUNCT_1:49
.=p4 by A18,Th113;
hence thesis by A10,A16,A17,A19,A20,XBOOLE_0:def 4;
end;
p in the carrier of (TOP-REAL 2)|K1;
then consider q being Point of TOP-REAL 2 such that
A21: q=p and
A22: q`2>=0 and
q <>0.TOP-REAL 2 by A2,A11;
(cn-FanMorphS).q=q by A22,Th113;
then p in V2 by A7,A10,A12,A21,XBOOLE_0:def 4;
then
A23: p in W2 by XBOOLE_0:def 4;
W2 is open by A9,TOPS_2:24;
hence thesis by A23,A13;
end;
hence thesis by JGRAPH_2:10;
end;
theorem Th126:
for cn being Real, 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 -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p
`2<=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2<=0 & p<>0.TOP-REAL 2 };
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2<=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th124;
end;
theorem Th127:
for cn being Real, 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 -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0=NonZero TOP-REAL 2 & K0={p: p
`2>=0 & p<>0.TOP-REAL 2} holds f is continuous
proof
let cn be Real, B0 be Subset of TOP-REAL 2,
K0 be Subset of (TOP-REAL 2)|B0,f
be Function of ((TOP-REAL 2)|B0)|K0,((TOP-REAL 2)|B0);
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;
assume
A1: -1<cn & cn<1 & f=(cn-FanMorphS)|K0 & B0=NonZero TOP-REAL 2 & K0={p:
p`2>=0 & p<>0.TOP-REAL 2};
K0 c= B0
proof
let x be object;
assume x in K0;
then
A2: ex p8 being Point of TOP-REAL 2 st x=p8 & p8`2>=0 & p8 <>0.TOP-REAL 2
by A1;
then not x in {0.TOP-REAL 2} by TARSKI:def 1;
hence thesis by A1,A2,XBOOLE_0:def 5;
end;
then ((TOP-REAL 2)|B0)|K0=(TOP-REAL 2)|K1 by PRE_TOPC:7;
hence thesis by A1,Th125;
end;
theorem Th128:
for cn being Real,p being Point of TOP-REAL 2 holds |.(cn
-FanMorphS).p.|=|.p.|
proof
let cn be Real,p be Point of TOP-REAL 2;
set f=cn-FanMorphS;
set z=f.p;
set q=p;
reconsider qz=z as Point of TOP-REAL 2;
per cases;
suppose
A1: q`1/|.q.|>=cn & q`2<0;
then
A2: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-(
(q`1/|.q.|-cn)/(1-cn))^2))]| by Th113;
then
A3: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
A4: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A2,EUCLID:52;
A5: (q`1/|.q.|-cn)>=0 by A1,XREAL_1:48;
A6: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
|.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24;
then
A7: (|.q.|)^2>0 by SQUARE_1:12;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A6,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A7,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then
A8: 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
per cases;
suppose
A9: 1-cn=0;
A10: ((q`1/|.q.|-cn)/(1-cn))=(q`1/|.q.|-cn)*(1-cn)" by XCMPLX_0:def 9
.= (q`1/|.q.|-cn)*0 by A9
.=0;
then 1-((q`1/|.q.|-cn)/(1-cn))^2=1;
then (cn-FanMorphS).q= |[ |.q.|*0,|.q.|*(-1)]| by A1,A10,Th113,
SQUARE_1:18
.=|[0,-(|.q.|)]|;
then ((cn-FanMorphS).q)`2=(- |.q.|) & ((cn-FanMorphS).q)`1=0 by EUCLID:52
;
then |.(cn-FanMorphS).p.|=sqrt((-(|.q.|))^2+0^2) by JGRAPH_3:1
.=sqrt(((|.q.|))^2)
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A11: 1-cn<>0;
per cases by A11;
suppose
A12: 1-cn>0;
-(1-cn)<= -( q`1/|.q.|-cn) by A8,XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A12,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A5,A12,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A13: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
A14: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2 by A3
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A13,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A4,A14;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A15: 1-cn<0;
0+(q`1)^2<(q`1)^2+(q`2)^2 by A1,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A7,A6,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A7,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then
A16: 1 > q`1/|.p.| by SQUARE_1:52;
q`1/|.q.|-cn>=0 by A1,XREAL_1:48;
hence thesis by A15,A16,XREAL_1:9;
end;
end;
end;
suppose
A17: q`1/|.q.|<cn & q`2<0;
then |.q.|<>0 by JGRAPH_2:3,TOPRNS_1:24;
then
A18: (|.q.|)^2>0 by SQUARE_1:12;
A19: (q`1/|.q.|-cn)<0 by A17,XREAL_1:49;
A20: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A20,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A18,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then
A21: -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
A22: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-
((q`1/|.q.|-cn)/(1+cn))^2))]| by A17,Th114;
then
A23: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
A24: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A22,EUCLID:52;
per cases;
suppose
A25: 1+cn=0;
((q`1/|.q.|-cn)/(1+cn))=(q`1/|.q.|-cn)*(1+cn)" by XCMPLX_0:def 9
.= (q`1/|.q.|-cn)*0 by A25
.=0;
then ((cn-FanMorphS).q)`2=-(|.q.|) & ((cn-FanMorphS).q)`1=0 by A22,
EUCLID:52;
then |.(cn-FanMorphS).p.|=sqrt((-(|.q.|))^2+0^2) by JGRAPH_3:1
.=sqrt(((|.q.|))^2)
.=|.q.| by SQUARE_1:22;
hence thesis;
end;
suppose
A26: 1+cn<>0;
per cases by A26;
suppose
A27: 1+cn>0;
then (-(1+cn))/(1+cn)<=(( q`1/|.q.|-cn))/(1+cn) by A21,XREAL_1:72;
then -1<=(( q`1/|.q.|-cn))/(1+cn) by A27,XCMPLX_1:197;
then ( (q`1/|.q.|-cn) /(1+cn))^2<=1^2 by A19,A27,SQUARE_1:49;
then
A28: 1-(((q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
A29: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2 by A23
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A28,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A24,A29;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
hence thesis by SQUARE_1:22;
end;
suppose
A30: 1+cn<0;
0+(q`1)^2<(q`1)^2+(q`2)^2 by A17,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A18,A20,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A18,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1 < q`1/|.p.| by SQUARE_1:52;
then
A31: q`1/|.q.|-cn>-1-cn by XREAL_1:9;
-(1+cn)>-0 by A30,XREAL_1:24;
hence thesis by A17,A31,XREAL_1:49;
end;
end;
end;
suppose
q`2>=0;
hence thesis by Th113;
end;
end;
theorem Th129:
for cn being Real,x,K0 being set st -1<cn & cn<1 & x in K0 & K0
={p: p`2<=0 & p<>0.TOP-REAL 2} holds (cn-FanMorphS).x in K0
proof
let cn be Real,x,K0 be set;
assume
A1: -1<cn & cn<1 & x in K0 & K0={p: p`2<=0 & p<>0.TOP-REAL 2};
then consider p such that
A2: p=x and
A3: p`2<=0 and
A4: p<>0.TOP-REAL 2;
A5: now
assume |.p.|<=0;
then |.p.|=0;
hence contradiction by A4,TOPRNS_1:24;
end;
then
A6: (|.p.|)^2>0 by SQUARE_1:12;
per cases;
suppose
A7: p`1/|.p.|<=cn;
reconsider p9= (cn-FanMorphS).p as Point of TOP-REAL 2;
(cn-FanMorphS).p= |[ |.p.|*((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-(
(p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A3,A4,A7,Th115;
then
A8: p9`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
A9: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A10: 1+cn>0 by A1,XREAL_1:148;
per cases;
suppose
p`2=0;
hence thesis by A1,A2,Th113;
end;
suppose
p`2<>0;
then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A9,XREAL_1:74;
then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76;
then -1 < p`1/|.p.| by SQUARE_1:52;
then -1-cn< p`1/|.p.|-cn by XREAL_1:9;
then (-1)*(1+cn)/(1+cn)< (p`1/|.p.|-cn)/(1+cn) by A10,XREAL_1:74;
then
A11: -1< (p`1/|.p.|-cn)/(1+cn) by A10,XCMPLX_1:89;
p`1/|.p.|-cn<=0 by A7,XREAL_1:47;
then 1^2> ((p`1/|.p.|-cn)/(1+cn))^2 by A10,A11,SQUARE_1:50;
then 1-((p`1/|.p.|-cn)/(1+cn))^2>0 by XREAL_1:50;
then --sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>0 by SQUARE_1:25;
then -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)<0;
then |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))<0 by A5,XREAL_1:132;
hence thesis by A1,A2,A8,JGRAPH_2:3;
end;
end;
suppose
A12: p`1/|.p.|>cn;
reconsider p9= (cn-FanMorphS).p as Point of TOP-REAL 2;
(cn-FanMorphS).p= |[ |.p.|*((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-(
(p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A3,A4,A12,Th115;
then
A13: p9`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
A14: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A15: 1-cn>0 by A1,XREAL_1:149;
per cases;
suppose
p`2=0;
hence thesis by A1,A2,Th113;
end;
suppose
p`2<>0;
then 0+(p`1)^2<(p`1)^2+(p`2)^2 by SQUARE_1:12,XREAL_1:8;
then (p`1)^2/(|.p.|)^2 < (|.p.|)^2/(|.p.|)^2 by A6,A14,XREAL_1:74;
then (p`1)^2/(|.p.|)^2 < 1 by A6,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 < 1 by XCMPLX_1:76;
then p`1/|.p.|<1 by SQUARE_1:52;
then (p`1/|.p.|-cn)<1-cn by XREAL_1:9;
then (p`1/|.p.|-cn)/(1-cn)<(1-cn)/(1-cn) by A15,XREAL_1:74;
then
A16: (p`1/|.p.|-cn)/(1-cn)<1 by A15,XCMPLX_1:60;
-(1-cn)< -0 & p`1/|.p.|-cn>=cn-cn by A12,A15,XREAL_1:9,24;
then (-1)*(1-cn)/(1-cn)< (p`1/|.p.|-cn)/(1-cn) by A15,XREAL_1:74;
then -1< (p`1/|.p.|-cn)/(1-cn) by A15,XCMPLX_1:89;
then 1^2> ((p`1/|.p.|-cn)/(1-cn))^2 by A16,SQUARE_1:50;
then 1-((p`1/|.p.|-cn)/(1-cn))^2>0 by XREAL_1:50;
then --sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>0 by SQUARE_1:25;
then -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)<0;
then p9`2<0 by A5,A13,XREAL_1:132;
hence thesis by A1,A2,JGRAPH_2:3;
end;
end;
end;
theorem Th130:
for cn being Real,x,K0 being set st -1<cn & cn<1 & x in K0 & K0
={p: p`2>=0 & p<>0.TOP-REAL 2} holds (cn-FanMorphS).x in K0
proof
let cn be Real,x,K0 be set;
assume
A1: -1<cn & cn<1 & x in K0 & K0={p: p`2>=0 & p<>0.TOP-REAL 2};
then ex p st p=x & p`2>=0 & p<>0.TOP-REAL 2;
hence thesis by A1,Th113;
end;
theorem Th131:
for cn being Real, D being non empty Subset of TOP-REAL 2 st -1
<cn & cn<1 & D`={0.TOP-REAL 2} holds ex h being Function of (TOP-REAL 2)|D,(
TOP-REAL 2)|D st h=(cn-FanMorphS)|D & h is continuous
proof
set Y1=|[0,1]|;
defpred P[Point of TOP-REAL 2] means $1`2<=0;
reconsider B0= {0.TOP-REAL 2} as Subset of TOP-REAL 2;
let cn be Real,D be non empty Subset of TOP-REAL 2;
assume that
A1: -1<cn & cn<1 and
A2: D`={0.TOP-REAL 2};
A3: the carrier of ((TOP-REAL 2)|D)=D by PRE_TOPC:8;
dom (cn-FanMorphS)=(the carrier of (TOP-REAL 2)) by FUNCT_2:def 1;
then
A4: dom ((cn-FanMorphS)|D)=(the carrier of (TOP-REAL 2))/\ D by RELAT_1:61
.=the carrier of ((TOP-REAL 2)|D) by A3,XBOOLE_1:28;
(|[0,-1]|)`2=-1 & |[0,-1]|<>0.TOP-REAL 2 by EUCLID:52,JGRAPH_2:3;
then
A5: |[0,-1]| in {p where p is Point of TOP-REAL 2: p`2<=0 & p<>0.TOP-REAL 2};
Y1`2=1 by EUCLID:52;
then
A6: Y1 in {p where p is Point of TOP-REAL 2: p`2>=0 & p<>0.TOP-REAL 2} by
JGRAPH_2:3;
A7: D =B0` by A2
.=(NonZero TOP-REAL 2) by SUBSET_1:def 4;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A7);
then reconsider
K0={p:p`2<=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A5;
A8: K0=the carrier of ((TOP-REAL 2)|D)|K0 by PRE_TOPC:8;
defpred P[Point of TOP-REAL 2] means $1`2>=0;
{p: P[p] & p<>0.TOP-REAL 2} c= the carrier of (TOP-REAL 2)|D from
InclSub(A7);
then reconsider
K1={p: p`2>=0 & p<>0.TOP-REAL 2} as non empty Subset of (TOP-REAL
2)|D by A6;
A9: K0 is closed & K1 is closed by A7,Th62,Th63;
A10: the carrier of ((TOP-REAL 2)|D) =D by PRE_TOPC:8;
A11: rng ((cn-FanMorphS)|K0) c= the carrier of ((TOP-REAL 2)|D)|K0
proof
let y be object;
assume y in rng ((cn-FanMorphS)|K0);
then consider x being object such that
A12: x in dom ((cn-FanMorphS)|K0) and
A13: y=((cn-FanMorphS)|K0).x by FUNCT_1:def 3;
x in (dom (cn-FanMorphS)) /\ K0 by A12,RELAT_1:61;
then
A14: x in K0 by XBOOLE_0:def 4;
K0 c= the carrier of TOP-REAL 2 by A10,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A14;
(cn-FanMorphS).p=y by A13,A14,FUNCT_1:49;
then y in K0 by A1,A14,Th129;
hence thesis by PRE_TOPC:8;
end;
A15: 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<=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
dom ((cn-FanMorphS)|K0)= dom ((cn-FanMorphS)) /\ K0 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K0 by FUNCT_2:def 1
.=K0 by A15,XBOOLE_1:28;
then reconsider f=(cn-FanMorphS)|K0 as Function of ((TOP-REAL 2)|D)|K0, ((
TOP-REAL 2)|D) by A8,A11,FUNCT_2:2,XBOOLE_1:1;
A16: K1=the carrier of ((TOP-REAL 2)|D)|K1 by PRE_TOPC:8;
A17: rng ((cn-FanMorphS)|K1) c= the carrier of ((TOP-REAL 2)|D)|K1
proof
let y be object;
assume y in rng ((cn-FanMorphS)|K1);
then consider x being object such that
A18: x in dom ((cn-FanMorphS)|K1) and
A19: y=((cn-FanMorphS)|K1).x by FUNCT_1:def 3;
x in (dom (cn-FanMorphS)) /\ K1 by A18,RELAT_1:61;
then
A20: x in K1 by XBOOLE_0:def 4;
K1 c= the carrier of TOP-REAL 2 by A10,XBOOLE_1:1;
then reconsider p=x as Point of TOP-REAL 2 by A20;
(cn-FanMorphS).p=y by A19,A20,FUNCT_1:49;
then y in K1 by A1,A20,Th130;
hence thesis by PRE_TOPC:8;
end;
A21: 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`2>=0 & p8 <>0.TOP-REAL 2;
hence thesis;
end;
dom ((cn-FanMorphS)|K1)= dom ((cn-FanMorphS)) /\ K1 by RELAT_1:61
.=((the carrier of TOP-REAL 2)) /\ K1 by FUNCT_2:def 1
.=K1 by A21,XBOOLE_1:28;
then reconsider g=(cn-FanMorphS)|K1 as Function of ((TOP-REAL 2)|D)|K1, ((
TOP-REAL 2)|D) by A16,A17,FUNCT_2:2,XBOOLE_1:1;
A22: K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
A23: D c= K0 \/ K1
proof
let x be object;
assume
A24: x in D;
then reconsider px=x as Point of TOP-REAL 2;
not x in {0.TOP-REAL 2} by A7,A24,XBOOLE_0:def 5;
then px`2>=0 & px<>0.TOP-REAL 2 or px`2<=0 & px<>0.TOP-REAL 2 by
TARSKI:def 1;
then x in K1 or x in K0;
hence thesis by XBOOLE_0:def 3;
end;
A25: dom f=K0 by A8,FUNCT_2:def 1;
A26: K0=[#](((TOP-REAL 2)|D)|K0) by PRE_TOPC:def 5;
A27: 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
A28: x in ([#]((((TOP-REAL 2)|D)|K0))) /\ ([#] ((((TOP-REAL 2)|D)|K1)) );
then x in K0 by A26,XBOOLE_0:def 4;
then f.x=(cn-FanMorphS).x by FUNCT_1:49;
hence thesis by A22,A28,FUNCT_1:49;
end;
D= [#]((TOP-REAL 2)|D) by PRE_TOPC:def 5;
then
A29: ([#](((TOP-REAL 2)|D)|K0)) \/ ([#](((TOP-REAL 2)|D)|K1)) = [#]((
TOP-REAL 2)|D) by A26,A22,A23,XBOOLE_0:def 10;
A30: f is continuous & g is continuous by A1,A7,Th126,Th127;
then consider h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D such that
A31: h= f+*g and
h is continuous by A26,A22,A29,A9,A27,JGRAPH_2:1;
A32: dom g=K1 by A16,FUNCT_2:def 1;
K0=[#](((TOP-REAL 2)|D)|K0) & K1=[#](((TOP-REAL 2)|D)|K1) by PRE_TOPC:def 5;
then
A33: f tolerates g by A27,A25,A32,PARTFUN1:def 4;
A34: the carrier of ((TOP-REAL 2)|D) =(NonZero TOP-REAL 2) by A7,PRE_TOPC:8;
A35: for x being object st x in dom h holds h.x=((cn-FanMorphS)|D).x
proof
let x be object;
assume
A36: x in dom h;
then reconsider p=x as Point of TOP-REAL 2 by A34,XBOOLE_0:def 5;
not x in {0.TOP-REAL 2} by A7,A3,A36,XBOOLE_0:def 5;
then
A37: x <>0.TOP-REAL 2 by TARSKI:def 1;
per cases;
suppose
A38: x in K0;
A39: (cn-FanMorphS)|D.p=(cn-FanMorphS).p by A3,A36,FUNCT_1:49
.=f.p by A38,FUNCT_1:49;
h.p=(g+*f).p by A31,A33,FUNCT_4:34
.=f.p by A25,A38,FUNCT_4:13;
hence thesis by A39;
end;
suppose
not x in K0;
then not p`2<=0 by A37;
then
A40: x in K1 by A37;
(cn-FanMorphS)|D.p=(cn-FanMorphS).p by A3,A36,FUNCT_1:49
.=g.p by A40,FUNCT_1:49;
hence thesis by A31,A32,A40,FUNCT_4:13;
end;
end;
dom h=the carrier of ((TOP-REAL 2)|D) by FUNCT_2:def 1;
then f+*g=(cn-FanMorphS)|D by A31,A4,A35,FUNCT_1:2;
hence thesis by A26,A22,A29,A30,A9,A27,JGRAPH_2:1;
end;
theorem Th132:
for cn being Real st -1<cn & cn<1 holds cn-FanMorphS is continuous
proof
reconsider D=NonZero TOP-REAL 2 as non empty Subset of TOP-REAL 2 by
JGRAPH_2:9;
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
reconsider f=cn-FanMorphS as Function of TOP-REAL 2,TOP-REAL 2;
A3: f.(0.TOP-REAL 2)=0.TOP-REAL 2 by Th113,JGRAPH_2:3;
A4: 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;
A5: [#]((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 A5,XBOOLE_0:def 5;
then
A6: p<>0.TOP-REAL 2 by TARSKI:def 1;
per cases;
suppose
A7: q`1/|.q.|>=cn & q`2<=0;
set q9= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-
cn)/(1-cn))^2))]|;
A8: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
A9: q9`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
now
assume
A10: q9=0.TOP-REAL 2;
A11: |.q.|<>0^2 by A6,TOPRNS_1:24;
then -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)= -sqrt(1-0) by A8,A10,
JGRAPH_2:3,XCMPLX_1:6
.=-1;
hence contradiction by A9,A10,A11,JGRAPH_2:3,XCMPLX_1:6;
end;
hence thesis by A1,A2,A3,A6,A7,Th115;
end;
suppose
A12: q`1/|.q.|<cn & q`2<=0;
set q9=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn
)/(1+cn))^2))]|;
A13: q9`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
A14: q9`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
now
assume
A15: q9=0.TOP-REAL 2;
A16: |.q.|<>0^2 by A6,TOPRNS_1:24;
then -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)= -sqrt(1-0) by A13,A15,
JGRAPH_2:3,XCMPLX_1:6
.=-1;
hence contradiction by A14,A15,A16,JGRAPH_2:3,XCMPLX_1:6;
end;
hence thesis by A1,A2,A3,A6,A12,Th115;
end;
suppose
q`2>0;
then f.p=p by Th113;
hence thesis by A6,Th113,JGRAPH_2:3;
end;
end;
A17: 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 Lm11;
assume that
A18: f.(0.TOP-REAL 2) in V and
A19: V is open;
VV is open by A19,Lm11,PRE_TOPC:30;
then consider r being Real such that
A20: r>0 and
A21: Ball(u0,r) c= V by A3,A18,TOPMETR:15;
reconsider r as Real;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider W1=Ball(u0,r) as Subset of TOP-REAL 2;
A22: W1 is open by GOBOARD6:3;
A23: f.:W1 c= W1
proof
let z be object;
assume z in f.:W1;
then consider y being object such that
A24: y in dom f and
A25: y in W1 and
A26: z=f.y by FUNCT_1:def 6;
z in rng f by A24,A26,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 A24;
reconsider qy=q as Point of Euclid 2 by EUCLID:67;
dist(u0,qy)<r by A25,METRIC_1:11;
then
A27: |.(0.TOP-REAL 2) - q.|<r by JGRAPH_1:28;
per cases by JGRAPH_2:3;
suppose
q`2>=0;
hence thesis by A25,A26,Th113;
end;
suppose
A28: q<>0.TOP-REAL 2 & q`1/|.q.|>=cn & q`2<=0;
then
A29: (q`1/|.q.|-cn)>= 0 by XREAL_1:48;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A30: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A31: 1-cn>0 by A2,XREAL_1:149;
|.q.|<>0 by A28,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`1)^2/(|.q.|)^2 <= 1 by A30,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A31,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A31,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A31,A29,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A32: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
A33: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)) , |.q.|*( -
sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A28,Th115;
then
A34: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by A26,EUCLID:52;
qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by A26,A33,EUCLID:52;
then
A35: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A32,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A34,A35;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A36: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A27,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A36,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
suppose
A37: q<>0.TOP-REAL 2 & q`1/|.q.|<cn & q`2<=0;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A38: (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
A39: 1+cn>0 by A1,XREAL_1:148;
|.q.|<>0 by A37,TOPRNS_1:24;
then (|.q.|)^2>0 by SQUARE_1:12;
then (q`1)^2/(|.q.|)^2 <= 1 by A38,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then --1>=-q`1/|.q.| by XREAL_1:24;
then 1+cn>=-q`1/|.q.|+cn by XREAL_1:7;
then
A40: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A39,XREAL_1:185;
(cn-q`1/|.q.|)>=0 by A37,XREAL_1:48;
then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A39;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A40,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A41: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
A42: (cn-FanMorphS).q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -
sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A37,Th115;
then
A43: qz`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by A26,EUCLID:52;
qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by A26,A42,EUCLID:52;
then
A44: (qz`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A41,SQUARE_1:def 2;
(|.qz.|)^2=(qz`1)^2+(qz`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A43,A44;
then sqrt((|.qz.|)^2)=|.q.| by SQUARE_1:22;
then
A45: |.qz.|=|.q.| by SQUARE_1:22;
|.- q.|<r by A27,RLVECT_1:4;
then |.q.|<r by TOPRNS_1:26;
then |.- qz.|<r by A45,TOPRNS_1:26;
then |.(0.TOP-REAL 2) - qz.|<r by RLVECT_1:4;
then dist(u0,pz)<r by JGRAPH_1:28;
hence thesis by METRIC_1:11;
end;
end;
u0 in W1 by A20,GOBOARD6:1;
hence thesis by A21,A22,A23,XBOOLE_1:1;
end;
A46: D`= {0.TOP-REAL 2} by JGRAPH_3:20;
then
ex h being Function of (TOP-REAL 2)|D,(TOP-REAL 2)|D st h=(cn-FanMorphS
)|D & h is continuous by A1,A2,Th131;
hence thesis by A3,A46,A4,A17,JGRAPH_3:3;
end;
theorem Th133:
for cn being Real st -1<cn & cn<1 holds cn-FanMorphS is one-to-one
proof
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
for x1,x2 being object st x1 in dom (cn-FanMorphS) & x2 in dom (cn
-FanMorphS) & (cn-FanMorphS).x1=(cn-FanMorphS).x2 holds x1=x2
proof
let x1,x2 be object;
assume that
A3: x1 in dom (cn-FanMorphS) and
A4: x2 in dom (cn-FanMorphS) and
A5: (cn-FanMorphS).x1=(cn-FanMorphS).x2;
reconsider p2=x2 as Point of TOP-REAL 2 by A4;
reconsider p1=x1 as Point of TOP-REAL 2 by A3;
set q=p1,p=p2;
A6: 1-cn>0 by A2,XREAL_1:149;
per cases by JGRAPH_2:3;
suppose
A7: q`2>=0;
then
A8: (cn-FanMorphS).q=q by Th113;
per cases by JGRAPH_2:3;
suppose
p`2>=0;
hence thesis by A5,A8,Th113;
end;
suppose
A9: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2<=0;
then
A10: |.p.|<>0 by TOPRNS_1:24;
then
A11: (|.p.|)^2>0 by SQUARE_1:12;
A12: (p`1/|.p.|-cn)>=0 by A9,XREAL_1:48;
A13: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
0<=(p`2)^2 by XREAL_1:63;
then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A13,XREAL_1:72;
then (p`1)^2/(|.p.|)^2 <= 1 by A11,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`1/|.p.| by SQUARE_1:51;
then 1-cn>=p`1/|.p.|-cn by XREAL_1:9;
then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72;
then
A14: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197;
A15: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(
1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A9,Th115;
then
A16: q`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by A5,A8,EUCLID:52;
(p`1/|.p.|-cn)>= 0 by A9,XREAL_1:48;
then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A14,SQUARE_1:49;
then
A17: 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`1/|.p.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`1/|.p.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(p`1/|.p.|-cn)^2/(1-cn)^2)>=0;
then sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
then q`2=0 by A5,A7,A8,A15,EUCLID:52;
then
A18: -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)=-0 by A16,A10,XCMPLX_1:6;
1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by A17,XCMPLX_1:187;
then 1-((p`1/|.p.|-cn)/(1-cn))^2=0 by A18,SQUARE_1:24;
then 1= (p`1/|.p.|-cn)/(1-cn) by A6,A12,SQUARE_1:18,22;
then 1 *(1-cn)=(p`1/|.p.|-cn) by A6,XCMPLX_1:87;
then 1 *|.p.|=p`1 by A9,TOPRNS_1:24,XCMPLX_1:87;
then p`2=0 by A13,XCMPLX_1:6;
hence thesis by A5,A8,Th113;
end;
suppose
A19: p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2<=0;
then
A20: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(
1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,Th115;
then
A21: q`2=|.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by A5,A8,EUCLID:52;
A22: (|.p.|)^2 =(p`1)^2+(p`2)^2 by JGRAPH_3:1;
A23: |.p.|<>0 by A19,TOPRNS_1:24;
then
A24: (|.p.|)^2>0 by SQUARE_1:12;
A25: 1+cn>0 by A1,XREAL_1:148;
A26: (p`1/|.p.|-cn)<=0 by A19,XREAL_1:47;
then
A27: -1<=(-( p`1/|.p.|-cn))/(1+cn) by A25;
0<=(p`2)^2 by XREAL_1:63;
then 0+(p`1)^2<=(p`1)^2+(p`2)^2 by XREAL_1:7;
then (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by A22,XREAL_1:72;
then (p`1)^2/(|.p.|)^2 <= 1 by A24,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then (-((p`1)/|.p.|))^2 <= 1;
then 1>= -p`1/|.p.| by SQUARE_1:51;
then (1+cn)>= -p`1/|.p.|+cn by XREAL_1:7;
then (-(p`1/|.p.|-cn))/(1+cn)<=1 by A25,XREAL_1:185;
then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A27,SQUARE_1:49;
then
A28: 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(p`1/|.p.|-cn))/(1+cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(p`1/|.p.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((p`1/|.p.|-cn))^2/(1+cn)^2)>=0;
then sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
then q`2=0 by A5,A7,A8,A20,EUCLID:52;
then
A29: -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)=-0 by A21,A23,XCMPLX_1:6;
1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by A28,XCMPLX_1:187;
then 1-((p`1/|.p.|-cn)/(1+cn))^2=0 by A29,SQUARE_1:24;
then 1=(-((p`1/|.p.|-cn)/(1+cn)))^2;
then 1= -((p`1/|.p.|-cn)/(1+cn)) by A25,A26,SQUARE_1:18,22;
then 1= ((-(p`1/|.p.|-cn))/(1+cn)) by XCMPLX_1:187;
then 1 *(1+cn)=-(p`1/|.p.|-cn) by A25,XCMPLX_1:87;
then 1+cn-cn=-p`1/|.p.|;
then 1=(-p`1)/|.p.| by XCMPLX_1:187;
then 1 *|.p.|=-p`1 by A19,TOPRNS_1:24,XCMPLX_1:87;
then (p`1)^2-(p`1)^2 =(p`2)^2 by A22,XCMPLX_1:26;
then p`2=0 by XCMPLX_1:6;
hence thesis by A5,A8,Th113;
end;
end;
suppose
A30: q`1/|.q.|>=cn & q`2<=0 & q<>0.TOP-REAL 2;
then |.q.|<>0 by TOPRNS_1:24;
then
A31: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-
cn)/(1-cn))^2))]|;
A32: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
A33: cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-
((q`1/|.q.|-cn)/(1-cn))^2))]| by A1,A2,A30,Th115;
per cases by JGRAPH_2:3;
suppose
A34: p`2>=0;
then
A35: (cn-FanMorphS).p=p by Th113;
then
A36: p`2=|.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by A5,A33,EUCLID:52;
A37: (q`1/|.q.|-cn)>=0 by A30,XREAL_1:48;
A38: 1-cn>0 by A2,XREAL_1:149;
A39: |.q.|<>0 by A30,TOPRNS_1:24;
then
A40: (|.q.|)^2>0 by SQUARE_1:12;
A41: (q`1/|.q.|-cn)>= 0 by A30,XREAL_1:48;
A42: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A42,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A40,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A38,XREAL_1:72;
then -1<=(-( q`1/|.q.|-cn))/(1-cn) by A38,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A38,A41,SQUARE_1:49;
then
A43: 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then sqrt(1-((-(q`1/|.q.|-cn))/(1-cn))^2)>=0 by SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1-cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1-cn)^2)>=0;
then sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)>=0 by XCMPLX_1:76;
then p`2=0 by A5,A33,A34,A35,EUCLID:52;
then
A44: -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)=-0 by A36,A39,XCMPLX_1:6;
1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by A43,XCMPLX_1:187;
then 1-((q`1/|.q.|-cn)/(1-cn))^2=0 by A44,SQUARE_1:24;
then 1= (q`1/|.q.|-cn)/(1-cn) by A38,A37,SQUARE_1:18,22;
then 1 *(1-cn)=(q`1/|.q.|-cn) by A38,XCMPLX_1:87;
then 1 *|.q.|=q`1 by A30,TOPRNS_1:24,XCMPLX_1:87;
then q`2=0 by A42,XCMPLX_1:6;
hence thesis by A5,A35,Th113;
end;
suppose
A45: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2<=0;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A31,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then 1>=q`1/|.q.| by SQUARE_1:51;
then 1-cn>=q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)<= -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XREAL_1:72;
then
A46: -1<=(-( q`1/|.q.|-cn))/(1-cn) by A6,XCMPLX_1:197;
(q`1/|.q.|-cn)>= 0 by A30,XREAL_1:48;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<=1^2 by A6,A46,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A47: 1-(-((q`1/|.q.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
then
A48: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1-cn))^2) by A47,SQUARE_1:def 2;
A49: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1-cn)) by EUCLID:52;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A49,A48;
then
A50: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A51: |.q4.|=|.q.| by SQUARE_1:22;
0<=(p`2)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A52: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
|.p.|<>0 by A45,TOPRNS_1:24;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`1)^2/(|.p.|)^2 <= 1 by A52,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then 1>=p`1/|.p.| by SQUARE_1:51;
then 1-cn>=p`1/|.p.|-cn by XREAL_1:9;
then -(1-cn)<= -( p`1/|.p.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XREAL_1:72;
then
A53: -1<=(-( p`1/|.p.|-cn))/(1-cn) by A6,XCMPLX_1:197;
(p`1/|.p.|-cn)>= 0 by A45,XREAL_1:48;
then ((-(p`1/|.p.|-cn))/(1-cn))^2<=1^2 by A6,A53,SQUARE_1:49;
then 1-((-(p`1/|.p.|-cn))/(1-cn))^2>=0 by XREAL_1:48;
then
A54: 1-(-((p`1/|.p.|-cn))/(1-cn))^2>=0 by XCMPLX_1:187;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.|
-cn)/(1-cn))^2))]|;
A55: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) by EUCLID:52;
p4`2= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2)) by EUCLID:52;
then
A56: (p4`2)^2= (|.p.|)^2*(sqrt(1-((p`1/|.p.|-cn)/(1-cn))^2))^2
.= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1-cn))^2) by A54,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A55,A56;
then
A57: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A58: |.p4.|=|.p.| by SQUARE_1:22;
A59: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt
(1-((p`1/|.p.|-cn)/(1-cn))^2))]| by A1,A2,A45,Th115;
then ((p`1/|.p.|-cn)/(1-cn)) =|.q.|* ((q`1/|.q.|-cn)/(1-cn))/|.p.| by
A5,A33,A32,A45,A55,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1-cn)=(q`1/|.q.|-cn)/(1-cn) by A5,A33,A45,A59,A50
,A57,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1-cn)*(1-cn)=q`1/|.q.|-cn by A6,XCMPLX_1:87;
then p`1/|.p.|-cn=q`1/|.q.|-cn by A6,XCMPLX_1:87;
then p`1/|.p.|*|.p.|=q`1 by A5,A33,A45,A59,A51,A58,TOPRNS_1:24
,XCMPLX_1:87;
then
A60: p`1=q`1 by A45,TOPRNS_1:24,XCMPLX_1:87;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then (-p`2)^2=(q`2)^2 by A5,A33,A59,A51,A58,A60;
then -p`2=sqrt((-q`2)^2) by A45,SQUARE_1:22;
then
A61: --p`2=--q`2 by A30,SQUARE_1:22;
p=|[p`1,p`2]| by EUCLID:53;
hence thesis by A60,A61,EUCLID:53;
end;
suppose
A62: p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2<=0;
then p`1/|.p.|-cn<0 by XREAL_1:49;
then
A63: ((p`1/|.p.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.|
-cn)/(1+cn))^2))]|;
A64: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) & q`1/|.q.|-cn>=0 by A30,EUCLID:52
,XREAL_1:48;
A65: 1-cn>0 by A2,XREAL_1:149;
cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt
(1-((p`1/|.p .|-cn)/(1+cn))^2))]| & |.p.|<>0 by A1,A2,A62,Th115,TOPRNS_1:24;
hence thesis by A5,A33,A32,A63,A64,A65,XREAL_1:132;
end;
end;
suppose
A66: q`1/|.q.|<cn & q`2<=0 & q<>0.TOP-REAL 2;
then
A67: |.q.|<>0 by TOPRNS_1:24;
then
A68: (|.q.|)^2>0 by SQUARE_1:12;
set q4= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-
cn)/(1+cn))^2))]|;
A69: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
A70: cn-FanMorphS.q= |[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1
-((q`1/|.q.|-cn)/(1+cn))^2))]| by A1,A2,A66,Th115;
per cases by JGRAPH_2:3;
suppose
A71: p`2>=0;
then
A72: (cn-FanMorphS).p=p by Th113;
then
A73: p`2=|.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by A5,A70,EUCLID:52;
A74: (|.q.|)^2 =(q`1)^2+(q`2)^2 by JGRAPH_3:1;
A75: 1+cn>0 by A1,XREAL_1:148;
0<=(q`2)^2 by XREAL_1:63;
then 0+(q`1)^2<=(q`1)^2+(q`2)^2 by XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by A74,XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A68,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then (-((q`1)/|.q.|))^2 <= 1;
then 1>= -q`1/|.q.| by SQUARE_1:51;
then (1+cn)>= -q`1/|.q.|+cn by XREAL_1:7;
then
A76: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A75,XREAL_1:185;
A77: (q`1/|.q.|-cn)<=0 by A66,XREAL_1:47;
then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A75;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A76,SQUARE_1:49;
then
A78: 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A79: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>=0 by A78,SQUARE_1:def 2;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)>=0 by XCMPLX_1:76;
then sqrt(1-((q`1/|.q.|-cn))^2/(1+cn)^2)>=0;
then sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)>=0 by XCMPLX_1:76;
then p`2=0 by A5,A70,A71,A72,EUCLID:52;
then -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)=-0 by A67,A73,XCMPLX_1:6;
then 1-((q`1/|.q.|-cn)/(1+cn))^2=0 by A79,SQUARE_1:24;
then 1=(-((q`1/|.q.|-cn)/(1+cn)))^2;
then 1= -((q`1/|.q.|-cn)/(1+cn)) by A75,A77,SQUARE_1:18,22;
then 1= ((-(q`1/|.q.|-cn))/(1+cn)) by XCMPLX_1:187;
then 1 *(1+cn)=-(q`1/|.q.|-cn) by A75,XCMPLX_1:87;
then 1+cn-cn=-q`1/|.q.|;
then 1=(-q`1)/|.q.| by XCMPLX_1:187;
then 1 *|.q.|=-q`1 by A66,TOPRNS_1:24,XCMPLX_1:87;
then (q`1)^2-(q`1)^2 =(q`2)^2 by A74,XCMPLX_1:26;
then q`2=0 by XCMPLX_1:6;
hence thesis by A5,A72,Th113;
end;
suppose
A80: p<>0.TOP-REAL 2 & p`1/|.p.|>=cn & p`2<=0;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt(1-((p`1/|.p.|
-cn)/(1-cn))^2))]|;
A81: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1-cn)) & |.q.|<>0 by A66,EUCLID:52
,TOPRNS_1:24;
q`1/|.q.|-cn<0 by A66,XREAL_1:49;
then
A82: ((q`1/|.q.|-cn)/(1+cn))<0 by A1,XREAL_1:141,148;
A83: 1-cn>0 by A2,XREAL_1:149;
cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1-cn)), |.p.|*( -sqrt
(1-((p`1/|.p .|-cn)/(1-cn))^2))]| & p`1/|.p.|-cn>=0 by A1,A2,A80,Th115,
XREAL_1:48;
hence thesis by A5,A70,A69,A82,A81,A83,XREAL_1:132;
end;
suppose
A84: p<>0.TOP-REAL 2 & p`1/|.p.|<cn & p`2<=0;
0<=(p`2)^2 by XREAL_1:63;
then (|.p.|)^2 =(p`1)^2+(p`2)^2 & 0+(p`1)^2<=(p`1)^2+(p`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then
A85: (p`1)^2/(|.p.|)^2 <= (|.p.|)^2/(|.p.|)^2 by XREAL_1:72;
A86: 1+cn>0 by A1,XREAL_1:148;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A68,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then -1<=q`1/|.q.| by SQUARE_1:51;
then -1-cn<=q`1/|.q.|-cn by XREAL_1:9;
then -(-1-cn)>= -(q`1/|.q.|-cn) by XREAL_1:24;
then
A87: (-(q`1/|.q.|-cn))/(1+cn)<=1 by A86,XREAL_1:185;
(q`1/|.q.|-cn)<=0 by A66,XREAL_1:47;
then -1<=(-( q`1/|.q.|-cn))/(1+cn) by A86;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<=1^2 by A87,SQUARE_1:49;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A88: 1-(-((q`1/|.q.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
q4`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) by EUCLID:52;
then
A89: (q4`2)^2= (|.q.|)^2*(sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2))^2
.= (|.q.|)^2*(1-((q`1/|.q.|-cn)/(1+cn))^2) by A88,SQUARE_1:def 2;
A90: q4`1= |.q.|* ((q`1/|.q.|-cn)/(1+cn)) by EUCLID:52;
set p4= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt(1-((p`1/|.p.|
-cn)/(1+cn))^2))]|;
A91: p4`1= |.p.|* ((p`1/|.p.|-cn)/(1+cn)) by EUCLID:52;
|.p.|<>0 by A84,TOPRNS_1:24;
then (|.p.|)^2>0 by SQUARE_1:12;
then (p`1)^2/(|.p.|)^2 <= 1 by A85,XCMPLX_1:60;
then ((p`1)/|.p.|)^2 <= 1 by XCMPLX_1:76;
then -1<=p`1/|.p.| by SQUARE_1:51;
then -1-cn<=p`1/|.p.|-cn by XREAL_1:9;
then -(-1-cn)>= -(p`1/|.p.|-cn) by XREAL_1:24;
then
A92: (-(p`1/|.p.|-cn))/(1+cn)<=1 by A86,XREAL_1:185;
(p`1/|.p.|-cn)<=0 by A84,XREAL_1:47;
then -1<=(-( p`1/|.p.|-cn))/(1+cn) by A86;
then ((-(p`1/|.p.|-cn))/(1+cn))^2<=1^2 by A92,SQUARE_1:49;
then 1-((-(p`1/|.p.|-cn))/(1+cn))^2>=0 by XREAL_1:48;
then
A93: 1-(-((p`1/|.p.|-cn))/(1+cn))^2>=0 by XCMPLX_1:187;
p4`2= |.p.|*( -sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2)) by EUCLID:52;
then
A94: (p4`2)^2= (|.p.|)^2*(sqrt(1-((p`1/|.p.|-cn)/(1+cn))^2))^2
.= (|.p.|)^2*(1-((p`1/|.p.|-cn)/(1+cn))^2) by A93,SQUARE_1:def 2;
(|.p4.|)^2=(p4`1)^2+(p4`2)^2 by JGRAPH_3:1
.=(|.p.|)^2 by A91,A94;
then
A95: sqrt((|.p4.|)^2)=|.p.| by SQUARE_1:22;
then
A96: |.p4.|=|.p.| by SQUARE_1:22;
(|.q4.|)^2=(q4`1)^2+(q4`2)^2 by JGRAPH_3:1
.=(|.q.|)^2 by A90,A89;
then
A97: sqrt((|.q4.|)^2)=|.q.| by SQUARE_1:22;
then
A98: |.q4.|=|.q.| by SQUARE_1:22;
A99: cn-FanMorphS.p= |[ |.p.|* ((p`1/|.p.|-cn)/(1+cn)), |.p.|*( -sqrt
(1-((p`1/|.p.|-cn)/(1+cn))^2))]| by A1,A2,A84,Th115;
then ((p`1/|.p.|-cn)/(1+cn)) =|.q.|* ((q`1/|.q.|-cn)/(1+cn))/|.p.| by
A5,A70,A69,A84,A91,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1+cn)=(q`1/|.q.|-cn)/(1+cn) by A5,A70,A84,A99,A97
,A95,TOPRNS_1:24,XCMPLX_1:89;
then (p`1/|.p.|-cn)/(1+cn)*(1+cn)=q`1/|.q.|-cn by A86,XCMPLX_1:87;
then p`1/|.p.|-cn=q`1/|.q.|-cn by A86,XCMPLX_1:87;
then p`1/|.p.|*|.p.|=q`1 by A5,A70,A84,A99,A98,A96,TOPRNS_1:24
,XCMPLX_1:87;
then
A100: p`1=q`1 by A84,TOPRNS_1:24,XCMPLX_1:87;
|.p.|^2=(p`1)^2+(p`2)^2 & |.q.|^2=(q`1)^2+(q`2)^2 by JGRAPH_3:1;
then (-p`2)^2=(q`2)^2 by A5,A70,A99,A98,A96,A100;
then -p`2=sqrt((-q`2)^2) by A84,SQUARE_1:22;
then
A101: --p`2=--q`2 by A66,SQUARE_1:22;
p=|[p`1,p`2]| by EUCLID:53;
hence thesis by A100,A101,EUCLID:53;
end;
end;
end;
hence thesis by FUNCT_1:def 4;
end;
theorem Th134:
for cn being Real st -1<cn & cn<1 holds (cn-FanMorphS) is
Function of TOP-REAL 2,TOP-REAL 2 & rng (cn-FanMorphS) = the carrier of
TOP-REAL 2
proof
let cn be Real;
assume that
A1: -1<cn and
A2: cn<1;
thus (cn-FanMorphS) is Function of TOP-REAL 2,TOP-REAL 2;
for f being Function of TOP-REAL 2,TOP-REAL 2 st f=cn-FanMorphS holds
rng (cn-FanMorphS)=the carrier of TOP-REAL 2
proof
let f be Function of TOP-REAL 2,TOP-REAL 2;
assume
A3: f=(cn-FanMorphS);
A4: 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 by JGRAPH_2:3;
case
q`2>=0;
then y=(cn-FanMorphS).q by Th113;
hence ex x being set st x in dom (cn-FanMorphS) & y=(cn-FanMorphS).x
by A3,A4;
end;
case
A5: q`1/|.q.|>=0 & q`2<=0 & q<>0.TOP-REAL 2;
--(1+cn)>0 by A1,XREAL_1:148;
then
A6: -(-1-cn)>0;
A7: 1-cn>=0 by A2,XREAL_1:149;
then q`1/|.q.|*(1-cn)>=0 by A5;
then -1-cn<= q`1/|.q.|*(1-cn) by A6;
then
A8: -1-cn+cn<= q`1/|.q.|*(1-cn)+cn by XREAL_1:7;
set px=|[ |.q.|*(q`1/|.q.|*(1-cn)+cn), -((|.q.|)*sqrt(1-(q`1/|.q.|*(
1-cn)+cn)^2))]|;
A9: px`1 = |.q.|*(q`1/|.q.|*(1-cn)+cn) by EUCLID:52;
|.q.|<>0 by A5,TOPRNS_1:24;
then
A10: |.q.|^2>0 by SQUARE_1:12;
A11: dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
A12: 1-cn>0 by A2,XREAL_1:149;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A10,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`1/|.q.|<=1 by SQUARE_1:51;
then q`1/|.q.|*(1-cn) <=1 *(1-cn) by A12,XREAL_1:64;
then q`1/|.q.|*(1-cn)+cn-cn <=1-cn;
then (q`1/|.q.|*(1-cn)+cn) <=1 by XREAL_1:9;
then 1^2>=(q`1/|.q.|*(1-cn)+cn)^2 by A8,SQUARE_1:49;
then
A13: 1-(q`1/|.q.|*(1-cn)+cn)^2>=0 by XREAL_1:48;
then
A14: sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2)>=0 by SQUARE_1:def 2;
A15: px`2 = -(|.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2) by EUCLID:52;
then
|.px.|^2=((- |.q.|)*sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|*
(q`1/|.q.|*(1-cn)+cn))^2 by A9,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1-cn)+cn)^2))^2 +(|.q.|)^2*((q`1
/|.q.|*(1-cn)+cn))^2;
then
A16: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1-cn)+cn)^2) +(|.q.|)^2*((q`1
/|.q.|*(1-cn)+cn))^2 by A13,SQUARE_1:def 2
.= (|.q.|)^2;
then
A17: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A18: px<>0.TOP-REAL 2 by A5,TOPRNS_1:23,24;
(q`1/|.q.|*(1-cn)+cn)>=0+cn by A5,A7,XREAL_1:7;
then px`1/|.px.| >=cn by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89;
then
A19: (cn-FanMorphS).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1-cn) ), |.px
.|*( -sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2))]| by A1,A2,A15,A14,A18,Th115;
|.px.|*( sqrt((-(q`2/|.q.|))^2))=|.q.|*(-(q`2/|.q.|)) by A5,A17,
SQUARE_1:22
.= (-q`2)/|.q.|*|.q.| by XCMPLX_1:187
.=-q`2 by A5,TOPRNS_1:24,XCMPLX_1:87;
then
A20: |.px.|*(- sqrt((-(q`2/|.q.|))^2))=q`2;
A21: |.px.|* ((px`1/|.px.|-cn)/(1-cn)) =|.q.|* (( ((q`1/|.q.|*(1-cn)
+cn))-cn)/(1-cn)) by A5,A9,A17,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`1/|.q.|) by A12,XCMPLX_1:89
.= q`1 by A5,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*( -sqrt(1-((px`1/|.px.|-cn)/(1-cn))^2)) = |.px.|*( -sqrt
(1-(q`1/|.px.|)^2)) by A5,A17,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*( -sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*( -sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by
A10,A16,XCMPLX_1:60
.= |.px.|*( -sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by
XCMPLX_1:120
.= |.px.|*( -sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A16,
JGRAPH_3:1
.= |.px.|*( -sqrt((q`2/|.q.|)^2)) by A17,XCMPLX_1:76;
hence ex x being set st x in dom (cn-FanMorphS) & y=(cn-FanMorphS).x
by A19,A21,A20,A11,EUCLID:53;
end;
case
A22: q`1/|.q.|<0 & q`2<=0 & q<>0.TOP-REAL 2;
A23: 1+cn>=0 by A1,XREAL_1:148;
1-cn>0 by A2,XREAL_1:149;
then
A24: 1-cn+cn>= q`1/|.q.|*(1+cn)+cn by A22,A23,XREAL_1:7;
A25: 1+cn>0 by A1,XREAL_1:148;
|.q.|<>0 by A22,TOPRNS_1:24;
then
A26: |.q.|^2>0 by SQUARE_1:12;
0<=(q`2)^2 by XREAL_1:63;
then (|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<=(q`1)^2+(q`2)^2 by
JGRAPH_3:1,XREAL_1:7;
then (q`1)^2/(|.q.|)^2 <= (|.q.|)^2/(|.q.|)^2 by XREAL_1:72;
then (q`1)^2/(|.q.|)^2 <= 1 by A26,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 <= 1 by XCMPLX_1:76;
then q`1/|.q.|>=-1 by SQUARE_1:51;
then q`1/|.q.|*(1+cn) >=(-1)*(1+cn) by A25,XREAL_1:64;
then q`1/|.q.|*(1+cn)+cn-cn >=-1-cn;
then (q`1/|.q.|*(1+cn)+cn) >=-1 by XREAL_1:9;
then 1^2>=(q`1/|.q.|*(1+cn)+cn)^2 by A24,SQUARE_1:49;
then
A27: 1-(q`1/|.q.|*(1+cn)+cn)^2>=0 by XREAL_1:48;
then
A28: sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2)>=0 by SQUARE_1:def 2;
A29: dom (cn-FanMorphS)=the carrier of TOP-REAL 2 by FUNCT_2:def 1;
set px=|[ |.q.|*(q`1/|.q.|*(1+cn)+cn), -(|.q.|)*(sqrt(1-(q`1/|.q.|*(
1+cn)+cn)^2))]|;
A30: px`1 = |.q.|*(q`1/|.q.|*(1+cn)+cn) by EUCLID:52;
A31: px`2 = -(|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2) by EUCLID:52;
then
|.px.|^2=(-(|.q.|)*sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|*(
q`1/|.q.|*(1+cn)+cn))^2 by A30,JGRAPH_3:1
.=(|.q.|)^2*(sqrt(1-(q`1/|.q.|*(1+cn)+cn)^2))^2 +(|.q.|)^2*((q`1
/|.q.|*(1+cn)+cn))^2;
then
A32: |.px.|^2=(|.q.|)^2*(1-(q`1/|.q.|*(1+cn)+cn)^2) +(|.q.|)^2*((q`1
/|.q.|*(1+cn)+cn))^2 by A27,SQUARE_1:def 2
.= (|.q.|)^2;
then
A33: |.px.|=sqrt(|.q.|^2) by SQUARE_1:22
.=|.q.| by SQUARE_1:22;
then
A34: px<>0.TOP-REAL 2 by A22,TOPRNS_1:23,24;
(q`1/|.q.|*(1+cn)+cn)<=0+cn by A22,A23,XREAL_1:7;
then px`1/|.px.| <=cn by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89;
then
A35: (cn-FanMorphS).px =|[ |.px.|* ((px`1/|.px.|-cn)/(1+cn) ), |.px
.|*( -sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2))]| by A1,A2,A31,A28,A34,Th115;
A36: |.px.|*( -sqrt((q`2/|.q.|)^2)) = |.px.|*( -sqrt((-q`2/|.q.|)^2) )
.=|.px.|*(--(q`2/|.q.|)) by A22,SQUARE_1:22
.=q`2 by A22,A33,TOPRNS_1:24,XCMPLX_1:87;
A37: |.px.|* ((px`1/|.px.|-cn)/(1+cn)) =|.q.|* (( ((q`1/|.q.|*(1+cn)
+cn))-cn)/(1+cn)) by A22,A30,A33,TOPRNS_1:24,XCMPLX_1:89
.=|.q.|* ( q`1/|.q.|) by A25,XCMPLX_1:89
.= q`1 by A22,TOPRNS_1:24,XCMPLX_1:87;
then
|.px.|*( -sqrt(1-((px`1/|.px.|-cn)/(1+cn))^2)) = |.px.|*( -sqrt
(1-(q`1/|.px.|)^2)) by A22,A33,TOPRNS_1:24,XCMPLX_1:89
.= |.px.|*( -sqrt(1-(q`1)^2/(|.px.|)^2)) by XCMPLX_1:76
.= |.px.|*( -sqrt( (|.px.|)^2/(|.px.|)^2-(q`1)^2/(|.px.|)^2)) by
A26,A32,XCMPLX_1:60
.= |.px.|*( -sqrt( ((|.px.|)^2-(q`1)^2)/(|.px.|)^2)) by
XCMPLX_1:120
.= |.px.|*( -sqrt( ((q`1)^2+(q`2)^2-(q`1)^2)/(|.px.|)^2)) by A32,
JGRAPH_3:1
.= |.px.|*( -sqrt((q`2/|.q.|)^2)) by A33,XCMPLX_1:76;
hence ex x being set st x in dom (cn-FanMorphS) & y=(cn-FanMorphS).x
by A35,A37,A36,A29,EUCLID:53;
end;
end;
hence thesis by A3,FUNCT_1:def 3;
end;
hence thesis by A3,XBOOLE_0:def 10;
end;
hence thesis;
end;
theorem Th135:
for cn being Real,p2 being Point of TOP-REAL 2 st -1<cn & cn<1
ex K being non empty compact Subset of TOP-REAL 2 st K = (cn-FanMorphS).:K & ex
V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is open & V2 c= K & (cn
-FanMorphS).p2 in V2
proof
reconsider O=0.TOP-REAL 2 as Point of Euclid 2 by EUCLID:67;
let cn be Real,p2 be Point of TOP-REAL 2;
A1: the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
the TopStruct of TOP-REAL 2 = TopSpaceMetr Euclid 2 by EUCLID:def 8;
then reconsider V0=Ball(O,|.p2.|+1) as Subset of TOP-REAL 2;
O in V0 & V0 c= cl_Ball(O,|.p2.|+1) by GOBOARD6:1,METRIC_1:14;
then reconsider
K0=cl_Ball(O,|.p2.|+1) as non empty compact Subset of TOP-REAL 2
by A1,Th15;
set q3= (cn-FanMorphS).p2;
reconsider VV0 = V0 as Subset of TopSpaceMetr Euclid 2;
reconsider u2=p2 as Point of Euclid 2 by EUCLID:67;
reconsider u3=q3 as Point of Euclid 2 by EUCLID:67;
A2: (cn-FanMorphS).:K0 c= K0
proof
let y be object;
assume y in (cn-FanMorphS).:K0;
then consider x being object such that
A3: x in dom (cn-FanMorphS) and
A4: x in K0 and
A5: y=(cn-FanMorphS).x by FUNCT_1:def 6;
reconsider q=x as Point of TOP-REAL 2 by A3;
reconsider uq=q as Point of Euclid 2 by EUCLID:67;
dist(O,uq)<= |.p2.|+1 by A4,METRIC_1:12;
then |.0.TOP-REAL 2 - q.|<= |.p2.|+1 by JGRAPH_1:28;
then |. -q.|<= |.p2.|+1 by RLVECT_1:4;
then
A6: |.q.|<= |.p2.|+1 by TOPRNS_1:26;
A7: y in rng (cn-FanMorphS) by A3,A5,FUNCT_1:def 3;
then reconsider u=y as Point of Euclid 2 by EUCLID:67;
reconsider q4=y as Point of TOP-REAL 2 by A7;
|.q4.|=|.q.| by A5,Th128;
then |. -q4.|<= |.p2.|+1 by A6,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q4.|<= |.p2.|+1 by RLVECT_1:4;
then dist(O,u)<= |.p2.|+1 by JGRAPH_1:28;
hence thesis by METRIC_1:12;
end;
VV0 is open by TOPMETR:14;
then
A8: V0 is open by Lm11,PRE_TOPC:30;
A9: |. p2.|<|.p2.|+1 by XREAL_1:29;
then |. -p2.|<|.p2.|+1 by TOPRNS_1:26;
then |.0.TOP-REAL 2 - p2.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u2)<|.p2.|+1 by JGRAPH_1:28;
then
A10: p2 in V0 by METRIC_1:11;
|.q3.|=|.p2.| by Th128;
then |.-q3.|<|.p2.|+1 by A9,TOPRNS_1:26;
then |.0.TOP-REAL 2 - q3.|<|.p2.|+1 by RLVECT_1:4;
then dist(O,u3)< |.p2.|+1 by JGRAPH_1:28;
then
A11: (cn-FanMorphS).p2 in V0 by METRIC_1:11;
assume
A12: -1<cn & cn<1;
K0 c= (cn-FanMorphS).:K0
proof
let y be object;
assume
A13: y in K0;
then reconsider q4=y as Point of TOP-REAL 2;
reconsider y as Point of Euclid 2 by A13;
the carrier of TOP-REAL 2 c= rng (cn-FanMorphS) by A12,Th134;
then q4 in rng (cn-FanMorphS);
then consider x being object such that
A14: x in dom (cn-FanMorphS) and
A15: y=(cn-FanMorphS).x by FUNCT_1:def 3;
reconsider x as Point of Euclid 2 by A14,Lm11;
reconsider q=x as Point of TOP-REAL 2 by A14;
|.q4.|=|.q.| by A15,Th128;
then q in K0 by A13,Lm12;
hence thesis by A14,A15,FUNCT_1:def 6;
end;
then K0= (cn-FanMorphS).:K0 by A2,XBOOLE_0:def 10;
hence thesis by A10,A8,A11,METRIC_1:14;
end;
theorem
for cn being Real st -1<cn & cn<1 ex f being Function of TOP-REAL 2,
TOP-REAL 2 st f=cn-FanMorphS & f is being_homeomorphism
proof
let cn be Real;
set f=cn-FanMorphS;
assume
A1: -1<cn & cn<1;
then
A2: for p2 being Point of TOP-REAL 2 ex K being non empty compact Subset of
TOP-REAL 2 st K = f.:K & ex V2 being Subset of TOP-REAL 2 st p2 in V2 & V2 is
open & V2 c= K & f.p2 in V2 by Th135;
rng (cn-FanMorphS) = the carrier of TOP-REAL 2 & cn-FanMorphS is
continuous by A1,Th132,Th134;
then f is being_homeomorphism by A1,A2,Th3,Th133;
hence thesis;
end;
theorem Th137:
for cn being Real,q being Point of TOP-REAL 2 st cn<1 & q`2<0 &
q`1/|.q.|>=cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q holds
p`2<0 & p`1>=0
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: cn<1 and
A2: q`2<0 and
A3: q`1/|.q.|>=cn;
A4: 1-cn>0 by A1,XREAL_1:149;
let p be Point of TOP-REAL 2;
set qz=p;
assume p=(cn-FanMorphS).q;
then
A5: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-
cn))^2))]| by A2,A3,Th113;
then
A6: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
A7: (q`1/|.q.|-cn)>= 0 by A3,XREAL_1:48;
A8: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24;
then
A9: (|.q.|)^2>0 by SQUARE_1:12;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A9,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A9,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then 1>q`1/|.q.| by SQUARE_1:52;
then 1-cn>q`1/|.q.|-cn by XREAL_1:9;
then -(1-cn)< -( q`1/|.q.|-cn) by XREAL_1:24;
then (-(1-cn))/(1-cn)<(-( q`1/|.q.|-cn))/(1-cn) by A4,XREAL_1:74;
then -1<(-( q`1/|.q.|-cn))/(1-cn) by A4,XCMPLX_1:197;
then ((-(q`1/|.q.|-cn))/(1-cn))^2<1^2 by A4,A7,SQUARE_1:50;
hence thesis by A5,A8,A4,A6,A7,Lm13,EUCLID:52,XREAL_1:132;
end;
theorem Th138:
for cn being Real,q being Point of TOP-REAL 2 st -1<cn & q`2<0
& q`1/|.q.|<cn holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q
holds p`2<0 & p`1<0
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: q`2<0 and
A3: q`1/|.q.|<cn;
A4: 1+cn>0 by A1,XREAL_1:148;
A5: (q`1/|.q.|-cn)< 0 by A3,XREAL_1:49;
then -( q`1/|.q.|-cn)>0 by XREAL_1:58;
then (-(1+cn))/(1+cn)<(-( q`1/|.q.|-cn))/(1+cn) by A4,XREAL_1:74;
then
A6: -1<(-( q`1/|.q.|-cn))/(1+cn) by A4,XCMPLX_1:197;
A7: |.q.|<>0 by A2,JGRAPH_2:3,TOPRNS_1:24;
then
A8: (|.q.|)^2>0 by SQUARE_1:12;
(|.q.|)^2 =(q`1)^2+(q`2)^2 & 0+(q`1)^2<(q`1)^2+(q`2)^2 by A2,JGRAPH_3:1
,SQUARE_1:12,XREAL_1:8;
then (q`1)^2/(|.q.|)^2 < (|.q.|)^2/(|.q.|)^2 by A8,XREAL_1:74;
then (q`1)^2/(|.q.|)^2 < 1 by A8,XCMPLX_1:60;
then ((q`1)/|.q.|)^2 < 1 by XCMPLX_1:76;
then -1<q`1/|.q.| by SQUARE_1:52;
then -1-cn<q`1/|.q.|-cn by XREAL_1:9;
then --(1+cn)> -(q`1/|.q.|-cn) by XREAL_1:24;
then (-(q`1/|.q.|-cn))/(1+cn)<1 by A4,XREAL_1:191;
then ((-(q`1/|.q.|-cn))/(1+cn))^2<1^2 by A6,SQUARE_1:50;
then 1-((-(q`1/|.q.|-cn))/(1+cn))^2>0 by XREAL_1:50;
then sqrt(1-((-(q`1/|.q.|-cn))/(1+cn))^2)>0 by SQUARE_1:25;
then sqrt(1-(-(q`1/|.q.|-cn))^2/(1+cn)^2)> 0 by XCMPLX_1:76;
then sqrt(1-(q`1/|.q.|-cn)^2/(1+cn)^2)> 0;
then --sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)> 0 by XCMPLX_1:76;
then
A9: -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)< 0;
let p be Point of TOP-REAL 2;
set qz=p;
assume p=(cn-FanMorphS).q;
then
p=|[ |.q.|* ((q`1/|.q.|-cn)/(1+cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+
cn))^2))]| by A2,A3,Th114;
then
A10: qz`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1+cn))^2)) & qz`1= |.q.|* ((q`1/
|.q.| -cn)/(1+cn)) by EUCLID:52;
((q`1/|.q.|-cn)/(1+cn))<0 by A1,A5,XREAL_1:141,148;
hence thesis by A7,A10,A9,XREAL_1:132;
end;
theorem Th139:
for cn being Real,q1,q2 being Point of TOP-REAL 2 st cn<1 & q1
`2<0 & q1`1/|.q1.|>=cn & q2`2<0 & q2`1/|.q2.|>=cn & q1`1/|.q1.|<q2`1/|.q2.|
holds for p1,p2 being Point of TOP-REAL 2 st p1=(cn-FanMorphS).q1 & p2=(cn
-FanMorphS).q2 holds p1`1/|.p1.|<p2`1/|.p2.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: cn<1 and
A2: q1`2<0 and
A3: q1`1/|.q1.|>=cn and
A4: q2`2<0 and
A5: q2`1/|.q2.|>=cn and
A6: q1`1/|.q1.|<q2`1/|.q2.|;
A7: q1`1/|.q1.|-cn< q2`1/|.q2.|-cn & 1-cn>0 by A1,A6,XREAL_1:9,149;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(cn-FanMorphS).q1 and
A9: p2=(cn-FanMorphS).q2;
A10: |.p2.|=|.q2.| by A9,Th128;
p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)), |.q2.|*( -sqrt(1-((q2`1/|.q2.|
-cn)/(1-cn))^2))]| by A4,A5,A9,Th113;
then
A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1-cn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`1/|.p2.|= ((q2`1/|.q2.|-cn)/(1-cn)) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)), |.q1.|*( -sqrt(1-((q1`1/|.q1.|-
cn)/(1-cn))^2))]| by A2,A3,A8,Th113;
then
A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1-cn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th128;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`1/|.p1.|= ((q1`1/|.q1.|-cn)/(1-cn)) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem Th140:
for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & q1
`2<0 & q1`1/|.q1.|<cn & q2`2<0 & q2`1/|.q2.|<cn & q1`1/|.q1.|<q2`1/|.q2.| holds
for p1,p2 being Point of TOP-REAL 2 st p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS)
.q2 holds p1`1/|.p1.|<p2`1/|.p2.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: q1`2<0 and
A3: q1`1/|.q1.|<cn and
A4: q2`2<0 and
A5: q2`1/|.q2.|<cn and
A6: q1`1/|.q1.|<q2`1/|.q2.|;
A7: q1`1/|.q1.|-cn< q2`1/|.q2.|-cn & 1+cn>0 by A1,A6,XREAL_1:9,148;
let p1,p2 be Point of TOP-REAL 2;
assume that
A8: p1=(cn-FanMorphS).q1 and
A9: p2=(cn-FanMorphS).q2;
A10: |.p2.|=|.q2.| by A9,Th128;
p2=|[ |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)), |.q2.|*( -sqrt(1-((q2`1/|.q2.|
-cn)/(1+cn))^2))]| by A4,A5,A9,Th114;
then
A11: p2`1= |.q2.|* ((q2`1/|.q2.|-cn)/(1+cn)) by EUCLID:52;
|.q2.|>0 by A4,Lm1,JGRAPH_2:3;
then
A12: p2`1/|.p2.|= (q2`1/|.q2.|-cn)/(1+cn) by A11,A10,XCMPLX_1:89;
p1=|[ |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)), |.q1.|*( -sqrt(1-((q1`1/|.q1.|-
cn)/(1+cn))^2))]| by A2,A3,A8,Th114;
then
A13: p1`1= |.q1.|* ((q1`1/|.q1.|-cn)/(1+cn)) by EUCLID:52;
A14: |.p1.|=|.q1.| by A8,Th128;
|.q1.|>0 by A2,Lm1,JGRAPH_2:3;
then p1`1/|.p1.|= (q1`1/|.q1.|-cn)/(1+cn) by A13,A14,XCMPLX_1:89;
hence thesis by A12,A7,XREAL_1:74;
end;
theorem
for cn being Real,q1,q2 being Point of TOP-REAL 2 st -1<cn & cn<1 & q1
`2<0 & q2`2<0 & q1`1/|.q1.|<q2`1/|.q2.| holds for p1,p2 being Point of TOP-REAL
2 st p1=(cn-FanMorphS).q1 & p2=(cn-FanMorphS).q2 holds p1`1/|.p1.|<p2`1/|.p2.|
proof
let cn be Real,q1,q2 be Point of TOP-REAL 2;
assume that
A1: -1<cn and
A2: cn<1 and
A3: q1`2<0 and
A4: q2`2<0 and
A5: q1`1/|.q1.|<q2`1/|.q2.|;
let p1,p2 be Point of TOP-REAL 2;
assume that
A6: p1=(cn-FanMorphS).q1 and
A7: p2=(cn-FanMorphS).q2;
per cases;
suppose
q1`1/|.q1.|>=cn & q2`1/|.q2.|>=cn;
hence thesis by A2,A3,A4,A5,A6,A7,Th139;
end;
suppose
q1`1/|.q1.|>=cn & q2`1/|.q2.|<cn;
hence thesis by A5,XXREAL_0:2;
end;
suppose
A8: q1`1/|.q1.|<cn & q2`1/|.q2.|>=cn;
then p2`1>=0 by A2,A4,A7,Th137;
then
A9: p2`1/|.p2.|>=0;
p1`1<0 by A1,A3,A6,A8,Th138;
hence thesis by A9,Lm1,JGRAPH_2:3,XREAL_1:141;
end;
suppose
q1`1/|.q1.|<cn & q2`1/|.q2.|<cn;
hence thesis by A1,A3,A4,A5,A6,A7,Th140;
end;
end;
theorem
for cn being Real,q being Point of TOP-REAL 2 st q`2<0 & q`1/|.q.|=cn
holds for p being Point of TOP-REAL 2 st p=(cn-FanMorphS).q holds p`2<0 & p`1=0
proof
let cn be Real,q be Point of TOP-REAL 2;
assume that
A1: q`2<0 and
A2: q`1/|.q.|=cn;
let p be Point of TOP-REAL 2;
A3: |.q.|<>0 by A1,JGRAPH_2:3,TOPRNS_1:24;
assume p=(cn-FanMorphS).q;
then
A4: p=|[ |.q.|* ((q`1/|.q.|-cn)/(1-cn)), |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-
cn))^2))]| by A1,A2,Th113;
then p`2= |.q.|*( -sqrt(1-((q`1/|.q.|-cn)/(1-cn))^2)) by EUCLID:52;
hence thesis by A2,A4,A3,EUCLID:52,XREAL_1:132;
end;
theorem
0.TOP-REAL 2 = (a-FanMorphS).(0.TOP-REAL 2) by Th113,JGRAPH_2:3;