:: JGRAPH_5 semantic presentation
begin
theorem Th1: :: JGRAPH_5:1
for p being Point of (TOP-REAL 2) st |.p.| <= 1 holds
( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( |.p.| <= 1 implies ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) )
set a = |.p.|;
A1: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1;
then (|.p.| ^2) - ((p `1) ^2) >= 0 by XREAL_1:63;
then ((|.p.| ^2) - ((p `1) ^2)) + ((p `1) ^2) >= 0 + ((p `1) ^2) by XREAL_1:7;
then A2: ( - |.p.| <= p `1 & p `1 <= |.p.| ) by SQUARE_1:47;
(|.p.| ^2) - ((p `2) ^2) >= 0 by A1, XREAL_1:63;
then ((|.p.| ^2) - ((p `2) ^2)) + ((p `2) ^2) >= 0 + ((p `2) ^2) by XREAL_1:7;
then A3: ( - |.p.| <= p `2 & p `2 <= |.p.| ) by SQUARE_1:47;
assume A4: |.p.| <= 1 ; ::_thesis: ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 )
then - |.p.| >= - 1 by XREAL_1:24;
hence ( - 1 <= p `1 & p `1 <= 1 & - 1 <= p `2 & p `2 <= 1 ) by A4, A2, A3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th2: :: JGRAPH_5:2
for p being Point of (TOP-REAL 2) st |.p.| <= 1 & p `1 <> 0 & p `2 <> 0 holds
( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( |.p.| <= 1 & p `1 <> 0 & p `2 <> 0 implies ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) )
assume that
A1: |.p.| <= 1 and
A2: p `1 <> 0 and
A3: p `2 <> 0 ; ::_thesis: ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 )
set a = |.p.|;
A4: |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2) by JGRAPH_3:1;
then ((|.p.| ^2) - ((p `1) ^2)) + ((p `1) ^2) > 0 + ((p `1) ^2) by A3, SQUARE_1:12, XREAL_1:8;
then A5: ( - |.p.| < p `1 & p `1 < |.p.| ) by SQUARE_1:48;
((|.p.| ^2) - ((p `2) ^2)) + ((p `2) ^2) > 0 + ((p `2) ^2) by A2, A4, SQUARE_1:12, XREAL_1:8;
then A6: ( - |.p.| < p `2 & p `2 < |.p.| ) by SQUARE_1:48;
- |.p.| >= - 1 by A1, XREAL_1:24;
hence ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) by A1, A5, A6, XXREAL_0:2; ::_thesis: verum
end;
theorem :: JGRAPH_5:3
for a, b, d, e, r3 being Real
for PM, PM2 being non empty MetrStruct
for x being Element of PM
for x2 being Element of PM2 st d <= a & a <= b & b <= e & PM = Closed-Interval-MSpace (a,b) & PM2 = Closed-Interval-MSpace (d,e) & x = x2 holds
Ball (x,r3) c= Ball (x2,r3)
proof
let a, b, d, e, r3 be Real; ::_thesis: for PM, PM2 being non empty MetrStruct
for x being Element of PM
for x2 being Element of PM2 st d <= a & a <= b & b <= e & PM = Closed-Interval-MSpace (a,b) & PM2 = Closed-Interval-MSpace (d,e) & x = x2 holds
Ball (x,r3) c= Ball (x2,r3)
let PM, PM2 be non empty MetrStruct ; ::_thesis: for x being Element of PM
for x2 being Element of PM2 st d <= a & a <= b & b <= e & PM = Closed-Interval-MSpace (a,b) & PM2 = Closed-Interval-MSpace (d,e) & x = x2 holds
Ball (x,r3) c= Ball (x2,r3)
let x be Element of PM; ::_thesis: for x2 being Element of PM2 st d <= a & a <= b & b <= e & PM = Closed-Interval-MSpace (a,b) & PM2 = Closed-Interval-MSpace (d,e) & x = x2 holds
Ball (x,r3) c= Ball (x2,r3)
let x2 be Element of PM2; ::_thesis: ( d <= a & a <= b & b <= e & PM = Closed-Interval-MSpace (a,b) & PM2 = Closed-Interval-MSpace (d,e) & x = x2 implies Ball (x,r3) c= Ball (x2,r3) )
assume that
A1: d <= a and
A2: a <= b and
A3: b <= e and
A4: PM = Closed-Interval-MSpace (a,b) and
A5: PM2 = Closed-Interval-MSpace (d,e) and
A6: x = x2 ; ::_thesis: Ball (x,r3) c= Ball (x2,r3)
a <= e by A2, A3, XXREAL_0:2;
then A7: a in [.d,e.] by A1, XXREAL_1:1;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Ball (x,r3) or z in Ball (x2,r3) )
assume z in Ball (x,r3) ; ::_thesis: z in Ball (x2,r3)
then z in { y where y is Element of PM : dist (x,y) < r3 } by METRIC_1:17;
then consider y being Element of PM such that
A8: ( y = z & dist (x,y) < r3 ) ;
the carrier of PM = [.a,b.] by A2, A4, TOPMETR:10;
then A9: y in [.a,b.] ;
A10: d <= b by A1, A2, XXREAL_0:2;
then b in [.d,e.] by A3, XXREAL_1:1;
then [.a,b.] c= [.d,e.] by A7, XXREAL_2:def_12;
then reconsider y3 = y as Element of PM2 by A3, A5, A10, A9, TOPMETR:10, XXREAL_0:2;
A11: dist (x,y) = the distance of PM . (x,y) by METRIC_1:def_1;
A12: the distance of PM . (x,y) = real_dist . (x,y) by A4, METRIC_1:def_13, TOPMETR:def_1;
real_dist . (x,y) = the distance of PM2 . (x2,y3) by A5, A6, METRIC_1:def_13, TOPMETR:def_1
.= dist (x2,y3) by METRIC_1:def_1 ;
then z in { y2 where y2 is Element of PM2 : dist (x2,y2) < r3 } by A8, A12, A11;
hence z in Ball (x2,r3) by METRIC_1:17; ::_thesis: verum
end;
theorem :: JGRAPH_5:4
for a, b being real number
for B being Subset of I[01] st 0 <= a & a <= b & b <= 1 & B = [.a,b.] holds
Closed-Interval-TSpace (a,b) = I[01] | B by TOPMETR:20, TOPMETR:23;
theorem Th5: :: JGRAPH_5:5
for X being TopStruct
for Y, Z being non empty TopStruct
for f being Function of X,Y
for h being Function of Y,Z st h is being_homeomorphism & f is continuous holds
h * f is continuous
proof
let X be TopStruct ; ::_thesis: for Y, Z being non empty TopStruct
for f being Function of X,Y
for h being Function of Y,Z st h is being_homeomorphism & f is continuous holds
h * f is continuous
let Y, Z be non empty TopStruct ; ::_thesis: for f being Function of X,Y
for h being Function of Y,Z st h is being_homeomorphism & f is continuous holds
h * f is continuous
let f be Function of X,Y; ::_thesis: for h being Function of Y,Z st h is being_homeomorphism & f is continuous holds
h * f is continuous
let h be Function of Y,Z; ::_thesis: ( h is being_homeomorphism & f is continuous implies h * f is continuous )
assume that
A1: h is being_homeomorphism and
A2: f is continuous ; ::_thesis: h * f is continuous
h is continuous by A1, TOPS_2:def_5;
hence h * f is continuous by A2, TOPS_2:46; ::_thesis: verum
end;
theorem Th6: :: JGRAPH_5:6
for X, Y, Z being TopStruct
for f being Function of X,Y
for h being Function of Y,Z st h is being_homeomorphism & f is one-to-one holds
h * f is one-to-one
proof
let X, Y, Z be TopStruct ; ::_thesis: for f being Function of X,Y
for h being Function of Y,Z st h is being_homeomorphism & f is one-to-one holds
h * f is one-to-one
let f be Function of X,Y; ::_thesis: for h being Function of Y,Z st h is being_homeomorphism & f is one-to-one holds
h * f is one-to-one
let h be Function of Y,Z; ::_thesis: ( h is being_homeomorphism & f is one-to-one implies h * f is one-to-one )
assume that
A1: h is being_homeomorphism and
A2: f is one-to-one ; ::_thesis: h * f is one-to-one
h is one-to-one by A1, TOPS_2:def_5;
hence h * f is one-to-one by A2; ::_thesis: verum
end;
theorem Th7: :: JGRAPH_5:7
for X being TopStruct
for S, V being non empty TopStruct
for B being non empty Subset of S
for f being Function of X,(S | B)
for g being Function of S,V
for h being Function of X,V st h = g * f & f is continuous & g is continuous holds
h is continuous
proof
let X be TopStruct ; ::_thesis: for S, V being non empty TopStruct
for B being non empty Subset of S
for f being Function of X,(S | B)
for g being Function of S,V
for h being Function of X,V st h = g * f & f is continuous & g is continuous holds
h is continuous
let S, V be non empty TopStruct ; ::_thesis: for B being non empty Subset of S
for f being Function of X,(S | B)
for g being Function of S,V
for h being Function of X,V st h = g * f & f is continuous & g is continuous holds
h is continuous
let B be non empty Subset of S; ::_thesis: for f being Function of X,(S | B)
for g being Function of S,V
for h being Function of X,V st h = g * f & f is continuous & g is continuous holds
h is continuous
let f be Function of X,(S | B); ::_thesis: for g being Function of S,V
for h being Function of X,V st h = g * f & f is continuous & g is continuous holds
h is continuous
let g be Function of S,V; ::_thesis: for h being Function of X,V st h = g * f & f is continuous & g is continuous holds
h is continuous
let h be Function of X,V; ::_thesis: ( h = g * f & f is continuous & g is continuous implies h is continuous )
assume that
A1: h = g * f and
A2: f is continuous and
A3: g is continuous ; ::_thesis: h is continuous
now__::_thesis:_for_P_being_Subset_of_V_st_P_is_closed_holds_
h_"_P_is_closed
let P be Subset of V; ::_thesis: ( P is closed implies h " P is closed )
A4: (g * f) " P = f " (g " P) by RELAT_1:146;
now__::_thesis:_(_P_is_closed_implies_h_"_P_is_closed_)
assume P is closed ; ::_thesis: h " P is closed
then A5: g " P is closed by A3, PRE_TOPC:def_6;
A6: the carrier of (S | B) = B by PRE_TOPC:8;
then reconsider F = B /\ (g " P) as Subset of (S | B) by XBOOLE_1:17;
A7: (rng f) /\ the carrier of (S | B) = rng f by XBOOLE_1:28;
[#] (S | B) = B by PRE_TOPC:def_5;
then A8: F is closed by A5, PRE_TOPC:13;
h " P = f " ((rng f) /\ (g " P)) by A1, A4, RELAT_1:133
.= f " ((rng f) /\ ( the carrier of (S | B) /\ (g " P))) by A7, XBOOLE_1:16
.= f " F by A6, RELAT_1:133 ;
hence h " P is closed by A2, A8, PRE_TOPC:def_6; ::_thesis: verum
end;
hence ( P is closed implies h " P is closed ) ; ::_thesis: verum
end;
hence h is continuous by PRE_TOPC:def_6; ::_thesis: verum
end;
theorem Th8: :: JGRAPH_5:8
for a, b, d, e, s1, s2, t1, t2 being Real
for h being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (d,e)) st h is being_homeomorphism & h . s1 = t1 & h . s2 = t2 & h . b = e & d <= e & t1 <= t2 & s1 in [.a,b.] & s2 in [.a,b.] holds
s1 <= s2
proof
let a, b, d, e, s1, s2, t1, t2 be Real; ::_thesis: for h being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (d,e)) st h is being_homeomorphism & h . s1 = t1 & h . s2 = t2 & h . b = e & d <= e & t1 <= t2 & s1 in [.a,b.] & s2 in [.a,b.] holds
s1 <= s2
let h be Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (d,e)); ::_thesis: ( h is being_homeomorphism & h . s1 = t1 & h . s2 = t2 & h . b = e & d <= e & t1 <= t2 & s1 in [.a,b.] & s2 in [.a,b.] implies s1 <= s2 )
assume that
A1: h is being_homeomorphism and
A2: h . s1 = t1 and
A3: h . s2 = t2 and
A4: h . b = e and
A5: d <= e and
A6: t1 <= t2 and
A7: s1 in [.a,b.] and
A8: s2 in [.a,b.] ; ::_thesis: s1 <= s2
A9: s1 <= b by A7, XXREAL_1:1;
reconsider C = [.d,e.] as non empty Subset of R^1 by A5, TOPMETR:17, XXREAL_1:1;
A10: R^1 | C = Closed-Interval-TSpace (d,e) by A5, TOPMETR:19;
A11: a <= s1 by A7, XXREAL_1:1;
then A12: the carrier of (Closed-Interval-TSpace (a,b)) = [.a,b.] by A9, TOPMETR:18, XXREAL_0:2;
then reconsider B1 = [.s1,b.] as Subset of (Closed-Interval-TSpace (a,b)) by A11, XXREAL_1:34;
A13: dom h = [#] (Closed-Interval-TSpace (a,b)) by A1, TOPS_2:def_5
.= [.a,b.] by A11, A9, TOPMETR:18, XXREAL_0:2 ;
A14: a <= s2 by A8, XXREAL_1:1;
then reconsider B = [.s2,s1.] as Subset of (Closed-Interval-TSpace (a,b)) by A9, A12, XXREAL_1:34;
reconsider Bb = [.s2,s1.] as Subset of (Closed-Interval-TSpace (a,b)) by A14, A9, A12, XXREAL_1:34;
reconsider f3 = h | Bb as Function of ((Closed-Interval-TSpace (a,b)) | B),(Closed-Interval-TSpace (d,e)) by PRE_TOPC:9;
assume A15: s1 > s2 ; ::_thesis: contradiction
then A16: Closed-Interval-TSpace (s2,s1) = (Closed-Interval-TSpace (a,b)) | B by A14, A9, TOPMETR:23;
then f3 is Function of (Closed-Interval-TSpace (s2,s1)),R^1 by A10, JORDAN6:3;
then reconsider f = h | B as Function of (Closed-Interval-TSpace (s2,s1)),R^1 ;
s2 in B by A15, XXREAL_1:1;
then A17: f . s2 = t2 by A3, FUNCT_1:49;
set t = (t1 + t2) / 2;
A18: the carrier of (Closed-Interval-TSpace (d,e)) = [.d,e.] by A5, TOPMETR:18;
h is one-to-one by A1, TOPS_2:def_5;
then t1 <> t2 by A2, A3, A7, A8, A13, A15, FUNCT_1:def_4;
then A19: t1 < t2 by A6, XXREAL_0:1;
then t1 + t1 < t1 + t2 by XREAL_1:8;
then A20: (2 * t1) / 2 < (t1 + t2) / 2 by XREAL_1:74;
dom f = the carrier of (Closed-Interval-TSpace (s2,s1)) by FUNCT_2:def_1;
then dom f = [.s2,s1.] by A15, TOPMETR:18;
then s2 in dom f by A15, XXREAL_1:1;
then t2 in rng f3 by A17, FUNCT_1:def_3;
then A21: t2 <= e by A18, XXREAL_1:1;
t1 + t2 < t2 + t2 by A19, XREAL_1:8;
then A22: (2 * t2) / 2 > (t1 + t2) / 2 by XREAL_1:74;
then A23: e > (t1 + t2) / 2 by A21, XXREAL_0:2;
reconsider B1b = [.s1,b.] as Subset of (Closed-Interval-TSpace (a,b)) by A11, A12, XXREAL_1:34;
reconsider f4 = h | B1b as Function of ((Closed-Interval-TSpace (a,b)) | B1),(Closed-Interval-TSpace (d,e)) by PRE_TOPC:9;
A24: Closed-Interval-TSpace (s1,b) = (Closed-Interval-TSpace (a,b)) | B1 by A11, A9, TOPMETR:23;
then f4 is Function of (Closed-Interval-TSpace (s1,b)),R^1 by A10, JORDAN6:3;
then reconsider f1 = h | B1 as Function of (Closed-Interval-TSpace (s1,b)),R^1 ;
A25: h is continuous by A1, TOPS_2:def_5;
then f4 is continuous by TOPMETR:7;
then A26: f1 is continuous by A10, A24, JORDAN6:3;
b in B1 by A9, XXREAL_1:1;
then A27: f1 . b = e by A4, FUNCT_1:49;
s1 in B1 by A9, XXREAL_1:1;
then A28: f1 . s1 = t1 by A2, FUNCT_1:49;
s1 < b by A2, A4, A9, A19, A21, XXREAL_0:1;
then consider r1 being Real such that
A29: f1 . r1 = (t1 + t2) / 2 and
A30: s1 < r1 and
A31: r1 < b by A20, A26, A28, A27, A23, TOPREAL5:6;
A32: r1 in B1 by A30, A31, XXREAL_1:1;
s1 in B by A15, XXREAL_1:1;
then A33: f . s1 = t1 by A2, FUNCT_1:49;
f3 is continuous by A25, TOPMETR:7;
then f is continuous by A10, A16, JORDAN6:3;
then consider r being Real such that
A34: f . r = (t1 + t2) / 2 and
A35: s2 < r and
A36: r < s1 by A15, A17, A33, A20, A22, TOPREAL5:7;
A37: a < r by A14, A35, XXREAL_0:2;
a < r1 by A11, A30, XXREAL_0:2;
then A38: r1 in [.a,b.] by A31, XXREAL_1:1;
A39: h is one-to-one by A1, TOPS_2:def_5;
r < b by A9, A36, XXREAL_0:2;
then A40: r in [.a,b.] by A37, XXREAL_1:1;
r in [.s2,s1.] by A35, A36, XXREAL_1:1;
then h . r = (t1 + t2) / 2 by A34, FUNCT_1:49
.= h . r1 by A29, A32, FUNCT_1:49 ;
hence contradiction by A13, A39, A36, A40, A30, A38, FUNCT_1:def_4; ::_thesis: verum
end;
theorem Th9: :: JGRAPH_5:9
for a, b, d, e, s1, s2, t1, t2 being Real
for h being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (d,e)) st h is being_homeomorphism & h . s1 = t1 & h . s2 = t2 & h . b = d & e >= d & t1 >= t2 & s1 in [.a,b.] & s2 in [.a,b.] holds
s1 <= s2
proof
let a, b, d, e, s1, s2, t1, t2 be Real; ::_thesis: for h being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (d,e)) st h is being_homeomorphism & h . s1 = t1 & h . s2 = t2 & h . b = d & e >= d & t1 >= t2 & s1 in [.a,b.] & s2 in [.a,b.] holds
s1 <= s2
let h be Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (d,e)); ::_thesis: ( h is being_homeomorphism & h . s1 = t1 & h . s2 = t2 & h . b = d & e >= d & t1 >= t2 & s1 in [.a,b.] & s2 in [.a,b.] implies s1 <= s2 )
assume that
A1: h is being_homeomorphism and
A2: h . s1 = t1 and
A3: h . s2 = t2 and
A4: h . b = d and
A5: e >= d and
A6: t1 >= t2 and
A7: s1 in [.a,b.] and
A8: s2 in [.a,b.] ; ::_thesis: s1 <= s2
A9: s1 <= b by A7, XXREAL_1:1;
reconsider C = [.d,e.] as non empty Subset of R^1 by A5, TOPMETR:17, XXREAL_1:1;
A10: R^1 | C = Closed-Interval-TSpace (d,e) by A5, TOPMETR:19;
A11: a <= s1 by A7, XXREAL_1:1;
then A12: the carrier of (Closed-Interval-TSpace (a,b)) = [.a,b.] by A9, TOPMETR:18, XXREAL_0:2;
then reconsider B1 = [.s1,b.] as Subset of (Closed-Interval-TSpace (a,b)) by A11, XXREAL_1:34;
A13: dom h = [#] (Closed-Interval-TSpace (a,b)) by A1, TOPS_2:def_5
.= [.a,b.] by A11, A9, TOPMETR:18, XXREAL_0:2 ;
A14: a <= s2 by A8, XXREAL_1:1;
then reconsider B = [.s2,s1.] as Subset of (Closed-Interval-TSpace (a,b)) by A9, A12, XXREAL_1:34;
reconsider Bb = [.s2,s1.] as Subset of (Closed-Interval-TSpace (a,b)) by A14, A9, A12, XXREAL_1:34;
reconsider f3 = h | Bb as Function of ((Closed-Interval-TSpace (a,b)) | B),(Closed-Interval-TSpace (d,e)) by PRE_TOPC:9;
assume A15: s1 > s2 ; ::_thesis: contradiction
then A16: Closed-Interval-TSpace (s2,s1) = (Closed-Interval-TSpace (a,b)) | B by A14, A9, TOPMETR:23;
then f3 is Function of (Closed-Interval-TSpace (s2,s1)),R^1 by A10, JORDAN6:3;
then reconsider f = h | B as Function of (Closed-Interval-TSpace (s2,s1)),R^1 ;
s2 in B by A15, XXREAL_1:1;
then A17: f . s2 = t2 by A3, FUNCT_1:49;
set t = (t1 + t2) / 2;
A18: the carrier of (Closed-Interval-TSpace (d,e)) = [.d,e.] by A5, TOPMETR:18;
h is one-to-one by A1, TOPS_2:def_5;
then t1 <> t2 by A2, A3, A7, A8, A13, A15, FUNCT_1:def_4;
then A19: t1 > t2 by A6, XXREAL_0:1;
then t1 + t1 > t1 + t2 by XREAL_1:8;
then A20: (2 * t1) / 2 > (t1 + t2) / 2 by XREAL_1:74;
dom f = the carrier of (Closed-Interval-TSpace (s2,s1)) by FUNCT_2:def_1;
then dom f = [.s2,s1.] by A15, TOPMETR:18;
then s2 in dom f by A15, XXREAL_1:1;
then t2 in rng f3 by A17, FUNCT_1:def_3;
then A21: d <= t2 by A18, XXREAL_1:1;
t1 + t2 > t2 + t2 by A19, XREAL_1:8;
then A22: (2 * t2) / 2 < (t1 + t2) / 2 by XREAL_1:74;
then A23: d < (t1 + t2) / 2 by A21, XXREAL_0:2;
reconsider B1b = [.s1,b.] as Subset of (Closed-Interval-TSpace (a,b)) by A11, A12, XXREAL_1:34;
reconsider f4 = h | B1b as Function of ((Closed-Interval-TSpace (a,b)) | B1),(Closed-Interval-TSpace (d,e)) by PRE_TOPC:9;
A24: Closed-Interval-TSpace (s1,b) = (Closed-Interval-TSpace (a,b)) | B1 by A11, A9, TOPMETR:23;
then f4 is Function of (Closed-Interval-TSpace (s1,b)),R^1 by A10, JORDAN6:3;
then reconsider f1 = h | B1 as Function of (Closed-Interval-TSpace (s1,b)),R^1 ;
A25: h is continuous by A1, TOPS_2:def_5;
then f4 is continuous by TOPMETR:7;
then A26: f1 is continuous by A10, A24, JORDAN6:3;
b in B1 by A9, XXREAL_1:1;
then A27: f1 . b = d by A4, FUNCT_1:49;
s1 in B1 by A9, XXREAL_1:1;
then A28: f1 . s1 = t1 by A2, FUNCT_1:49;
s1 < b by A2, A4, A9, A19, A21, XXREAL_0:1;
then consider r1 being Real such that
A29: f1 . r1 = (t1 + t2) / 2 and
A30: s1 < r1 and
A31: r1 < b by A20, A26, A28, A27, A23, TOPREAL5:7;
A32: r1 in B1 by A30, A31, XXREAL_1:1;
s1 in B by A15, XXREAL_1:1;
then A33: f . s1 = t1 by A2, FUNCT_1:49;
f3 is continuous by A25, TOPMETR:7;
then f is continuous by A10, A16, JORDAN6:3;
then consider r being Real such that
A34: f . r = (t1 + t2) / 2 and
A35: s2 < r and
A36: r < s1 by A15, A17, A33, A20, A22, TOPREAL5:6;
A37: a < r by A14, A35, XXREAL_0:2;
a < r1 by A11, A30, XXREAL_0:2;
then A38: r1 in [.a,b.] by A31, XXREAL_1:1;
A39: h is one-to-one by A1, TOPS_2:def_5;
r < b by A9, A36, XXREAL_0:2;
then A40: r in [.a,b.] by A37, XXREAL_1:1;
r in [.s2,s1.] by A35, A36, XXREAL_1:1;
then h . r = (t1 + t2) / 2 by A34, FUNCT_1:49
.= h . r1 by A29, A32, FUNCT_1:49 ;
hence contradiction by A13, A39, A36, A40, A30, A38, FUNCT_1:def_4; ::_thesis: verum
end;
theorem :: JGRAPH_5:10
for n being Element of NAT holds - (0. (TOP-REAL n)) = 0. (TOP-REAL n)
proof
let n be Element of NAT ; ::_thesis: - (0. (TOP-REAL n)) = 0. (TOP-REAL n)
(0. (TOP-REAL n)) + (0. (TOP-REAL n)) = 0. (TOP-REAL n) by EUCLID:27;
hence - (0. (TOP-REAL n)) = 0. (TOP-REAL n) by EUCLID:37; ::_thesis: verum
end;
begin
theorem Th11: :: JGRAPH_5:11
for f, g being Function of I[01],(TOP-REAL 2)
for a, b, c, d being Real
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) holds
rng f meets rng g
proof
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for a, b, c, d being Real
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) holds
rng f meets rng g
let a, b, c, d be Real; ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) holds
rng f meets rng g
let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) implies rng f meets rng g )
assume that
A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one ) and
A2: a <> b and
A3: c <> d and
A4: (f . O) `1 = a and
A5: ( c <= (f . O) `2 & (f . O) `2 <= d ) and
A6: ( (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c ) and
A7: ( a <= (g . O) `1 & (g . O) `1 <= b ) and
A8: ( (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) ) ; ::_thesis: rng f meets rng g
c <= d by A5, XXREAL_0:2;
then A9: c < d by A3, XXREAL_0:1;
a <= b by A7, XXREAL_0:2;
then a < b by A2, XXREAL_0:1;
hence rng f meets rng g by A1, A4, A5, A6, A7, A8, A9, JGRAPH_2:45; ::_thesis: verum
end;
Lm1: 0 in [.0,1.]
by XXREAL_1:1;
Lm2: 1 in [.0,1.]
by XXREAL_1:1;
theorem Th12: :: JGRAPH_5:12
for f being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one holds
ex f2 being Function of I[01],(TOP-REAL 2) st
( f2 . 0 = f . 1 & f2 . 1 = f . 0 & rng f2 = rng f & f2 is continuous & f2 is one-to-one )
proof
let f be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one implies ex f2 being Function of I[01],(TOP-REAL 2) st
( f2 . 0 = f . 1 & f2 . 1 = f . 0 & rng f2 = rng f & f2 is continuous & f2 is one-to-one ) )
A1: I[01] is compact by HEINE:4, TOPMETR:20;
A2: dom f = the carrier of I[01] by FUNCT_2:def_1;
then reconsider P = rng f as non empty Subset of (TOP-REAL 2) by Lm1, BORSUK_1:40, FUNCT_1:3;
( f . 1 in rng f & f . 0 in rng f ) by A2, Lm1, Lm2, BORSUK_1:40, FUNCT_1:3;
then reconsider p1 = f . 0, p2 = f . 1 as Point of (TOP-REAL 2) ;
assume ( f is continuous & f is one-to-one ) ; ::_thesis: ex f2 being Function of I[01],(TOP-REAL 2) st
( f2 . 0 = f . 1 & f2 . 1 = f . 0 & rng f2 = rng f & f2 is continuous & f2 is one-to-one )
then ex f1 being Function of I[01],((TOP-REAL 2) | P) st
( f1 = f & f1 is being_homeomorphism ) by A1, JGRAPH_1:46;
then P is_an_arc_of p1,p2 by TOPREAL1:def_1;
then P is_an_arc_of p2,p1 by JORDAN5B:14;
then consider f3 being Function of I[01],((TOP-REAL 2) | P) such that
A3: f3 is being_homeomorphism and
A4: ( f3 . 0 = p2 & f3 . 1 = p1 ) by TOPREAL1:def_1;
A5: ex f4 being Function of I[01],(TOP-REAL 2) st
( f3 = f4 & f4 is continuous & f4 is one-to-one ) by A3, JORDAN7:15;
rng f3 = [#] ((TOP-REAL 2) | P) by A3, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
hence ex f2 being Function of I[01],(TOP-REAL 2) st
( f2 . 0 = f . 1 & f2 . 1 = f . 0 & rng f2 = rng f & f2 is continuous & f2 is one-to-one ) by A4, A5; ::_thesis: verum
end;
theorem Th13: :: JGRAPH_5:13
for f, g being Function of I[01],(TOP-REAL 2)
for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0, KXP, KXN, KYP, KYN be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
then ex g2 being Function of I[01],(TOP-REAL 2) st
( g2 . 0 = g . 1 & g2 . 1 = g . 0 & rng g2 = rng g & g2 is continuous & g2 is one-to-one ) by Th12;
hence rng f meets rng g by A1, JGRAPH_3:44; ::_thesis: verum
end;
theorem Th14: :: JGRAPH_5:14
for f, g being Function of I[01],(TOP-REAL 2)
for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0, KXP, KXN, KYP, KYN be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYN & g . I in KYP & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
A2: dom g = the carrier of I[01] by FUNCT_2:def_1;
reconsider gg = (Sq_Circ ") * g as Function of I[01],(TOP-REAL 2) by FUNCT_2:13, JGRAPH_3:29;
reconsider ff = (Sq_Circ ") * f as Function of I[01],(TOP-REAL 2) by FUNCT_2:13, JGRAPH_3:29;
A3: dom gg = the carrier of I[01] by FUNCT_2:def_1;
A4: dom ff = the carrier of I[01] by FUNCT_2:def_1;
A5: ( (ff . O) `1 = - 1 & (ff . I) `1 = 1 & (gg . O) `2 = - 1 & (gg . I) `2 = 1 )
proof
reconsider pz = gg . O as Point of (TOP-REAL 2) ;
reconsider py = ff . I as Point of (TOP-REAL 2) ;
reconsider px = ff . O as Point of (TOP-REAL 2) ;
set q = px;
A6: |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `1 = (px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2))) by EUCLID:52;
reconsider pu = gg . I as Point of (TOP-REAL 2) ;
A7: |[((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1 = (py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2))) by EUCLID:52;
consider p2 being Point of (TOP-REAL 2) such that
A8: f . I = p2 and
A9: |.p2.| = 1 and
A10: p2 `2 <= p2 `1 and
A11: p2 `2 >= - (p2 `1) by A1;
A12: ff . I = (Sq_Circ ") . (f . I) by A4, FUNCT_1:12;
then A13: p2 = Sq_Circ . py by A8, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A14: p2 <> 0. (TOP-REAL 2) by A9, TOPRNS_1:23;
then A15: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A10, A11, JGRAPH_3:28;
then A16: py `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A12, A8, EUCLID:52;
((p2 `2) / (p2 `1)) ^2 >= 0 by XREAL_1:63;
then A17: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25;
A18: py `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A12, A8, A15, EUCLID:52;
A19: now__::_thesis:_(_py_`1_=_0_implies_not_py_`2_=_0_)
assume ( py `1 = 0 & py `2 = 0 ) ; ::_thesis: contradiction
then ( p2 `1 = 0 & p2 `2 = 0 ) by A16, A18, A17, XCMPLX_1:6;
hence contradiction by A14, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A20: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( py `2 >= py `1 & py `2 <= - (py `1) ) ) by A10, A11, A17, XREAL_1:64;
then ( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - (py `1) <= py `2 ) or ( py `2 >= py `1 & py `2 <= - (py `1) ) ) by A12, A8, A15, A16, A17, EUCLID:52, XREAL_1:64;
then A21: Sq_Circ . py = |[((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))))]| by A16, A18, A19, JGRAPH_2:3, JGRAPH_3:def_1;
A22: ((py `2) / (py `1)) ^2 >= 0 by XREAL_1:63;
then A23: sqrt (1 + (((py `2) / (py `1)) ^2)) > 0 by SQUARE_1:25;
A24: now__::_thesis:_not_py_`1_=_-_1
assume A25: py `1 = - 1 ; ::_thesis: contradiction
- (p2 `2) <= - (- (p2 `1)) by A11, XREAL_1:24;
then - (p2 `2) < 0 by A13, A21, A7, A22, A25, SQUARE_1:25, XREAL_1:141;
then - (- (p2 `2)) > - 0 ;
hence contradiction by A10, A13, A21, A23, A25, EUCLID:52; ::_thesis: verum
end;
|[((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 = (py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))) by EUCLID:52;
then |.p2.| ^2 = (((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))) ^2) + (((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2)))) ^2) by A13, A21, A7, JGRAPH_3:1
.= (((py `1) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) + (((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((py `1) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) + (((py `2) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((py `1) ^2) / (1 + (((py `2) / (py `1)) ^2))) + (((py `2) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) by A22, SQUARE_1:def_2
.= (((py `1) ^2) / (1 + (((py `2) / (py `1)) ^2))) + (((py `2) ^2) / (1 + (((py `2) / (py `1)) ^2))) by A22, SQUARE_1:def_2
.= (((py `1) ^2) + ((py `2) ^2)) / (1 + (((py `2) / (py `1)) ^2)) by XCMPLX_1:62 ;
then ((((py `1) ^2) + ((py `2) ^2)) / (1 + (((py `2) / (py `1)) ^2))) * (1 + (((py `2) / (py `1)) ^2)) = 1 * (1 + (((py `2) / (py `1)) ^2)) by A9;
then ((py `1) ^2) + ((py `2) ^2) = 1 + (((py `2) / (py `1)) ^2) by A22, XCMPLX_1:87;
then A26: (((py `1) ^2) + ((py `2) ^2)) - 1 = ((py `2) ^2) / ((py `1) ^2) by XCMPLX_1:76;
py `1 <> 0 by A16, A18, A17, A19, A20, XREAL_1:64;
then ((((py `1) ^2) + ((py `2) ^2)) - 1) * ((py `1) ^2) = (py `2) ^2 by A26, XCMPLX_1:6, XCMPLX_1:87;
then A27: (((py `1) ^2) - 1) * (((py `1) ^2) + ((py `2) ^2)) = 0 ;
((py `1) ^2) + ((py `2) ^2) <> 0 by A19, COMPLEX1:1;
then ((py `1) - 1) * ((py `1) + 1) = 0 by A27, XCMPLX_1:6;
then A28: ( (py `1) - 1 = 0 or (py `1) + 1 = 0 ) by XCMPLX_1:6;
A29: |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `2 = (pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) by EUCLID:52;
consider p1 being Point of (TOP-REAL 2) such that
A30: f . O = p1 and
A31: |.p1.| = 1 and
A32: p1 `2 >= p1 `1 and
A33: p1 `2 <= - (p1 `1) by A1;
((p1 `2) / (p1 `1)) ^2 >= 0 by XREAL_1:63;
then A34: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) > 0 by SQUARE_1:25;
A35: ff . O = (Sq_Circ ") . (f . O) by A4, FUNCT_1:12;
then A36: p1 = Sq_Circ . px by A30, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A37: p1 <> 0. (TOP-REAL 2) by A31, TOPRNS_1:23;
then (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A32, A33, JGRAPH_3:28;
then A38: ( px `1 = (p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) & px `2 = (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) by A35, A30, EUCLID:52;
A39: now__::_thesis:_(_px_`1_=_0_implies_not_px_`2_=_0_)
assume ( px `1 = 0 & px `2 = 0 ) ; ::_thesis: contradiction
then ( p1 `1 = 0 & p1 `2 = 0 ) by A38, A34, XCMPLX_1:6;
hence contradiction by A37, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) ) by A32, A33, A34, XREAL_1:64;
then A40: ( ( p1 `2 <= p1 `1 & (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A38, A34, XREAL_1:64;
then ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A38, A34, XREAL_1:64;
then A41: Sq_Circ . px = |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| by A39, JGRAPH_2:3, JGRAPH_3:def_1;
A42: ((px `2) / (px `1)) ^2 >= 0 by XREAL_1:63;
then A43: sqrt (1 + (((px `2) / (px `1)) ^2)) > 0 by SQUARE_1:25;
A44: now__::_thesis:_not_px_`1_=_1
assume A45: px `1 = 1 ; ::_thesis: contradiction
- (p1 `2) >= - (- (p1 `1)) by A33, XREAL_1:24;
then - (p1 `2) > 0 by A36, A41, A6, A43, A45, XREAL_1:139;
then - (- (p1 `2)) < - 0 ;
hence contradiction by A32, A36, A41, A43, A45, EUCLID:52; ::_thesis: verum
end;
|[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `2 = (px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))) by EUCLID:52;
then |.p1.| ^2 = (((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by A36, A41, A6, JGRAPH_3:1
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by A42, SQUARE_1:def_2
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / (1 + (((px `2) / (px `1)) ^2))) by A42, SQUARE_1:def_2
.= (((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2)) by XCMPLX_1:62 ;
then ((((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2))) * (1 + (((px `2) / (px `1)) ^2)) = 1 * (1 + (((px `2) / (px `1)) ^2)) by A31;
then ((px `1) ^2) + ((px `2) ^2) = 1 + (((px `2) / (px `1)) ^2) by A42, XCMPLX_1:87;
then A46: (((px `1) ^2) + ((px `2) ^2)) - 1 = ((px `2) ^2) / ((px `1) ^2) by XCMPLX_1:76;
px `1 <> 0 by A38, A34, A39, A40, XREAL_1:64;
then ((((px `1) ^2) + ((px `2) ^2)) - 1) * ((px `1) ^2) = (px `2) ^2 by A46, XCMPLX_1:6, XCMPLX_1:87;
then A47: (((px `1) ^2) - 1) * (((px `1) ^2) + ((px `2) ^2)) = 0 ;
A48: |[((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))),((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))))]| `2 = (pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))) by EUCLID:52;
consider p4 being Point of (TOP-REAL 2) such that
A49: g . I = p4 and
A50: |.p4.| = 1 and
A51: p4 `2 >= p4 `1 and
A52: p4 `2 >= - (p4 `1) by A1;
((p4 `1) / (p4 `2)) ^2 >= 0 by XREAL_1:63;
then A53: sqrt (1 + (((p4 `1) / (p4 `2)) ^2)) > 0 by SQUARE_1:25;
A54: - (p4 `2) <= - (- (p4 `1)) by A52, XREAL_1:24;
then A55: ( ( p4 `1 <= p4 `2 & (- (p4 `2)) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) <= (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) ) or ( pu `1 >= pu `2 & pu `1 <= - (pu `2) ) ) by A51, A53, XREAL_1:64;
A56: gg . I = (Sq_Circ ") . (g . I) by A3, FUNCT_1:12;
then A57: p4 = Sq_Circ . pu by A49, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A58: p4 <> 0. (TOP-REAL 2) by A50, TOPRNS_1:23;
then A59: (Sq_Circ ") . p4 = |[((p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2)))),((p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))))]| by A51, A54, JGRAPH_3:30;
then A60: pu `2 = (p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) by A56, A49, EUCLID:52;
A61: pu `1 = (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) by A56, A49, A59, EUCLID:52;
A62: now__::_thesis:_(_pu_`2_=_0_implies_not_pu_`1_=_0_)
assume ( pu `2 = 0 & pu `1 = 0 ) ; ::_thesis: contradiction
then ( p4 `2 = 0 & p4 `1 = 0 ) by A60, A61, A53, XCMPLX_1:6;
hence contradiction by A58, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) <= (p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) & - (pu `2) <= pu `1 ) or ( pu `1 >= pu `2 & pu `1 <= - (pu `2) ) ) by A56, A49, A59, A60, A53, A55, EUCLID:52, XREAL_1:64;
then A63: Sq_Circ . pu = |[((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))),((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))))]| by A60, A61, A62, JGRAPH_2:3, JGRAPH_3:4;
A64: ((pu `1) / (pu `2)) ^2 >= 0 by XREAL_1:63;
then A65: sqrt (1 + (((pu `1) / (pu `2)) ^2)) > 0 by SQUARE_1:25;
A66: now__::_thesis:_not_pu_`2_=_-_1
assume A67: pu `2 = - 1 ; ::_thesis: contradiction
then - (p4 `1) < 0 by A52, A57, A63, A48, A64, SQUARE_1:25, XREAL_1:141;
then - (- (p4 `1)) > - 0 ;
hence contradiction by A51, A57, A63, A65, A67, EUCLID:52; ::_thesis: verum
end;
|[((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))),((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))))]| `1 = (pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))) by EUCLID:52;
then |.p4.| ^2 = (((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))) ^2) + (((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))) ^2) by A57, A63, A48, JGRAPH_3:1
.= (((pu `2) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) + (((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((pu `2) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) + (((pu `1) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((pu `2) ^2) / (1 + (((pu `1) / (pu `2)) ^2))) + (((pu `1) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) by A64, SQUARE_1:def_2
.= (((pu `2) ^2) / (1 + (((pu `1) / (pu `2)) ^2))) + (((pu `1) ^2) / (1 + (((pu `1) / (pu `2)) ^2))) by A64, SQUARE_1:def_2
.= (((pu `2) ^2) + ((pu `1) ^2)) / (1 + (((pu `1) / (pu `2)) ^2)) by XCMPLX_1:62 ;
then ((((pu `2) ^2) + ((pu `1) ^2)) / (1 + (((pu `1) / (pu `2)) ^2))) * (1 + (((pu `1) / (pu `2)) ^2)) = 1 * (1 + (((pu `1) / (pu `2)) ^2)) by A50;
then ((pu `2) ^2) + ((pu `1) ^2) = 1 + (((pu `1) / (pu `2)) ^2) by A64, XCMPLX_1:87;
then A68: (((pu `2) ^2) + ((pu `1) ^2)) - 1 = ((pu `1) ^2) / ((pu `2) ^2) by XCMPLX_1:76;
pu `2 <> 0 by A60, A61, A53, A62, A55, XREAL_1:64;
then ((((pu `2) ^2) + ((pu `1) ^2)) - 1) * ((pu `2) ^2) = (pu `1) ^2 by A68, XCMPLX_1:6, XCMPLX_1:87;
then A69: (((pu `2) ^2) - 1) * (((pu `2) ^2) + ((pu `1) ^2)) = 0 ;
((pu `2) ^2) + ((pu `1) ^2) <> 0 by A62, COMPLEX1:1;
then ((pu `2) - 1) * ((pu `2) + 1) = 0 by A69, XCMPLX_1:6;
then A70: ( (pu `2) - 1 = 0 or (pu `2) + 1 = 0 ) by XCMPLX_1:6;
consider p3 being Point of (TOP-REAL 2) such that
A71: g . O = p3 and
A72: |.p3.| = 1 and
A73: p3 `2 <= p3 `1 and
A74: p3 `2 <= - (p3 `1) by A1;
A75: p3 <> 0. (TOP-REAL 2) by A72, TOPRNS_1:23;
A76: gg . O = (Sq_Circ ") . (g . O) by A3, FUNCT_1:12;
then A77: p3 = Sq_Circ . pz by A71, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A78: - (p3 `2) >= - (- (p3 `1)) by A74, XREAL_1:24;
then A79: (Sq_Circ ") . p3 = |[((p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A73, A75, JGRAPH_3:30;
then A80: pz `2 = (p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) by A76, A71, EUCLID:52;
((p3 `1) / (p3 `2)) ^2 >= 0 by XREAL_1:63;
then A81: sqrt (1 + (((p3 `1) / (p3 `2)) ^2)) > 0 by SQUARE_1:25;
A82: pz `1 = (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) by A76, A71, A79, EUCLID:52;
A83: now__::_thesis:_(_pz_`2_=_0_implies_not_pz_`1_=_0_)
assume ( pz `2 = 0 & pz `1 = 0 ) ; ::_thesis: contradiction
then ( p3 `2 = 0 & p3 `1 = 0 ) by A80, A82, A81, XCMPLX_1:6;
hence contradiction by A75, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (- (p3 `2)) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) ) ) by A73, A78, A81, XREAL_1:64;
then A84: ( ( p3 `1 <= p3 `2 & (- (p3 `2)) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A80, A82, A81, XREAL_1:64;
then ( ( (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) & - (pz `2) <= pz `1 ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A76, A71, A79, A80, A81, EUCLID:52, XREAL_1:64;
then A85: Sq_Circ . pz = |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| by A80, A82, A83, JGRAPH_2:3, JGRAPH_3:4;
A86: ((pz `1) / (pz `2)) ^2 >= 0 by XREAL_1:63;
then A87: sqrt (1 + (((pz `1) / (pz `2)) ^2)) > 0 by SQUARE_1:25;
A88: now__::_thesis:_not_pz_`2_=_1
assume A89: pz `2 = 1 ; ::_thesis: contradiction
then - (p3 `1) > 0 by A74, A77, A85, A29, A87, XREAL_1:139;
then - (- (p3 `1)) < - 0 ;
hence contradiction by A73, A77, A85, A87, A89, EUCLID:52; ::_thesis: verum
end;
|[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `1 = (pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) by EUCLID:52;
then |.p3.| ^2 = (((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by A77, A85, A29, JGRAPH_3:1
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by A86, SQUARE_1:def_2
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) by A86, SQUARE_1:def_2
.= (((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2)) by XCMPLX_1:62 ;
then ((((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2))) * (1 + (((pz `1) / (pz `2)) ^2)) = 1 * (1 + (((pz `1) / (pz `2)) ^2)) by A72;
then ((pz `2) ^2) + ((pz `1) ^2) = 1 + (((pz `1) / (pz `2)) ^2) by A86, XCMPLX_1:87;
then A90: (((pz `2) ^2) + ((pz `1) ^2)) - 1 = ((pz `1) ^2) / ((pz `2) ^2) by XCMPLX_1:76;
pz `2 <> 0 by A80, A82, A81, A83, A84, XREAL_1:64;
then ((((pz `2) ^2) + ((pz `1) ^2)) - 1) * ((pz `2) ^2) = (pz `1) ^2 by A90, XCMPLX_1:6, XCMPLX_1:87;
then A91: (((pz `2) ^2) - 1) * (((pz `2) ^2) + ((pz `1) ^2)) = 0 ;
((pz `2) ^2) + ((pz `1) ^2) <> 0 by A83, COMPLEX1:1;
then ((pz `2) - 1) * ((pz `2) + 1) = 0 by A91, XCMPLX_1:6;
then A92: ( (pz `2) - 1 = 0 or (pz `2) + 1 = 0 ) by XCMPLX_1:6;
((px `1) ^2) + ((px `2) ^2) <> 0 by A39, COMPLEX1:1;
then ((px `1) - 1) * ((px `1) + 1) = 0 by A47, XCMPLX_1:6;
then ( (px `1) - 1 = 0 or (px `1) + 1 = 0 ) by XCMPLX_1:6;
hence ( (ff . O) `1 = - 1 & (ff . I) `1 = 1 & (gg . O) `2 = - 1 & (gg . I) `2 = 1 ) by A44, A28, A24, A92, A88, A70, A66; ::_thesis: verum
end;
A93: dom f = the carrier of I[01] by FUNCT_2:def_1;
A94: for r being Point of I[01] holds
( ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 ) & ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 ) )
proof
let r be Point of I[01]; ::_thesis: ( ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 ) & ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 ) )
f . r in rng f by A93, FUNCT_1:3;
then f . r in C0 by A1;
then consider p1 being Point of (TOP-REAL 2) such that
A95: f . r = p1 and
A96: |.p1.| >= 1 by A1;
g . r in rng g by A2, FUNCT_1:3;
then g . r in C0 by A1;
then consider p2 being Point of (TOP-REAL 2) such that
A97: g . r = p2 and
A98: |.p2.| >= 1 by A1;
A99: gg . r = (Sq_Circ ") . (g . r) by A3, FUNCT_1:12;
A100: now__::_thesis:_(_(_p2_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p2_<>_0._(TOP-REAL_2)_&_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_)_&_(_-_1_>=_(gg_._r)_`1_or_(gg_._r)_`1_>=_1_or_-_1_>=_(gg_._r)_`2_or_(gg_._r)_`2_>=_1_)_)_or_(_p2_<>_0._(TOP-REAL_2)_&_not_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_)_&_not_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_)_&_(_-_1_>=_(gg_._r)_`1_or_(gg_._r)_`1_>=_1_or_-_1_>=_(gg_._r)_`2_or_(gg_._r)_`2_>=_1_)_)_)
percases ( p2 = 0. (TOP-REAL 2) or ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) or ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ;
case p2 = 0. (TOP-REAL 2) ; ::_thesis: contradiction
hence contradiction by A98, TOPRNS_1:23; ::_thesis: verum
end;
caseA101: ( p2 <> 0. (TOP-REAL 2) & ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ) ; ::_thesis: ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 )
reconsider px = gg . r as Point of (TOP-REAL 2) ;
A102: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A101, JGRAPH_3:28;
then A103: px `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A99, A97, EUCLID:52;
set q = px;
A104: (px `1) ^2 >= 0 by XREAL_1:63;
|.p2.| ^2 >= |.p2.| by A98, XREAL_1:151;
then A105: |.p2.| ^2 >= 1 by A98, XXREAL_0:2;
A106: (px `2) ^2 >= 0 by XREAL_1:63;
((p2 `2) / (p2 `1)) ^2 >= 0 by XREAL_1:63;
then A107: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25;
A108: px `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A99, A97, A102, EUCLID:52;
A109: now__::_thesis:_(_px_`1_=_0_implies_not_px_`2_=_0_)
assume ( px `1 = 0 & px `2 = 0 ) ; ::_thesis: contradiction
then ( p2 `1 = 0 & p2 `2 = 0 ) by A103, A108, A107, XCMPLX_1:6;
hence contradiction by A101, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) ) by A101, A107, XREAL_1:64;
then A110: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A103, A108, A107, XREAL_1:64;
then A111: px `1 <> 0 by A103, A108, A107, A109, XREAL_1:64;
( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A99, A97, A102, A103, A107, A110, EUCLID:52, XREAL_1:64;
then A112: Sq_Circ . px = |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| by A103, A108, A109, JGRAPH_2:3, JGRAPH_3:def_1;
(Sq_Circ ") . p2 = px by A3, A97, FUNCT_1:12;
then A113: p2 = Sq_Circ . px by FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A114: ((px `2) / (px `1)) ^2 >= 0 by XREAL_1:63;
( |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `1 = (px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2))) & |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `2 = (px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))) ) by EUCLID:52;
then |.p2.| ^2 = (((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by A113, A112, JGRAPH_3:1
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by A114, SQUARE_1:def_2
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / (1 + (((px `2) / (px `1)) ^2))) by A114, SQUARE_1:def_2
.= (((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2)) by XCMPLX_1:62 ;
then ((((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2))) * (1 + (((px `2) / (px `1)) ^2)) >= 1 * (1 + (((px `2) / (px `1)) ^2)) by A114, A105, XREAL_1:64;
then ((px `1) ^2) + ((px `2) ^2) >= 1 + (((px `2) / (px `1)) ^2) by A114, XCMPLX_1:87;
then ((px `1) ^2) + ((px `2) ^2) >= 1 + (((px `2) ^2) / ((px `1) ^2)) by XCMPLX_1:76;
then (((px `1) ^2) + ((px `2) ^2)) - 1 >= (1 + (((px `2) ^2) / ((px `1) ^2))) - 1 by XREAL_1:9;
then ((((px `1) ^2) + ((px `2) ^2)) - 1) * ((px `1) ^2) >= (((px `2) ^2) / ((px `1) ^2)) * ((px `1) ^2) by A104, XREAL_1:64;
then (((px `1) ^2) + (((px `2) ^2) - 1)) * ((px `1) ^2) >= (px `2) ^2 by A111, XCMPLX_1:6, XCMPLX_1:87;
then ((((px `1) ^2) * ((px `1) ^2)) + (((px `1) ^2) * (((px `2) ^2) - 1))) - ((px `2) ^2) >= ((px `2) ^2) - ((px `2) ^2) by XREAL_1:9;
then A115: (((px `1) ^2) - 1) * (((px `1) ^2) + ((px `2) ^2)) >= 0 ;
((px `1) ^2) + ((px `2) ^2) <> 0 by A109, COMPLEX1:1;
then ((px `1) - 1) * ((px `1) + 1) >= 0 by A104, A115, A106, XREAL_1:132;
hence ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 ) by XREAL_1:95; ::_thesis: verum
end;
caseA116: ( p2 <> 0. (TOP-REAL 2) & not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) ; ::_thesis: ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 )
reconsider pz = gg . r as Point of (TOP-REAL 2) ;
A117: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))))]| by A116, JGRAPH_3:28;
then A118: pz `2 = (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A99, A97, EUCLID:52;
((p2 `1) / (p2 `2)) ^2 >= 0 by XREAL_1:63;
then A119: sqrt (1 + (((p2 `1) / (p2 `2)) ^2)) > 0 by SQUARE_1:25;
A120: now__::_thesis:_(_pz_`2_=_0_implies_not_pz_`1_=_0_)
assume that
A121: pz `2 = 0 and
pz `1 = 0 ; ::_thesis: contradiction
p2 `2 = 0 by A118, A119, A121, XCMPLX_1:6;
hence contradiction by A116; ::_thesis: verum
end;
A122: pz `1 = (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) by A99, A97, A117, EUCLID:52;
( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A116, JGRAPH_2:13;
then ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) ) by A119, XREAL_1:64;
then A123: ( ( p2 `1 <= p2 `2 & (- (p2 `2)) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A118, A122, A119, XREAL_1:64;
then A124: pz `2 <> 0 by A118, A122, A119, A120, XREAL_1:64;
( ( (p2 `1) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `1) / (p2 `2)) ^2))) & - (pz `2) <= pz `1 ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A99, A97, A117, A118, A119, A123, EUCLID:52, XREAL_1:64;
then A125: Sq_Circ . pz = |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| by A118, A122, A120, JGRAPH_2:3, JGRAPH_3:4;
A126: ((pz `1) / (pz `2)) ^2 >= 0 by XREAL_1:63;
|.p2.| ^2 >= |.p2.| by A98, XREAL_1:151;
then A127: |.p2.| ^2 >= 1 by A98, XXREAL_0:2;
A128: |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `1 = (pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) by EUCLID:52;
A129: (pz `1) ^2 >= 0 by XREAL_1:63;
A130: (pz `2) ^2 >= 0 by XREAL_1:63;
( p2 = Sq_Circ . pz & |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `2 = (pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) ) by A99, A97, EUCLID:52, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
then |.p2.| ^2 = (((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by A125, A128, JGRAPH_3:1
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by A126, SQUARE_1:def_2
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) by A126, SQUARE_1:def_2
.= (((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2)) by XCMPLX_1:62 ;
then ((((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2))) * (1 + (((pz `1) / (pz `2)) ^2)) >= 1 * (1 + (((pz `1) / (pz `2)) ^2)) by A126, A127, XREAL_1:64;
then ((pz `2) ^2) + ((pz `1) ^2) >= 1 + (((pz `1) / (pz `2)) ^2) by A126, XCMPLX_1:87;
then ((pz `2) ^2) + ((pz `1) ^2) >= 1 + (((pz `1) ^2) / ((pz `2) ^2)) by XCMPLX_1:76;
then (((pz `2) ^2) + ((pz `1) ^2)) - 1 >= (1 + (((pz `1) ^2) / ((pz `2) ^2))) - 1 by XREAL_1:9;
then ((((pz `2) ^2) + ((pz `1) ^2)) - 1) * ((pz `2) ^2) >= (((pz `1) ^2) / ((pz `2) ^2)) * ((pz `2) ^2) by A130, XREAL_1:64;
then (((pz `2) ^2) + (((pz `1) ^2) - 1)) * ((pz `2) ^2) >= (pz `1) ^2 by A124, XCMPLX_1:6, XCMPLX_1:87;
then ((((pz `2) ^2) * ((pz `2) ^2)) + (((pz `2) ^2) * (((pz `1) ^2) - 1))) - ((pz `1) ^2) >= ((pz `1) ^2) - ((pz `1) ^2) by XREAL_1:9;
then A131: (((pz `2) ^2) - 1) * (((pz `2) ^2) + ((pz `1) ^2)) >= 0 ;
((pz `2) ^2) + ((pz `1) ^2) <> 0 by A120, COMPLEX1:1;
then ((pz `2) - 1) * ((pz `2) + 1) >= 0 by A130, A131, A129, XREAL_1:132;
hence ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 ) by XREAL_1:95; ::_thesis: verum
end;
end;
end;
A132: ff . r = (Sq_Circ ") . (f . r) by A4, FUNCT_1:12;
now__::_thesis:_(_(_p1_=_0._(TOP-REAL_2)_&_contradiction_)_or_(_p1_<>_0._(TOP-REAL_2)_&_(_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_)_or_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_)_)_&_(_-_1_>=_(ff_._r)_`1_or_(ff_._r)_`1_>=_1_or_-_1_>=_(ff_._r)_`2_or_(ff_._r)_`2_>=_1_)_)_or_(_p1_<>_0._(TOP-REAL_2)_&_not_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_)_&_not_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_)_&_(_-_1_>=_(ff_._r)_`1_or_(ff_._r)_`1_>=_1_or_-_1_>=_(ff_._r)_`2_or_(ff_._r)_`2_>=_1_)_)_)
percases ( p1 = 0. (TOP-REAL 2) or ( p1 <> 0. (TOP-REAL 2) & ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) or ( p1 <> 0. (TOP-REAL 2) & not ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) & not ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) ;
case p1 = 0. (TOP-REAL 2) ; ::_thesis: contradiction
hence contradiction by A96, TOPRNS_1:23; ::_thesis: verum
end;
caseA133: ( p1 <> 0. (TOP-REAL 2) & ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ) ; ::_thesis: ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 )
reconsider px = ff . r as Point of (TOP-REAL 2) ;
A134: (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A133, JGRAPH_3:28;
then A135: px `1 = (p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A132, A95, EUCLID:52;
((p1 `2) / (p1 `1)) ^2 >= 0 by XREAL_1:63;
then A136: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) > 0 by SQUARE_1:25;
A137: px `2 = (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) by A132, A95, A134, EUCLID:52;
A138: now__::_thesis:_(_px_`1_=_0_implies_not_px_`2_=_0_)
assume ( px `1 = 0 & px `2 = 0 ) ; ::_thesis: contradiction
then ( p1 `1 = 0 & p1 `2 = 0 ) by A135, A137, A136, XCMPLX_1:6;
hence contradiction by A133, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) ) by A133, A136, XREAL_1:64;
then A139: ( ( p1 `2 <= p1 `1 & (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A135, A137, A136, XREAL_1:64;
then A140: px `1 <> 0 by A135, A137, A136, A138, XREAL_1:64;
|.p1.| ^2 >= |.p1.| by A96, XREAL_1:151;
then A141: |.p1.| ^2 >= 1 by A96, XXREAL_0:2;
A142: (px `1) ^2 >= 0 by XREAL_1:63;
set q = px;
A143: |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `2 = (px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))) by EUCLID:52;
A144: (px `2) ^2 >= 0 by XREAL_1:63;
A145: ((px `2) / (px `1)) ^2 >= 0 by XREAL_1:63;
( ( (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A132, A95, A134, A135, A136, A139, EUCLID:52, XREAL_1:64;
then A146: Sq_Circ . px = |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| by A135, A137, A138, JGRAPH_2:3, JGRAPH_3:def_1;
( p1 = Sq_Circ . px & |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `1 = (px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2))) ) by A132, A95, EUCLID:52, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
then |.p1.| ^2 = (((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by A146, A143, JGRAPH_3:1
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by A145, SQUARE_1:def_2
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / (1 + (((px `2) / (px `1)) ^2))) by A145, SQUARE_1:def_2
.= (((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2)) by XCMPLX_1:62 ;
then ((((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2))) * (1 + (((px `2) / (px `1)) ^2)) >= 1 * (1 + (((px `2) / (px `1)) ^2)) by A145, A141, XREAL_1:64;
then ((px `1) ^2) + ((px `2) ^2) >= 1 + (((px `2) / (px `1)) ^2) by A145, XCMPLX_1:87;
then ((px `1) ^2) + ((px `2) ^2) >= 1 + (((px `2) ^2) / ((px `1) ^2)) by XCMPLX_1:76;
then (((px `1) ^2) + ((px `2) ^2)) - 1 >= (1 + (((px `2) ^2) / ((px `1) ^2))) - 1 by XREAL_1:9;
then ((((px `1) ^2) + ((px `2) ^2)) - 1) * ((px `1) ^2) >= (((px `2) ^2) / ((px `1) ^2)) * ((px `1) ^2) by A142, XREAL_1:64;
then (((px `1) ^2) + (((px `2) ^2) - 1)) * ((px `1) ^2) >= (px `2) ^2 by A140, XCMPLX_1:6, XCMPLX_1:87;
then ((((px `1) ^2) * ((px `1) ^2)) + (((px `1) ^2) * (((px `2) ^2) - 1))) - ((px `2) ^2) >= ((px `2) ^2) - ((px `2) ^2) by XREAL_1:9;
then A147: (((px `1) ^2) - 1) * (((px `1) ^2) + ((px `2) ^2)) >= 0 ;
((px `1) ^2) + ((px `2) ^2) <> 0 by A138, COMPLEX1:1;
then ((px `1) - 1) * ((px `1) + 1) >= 0 by A142, A147, A144, XREAL_1:132;
hence ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 ) by XREAL_1:95; ::_thesis: verum
end;
caseA148: ( p1 <> 0. (TOP-REAL 2) & not ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) & not ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) ; ::_thesis: ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 )
reconsider pz = ff . r as Point of (TOP-REAL 2) ;
A149: (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))))]| by A148, JGRAPH_3:28;
then A150: pz `2 = (p1 `2) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A132, A95, EUCLID:52;
((p1 `1) / (p1 `2)) ^2 >= 0 by XREAL_1:63;
then A151: sqrt (1 + (((p1 `1) / (p1 `2)) ^2)) > 0 by SQUARE_1:25;
A152: now__::_thesis:_(_pz_`2_=_0_implies_not_pz_`1_=_0_)
assume that
A153: pz `2 = 0 and
pz `1 = 0 ; ::_thesis: contradiction
p1 `2 = 0 by A150, A151, A153, XCMPLX_1:6;
hence contradiction by A148; ::_thesis: verum
end;
A154: pz `1 = (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) by A132, A95, A149, EUCLID:52;
( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A148, JGRAPH_2:13;
then ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) <= (- (p1 `2)) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ) ) by A151, XREAL_1:64;
then A155: ( ( p1 `1 <= p1 `2 & (- (p1 `2)) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) <= (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A150, A154, A151, XREAL_1:64;
then A156: pz `2 <> 0 by A150, A154, A151, A152, XREAL_1:64;
( ( (p1 `1) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `1) / (p1 `2)) ^2))) & - (pz `2) <= pz `1 ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A132, A95, A149, A150, A151, A155, EUCLID:52, XREAL_1:64;
then A157: Sq_Circ . pz = |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| by A150, A154, A152, JGRAPH_2:3, JGRAPH_3:4;
A158: ((pz `1) / (pz `2)) ^2 >= 0 by XREAL_1:63;
|.p1.| ^2 >= |.p1.| by A96, XREAL_1:151;
then A159: |.p1.| ^2 >= 1 by A96, XXREAL_0:2;
A160: |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `1 = (pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) by EUCLID:52;
A161: (pz `1) ^2 >= 0 by XREAL_1:63;
A162: (pz `2) ^2 >= 0 by XREAL_1:63;
( p1 = Sq_Circ . pz & |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `2 = (pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) ) by A132, A95, EUCLID:52, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
then |.p1.| ^2 = (((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by A157, A160, JGRAPH_3:1
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by A158, SQUARE_1:def_2
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) by A158, SQUARE_1:def_2
.= (((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2)) by XCMPLX_1:62 ;
then ((((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2))) * (1 + (((pz `1) / (pz `2)) ^2)) >= 1 * (1 + (((pz `1) / (pz `2)) ^2)) by A158, A159, XREAL_1:64;
then ((pz `2) ^2) + ((pz `1) ^2) >= 1 + (((pz `1) / (pz `2)) ^2) by A158, XCMPLX_1:87;
then ((pz `2) ^2) + ((pz `1) ^2) >= 1 + (((pz `1) ^2) / ((pz `2) ^2)) by XCMPLX_1:76;
then (((pz `2) ^2) + ((pz `1) ^2)) - 1 >= (1 + (((pz `1) ^2) / ((pz `2) ^2))) - 1 by XREAL_1:9;
then ((((pz `2) ^2) + ((pz `1) ^2)) - 1) * ((pz `2) ^2) >= (((pz `1) ^2) / ((pz `2) ^2)) * ((pz `2) ^2) by A162, XREAL_1:64;
then (((pz `2) ^2) + (((pz `1) ^2) - 1)) * ((pz `2) ^2) >= (pz `1) ^2 by A156, XCMPLX_1:6, XCMPLX_1:87;
then ((((pz `2) ^2) * ((pz `2) ^2)) + (((pz `2) ^2) * (((pz `1) ^2) - 1))) - ((pz `1) ^2) >= ((pz `1) ^2) - ((pz `1) ^2) by XREAL_1:9;
then A163: (((pz `2) ^2) - 1) * (((pz `2) ^2) + ((pz `1) ^2)) >= 0 ;
((pz `2) ^2) + ((pz `1) ^2) <> 0 by A152, COMPLEX1:1;
then ((pz `2) - 1) * ((pz `2) + 1) >= 0 by A162, A163, A161, XREAL_1:132;
hence ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 ) by XREAL_1:95; ::_thesis: verum
end;
end;
end;
hence ( ( - 1 >= (ff . r) `1 or (ff . r) `1 >= 1 or - 1 >= (ff . r) `2 or (ff . r) `2 >= 1 ) & ( - 1 >= (gg . r) `1 or (gg . r) `1 >= 1 or - 1 >= (gg . r) `2 or (gg . r) `2 >= 1 ) ) by A100; ::_thesis: verum
end;
( - 1 <= (ff . O) `2 & (ff . O) `2 <= 1 & - 1 <= (ff . I) `2 & (ff . I) `2 <= 1 & - 1 <= (gg . O) `1 & (gg . O) `1 <= 1 & - 1 <= (gg . I) `1 & (gg . I) `1 <= 1 )
proof
reconsider pz = gg . O as Point of (TOP-REAL 2) ;
reconsider py = ff . I as Point of (TOP-REAL 2) ;
reconsider px = ff . O as Point of (TOP-REAL 2) ;
set q = px;
A164: ( |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `1 = (px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2))) & |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| `2 = (px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))) ) by EUCLID:52;
A165: ((px `2) / (px `1)) ^2 >= 0 by XREAL_1:63;
consider p1 being Point of (TOP-REAL 2) such that
A166: f . O = p1 and
A167: |.p1.| = 1 and
A168: ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) by A1;
A169: ff . O = (Sq_Circ ") . (f . O) by A4, FUNCT_1:12;
then A170: p1 = Sq_Circ . px by A166, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
((p1 `2) / (p1 `1)) ^2 >= 0 by XREAL_1:63;
then A171: sqrt (1 + (((p1 `2) / (p1 `1)) ^2)) > 0 by SQUARE_1:25;
A172: p1 <> 0. (TOP-REAL 2) by A167, TOPRNS_1:23;
then (Sq_Circ ") . p1 = |[((p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2)))),((p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))))]| by A168, JGRAPH_3:28;
then A173: ( px `1 = (p1 `1) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) & px `2 = (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) by A169, A166, EUCLID:52;
A174: now__::_thesis:_(_px_`1_=_0_implies_not_px_`2_=_0_)
assume ( px `1 = 0 & px `2 = 0 ) ; ::_thesis: contradiction
then ( p1 `1 = 0 & p1 `2 = 0 ) by A173, A171, XCMPLX_1:6;
hence contradiction by A172, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) ) by A168, A171, XREAL_1:64;
then A175: ( ( p1 `2 <= p1 `1 & (- (p1 `1)) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) <= (p1 `2) * (sqrt (1 + (((p1 `2) / (p1 `1)) ^2))) ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A173, A171, XREAL_1:64;
then ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) by A173, A171, XREAL_1:64;
then Sq_Circ . px = |[((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))),((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2))))]| by A174, JGRAPH_2:3, JGRAPH_3:def_1;
then |.p1.| ^2 = (((px `1) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by A170, A164, JGRAPH_3:1
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) / (sqrt (1 + (((px `2) / (px `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((px `1) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / ((sqrt (1 + (((px `2) / (px `1)) ^2))) ^2)) by A165, SQUARE_1:def_2
.= (((px `1) ^2) / (1 + (((px `2) / (px `1)) ^2))) + (((px `2) ^2) / (1 + (((px `2) / (px `1)) ^2))) by A165, SQUARE_1:def_2
.= (((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2)) by XCMPLX_1:62 ;
then ((((px `1) ^2) + ((px `2) ^2)) / (1 + (((px `2) / (px `1)) ^2))) * (1 + (((px `2) / (px `1)) ^2)) = 1 * (1 + (((px `2) / (px `1)) ^2)) by A167;
then ((px `1) ^2) + ((px `2) ^2) = 1 + (((px `2) / (px `1)) ^2) by A165, XCMPLX_1:87;
then A176: (((px `1) ^2) + ((px `2) ^2)) - 1 = ((px `2) ^2) / ((px `1) ^2) by XCMPLX_1:76;
px `1 <> 0 by A173, A171, A174, A175, XREAL_1:64;
then ((((px `1) ^2) + ((px `2) ^2)) - 1) * ((px `1) ^2) = (px `2) ^2 by A176, XCMPLX_1:6, XCMPLX_1:87;
then A177: (((px `1) ^2) - 1) * (((px `1) ^2) + ((px `2) ^2)) = 0 ;
((px `1) ^2) + ((px `2) ^2) <> 0 by A174, COMPLEX1:1;
then ((px `1) - 1) * ((px `1) + 1) = 0 by A177, XCMPLX_1:6;
then ( (px `1) - 1 = 0 or (px `1) + 1 = 0 ) by XCMPLX_1:6;
then ( px `1 = 1 or px `1 = 0 - 1 ) ;
hence ( - 1 <= (ff . O) `2 & (ff . O) `2 <= 1 ) by A173, A171, A175, XREAL_1:64; ::_thesis: ( - 1 <= (ff . I) `2 & (ff . I) `2 <= 1 & - 1 <= (gg . O) `1 & (gg . O) `1 <= 1 & - 1 <= (gg . I) `1 & (gg . I) `1 <= 1 )
A178: ((py `2) / (py `1)) ^2 >= 0 by XREAL_1:63;
reconsider pu = gg . I as Point of (TOP-REAL 2) ;
A179: ( |[((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `1 = (py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2))) & |[((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))))]| `2 = (py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))) ) by EUCLID:52;
A180: ((pz `1) / (pz `2)) ^2 >= 0 by XREAL_1:63;
consider p2 being Point of (TOP-REAL 2) such that
A181: f . I = p2 and
A182: |.p2.| = 1 and
A183: ( p2 `2 <= p2 `1 & p2 `2 >= - (p2 `1) ) by A1;
A184: ff . I = (Sq_Circ ") . (f . I) by A4, FUNCT_1:12;
then A185: p2 = Sq_Circ . py by A181, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A186: p2 <> 0. (TOP-REAL 2) by A182, TOPRNS_1:23;
then A187: (Sq_Circ ") . p2 = |[((p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2)))),((p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))))]| by A183, JGRAPH_3:28;
then A188: py `1 = (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A184, A181, EUCLID:52;
A189: py `2 = (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) by A184, A181, A187, EUCLID:52;
((p2 `2) / (p2 `1)) ^2 >= 0 by XREAL_1:63;
then A190: sqrt (1 + (((p2 `2) / (p2 `1)) ^2)) > 0 by SQUARE_1:25;
A191: now__::_thesis:_(_py_`1_=_0_implies_not_py_`2_=_0_)
assume ( py `1 = 0 & py `2 = 0 ) ; ::_thesis: contradiction
then ( p2 `1 = 0 & p2 `2 = 0 ) by A188, A189, A190, XCMPLX_1:6;
hence contradiction by A186, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A192: ( ( p2 `2 <= p2 `1 & (- (p2 `1)) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) ) or ( py `2 >= py `1 & py `2 <= - (py `1) ) ) by A183, A190, XREAL_1:64;
then A193: ( ( (p2 `2) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) <= (p2 `1) * (sqrt (1 + (((p2 `2) / (p2 `1)) ^2))) & - (py `1) <= py `2 ) or ( py `2 >= py `1 & py `2 <= - (py `1) ) ) by A184, A181, A187, A188, A190, EUCLID:52, XREAL_1:64;
then Sq_Circ . py = |[((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))),((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2))))]| by A188, A189, A191, JGRAPH_2:3, JGRAPH_3:def_1;
then |.p2.| ^2 = (((py `1) / (sqrt (1 + (((py `2) / (py `1)) ^2)))) ^2) + (((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2)))) ^2) by A185, A179, JGRAPH_3:1
.= (((py `1) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) + (((py `2) / (sqrt (1 + (((py `2) / (py `1)) ^2)))) ^2) by XCMPLX_1:76
.= (((py `1) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) + (((py `2) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) by XCMPLX_1:76
.= (((py `1) ^2) / (1 + (((py `2) / (py `1)) ^2))) + (((py `2) ^2) / ((sqrt (1 + (((py `2) / (py `1)) ^2))) ^2)) by A178, SQUARE_1:def_2
.= (((py `1) ^2) / (1 + (((py `2) / (py `1)) ^2))) + (((py `2) ^2) / (1 + (((py `2) / (py `1)) ^2))) by A178, SQUARE_1:def_2
.= (((py `1) ^2) + ((py `2) ^2)) / (1 + (((py `2) / (py `1)) ^2)) by XCMPLX_1:62 ;
then ((((py `1) ^2) + ((py `2) ^2)) / (1 + (((py `2) / (py `1)) ^2))) * (1 + (((py `2) / (py `1)) ^2)) = 1 * (1 + (((py `2) / (py `1)) ^2)) by A182;
then ((py `1) ^2) + ((py `2) ^2) = 1 + (((py `2) / (py `1)) ^2) by A178, XCMPLX_1:87;
then A194: (((py `1) ^2) + ((py `2) ^2)) - 1 = ((py `2) ^2) / ((py `1) ^2) by XCMPLX_1:76;
py `1 <> 0 by A188, A189, A190, A191, A192, XREAL_1:64;
then ((((py `1) ^2) + ((py `2) ^2)) - 1) * ((py `1) ^2) = (py `2) ^2 by A194, XCMPLX_1:6, XCMPLX_1:87;
then A195: (((py `1) ^2) - 1) * (((py `1) ^2) + ((py `2) ^2)) = 0 ;
((py `1) ^2) + ((py `2) ^2) <> 0 by A191, COMPLEX1:1;
then ((py `1) - 1) * ((py `1) + 1) = 0 by A195, XCMPLX_1:6;
then ( (py `1) - 1 = 0 or (py `1) + 1 = 0 ) by XCMPLX_1:6;
hence ( - 1 <= (ff . I) `2 & (ff . I) `2 <= 1 ) by A188, A189, A193; ::_thesis: ( - 1 <= (gg . O) `1 & (gg . O) `1 <= 1 & - 1 <= (gg . I) `1 & (gg . I) `1 <= 1 )
A196: ( |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `2 = (pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) & |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| `1 = (pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))) ) by EUCLID:52;
consider p3 being Point of (TOP-REAL 2) such that
A197: g . O = p3 and
A198: |.p3.| = 1 and
A199: p3 `2 <= p3 `1 and
A200: p3 `2 <= - (p3 `1) by A1;
A201: p3 <> 0. (TOP-REAL 2) by A198, TOPRNS_1:23;
A202: gg . O = (Sq_Circ ") . (g . O) by A3, FUNCT_1:12;
then A203: p3 = Sq_Circ . pz by A197, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A204: - (p3 `2) >= - (- (p3 `1)) by A200, XREAL_1:24;
then A205: (Sq_Circ ") . p3 = |[((p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2)))),((p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))))]| by A199, A201, JGRAPH_3:30;
then A206: pz `2 = (p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) by A202, A197, EUCLID:52;
A207: pz `1 = (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) by A202, A197, A205, EUCLID:52;
((p3 `1) / (p3 `2)) ^2 >= 0 by XREAL_1:63;
then A208: sqrt (1 + (((p3 `1) / (p3 `2)) ^2)) > 0 by SQUARE_1:25;
A209: now__::_thesis:_(_pz_`2_=_0_implies_not_pz_`1_=_0_)
assume ( pz `2 = 0 & pz `1 = 0 ) ; ::_thesis: contradiction
then ( p3 `2 = 0 & p3 `1 = 0 ) by A206, A207, A208, XCMPLX_1:6;
hence contradiction by A201, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (- (p3 `2)) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) ) ) by A199, A204, A208, XREAL_1:64;
then A210: ( ( p3 `1 <= p3 `2 & (- (p3 `2)) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A206, A207, A208, XREAL_1:64;
then A211: ( ( (p3 `1) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) <= (p3 `2) * (sqrt (1 + (((p3 `1) / (p3 `2)) ^2))) & - (pz `2) <= pz `1 ) or ( pz `1 >= pz `2 & pz `1 <= - (pz `2) ) ) by A202, A197, A205, A206, A208, EUCLID:52, XREAL_1:64;
then Sq_Circ . pz = |[((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))),((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2))))]| by A206, A207, A209, JGRAPH_2:3, JGRAPH_3:4;
then |.p3.| ^2 = (((pz `2) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by A203, A196, JGRAPH_3:1
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) / (sqrt (1 + (((pz `1) / (pz `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((pz `2) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / ((sqrt (1 + (((pz `1) / (pz `2)) ^2))) ^2)) by A180, SQUARE_1:def_2
.= (((pz `2) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) + (((pz `1) ^2) / (1 + (((pz `1) / (pz `2)) ^2))) by A180, SQUARE_1:def_2
.= (((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2)) by XCMPLX_1:62 ;
then ((((pz `2) ^2) + ((pz `1) ^2)) / (1 + (((pz `1) / (pz `2)) ^2))) * (1 + (((pz `1) / (pz `2)) ^2)) = 1 * (1 + (((pz `1) / (pz `2)) ^2)) by A198;
then ((pz `2) ^2) + ((pz `1) ^2) = 1 + (((pz `1) / (pz `2)) ^2) by A180, XCMPLX_1:87;
then A212: (((pz `2) ^2) + ((pz `1) ^2)) - 1 = ((pz `1) ^2) / ((pz `2) ^2) by XCMPLX_1:76;
pz `2 <> 0 by A206, A207, A208, A209, A210, XREAL_1:64;
then ((((pz `2) ^2) + ((pz `1) ^2)) - 1) * ((pz `2) ^2) = (pz `1) ^2 by A212, XCMPLX_1:6, XCMPLX_1:87;
then A213: (((pz `2) ^2) - 1) * (((pz `2) ^2) + ((pz `1) ^2)) = 0 ;
((pz `2) ^2) + ((pz `1) ^2) <> 0 by A209, COMPLEX1:1;
then ((pz `2) - 1) * ((pz `2) + 1) = 0 by A213, XCMPLX_1:6;
then ( (pz `2) - 1 = 0 or (pz `2) + 1 = 0 ) by XCMPLX_1:6;
hence ( - 1 <= (gg . O) `1 & (gg . O) `1 <= 1 ) by A206, A207, A211; ::_thesis: ( - 1 <= (gg . I) `1 & (gg . I) `1 <= 1 )
A214: ( |[((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))),((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))))]| `2 = (pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))) & |[((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))),((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))))]| `1 = (pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))) ) by EUCLID:52;
A215: ((pu `1) / (pu `2)) ^2 >= 0 by XREAL_1:63;
consider p4 being Point of (TOP-REAL 2) such that
A216: g . I = p4 and
A217: |.p4.| = 1 and
A218: p4 `2 >= p4 `1 and
A219: p4 `2 >= - (p4 `1) by A1;
A220: - (p4 `2) <= - (- (p4 `1)) by A219, XREAL_1:24;
A221: gg . I = (Sq_Circ ") . (g . I) by A3, FUNCT_1:12;
then A222: p4 = Sq_Circ . pu by A216, FUNCT_1:32, JGRAPH_3:22, JGRAPH_3:43;
A223: p4 <> 0. (TOP-REAL 2) by A217, TOPRNS_1:23;
then A224: (Sq_Circ ") . p4 = |[((p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2)))),((p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))))]| by A218, A220, JGRAPH_3:30;
then A225: pu `2 = (p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) by A221, A216, EUCLID:52;
A226: pu `1 = (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) by A221, A216, A224, EUCLID:52;
((p4 `1) / (p4 `2)) ^2 >= 0 by XREAL_1:63;
then A227: sqrt (1 + (((p4 `1) / (p4 `2)) ^2)) > 0 by SQUARE_1:25;
A228: now__::_thesis:_(_pu_`2_=_0_implies_not_pu_`1_=_0_)
assume ( pu `2 = 0 & pu `1 = 0 ) ; ::_thesis: contradiction
then ( p4 `2 = 0 & p4 `1 = 0 ) by A225, A226, A227, XCMPLX_1:6;
hence contradiction by A223, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A229: ( ( p4 `1 <= p4 `2 & (- (p4 `2)) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) <= (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) ) or ( pu `1 >= pu `2 & pu `1 <= - (pu `2) ) ) by A218, A220, A227, XREAL_1:64;
then A230: ( ( (p4 `1) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) <= (p4 `2) * (sqrt (1 + (((p4 `1) / (p4 `2)) ^2))) & - (pu `2) <= pu `1 ) or ( pu `1 >= pu `2 & pu `1 <= - (pu `2) ) ) by A221, A216, A224, A225, A227, EUCLID:52, XREAL_1:64;
then Sq_Circ . pu = |[((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))),((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2))))]| by A225, A226, A228, JGRAPH_2:3, JGRAPH_3:4;
then |.p4.| ^2 = (((pu `2) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))) ^2) + (((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))) ^2) by A222, A214, JGRAPH_3:1
.= (((pu `2) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) + (((pu `1) / (sqrt (1 + (((pu `1) / (pu `2)) ^2)))) ^2) by XCMPLX_1:76
.= (((pu `2) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) + (((pu `1) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) by XCMPLX_1:76
.= (((pu `2) ^2) / (1 + (((pu `1) / (pu `2)) ^2))) + (((pu `1) ^2) / ((sqrt (1 + (((pu `1) / (pu `2)) ^2))) ^2)) by A215, SQUARE_1:def_2
.= (((pu `2) ^2) / (1 + (((pu `1) / (pu `2)) ^2))) + (((pu `1) ^2) / (1 + (((pu `1) / (pu `2)) ^2))) by A215, SQUARE_1:def_2
.= (((pu `2) ^2) + ((pu `1) ^2)) / (1 + (((pu `1) / (pu `2)) ^2)) by XCMPLX_1:62 ;
then ((((pu `2) ^2) + ((pu `1) ^2)) / (1 + (((pu `1) / (pu `2)) ^2))) * (1 + (((pu `1) / (pu `2)) ^2)) = 1 * (1 + (((pu `1) / (pu `2)) ^2)) by A217;
then ((pu `2) ^2) + ((pu `1) ^2) = 1 + (((pu `1) / (pu `2)) ^2) by A215, XCMPLX_1:87;
then A231: (((pu `2) ^2) + ((pu `1) ^2)) - 1 = ((pu `1) ^2) / ((pu `2) ^2) by XCMPLX_1:76;
pu `2 <> 0 by A225, A226, A227, A228, A229, XREAL_1:64;
then ((((pu `2) ^2) + ((pu `1) ^2)) - 1) * ((pu `2) ^2) = (pu `1) ^2 by A231, XCMPLX_1:6, XCMPLX_1:87;
then A232: (((pu `2) ^2) - 1) * (((pu `2) ^2) + ((pu `1) ^2)) = 0 ;
((pu `2) ^2) + ((pu `1) ^2) <> 0 by A228, COMPLEX1:1;
then ((pu `2) - 1) * ((pu `2) + 1) = 0 by A232, XCMPLX_1:6;
then ( (pu `2) - 1 = 0 or (pu `2) + 1 = 0 ) by XCMPLX_1:6;
hence ( - 1 <= (gg . I) `1 & (gg . I) `1 <= 1 ) by A225, A226, A230; ::_thesis: verum
end;
then rng ff meets rng gg by A1, A5, A94, Th11, JGRAPH_3:22, JGRAPH_3:42;
then consider y being set such that
A233: y in rng ff and
A234: y in rng gg by XBOOLE_0:3;
consider x1 being set such that
A235: x1 in dom ff and
A236: y = ff . x1 by A233, FUNCT_1:def_3;
consider x2 being set such that
A237: x2 in dom gg and
A238: y = gg . x2 by A234, FUNCT_1:def_3;
A239: ( dom (Sq_Circ ") = the carrier of (TOP-REAL 2) & gg . x2 = (Sq_Circ ") . (g . x2) ) by A237, FUNCT_1:12, FUNCT_2:def_1, JGRAPH_3:29;
x1 in dom f by A235, FUNCT_1:11;
then A240: f . x1 in rng f by FUNCT_1:def_3;
x2 in dom g by A237, FUNCT_1:11;
then A241: g . x2 in rng g by FUNCT_1:def_3;
ff . x1 = (Sq_Circ ") . (f . x1) by A235, FUNCT_1:12;
then f . x1 = g . x2 by A236, A238, A240, A241, A239, FUNCT_1:def_4, JGRAPH_3:22;
hence rng f meets rng g by A240, A241, XBOOLE_0:3; ::_thesis: verum
end;
theorem Th15: :: JGRAPH_5:15
for f, g being Function of I[01],(TOP-REAL 2)
for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0, KXP, KXN, KYP, KYN being Subset of (TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0, KXP, KXN, KYP, KYN be Subset of (TOP-REAL 2); ::_thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let O, I be Point of I[01]; ::_thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } & KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } & KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } & KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } & f . O in KXN & f . I in KXP & g . O in KYP & g . I in KYN & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
then ex g2 being Function of I[01],(TOP-REAL 2) st
( g2 . 0 = g . 1 & g2 . 1 = g . 0 & rng g2 = rng g & g2 is continuous & g2 is one-to-one ) by Th12;
hence rng f meets rng g by A1, Th14; ::_thesis: verum
end;
theorem Th16: :: JGRAPH_5:16
for f, g being Function of I[01],(TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st C0 = { q where q is Point of (TOP-REAL 2) : |.q.| >= 1 } & f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = |[(- 1),0]| & f . 1 = |[1,0]| & g . 1 = |[0,1]| & g . 0 = |[0,(- 1)]| & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
reconsider I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
reconsider O = 0 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) );
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st C0 = { q where q is Point of (TOP-REAL 2) : |.q.| >= 1 } & f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = |[(- 1),0]| & f . 1 = |[1,0]| & g . 1 = |[0,1]| & g . 0 = |[0,(- 1)]| & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0 be Subset of (TOP-REAL 2); ::_thesis: ( C0 = { q where q is Point of (TOP-REAL 2) : |.q.| >= 1 } & f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = |[(- 1),0]| & f . 1 = |[1,0]| & g . 1 = |[0,1]| & g . 0 = |[0,(- 1)]| & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A1: ( C0 = { q where q is Point of (TOP-REAL 2) : |.q.| >= 1 } & f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = |[(- 1),0]| & f . 1 = |[1,0]| & g . 1 = |[0,1]| & g . 0 = |[0,(- 1)]| & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
{ q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ;
A2: |[0,1]| `1 = 0 by EUCLID:52;
defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) );
{ q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) );
{ q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) );
{ q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ;
A3: |[0,(- 1)]| `1 = 0 by EUCLID:52;
|[0,(- 1)]| `2 = - 1 by EUCLID:52;
then A4: |.|[0,(- 1)]|.| = sqrt ((0 ^2) + ((- 1) ^2)) by A3, JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
|[0,(- 1)]| `2 <= - (|[0,(- 1)]| `1) by A3, EUCLID:52;
then A5: g . O in KYN by A1, A3, A4;
A6: |[(- 1),0]| `1 = - 1 by EUCLID:52;
then A7: |[(- 1),0]| `2 <= - (|[(- 1),0]| `1) by EUCLID:52;
|[0,1]| `2 = 1 by EUCLID:52;
then A8: |.|[0,1]|.| = sqrt ((0 ^2) + (1 ^2)) by A2, JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
|[0,1]| `2 >= - (|[0,1]| `1) by A2, EUCLID:52;
then A9: g . I in KYP by A1, A2, A8;
A10: ( |[1,0]| `1 = 1 & |[1,0]| `2 = 0 ) by EUCLID:52;
then |.|[1,0]|.| = sqrt ((1 ^2) + (0 ^2)) by JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
then A11: f . I in KXP by A1, A10;
A12: |[(- 1),0]| `2 = 0 by EUCLID:52;
then |.|[(- 1),0]|.| = sqrt (((- 1) ^2) + (0 ^2)) by A6, JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
then f . O in KXN by A1, A6, A12, A7;
hence rng f meets rng g by A1, A11, A5, A9, Th14; ::_thesis: verum
end;
theorem :: JGRAPH_5:17
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & |.p1.| = 1 & |.p2.| = 1 & |.p3.| = 1 & |.p4.| = 1 & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h is being_homeomorphism & h .: C0 c= C0 & h . p1 = |[(- 1),0]| & h . p2 = |[0,1]| & h . p3 = |[1,0]| & h . p4 = |[0,(- 1)]| ) holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & |.p1.| = 1 & |.p2.| = 1 & |.p3.| = 1 & |.p4.| = 1 & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h is being_homeomorphism & h .: C0 c= C0 & h . p1 = |[(- 1),0]| & h . p2 = |[0,1]| & h . p3 = |[1,0]| & h . p4 = |[0,(- 1)]| ) holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0 be Subset of (TOP-REAL 2); ::_thesis: ( C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & |.p1.| = 1 & |.p2.| = 1 & |.p3.| = 1 & |.p4.| = 1 & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h is being_homeomorphism & h .: C0 c= C0 & h . p1 = |[(- 1),0]| & h . p2 = |[0,1]| & h . p3 = |[1,0]| & h . p4 = |[0,(- 1)]| ) implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g )
assume A1: ( C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & |.p1.| = 1 & |.p2.| = 1 & |.p3.| = 1 & |.p4.| = 1 & ex h being Function of (TOP-REAL 2),(TOP-REAL 2) st
( h is being_homeomorphism & h .: C0 c= C0 & h . p1 = |[(- 1),0]| & h . p2 = |[0,1]| & h . p3 = |[1,0]| & h . p4 = |[0,(- 1)]| ) ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A2: h is being_homeomorphism and
A3: h .: C0 c= C0 and
A4: h . p1 = |[(- 1),0]| and
A5: h . p2 = |[0,1]| and
A6: h . p3 = |[1,0]| and
A7: h . p4 = |[0,(- 1)]| ;
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume that
A8: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one ) and
A9: f . 0 = p1 and
A10: f . 1 = p3 and
A11: g . 0 = p4 and
A12: g . 1 = p2 and
A13: rng f c= C0 and
A14: rng g c= C0 ; ::_thesis: rng f meets rng g
reconsider f2 = h * f as Function of I[01],(TOP-REAL 2) ;
0 in dom f2 by Lm1, BORSUK_1:40, FUNCT_2:def_1;
then A15: f2 . 0 = |[(- 1),0]| by A4, A9, FUNCT_1:12;
reconsider g2 = h * g as Function of I[01],(TOP-REAL 2) ;
0 in dom g2 by Lm1, BORSUK_1:40, FUNCT_2:def_1;
then A16: g2 . 0 = |[0,(- 1)]| by A7, A11, FUNCT_1:12;
1 in dom g2 by Lm2, BORSUK_1:40, FUNCT_2:def_1;
then A17: g2 . 1 = |[0,1]| by A5, A12, FUNCT_1:12;
A18: rng f2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 )
A19: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
assume y in rng f2 ; ::_thesis: y in C0
then consider x being set such that
A20: x in dom f2 and
A21: y = f2 . x by FUNCT_1:def_3;
x in dom f by A20, FUNCT_1:11;
then A22: f . x in rng f by FUNCT_1:def_3;
y = h . (f . x) by A20, A21, FUNCT_1:12;
then y in h .: C0 by A13, A22, A19, FUNCT_1:def_6;
hence y in C0 by A3; ::_thesis: verum
end;
A23: rng g2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 )
A24: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
assume y in rng g2 ; ::_thesis: y in C0
then consider x being set such that
A25: x in dom g2 and
A26: y = g2 . x by FUNCT_1:def_3;
x in dom g by A25, FUNCT_1:11;
then A27: g . x in rng g by FUNCT_1:def_3;
y = h . (g . x) by A25, A26, FUNCT_1:12;
then y in h .: C0 by A14, A27, A24, FUNCT_1:def_6;
hence y in C0 by A3; ::_thesis: verum
end;
1 in dom f2 by Lm2, BORSUK_1:40, FUNCT_2:def_1;
then A28: f2 . 1 = |[1,0]| by A6, A10, FUNCT_1:12;
( h is continuous & h is one-to-one ) by A2, TOPS_2:def_5;
then rng f2 meets rng g2 by A1, A8, A15, A28, A16, A17, A18, A23, Th16;
then consider q5 being set such that
A29: q5 in rng f2 and
A30: q5 in rng g2 by XBOOLE_0:3;
consider x being set such that
A31: x in dom f2 and
A32: q5 = f2 . x by A29, FUNCT_1:def_3;
x in dom f by A31, FUNCT_1:11;
then A33: f . x in rng f by FUNCT_1:def_3;
consider u being set such that
A34: u in dom g2 and
A35: q5 = g2 . u by A30, FUNCT_1:def_3;
A36: ( q5 = h . (g . u) & g . u in dom h ) by A34, A35, FUNCT_1:11, FUNCT_1:12;
A37: h is one-to-one by A2, TOPS_2:def_5;
u in dom g by A34, FUNCT_1:11;
then A38: g . u in rng g by FUNCT_1:def_3;
( q5 = h . (f . x) & f . x in dom h ) by A31, A32, FUNCT_1:11, FUNCT_1:12;
then f . x = g . u by A37, A36, FUNCT_1:def_4;
hence rng f meets rng g by A33, A38, XBOOLE_0:3; ::_thesis: verum
end;
begin
theorem Th18: :: JGRAPH_5:18
for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 > 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0
proof
let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 > 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q `2 > 0 implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0 )
assume that
A1: - 1 < cn and
A2: cn < 1 and
A3: q `2 > 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0
now__::_thesis:_(_(_(q_`1)_/_|.q.|_>=_cn_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(cn_-FanMorphN)_._q_holds_
p_`2_>_0_)_)_or_(_(q_`1)_/_|.q.|_<_cn_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(cn_-FanMorphN)_._q_holds_
p_`2_>_0_)_)_)
percases ( (q `1) / |.q.| >= cn or (q `1) / |.q.| < cn ) ;
case (q `1) / |.q.| >= cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0 by A2, A3, JGRAPH_4:75; ::_thesis: verum
end;
case (q `1) / |.q.| < cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0 by A1, A3, JGRAPH_4:76; ::_thesis: verum
end;
end;
end;
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 > 0 ; ::_thesis: verum
end;
theorem :: JGRAPH_5:19
for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 >= 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0
proof
let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 >= 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q `2 >= 0 implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0 )
assume that
A1: ( - 1 < cn & cn < 1 ) and
A2: q `2 >= 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0
now__::_thesis:_(_(_q_`2_>_0_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(cn_-FanMorphN)_._q_holds_
p_`2_>=_0_)_)_or_(_q_`2_=_0_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(cn_-FanMorphN)_._q_holds_
p_`2_>=_0_)_)_)
percases ( q `2 > 0 or q `2 = 0 ) by A2;
case q `2 > 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0 by A1, Th18; ::_thesis: verum
end;
case q `2 = 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0 by JGRAPH_4:49; ::_thesis: verum
end;
end;
end;
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
p `2 >= 0 ; ::_thesis: verum
end;
theorem Th20: :: JGRAPH_5:20
for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 >= 0 & (q `1) / |.q.| < cn & |.q.| <> 0 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; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 >= 0 & (q `1) / |.q.| < cn & |.q.| <> 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
( p `2 >= 0 & p `1 < 0 )
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q `2 >= 0 & (q `1) / |.q.| < cn & |.q.| <> 0 implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
( p `2 >= 0 & p `1 < 0 ) )
assume that
A1: - 1 < cn and
A2: cn < 1 and
A3: ( q `2 >= 0 & (q `1) / |.q.| < cn ) and
A4: |.q.| <> 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphN) . q holds
( p `2 >= 0 & p `1 < 0 )
let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphN) . q implies ( p `2 >= 0 & p `1 < 0 ) )
assume A5: p = (cn -FanMorphN) . q ; ::_thesis: ( p `2 >= 0 & p `1 < 0 )
now__::_thesis:_(_(_q_`2_=_0_&_p_`2_>=_0_&_p_`1_<_0_)_or_(_q_`2_<>_0_&_p_`2_>=_0_&_p_`1_<_0_)_)
percases ( q `2 = 0 or q `2 <> 0 ) ;
caseA6: q `2 = 0 ; ::_thesis: ( p `2 >= 0 & p `1 < 0 )
then |.q.| ^2 = ((q `1) ^2) + (0 ^2) by JGRAPH_3:1
.= (q `1) ^2 ;
then A7: ( |.q.| = q `1 or |.q.| = - (q `1) ) by SQUARE_1:40;
q = p by A5, A6, JGRAPH_4:49;
hence ( p `2 >= 0 & p `1 < 0 ) by A2, A3, A4, A7, XCMPLX_1:60; ::_thesis: verum
end;
case q `2 <> 0 ; ::_thesis: ( p `2 >= 0 & p `1 < 0 )
hence ( p `2 >= 0 & p `1 < 0 ) by A1, A3, A5, JGRAPH_4:76; ::_thesis: verum
end;
end;
end;
hence ( p `2 >= 0 & p `1 < 0 ) ; ::_thesis: verum
end;
theorem Th21: :: JGRAPH_5:21
for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 >= 0 & q2 `2 >= 0 & |.q1.| <> 0 & |.q2.| <> 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; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 >= 0 & q2 `2 >= 0 & |.q1.| <> 0 & |.q2.| <> 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.|
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q1 `2 >= 0 & q2 `2 >= 0 & |.q1.| <> 0 & |.q2.| <> 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies 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.| )
assume that
A1: ( - 1 < cn & cn < 1 ) and
A2: q1 `2 >= 0 and
A3: q2 `2 >= 0 and
A4: |.q1.| <> 0 and
A5: |.q2.| <> 0 and
A6: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: 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.|
now__::_thesis:_(_(_q1_`2_>_0_&_(_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.|_)_)_or_(_q1_`2_=_0_&_(_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.|_)_)_)
percases ( q1 `2 > 0 or q1 `2 = 0 ) by A2;
caseA7: q1 `2 > 0 ; ::_thesis: 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.|
now__::_thesis:_(_(_q2_`2_>_0_&_(_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.|_)_)_or_(_q2_`2_=_0_&_(_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.|_)_)_)
percases ( q2 `2 > 0 or q2 `2 = 0 ) by A3;
case q2 `2 > 0 ; ::_thesis: 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.|
hence 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.| by A1, A6, A7, JGRAPH_4:79; ::_thesis: verum
end;
caseA8: q2 `2 = 0 ; ::_thesis: 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.|
A9: now__::_thesis:_not_|.q2.|_=_-_(q2_`1)
|.q1.| ^2 = ((q1 `1) ^2) + ((q1 `2) ^2) by JGRAPH_3:1;
then (|.q1.| ^2) - ((q1 `1) ^2) >= 0 by XREAL_1:63;
then ((|.q1.| ^2) - ((q1 `1) ^2)) + ((q1 `1) ^2) >= 0 + ((q1 `1) ^2) by XREAL_1:7;
then - |.q1.| <= q1 `1 by SQUARE_1:47;
then A10: (- |.q1.|) / |.q1.| <= (q1 `1) / |.q1.| by XREAL_1:72;
assume |.q2.| = - (q2 `1) ; ::_thesis: contradiction
then 1 = (- (q2 `1)) / |.q2.| by A5, XCMPLX_1:60;
then (q1 `1) / |.q1.| < - 1 by A6, XCMPLX_1:190;
hence contradiction by A4, A10, XCMPLX_1:197; ::_thesis: verum
end;
|.q2.| ^2 = ((q2 `1) ^2) + (0 ^2) by A8, JGRAPH_3:1
.= (q2 `1) ^2 ;
then ( |.q2.| = q2 `1 or |.q2.| = - (q2 `1) ) by SQUARE_1:40;
then A11: (q2 `1) / |.q2.| = 1 by A5, A9, XCMPLX_1:60;
thus 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.| ::_thesis: verum
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| )
assume that
A12: p1 = (cn -FanMorphN) . q1 and
A13: p2 = (cn -FanMorphN) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.|
A14: |.p1.| = |.q1.| by A12, JGRAPH_4:66;
A15: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
A16: p1 `2 > 0 by A1, A7, A12, Th18;
A17: now__::_thesis:_not_1_=_(p1_`1)_/_|.p1.|
assume 1 = (p1 `1) / |.p1.| ; ::_thesis: contradiction
then 1 * |.p1.| = p1 `1 by A4, A14, XCMPLX_1:87;
hence contradiction by A15, A16, XCMPLX_1:6; ::_thesis: verum
end;
A18: p2 = q2 by A8, A13, JGRAPH_4:49;
(|.p1.| ^2) - ((p1 `1) ^2) >= 0 by A15, XREAL_1:63;
then ((|.p1.| ^2) - ((p1 `1) ^2)) + ((p1 `1) ^2) >= 0 + ((p1 `1) ^2) by XREAL_1:7;
then p1 `1 <= |.p1.| by SQUARE_1:47;
then |.p1.| / |.p1.| >= (p1 `1) / |.p1.| by XREAL_1:72;
then 1 >= (p1 `1) / |.p1.| by A4, A14, XCMPLX_1:60;
hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A11, A18, A17, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
hence 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.| ; ::_thesis: verum
end;
caseA19: q1 `2 = 0 ; ::_thesis: 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.|
A20: now__::_thesis:_not_|.q1.|_=_q1_`1
|.q2.| ^2 = ((q2 `1) ^2) + ((q2 `2) ^2) by JGRAPH_3:1;
then (|.q2.| ^2) - ((q2 `1) ^2) >= 0 by XREAL_1:63;
then ((|.q2.| ^2) - ((q2 `1) ^2)) + ((q2 `1) ^2) >= 0 + ((q2 `1) ^2) by XREAL_1:7;
then q2 `1 <= |.q2.| by SQUARE_1:47;
then A21: |.q2.| / |.q2.| >= (q2 `1) / |.q2.| by XREAL_1:72;
assume |.q1.| = q1 `1 ; ::_thesis: contradiction
then (q2 `1) / |.q2.| > 1 by A4, A6, XCMPLX_1:60;
hence contradiction by A5, A21, XCMPLX_1:60; ::_thesis: verum
end;
|.q1.| ^2 = ((q1 `1) ^2) + (0 ^2) by A19, JGRAPH_3:1
.= (q1 `1) ^2 ;
then ( |.q1.| = q1 `1 or |.q1.| = - (q1 `1) ) by SQUARE_1:40;
then (- (q1 `1)) / |.q1.| = 1 by A4, A20, XCMPLX_1:60;
then A22: - ((q1 `1) / |.q1.|) = 1 by XCMPLX_1:187;
thus 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.| ::_thesis: verum
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphN) . q1 & p2 = (cn -FanMorphN) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| )
assume that
A23: p1 = (cn -FanMorphN) . q1 and
A24: p2 = (cn -FanMorphN) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.|
A25: |.p2.| = |.q2.| by A24, JGRAPH_4:66;
A26: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
then (|.p2.| ^2) - ((p2 `1) ^2) >= 0 by XREAL_1:63;
then ((|.p2.| ^2) - ((p2 `1) ^2)) + ((p2 `1) ^2) >= 0 + ((p2 `1) ^2) by XREAL_1:7;
then - |.p2.| <= p2 `1 by SQUARE_1:47;
then (- |.p2.|) / |.p2.| <= (p2 `1) / |.p2.| by XREAL_1:72;
then A27: - 1 <= (p2 `1) / |.p2.| by A5, A25, XCMPLX_1:197;
A28: now__::_thesis:_(_(_q2_`2_=_0_&_(p2_`1)_/_|.p2.|_>_-_1_)_or_(_q2_`2_<>_0_&_(p2_`1)_/_|.p2.|_>_-_1_)_)
percases ( q2 `2 = 0 or q2 `2 <> 0 ) ;
case q2 `2 = 0 ; ::_thesis: (p2 `1) / |.p2.| > - 1
then p2 = q2 by A24, JGRAPH_4:49;
hence (p2 `1) / |.p2.| > - 1 by A6, A22; ::_thesis: verum
end;
case q2 `2 <> 0 ; ::_thesis: (p2 `1) / |.p2.| > - 1
then A29: p2 `2 > 0 by A1, A3, A24, Th18;
now__::_thesis:_not_-_1_=_(p2_`1)_/_|.p2.|
assume - 1 = (p2 `1) / |.p2.| ; ::_thesis: contradiction
then (- 1) * |.p2.| = p2 `1 by A5, A25, XCMPLX_1:87;
then |.p2.| ^2 = (p2 `1) ^2 ;
hence contradiction by A26, A29, XCMPLX_1:6; ::_thesis: verum
end;
hence (p2 `1) / |.p2.| > - 1 by A27, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
p1 = q1 by A19, A23, JGRAPH_4:49;
hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A22, A28; ::_thesis: verum
end;
end;
end;
end;
hence 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.| ; ::_thesis: verum
end;
theorem Th22: :: JGRAPH_5:22
for sn being Real
for q being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q `1 > 0 holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0
proof
let sn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q `1 > 0 holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & q `1 > 0 implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0 )
assume that
A1: - 1 < sn and
A2: sn < 1 and
A3: q `1 > 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0
now__::_thesis:_(_(_(q_`2)_/_|.q.|_>=_sn_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(sn_-FanMorphE)_._q_holds_
p_`1_>_0_)_)_or_(_(q_`2)_/_|.q.|_<_sn_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(sn_-FanMorphE)_._q_holds_
p_`1_>_0_)_)_)
percases ( (q `2) / |.q.| >= sn or (q `2) / |.q.| < sn ) ;
case (q `2) / |.q.| >= sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0
hence for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0 by A2, A3, JGRAPH_4:106; ::_thesis: verum
end;
case (q `2) / |.q.| < sn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0
hence for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0 by A1, A3, JGRAPH_4:107; ::_thesis: verum
end;
end;
end;
hence for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
p `1 > 0 ; ::_thesis: verum
end;
theorem :: JGRAPH_5:23
for sn being Real
for q being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q `1 >= 0 & (q `2) / |.q.| < sn & |.q.| <> 0 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; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q `1 >= 0 & (q `2) / |.q.| < sn & |.q.| <> 0 holds
for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
( p `1 >= 0 & p `2 < 0 )
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & q `1 >= 0 & (q `2) / |.q.| < sn & |.q.| <> 0 implies for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
( p `1 >= 0 & p `2 < 0 ) )
assume that
A1: - 1 < sn and
A2: sn < 1 and
A3: ( q `1 >= 0 & (q `2) / |.q.| < sn ) and
A4: |.q.| <> 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (sn -FanMorphE) . q holds
( p `1 >= 0 & p `2 < 0 )
let p be Point of (TOP-REAL 2); ::_thesis: ( p = (sn -FanMorphE) . q implies ( p `1 >= 0 & p `2 < 0 ) )
assume A5: p = (sn -FanMorphE) . q ; ::_thesis: ( p `1 >= 0 & p `2 < 0 )
now__::_thesis:_(_(_q_`1_=_0_&_p_`1_>=_0_&_p_`2_<_0_)_or_(_q_`1_<>_0_&_p_`1_>=_0_&_p_`2_<_0_)_)
percases ( q `1 = 0 or q `1 <> 0 ) ;
caseA6: q `1 = 0 ; ::_thesis: ( p `1 >= 0 & p `2 < 0 )
then |.q.| ^2 = ((q `2) ^2) + (0 ^2) by JGRAPH_3:1
.= (q `2) ^2 ;
then A7: ( |.q.| = q `2 or |.q.| = - (q `2) ) by SQUARE_1:40;
q = p by A5, A6, JGRAPH_4:82;
hence ( p `1 >= 0 & p `2 < 0 ) by A2, A3, A4, A7, XCMPLX_1:60; ::_thesis: verum
end;
case q `1 <> 0 ; ::_thesis: ( p `1 >= 0 & p `2 < 0 )
hence ( p `1 >= 0 & p `2 < 0 ) by A1, A3, A5, JGRAPH_4:107; ::_thesis: verum
end;
end;
end;
hence ( p `1 >= 0 & p `2 < 0 ) ; ::_thesis: verum
end;
theorem Th24: :: JGRAPH_5:24
for sn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 >= 0 & q2 `1 >= 0 & |.q1.| <> 0 & |.q2.| <> 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; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < sn & sn < 1 & q1 `1 >= 0 & q2 `1 >= 0 & |.q1.| <> 0 & |.q2.| <> 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.|
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < sn & sn < 1 & q1 `1 >= 0 & q2 `1 >= 0 & |.q1.| <> 0 & |.q2.| <> 0 & (q1 `2) / |.q1.| < (q2 `2) / |.q2.| implies 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.| )
assume that
A1: ( - 1 < sn & sn < 1 ) and
A2: q1 `1 >= 0 and
A3: q2 `1 >= 0 and
A4: |.q1.| <> 0 and
A5: |.q2.| <> 0 and
A6: (q1 `2) / |.q1.| < (q2 `2) / |.q2.| ; ::_thesis: 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.|
now__::_thesis:_(_(_q1_`1_>_0_&_(_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.|_)_)_or_(_q1_`1_=_0_&_(_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.|_)_)_)
percases ( q1 `1 > 0 or q1 `1 = 0 ) by A2;
caseA7: q1 `1 > 0 ; ::_thesis: 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.|
now__::_thesis:_(_(_q2_`1_>_0_&_(_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.|_)_)_or_(_q2_`1_=_0_&_(_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.|_)_)_)
percases ( q2 `1 > 0 or q2 `1 = 0 ) by A3;
case q2 `1 > 0 ; ::_thesis: 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.|
hence 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.| by A1, A6, A7, JGRAPH_4:110; ::_thesis: verum
end;
caseA8: q2 `1 = 0 ; ::_thesis: 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.|
A9: now__::_thesis:_not_|.q2.|_=_-_(q2_`2)
|.q1.| ^2 = ((q1 `2) ^2) + ((q1 `1) ^2) by JGRAPH_3:1;
then (|.q1.| ^2) - ((q1 `2) ^2) >= 0 by XREAL_1:63;
then ((|.q1.| ^2) - ((q1 `2) ^2)) + ((q1 `2) ^2) >= 0 + ((q1 `2) ^2) by XREAL_1:7;
then - |.q1.| <= q1 `2 by SQUARE_1:47;
then A10: (- |.q1.|) / |.q1.| <= (q1 `2) / |.q1.| by XREAL_1:72;
assume |.q2.| = - (q2 `2) ; ::_thesis: contradiction
then 1 = (- (q2 `2)) / |.q2.| by A5, XCMPLX_1:60;
then (q1 `2) / |.q1.| < - 1 by A6, XCMPLX_1:190;
hence contradiction by A4, A10, XCMPLX_1:197; ::_thesis: verum
end;
|.q2.| ^2 = ((q2 `2) ^2) + (0 ^2) by A8, JGRAPH_3:1
.= (q2 `2) ^2 ;
then ( |.q2.| = q2 `2 or |.q2.| = - (q2 `2) ) by SQUARE_1:40;
then A11: (q2 `2) / |.q2.| = 1 by A5, A9, XCMPLX_1:60;
thus 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.| ::_thesis: verum
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| )
assume that
A12: p1 = (sn -FanMorphE) . q1 and
A13: p2 = (sn -FanMorphE) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.|
A14: |.p1.| = |.q1.| by A12, JGRAPH_4:97;
A15: |.p1.| ^2 = ((p1 `2) ^2) + ((p1 `1) ^2) by JGRAPH_3:1;
A16: p1 `1 > 0 by A1, A7, A12, Th22;
A17: now__::_thesis:_not_1_=_(p1_`2)_/_|.p1.|
assume 1 = (p1 `2) / |.p1.| ; ::_thesis: contradiction
then 1 * |.p1.| = p1 `2 by A4, A14, XCMPLX_1:87;
hence contradiction by A15, A16, XCMPLX_1:6; ::_thesis: verum
end;
A18: p2 = q2 by A8, A13, JGRAPH_4:82;
(|.p1.| ^2) - ((p1 `2) ^2) >= 0 by A15, XREAL_1:63;
then ((|.p1.| ^2) - ((p1 `2) ^2)) + ((p1 `2) ^2) >= 0 + ((p1 `2) ^2) by XREAL_1:7;
then p1 `2 <= |.p1.| by SQUARE_1:47;
then |.p1.| / |.p1.| >= (p1 `2) / |.p1.| by XREAL_1:72;
then 1 >= (p1 `2) / |.p1.| by A4, A14, XCMPLX_1:60;
hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A11, A18, A17, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
hence 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.| ; ::_thesis: verum
end;
caseA19: q1 `1 = 0 ; ::_thesis: 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.|
A20: now__::_thesis:_not_|.q1.|_=_q1_`2
|.q2.| ^2 = ((q2 `2) ^2) + ((q2 `1) ^2) by JGRAPH_3:1;
then (|.q2.| ^2) - ((q2 `2) ^2) >= 0 by XREAL_1:63;
then ((|.q2.| ^2) - ((q2 `2) ^2)) + ((q2 `2) ^2) >= 0 + ((q2 `2) ^2) by XREAL_1:7;
then q2 `2 <= |.q2.| by SQUARE_1:47;
then A21: |.q2.| / |.q2.| >= (q2 `2) / |.q2.| by XREAL_1:72;
assume |.q1.| = q1 `2 ; ::_thesis: contradiction
then (q2 `2) / |.q2.| > 1 by A4, A6, XCMPLX_1:60;
hence contradiction by A5, A21, XCMPLX_1:60; ::_thesis: verum
end;
|.q1.| ^2 = ((q1 `2) ^2) + (0 ^2) by A19, JGRAPH_3:1
.= (q1 `2) ^2 ;
then ( |.q1.| = q1 `2 or |.q1.| = - (q1 `2) ) by SQUARE_1:40;
then (- (q1 `2)) / |.q1.| = 1 by A4, A20, XCMPLX_1:60;
then A22: - ((q1 `2) / |.q1.|) = 1 by XCMPLX_1:187;
thus 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.| ::_thesis: verum
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (sn -FanMorphE) . q1 & p2 = (sn -FanMorphE) . q2 implies (p1 `2) / |.p1.| < (p2 `2) / |.p2.| )
assume that
A23: p1 = (sn -FanMorphE) . q1 and
A24: p2 = (sn -FanMorphE) . q2 ; ::_thesis: (p1 `2) / |.p1.| < (p2 `2) / |.p2.|
A25: |.p2.| = |.q2.| by A24, JGRAPH_4:97;
A26: |.p2.| ^2 = ((p2 `2) ^2) + ((p2 `1) ^2) by JGRAPH_3:1;
then (|.p2.| ^2) - ((p2 `2) ^2) >= 0 by XREAL_1:63;
then ((|.p2.| ^2) - ((p2 `2) ^2)) + ((p2 `2) ^2) >= 0 + ((p2 `2) ^2) by XREAL_1:7;
then - |.p2.| <= p2 `2 by SQUARE_1:47;
then (- |.p2.|) / |.p2.| <= (p2 `2) / |.p2.| by XREAL_1:72;
then A27: - 1 <= (p2 `2) / |.p2.| by A5, A25, XCMPLX_1:197;
A28: now__::_thesis:_(_(_q2_`1_=_0_&_(p2_`2)_/_|.p2.|_>_-_1_)_or_(_q2_`1_<>_0_&_(p2_`2)_/_|.p2.|_>_-_1_)_)
percases ( q2 `1 = 0 or q2 `1 <> 0 ) ;
case q2 `1 = 0 ; ::_thesis: (p2 `2) / |.p2.| > - 1
then p2 = q2 by A24, JGRAPH_4:82;
hence (p2 `2) / |.p2.| > - 1 by A6, A22; ::_thesis: verum
end;
case q2 `1 <> 0 ; ::_thesis: (p2 `2) / |.p2.| > - 1
then A29: p2 `1 > 0 by A1, A3, A24, Th22;
now__::_thesis:_not_-_1_=_(p2_`2)_/_|.p2.|
assume - 1 = (p2 `2) / |.p2.| ; ::_thesis: contradiction
then (- 1) * |.p2.| = p2 `2 by A5, A25, XCMPLX_1:87;
then |.p2.| ^2 = (p2 `2) ^2 ;
hence contradiction by A26, A29, XCMPLX_1:6; ::_thesis: verum
end;
hence (p2 `2) / |.p2.| > - 1 by A27, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
p1 = q1 by A19, A23, JGRAPH_4:82;
hence (p1 `2) / |.p1.| < (p2 `2) / |.p2.| by A22, A28; ::_thesis: verum
end;
end;
end;
end;
hence 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.| ; ::_thesis: verum
end;
theorem Th25: :: JGRAPH_5:25
for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 < 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0
proof
let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 < 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q `2 < 0 implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0 )
assume that
A1: - 1 < cn and
A2: cn < 1 and
A3: q `2 < 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0
now__::_thesis:_(_(_(q_`1)_/_|.q.|_>=_cn_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(cn_-FanMorphS)_._q_holds_
p_`2_<_0_)_)_or_(_(q_`1)_/_|.q.|_<_cn_&_(_for_p_being_Point_of_(TOP-REAL_2)_st_p_=_(cn_-FanMorphS)_._q_holds_
p_`2_<_0_)_)_)
percases ( (q `1) / |.q.| >= cn or (q `1) / |.q.| < cn ) ;
case (q `1) / |.q.| >= cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0 by A2, A3, JGRAPH_4:137; ::_thesis: verum
end;
case (q `1) / |.q.| < cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0 by A1, A3, JGRAPH_4:138; ::_thesis: verum
end;
end;
end;
hence for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 < 0 ; ::_thesis: verum
end;
theorem Th26: :: JGRAPH_5:26
for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & 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; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & 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 )
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q `2 < 0 & (q `1) / |.q.| > cn implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
( p `2 < 0 & p `1 > 0 ) )
assume that
A1: - 1 < cn and
A2: cn < 1 and
A3: q `2 < 0 and
A4: (q `1) / |.q.| > cn ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
( p `2 < 0 & p `1 > 0 )
let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphS) . q implies ( p `2 < 0 & p `1 > 0 ) )
assume A5: p = (cn -FanMorphS) . q ; ::_thesis: ( p `2 < 0 & p `1 > 0 )
now__::_thesis:_not_p_`1_=_0
set q1 = |.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|;
set p1 = (1 / |.p.|) * p;
set p2 = (cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|);
( |[0,(- 1)]| `1 = 0 & |[0,(- 1)]| `2 = - 1 ) by EUCLID:52;
then A6: |.p.| * |[0,(- 1)]| = |[(|.p.| * 0),(|.p.| * (- 1))]| by EUCLID:57
.= |[0,(- |.p.|)]| ;
A7: ( |[cn,(- (sqrt (1 - (cn ^2))))]| `1 = cn & |[cn,(- (sqrt (1 - (cn ^2))))]| `2 = - (sqrt (1 - (cn ^2))) ) by EUCLID:52;
then A8: |.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]| = |[(|.p.| * cn),(|.p.| * (- (sqrt (1 - (cn ^2)))))]| by EUCLID:57;
then A9: (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|) `1 = |.p.| * cn by EUCLID:52;
assume A10: p `1 = 0 ; ::_thesis: contradiction
then |.p.| ^2 = ((p `2) ^2) + (0 ^2) by JGRAPH_3:1
.= (p `2) ^2 ;
then A11: ( p `2 = |.p.| or p `2 = - |.p.| ) by SQUARE_1:40;
then A12: |.p.| <> 0 by A2, A3, A4, A5, JGRAPH_4:137;
A13: (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|) `2 = - ((sqrt (1 - (cn ^2))) * |.p.|) by A8, EUCLID:52;
1 ^2 > cn ^2 by A1, A2, SQUARE_1:50;
then A14: 1 - (cn ^2) > 0 by XREAL_1:50;
then sqrt (1 - (cn ^2)) > 0 by SQUARE_1:25;
then - (- ((sqrt (1 - (cn ^2))) * |.p.|)) > 0 by A12, XREAL_1:129;
then A15: (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|) `2 < 0 by A13;
A16: |.p.| * ((1 / |.p.|) * p) = (|.p.| * (1 / |.p.|)) * p by EUCLID:30
.= 1 * p by A12, XCMPLX_1:106
.= p by EUCLID:29 ;
A17: (1 / |.p.|) * p = |[((1 / |.p.|) * (p `1)),((1 / |.p.|) * (p `2))]| by EUCLID:57;
then ((1 / |.p.|) * p) `2 = - (|.p.| * (1 / |.p.|)) by A2, A3, A4, A5, A11, EUCLID:52, JGRAPH_4:137
.= - 1 by A12, XCMPLX_1:106 ;
then A18: p = |.p.| * |[0,(- 1)]| by A10, A16, A17, EUCLID:52;
A19: |.(|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|).| = (abs |.p.|) * |.|[cn,(- (sqrt (1 - (cn ^2))))]|.| by TOPRNS_1:7
.= (abs |.p.|) * (sqrt ((cn ^2) + ((- (sqrt (1 - (cn ^2)))) ^2))) by A7, JGRAPH_3:1
.= (abs |.p.|) * (sqrt ((cn ^2) + ((sqrt (1 - (cn ^2))) ^2)))
.= (abs |.p.|) * (sqrt ((cn ^2) + (1 - (cn ^2)))) by A14, SQUARE_1:def_2
.= |.p.| by ABSVALUE:def_1, SQUARE_1:18 ;
then A20: |.((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)).| = |.p.| by JGRAPH_4:128;
A21: ((|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|) `1) / |.(|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|).| = cn by A12, A9, A19, XCMPLX_1:89;
then A22: ((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)) `1 = 0 by A15, JGRAPH_4:142;
then |.((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)).| ^2 = ((((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)) `2) ^2) + (0 ^2) by JGRAPH_3:1
.= (((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)) `2) ^2 ;
then ( ((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)) `2 = |.((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)).| or ((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)) `2 = - |.((cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|)).| ) by SQUARE_1:40;
then A23: (cn -FanMorphS) . (|.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]|) = |[0,(- |.p.|)]| by A15, A21, A22, A20, EUCLID:53, JGRAPH_4:142;
( cn -FanMorphS is one-to-one & dom (cn -FanMorphS) = the carrier of (TOP-REAL 2) ) by A1, A2, FUNCT_2:def_1, JGRAPH_4:133;
then |.p.| * |[cn,(- (sqrt (1 - (cn ^2))))]| = q by A5, A18, A23, A6, FUNCT_1:def_4;
hence contradiction by A4, A12, A9, A19, XCMPLX_1:89; ::_thesis: verum
end;
hence ( p `2 < 0 & p `1 > 0 ) by A2, A3, A4, A5, JGRAPH_4:137; ::_thesis: verum
end;
theorem Th27: :: JGRAPH_5:27
for cn being Real
for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 <= 0 & q2 `2 <= 0 & |.q1.| <> 0 & |.q2.| <> 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; ::_thesis: for q1, q2 being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q1 `2 <= 0 & q2 `2 <= 0 & |.q1.| <> 0 & |.q2.| <> 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.|
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q1 `2 <= 0 & q2 `2 <= 0 & |.q1.| <> 0 & |.q2.| <> 0 & (q1 `1) / |.q1.| < (q2 `1) / |.q2.| implies 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.| )
assume that
A1: ( - 1 < cn & cn < 1 ) and
A2: q1 `2 <= 0 and
A3: q2 `2 <= 0 and
A4: |.q1.| <> 0 and
A5: |.q2.| <> 0 and
A6: (q1 `1) / |.q1.| < (q2 `1) / |.q2.| ; ::_thesis: 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.|
now__::_thesis:_(_(_q1_`2_<_0_&_(_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.|_)_)_or_(_q1_`2_=_0_&_(_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.|_)_)_)
percases ( q1 `2 < 0 or q1 `2 = 0 ) by A2;
caseA7: q1 `2 < 0 ; ::_thesis: 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.|
now__::_thesis:_(_(_q2_`2_<_0_&_(_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.|_)_)_or_(_q2_`2_=_0_&_(_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.|_)_)_)
percases ( q2 `2 < 0 or q2 `2 = 0 ) by A3;
case q2 `2 < 0 ; ::_thesis: 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.|
hence 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.| by A1, A6, A7, JGRAPH_4:141; ::_thesis: verum
end;
caseA8: q2 `2 = 0 ; ::_thesis: 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.|
A9: now__::_thesis:_not_|.q2.|_=_-_(q2_`1)
|.q1.| ^2 = ((q1 `1) ^2) + ((q1 `2) ^2) by JGRAPH_3:1;
then (|.q1.| ^2) - ((q1 `1) ^2) >= 0 by XREAL_1:63;
then ((|.q1.| ^2) - ((q1 `1) ^2)) + ((q1 `1) ^2) >= 0 + ((q1 `1) ^2) by XREAL_1:7;
then - |.q1.| <= q1 `1 by SQUARE_1:47;
then A10: (- |.q1.|) / |.q1.| <= (q1 `1) / |.q1.| by XREAL_1:72;
assume |.q2.| = - (q2 `1) ; ::_thesis: contradiction
then 1 = (- (q2 `1)) / |.q2.| by A5, XCMPLX_1:60;
then (q1 `1) / |.q1.| < - 1 by A6, XCMPLX_1:190;
hence contradiction by A4, A10, XCMPLX_1:197; ::_thesis: verum
end;
|.q2.| ^2 = ((q2 `1) ^2) + (0 ^2) by A8, JGRAPH_3:1
.= (q2 `1) ^2 ;
then ( |.q2.| = q2 `1 or |.q2.| = - (q2 `1) ) by SQUARE_1:40;
then A11: (q2 `1) / |.q2.| = 1 by A5, A9, XCMPLX_1:60;
thus 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.| ::_thesis: verum
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| )
assume that
A12: p1 = (cn -FanMorphS) . q1 and
A13: p2 = (cn -FanMorphS) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.|
A14: |.p1.| = |.q1.| by A12, JGRAPH_4:128;
A15: |.p1.| ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
A16: p1 `2 < 0 by A1, A7, A12, Th25;
A17: now__::_thesis:_not_1_=_(p1_`1)_/_|.p1.|
assume 1 = (p1 `1) / |.p1.| ; ::_thesis: contradiction
then 1 * |.p1.| = p1 `1 by A4, A14, XCMPLX_1:87;
hence contradiction by A15, A16, XCMPLX_1:6; ::_thesis: verum
end;
A18: p2 = q2 by A8, A13, JGRAPH_4:113;
(|.p1.| ^2) - ((p1 `1) ^2) >= 0 by A15, XREAL_1:63;
then ((|.p1.| ^2) - ((p1 `1) ^2)) + ((p1 `1) ^2) >= 0 + ((p1 `1) ^2) by XREAL_1:7;
then p1 `1 <= |.p1.| by SQUARE_1:47;
then |.p1.| / |.p1.| >= (p1 `1) / |.p1.| by XREAL_1:72;
then 1 >= (p1 `1) / |.p1.| by A4, A14, XCMPLX_1:60;
hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A11, A18, A17, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
hence 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.| ; ::_thesis: verum
end;
caseA19: q1 `2 = 0 ; ::_thesis: 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.|
A20: now__::_thesis:_not_|.q1.|_=_q1_`1
|.q2.| ^2 = ((q2 `1) ^2) + ((q2 `2) ^2) by JGRAPH_3:1;
then (|.q2.| ^2) - ((q2 `1) ^2) >= 0 by XREAL_1:63;
then ((|.q2.| ^2) - ((q2 `1) ^2)) + ((q2 `1) ^2) >= 0 + ((q2 `1) ^2) by XREAL_1:7;
then q2 `1 <= |.q2.| by SQUARE_1:47;
then A21: |.q2.| / |.q2.| >= (q2 `1) / |.q2.| by XREAL_1:72;
assume |.q1.| = q1 `1 ; ::_thesis: contradiction
then (q2 `1) / |.q2.| > 1 by A4, A6, XCMPLX_1:60;
hence contradiction by A5, A21, XCMPLX_1:60; ::_thesis: verum
end;
|.q1.| ^2 = ((q1 `1) ^2) + (0 ^2) by A19, JGRAPH_3:1
.= (q1 `1) ^2 ;
then ( |.q1.| = q1 `1 or |.q1.| = - (q1 `1) ) by SQUARE_1:40;
then (- (q1 `1)) / |.q1.| = 1 by A4, A20, XCMPLX_1:60;
then A22: - ((q1 `1) / |.q1.|) = 1 by XCMPLX_1:187;
thus 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.| ::_thesis: verum
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 = (cn -FanMorphS) . q1 & p2 = (cn -FanMorphS) . q2 implies (p1 `1) / |.p1.| < (p2 `1) / |.p2.| )
assume that
A23: p1 = (cn -FanMorphS) . q1 and
A24: p2 = (cn -FanMorphS) . q2 ; ::_thesis: (p1 `1) / |.p1.| < (p2 `1) / |.p2.|
A25: |.p2.| = |.q2.| by A24, JGRAPH_4:128;
A26: |.p2.| ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
then (|.p2.| ^2) - ((p2 `1) ^2) >= 0 by XREAL_1:63;
then ((|.p2.| ^2) - ((p2 `1) ^2)) + ((p2 `1) ^2) >= 0 + ((p2 `1) ^2) by XREAL_1:7;
then - |.p2.| <= p2 `1 by SQUARE_1:47;
then (- |.p2.|) / |.p2.| <= (p2 `1) / |.p2.| by XREAL_1:72;
then A27: - 1 <= (p2 `1) / |.p2.| by A5, A25, XCMPLX_1:197;
A28: now__::_thesis:_(_(_q2_`2_=_0_&_(p2_`1)_/_|.p2.|_>_-_1_)_or_(_q2_`2_<>_0_&_(p2_`1)_/_|.p2.|_>_-_1_)_)
percases ( q2 `2 = 0 or q2 `2 <> 0 ) ;
case q2 `2 = 0 ; ::_thesis: (p2 `1) / |.p2.| > - 1
then p2 = q2 by A24, JGRAPH_4:113;
hence (p2 `1) / |.p2.| > - 1 by A6, A22; ::_thesis: verum
end;
case q2 `2 <> 0 ; ::_thesis: (p2 `1) / |.p2.| > - 1
then A29: p2 `2 < 0 by A1, A3, A24, Th25;
now__::_thesis:_not_-_1_=_(p2_`1)_/_|.p2.|
assume - 1 = (p2 `1) / |.p2.| ; ::_thesis: contradiction
then (- 1) * |.p2.| = p2 `1 by A5, A25, XCMPLX_1:87;
then |.p2.| ^2 = (p2 `1) ^2 ;
hence contradiction by A26, A29, XCMPLX_1:6; ::_thesis: verum
end;
hence (p2 `1) / |.p2.| > - 1 by A27, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
p1 = q1 by A19, A23, JGRAPH_4:113;
hence (p1 `1) / |.p1.| < (p2 `1) / |.p2.| by A22, A28; ::_thesis: verum
end;
end;
end;
end;
hence 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.| ; ::_thesis: verum
end;
begin
Lm3: now__::_thesis:_for_P_being_non_empty_compact_Subset_of_(TOP-REAL_2)_st_P_=__{__q_where_q_is_Point_of_(TOP-REAL_2)_:_|.q.|_=_1__}__holds_
(_proj1_.:_P_=_[.(-_1),1.]_&_proj2_.:_P_=_[.(-_1),1.]_)
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies ( proj1 .: P = [.(- 1),1.] & proj2 .: P = [.(- 1),1.] ) )
assume A1: P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: ( proj1 .: P = [.(- 1),1.] & proj2 .: P = [.(- 1),1.] )
A2: [.(- 1),1.] c= proj1 .: P
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [.(- 1),1.] or y in proj1 .: P )
assume y in [.(- 1),1.] ; ::_thesis: y in proj1 .: P
then y in { r where r is Real : ( - 1 <= r & r <= 1 ) } by RCOMP_1:def_1;
then consider r being Real such that
A3: y = r and
A4: ( - 1 <= r & r <= 1 ) ;
set q = |[r,(sqrt (1 - (r ^2)))]|;
1 ^2 >= r ^2 by A4, SQUARE_1:49;
then A5: 1 - (r ^2) >= 0 by XREAL_1:48;
( |[r,(sqrt (1 - (r ^2)))]| `1 = r & |[r,(sqrt (1 - (r ^2)))]| `2 = sqrt (1 - (r ^2)) ) by EUCLID:52;
then |.|[r,(sqrt (1 - (r ^2)))]|.| = sqrt ((r ^2) + ((sqrt (1 - (r ^2))) ^2)) by JGRAPH_3:1
.= sqrt ((r ^2) + (1 - (r ^2))) by A5, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then A6: ( dom proj1 = the carrier of (TOP-REAL 2) & |[r,(sqrt (1 - (r ^2)))]| in P ) by A1, FUNCT_2:def_1;
proj1 . |[r,(sqrt (1 - (r ^2)))]| = |[r,(sqrt (1 - (r ^2)))]| `1 by PSCOMP_1:def_5
.= r by EUCLID:52 ;
hence y in proj1 .: P by A3, A6, FUNCT_1:def_6; ::_thesis: verum
end;
proj1 .: P c= [.(- 1),1.]
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in proj1 .: P or y in [.(- 1),1.] )
assume y in proj1 .: P ; ::_thesis: y in [.(- 1),1.]
then consider x being set such that
A7: x in dom proj1 and
A8: x in P and
A9: y = proj1 . x by FUNCT_1:def_6;
reconsider q = x as Point of (TOP-REAL 2) by A7;
ex q2 being Point of (TOP-REAL 2) st
( q2 = x & |.q2.| = 1 ) by A1, A8;
then A10: ((q `1) ^2) + ((q `2) ^2) = 1 ^2 by JGRAPH_3:1;
0 <= (q `2) ^2 by XREAL_1:63;
then (1 - ((q `1) ^2)) + ((q `1) ^2) >= 0 + ((q `1) ^2) by A10, XREAL_1:7;
then A11: ( - 1 <= q `1 & q `1 <= 1 ) by SQUARE_1:51;
y = q `1 by A9, PSCOMP_1:def_5;
hence y in [.(- 1),1.] by A11, XXREAL_1:1; ::_thesis: verum
end;
hence proj1 .: P = [.(- 1),1.] by A2, XBOOLE_0:def_10; ::_thesis: proj2 .: P = [.(- 1),1.]
A12: [.(- 1),1.] c= proj2 .: P
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [.(- 1),1.] or y in proj2 .: P )
assume y in [.(- 1),1.] ; ::_thesis: y in proj2 .: P
then y in { r where r is Real : ( - 1 <= r & r <= 1 ) } by RCOMP_1:def_1;
then consider r being Real such that
A13: y = r and
A14: ( - 1 <= r & r <= 1 ) ;
set q = |[(sqrt (1 - (r ^2))),r]|;
1 ^2 >= r ^2 by A14, SQUARE_1:49;
then A15: 1 - (r ^2) >= 0 by XREAL_1:48;
( |[(sqrt (1 - (r ^2))),r]| `2 = r & |[(sqrt (1 - (r ^2))),r]| `1 = sqrt (1 - (r ^2)) ) by EUCLID:52;
then |.|[(sqrt (1 - (r ^2))),r]|.| = sqrt (((sqrt (1 - (r ^2))) ^2) + (r ^2)) by JGRAPH_3:1
.= sqrt ((1 - (r ^2)) + (r ^2)) by A15, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then A16: ( dom proj2 = the carrier of (TOP-REAL 2) & |[(sqrt (1 - (r ^2))),r]| in P ) by A1, FUNCT_2:def_1;
proj2 . |[(sqrt (1 - (r ^2))),r]| = |[(sqrt (1 - (r ^2))),r]| `2 by PSCOMP_1:def_6
.= r by EUCLID:52 ;
hence y in proj2 .: P by A13, A16, FUNCT_1:def_6; ::_thesis: verum
end;
proj2 .: P c= [.(- 1),1.]
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in proj2 .: P or y in [.(- 1),1.] )
assume y in proj2 .: P ; ::_thesis: y in [.(- 1),1.]
then consider x being set such that
A17: x in dom proj2 and
A18: x in P and
A19: y = proj2 . x by FUNCT_1:def_6;
reconsider q = x as Point of (TOP-REAL 2) by A17;
ex q2 being Point of (TOP-REAL 2) st
( q2 = x & |.q2.| = 1 ) by A1, A18;
then A20: ((q `1) ^2) + ((q `2) ^2) = 1 ^2 by JGRAPH_3:1;
0 <= (q `1) ^2 by XREAL_1:63;
then (1 - ((q `2) ^2)) + ((q `2) ^2) >= 0 + ((q `2) ^2) by A20, XREAL_1:7;
then A21: ( - 1 <= q `2 & q `2 <= 1 ) by SQUARE_1:51;
y = q `2 by A19, PSCOMP_1:def_6;
hence y in [.(- 1),1.] by A21, XXREAL_1:1; ::_thesis: verum
end;
hence proj2 .: P = [.(- 1),1.] by A12, XBOOLE_0:def_10; ::_thesis: verum
end;
Lm4: for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
W-bound P = - 1
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies W-bound P = - 1 )
assume P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: W-bound P = - 1
then proj1 .: P = [.(- 1),1.] by Lm3;
then (proj1 | P) .: P = [.(- 1),1.] by RELAT_1:129;
then ( the carrier of ((TOP-REAL 2) | P) = P & lower_bound ((proj1 | P) .: P) = - 1 ) by JORDAN5A:19, PRE_TOPC:8;
then lower_bound (proj1 | P) = - 1 by PSCOMP_1:def_1;
hence W-bound P = - 1 by PSCOMP_1:def_7; ::_thesis: verum
end;
theorem Th28: :: JGRAPH_5:28
for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
( W-bound P = - 1 & E-bound P = 1 & S-bound P = - 1 & N-bound P = 1 )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies ( W-bound P = - 1 & E-bound P = 1 & S-bound P = - 1 & N-bound P = 1 ) )
A1: the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
assume A2: P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: ( W-bound P = - 1 & E-bound P = 1 & S-bound P = - 1 & N-bound P = 1 )
hence W-bound P = - 1 by Lm4; ::_thesis: ( E-bound P = 1 & S-bound P = - 1 & N-bound P = 1 )
proj1 .: P = [.(- 1),1.] by A2, Lm3;
then (proj1 | P) .: P = [.(- 1),1.] by RELAT_1:129;
then upper_bound ((proj1 | P) .: the carrier of ((TOP-REAL 2) | P)) = 1 by A1, JORDAN5A:19;
then upper_bound (proj1 | P) = 1 by PSCOMP_1:def_2;
hence E-bound P = 1 by PSCOMP_1:def_9; ::_thesis: ( S-bound P = - 1 & N-bound P = 1 )
proj2 .: P = [.(- 1),1.] by A2, Lm3;
then A3: (proj2 | P) .: P = [.(- 1),1.] by RELAT_1:129;
then lower_bound ((proj2 | P) .: P) = - 1 by JORDAN5A:19;
then lower_bound (proj2 | P) = - 1 by A1, PSCOMP_1:def_1;
hence S-bound P = - 1 by PSCOMP_1:def_10; ::_thesis: N-bound P = 1
upper_bound ((proj2 | P) .: P) = 1 by A3, JORDAN5A:19;
then upper_bound (proj2 | P) = 1 by A1, PSCOMP_1:def_2;
hence N-bound P = 1 by PSCOMP_1:def_8; ::_thesis: verum
end;
theorem Th29: :: JGRAPH_5:29
for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
W-min P = |[(- 1),0]|
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies W-min P = |[(- 1),0]| )
A1: the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
assume A2: P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: W-min P = |[(- 1),0]|
then A3: W-bound P = - 1 by Lm4;
proj2 .: P = [.(- 1),1.] by A2, Lm3;
then A4: (proj2 | P) .: P = [.(- 1),1.] by RELAT_1:129;
then upper_bound ((proj2 | P) .: P) = 1 by JORDAN5A:19;
then upper_bound (proj2 | P) = 1 by A1, PSCOMP_1:def_2;
then N-bound P = 1 by PSCOMP_1:def_8;
then A5: NW-corner P = |[(- 1),1]| by A3, PSCOMP_1:def_12;
lower_bound ((proj2 | P) .: P) = - 1 by A4, JORDAN5A:19;
then lower_bound (proj2 | P) = - 1 by A1, PSCOMP_1:def_1;
then S-bound P = - 1 by PSCOMP_1:def_10;
then A6: SW-corner P = |[(- 1),(- 1)]| by A3, PSCOMP_1:def_11;
A7: (LSeg ((SW-corner P),(NW-corner P))) /\ P c= {|[(- 1),0]|}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg ((SW-corner P),(NW-corner P))) /\ P or x in {|[(- 1),0]|} )
assume A8: x in (LSeg ((SW-corner P),(NW-corner P))) /\ P ; ::_thesis: x in {|[(- 1),0]|}
then A9: x in { (((1 - l) * (SW-corner P)) + (l * (NW-corner P))) where l is Real : ( 0 <= l & l <= 1 ) } by XBOOLE_0:def_4;
x in P by A8, XBOOLE_0:def_4;
then A10: ex q2 being Point of (TOP-REAL 2) st
( q2 = x & |.q2.| = 1 ) by A2;
consider l being Real such that
A11: x = ((1 - l) * (SW-corner P)) + (l * (NW-corner P)) and
0 <= l and
l <= 1 by A9;
reconsider q3 = x as Point of (TOP-REAL 2) by A11;
x = |[((1 - l) * (- 1)),((1 - l) * (- 1))]| + (l * |[(- 1),1]|) by A6, A5, A11, EUCLID:58;
then x = |[((1 - l) * (- 1)),((1 - l) * (- 1))]| + |[(l * (- 1)),(l * 1)]| by EUCLID:58;
then A12: x = |[(((1 - l) * (- 1)) + (l * (- 1))),(((1 - l) * (- 1)) + (l * 1))]| by EUCLID:56;
then q3 `1 = - 1 by EUCLID:52;
then A13: 1 = sqrt (((- 1) ^2) + ((q3 `2) ^2)) by A10, JGRAPH_3:1
.= sqrt (1 + ((q3 `2) ^2)) ;
now__::_thesis:_not_(q3_`2)_^2_>_0
assume (q3 `2) ^2 > 0 ; ::_thesis: contradiction
then 1 < 1 + ((q3 `2) ^2) by XREAL_1:29;
hence contradiction by A13, SQUARE_1:18, SQUARE_1:27; ::_thesis: verum
end;
then (q3 `2) ^2 = 0 by XREAL_1:63;
then A14: q3 `2 = 0 by XCMPLX_1:6;
q3 `2 = ((1 - l) * (- 1)) + l by A12, EUCLID:52;
hence x in {|[(- 1),0]|} by A12, A14, TARSKI:def_1; ::_thesis: verum
end;
{|[(- 1),0]|} c= (LSeg ((SW-corner P),(NW-corner P))) /\ P
proof
set q = |[(- 1),0]|;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[(- 1),0]|} or x in (LSeg ((SW-corner P),(NW-corner P))) /\ P )
assume x in {|[(- 1),0]|} ; ::_thesis: x in (LSeg ((SW-corner P),(NW-corner P))) /\ P
then A15: x = |[(- 1),0]| by TARSKI:def_1;
( |[(- 1),0]| `2 = 0 & |[(- 1),0]| `1 = - 1 ) by EUCLID:52;
then |.|[(- 1),0]|.| = sqrt (((- 1) ^2) + (0 ^2)) by JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
then A16: x in P by A2, A15;
|[(- 1),0]| = |[(((1 / 2) * (- 1)) + ((1 / 2) * (- 1))),(((1 / 2) * (- 1)) + ((1 / 2) * 1))]| ;
then |[(- 1),0]| = |[((1 / 2) * (- 1)),((1 / 2) * (- 1))]| + |[((1 / 2) * (- 1)),((1 / 2) * 1)]| by EUCLID:56;
then |[(- 1),0]| = |[((1 / 2) * (- 1)),((1 / 2) * (- 1))]| + ((1 / 2) * |[(- 1),1]|) by EUCLID:58;
then |[(- 1),0]| = ((1 / 2) * |[(- 1),(- 1)]|) + ((1 - (1 / 2)) * |[(- 1),1]|) by EUCLID:58;
then x in LSeg ((SW-corner P),(NW-corner P)) by A6, A5, A15;
hence x in (LSeg ((SW-corner P),(NW-corner P))) /\ P by A16, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg ((SW-corner P),(NW-corner P))) /\ P = {|[(- 1),0]|} by A7, XBOOLE_0:def_10;
then A17: W-most P = {|[(- 1),0]|} by PSCOMP_1:def_15;
(proj2 | (W-most P)) .: the carrier of ((TOP-REAL 2) | (W-most P)) = (proj2 | (W-most P)) .: (W-most P) by PRE_TOPC:8
.= Im (proj2,|[(- 1),0]|) by A17, RELAT_1:129
.= {(proj2 . |[(- 1),0]|)} by SETWISEO:8
.= {(|[(- 1),0]| `2)} by PSCOMP_1:def_6
.= {0} by EUCLID:52 ;
then lower_bound ((proj2 | (W-most P)) .: the carrier of ((TOP-REAL 2) | (W-most P))) = 0 by SEQ_4:9;
then lower_bound (proj2 | (W-most P)) = 0 by PSCOMP_1:def_1;
hence W-min P = |[(- 1),0]| by A3, PSCOMP_1:def_19; ::_thesis: verum
end;
theorem Th30: :: JGRAPH_5:30
for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
E-max P = |[1,0]|
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies E-max P = |[1,0]| )
A1: the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
assume A2: P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: E-max P = |[1,0]|
then A3: E-bound P = 1 by Th28;
proj2 .: P = [.(- 1),1.] by A2, Lm3;
then A4: (proj2 | P) .: P = [.(- 1),1.] by RELAT_1:129;
then upper_bound ((proj2 | P) .: P) = 1 by JORDAN5A:19;
then upper_bound (proj2 | P) = 1 by A1, PSCOMP_1:def_2;
then N-bound P = 1 by PSCOMP_1:def_8;
then A5: NE-corner P = |[1,1]| by A3, PSCOMP_1:def_13;
lower_bound ((proj2 | P) .: P) = - 1 by A4, JORDAN5A:19;
then lower_bound (proj2 | P) = - 1 by A1, PSCOMP_1:def_1;
then S-bound P = - 1 by PSCOMP_1:def_10;
then A6: SE-corner P = |[1,(- 1)]| by A3, PSCOMP_1:def_14;
A7: (LSeg ((SE-corner P),(NE-corner P))) /\ P c= {|[1,0]|}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg ((SE-corner P),(NE-corner P))) /\ P or x in {|[1,0]|} )
assume A8: x in (LSeg ((SE-corner P),(NE-corner P))) /\ P ; ::_thesis: x in {|[1,0]|}
then A9: x in { (((1 - l) * (SE-corner P)) + (l * (NE-corner P))) where l is Real : ( 0 <= l & l <= 1 ) } by XBOOLE_0:def_4;
x in P by A8, XBOOLE_0:def_4;
then A10: ex q2 being Point of (TOP-REAL 2) st
( q2 = x & |.q2.| = 1 ) by A2;
consider l being Real such that
A11: x = ((1 - l) * (SE-corner P)) + (l * (NE-corner P)) and
0 <= l and
l <= 1 by A9;
reconsider q3 = x as Point of (TOP-REAL 2) by A11;
x = |[((1 - l) * 1),((1 - l) * (- 1))]| + (l * |[1,1]|) by A6, A5, A11, EUCLID:58;
then x = |[((1 - l) * 1),((1 - l) * (- 1))]| + |[(l * 1),(l * 1)]| by EUCLID:58;
then A12: x = |[(((1 - l) + l) * 1),(((1 - l) * (- 1)) + (l * 1))]| by EUCLID:56;
then A13: q3 `1 = 1 by EUCLID:52;
now__::_thesis:_not_(q3_`2)_^2_>_0
assume (q3 `2) ^2 > 0 ; ::_thesis: contradiction
then 1 ^2 < 1 + ((q3 `2) ^2) by XREAL_1:29;
hence contradiction by A13, A10, JGRAPH_3:1; ::_thesis: verum
end;
then (q3 `2) ^2 = 0 by XREAL_1:63;
then A14: q3 `2 = 0 by XCMPLX_1:6;
q3 `2 = ((1 - l) * (- 1)) + l by A12, EUCLID:52;
hence x in {|[1,0]|} by A12, A14, TARSKI:def_1; ::_thesis: verum
end;
{|[1,0]|} c= (LSeg ((SE-corner P),(NE-corner P))) /\ P
proof
set q = |[1,0]|;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[1,0]|} or x in (LSeg ((SE-corner P),(NE-corner P))) /\ P )
assume x in {|[1,0]|} ; ::_thesis: x in (LSeg ((SE-corner P),(NE-corner P))) /\ P
then A15: x = |[1,0]| by TARSKI:def_1;
( |[1,0]| `2 = 0 & |[1,0]| `1 = 1 ) by EUCLID:52;
then |.|[1,0]|.| = sqrt ((1 ^2) + (0 ^2)) by JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
then A16: x in P by A2, A15;
|[1,0]| = |[(((1 / 2) * 1) + ((1 / 2) * 1)),(((1 / 2) * (- 1)) + ((1 / 2) * 1))]| ;
then |[1,0]| = |[((1 / 2) * 1),((1 / 2) * (- 1))]| + |[((1 / 2) * 1),((1 / 2) * 1)]| by EUCLID:56;
then |[1,0]| = |[((1 / 2) * 1),((1 / 2) * (- 1))]| + ((1 / 2) * |[1,1]|) by EUCLID:58;
then |[1,0]| = ((1 / 2) * |[1,(- 1)]|) + ((1 - (1 / 2)) * |[1,1]|) by EUCLID:58;
then x in LSeg ((SE-corner P),(NE-corner P)) by A6, A5, A15;
hence x in (LSeg ((SE-corner P),(NE-corner P))) /\ P by A16, XBOOLE_0:def_4; ::_thesis: verum
end;
then (LSeg ((SE-corner P),(NE-corner P))) /\ P = {|[1,0]|} by A7, XBOOLE_0:def_10;
then A17: E-most P = {|[1,0]|} by PSCOMP_1:def_17;
(proj2 | (E-most P)) .: the carrier of ((TOP-REAL 2) | (E-most P)) = (proj2 | (E-most P)) .: (E-most P) by PRE_TOPC:8
.= Im (proj2,|[1,0]|) by A17, RELAT_1:129
.= {(proj2 . |[1,0]|)} by SETWISEO:8
.= {(|[1,0]| `2)} by PSCOMP_1:def_6
.= {0} by EUCLID:52 ;
then upper_bound ((proj2 | (E-most P)) .: the carrier of ((TOP-REAL 2) | (E-most P))) = 0 by SEQ_4:9;
then upper_bound (proj2 | (E-most P)) = 0 by PSCOMP_1:def_2;
hence E-max P = |[1,0]| by A3, PSCOMP_1:def_23; ::_thesis: verum
end;
theorem :: JGRAPH_5:31
for f being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds f . p = proj1 . p ) holds
f is continuous
proof
let f be Function of (TOP-REAL 2),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) holds f . p = proj1 . p ) implies f is continuous )
assume A1: for p being Point of (TOP-REAL 2) holds f . p = proj1 . p ; ::_thesis: f is continuous
reconsider f = f as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ;
(TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93;
then f is continuous by A1, JGRAPH_2:29;
hence f is continuous by PRE_TOPC:32; ::_thesis: verum
end;
theorem Th32: :: JGRAPH_5:32
for f being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds f . p = proj2 . p ) holds
f is continuous
proof
let f be Function of (TOP-REAL 2),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) holds f . p = proj2 . p ) implies f is continuous )
assume A1: for p being Point of (TOP-REAL 2) holds f . p = proj2 . p ; ::_thesis: f is continuous
reconsider f = f as Function of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #),R^1 ;
(TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93;
then f is continuous by A1, JGRAPH_2:30;
hence f is continuous by PRE_TOPC:32; ::_thesis: verum
end;
theorem Th33: :: JGRAPH_5:33
for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
( Upper_Arc P c= P & Lower_Arc P c= P )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies ( Upper_Arc P c= P & Lower_Arc P c= P ) )
assume P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: ( Upper_Arc P c= P & Lower_Arc P c= P )
then P is being_simple_closed_curve by JGRAPH_3:26;
hence ( Upper_Arc P c= P & Lower_Arc P c= P ) by JORDAN6:61; ::_thesis: verum
end;
theorem Th34: :: JGRAPH_5:34
for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) }
proof
reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } )
set P4 = Lower_Arc P;
set P1 = Upper_Arc P;
set P2 = Lower_Arc P;
set Q = Vertical_Line 0;
set p8 = First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0));
set pj = Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0));
A1: LSeg (|[0,(- 1)]|,|[0,1]|) c= Vertical_Line 0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (|[0,(- 1)]|,|[0,1]|) or x in Vertical_Line 0 )
assume x in LSeg (|[0,(- 1)]|,|[0,1]|) ; ::_thesis: x in Vertical_Line 0
then consider l being Real such that
A2: x = ((1 - l) * |[0,(- 1)]|) + (l * |[0,1]|) and
0 <= l and
l <= 1 ;
(((1 - l) * |[0,(- 1)]|) + (l * |[0,1]|)) `1 = (((1 - l) * |[0,(- 1)]|) `1) + ((l * |[0,1]|) `1) by TOPREAL3:2
.= ((1 - l) * (|[0,(- 1)]| `1)) + ((l * |[0,1]|) `1) by TOPREAL3:4
.= ((1 - l) * (|[0,(- 1)]| `1)) + (l * (|[0,1]| `1)) by TOPREAL3:4
.= ((1 - l) * 0) + (l * (|[0,1]| `1)) by EUCLID:52
.= ((1 - l) * 0) + (l * 0) by EUCLID:52
.= 0 ;
hence x in Vertical_Line 0 by A2; ::_thesis: verum
end;
reconsider R = Upper_Arc P as non empty Subset of (TOP-REAL 2) ;
assume A3: P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) }
then A4: P is being_simple_closed_curve by JGRAPH_3:26;
then A5: Upper_Arc P is_an_arc_of W-min P, E-max P by JORDAN6:def_8;
then consider f being Function of I[01],((TOP-REAL 2) | R) such that
A6: f is being_homeomorphism and
A7: f . 0 = W-min P and
A8: f . 1 = E-max P by TOPREAL1:def_1;
A9: ( dom f = the carrier of I[01] & dom h2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
A10: ex P2 being non empty Subset of (TOP-REAL 2) st
( P2 is_an_arc_of E-max P, W-min P & (Upper_Arc P) /\ P2 = {(W-min P),(E-max P)} & (Upper_Arc P) \/ P2 = P & (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point (P2,(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 ) by A4, JORDAN6:def_8;
then A11: Upper_Arc P c= P by XBOOLE_1:7;
A12: rng f = [#] ((TOP-REAL 2) | R) by A6, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
A13: ( S-bound P = - 1 & N-bound P = 1 ) by A3, Th28;
A14: Vertical_Line 0 is closed by JORDAN6:30;
A15: for p being Point of (TOP-REAL 2) holds h2 . p = proj2 . p ;
A16: ( W-bound P = - 1 & E-bound P = 1 ) by A3, Th28;
then A17: Upper_Arc P meets Vertical_Line 0 by A4, A13, A1, JORDAN6:69, XBOOLE_1:64;
A18: Lower_Arc P meets Vertical_Line 0 by A4, A16, A13, A1, JORDAN6:70, XBOOLE_1:64;
A19: (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 by A4, JORDAN6:def_9;
Upper_Arc P is closed by A5, JORDAN6:11;
then (Upper_Arc P) /\ (Vertical_Line 0) is closed by A14, TOPS_1:8;
then A20: First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) in (Upper_Arc P) /\ (Vertical_Line 0) by A5, A17, JORDAN5C:def_1;
then First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) in Upper_Arc P by XBOOLE_0:def_4;
then consider x8 being set such that
A21: x8 in dom f and
A22: First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = f . x8 by A12, FUNCT_1:def_3;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then x8 in { r where r is Real : ( 0 <= r & r <= 1 ) } by A21, RCOMP_1:def_1;
then consider r8 being Real such that
A23: x8 = r8 and
A24: 0 <= r8 and
A25: r8 <= 1 ;
A26: Vertical_Line 0 is closed by JORDAN6:30;
(Upper_Arc P) /\ (Vertical_Line 0) c= {|[0,(- 1)]|,|[0,1]|}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Upper_Arc P) /\ (Vertical_Line 0) or x in {|[0,(- 1)]|,|[0,1]|} )
assume A27: x in (Upper_Arc P) /\ (Vertical_Line 0) ; ::_thesis: x in {|[0,(- 1)]|,|[0,1]|}
then x in Upper_Arc P by XBOOLE_0:def_4;
then x in P by A10, XBOOLE_0:def_3;
then consider q being Point of (TOP-REAL 2) such that
A28: q = x and
A29: |.q.| = 1 by A3;
x in Vertical_Line 0 by A27, XBOOLE_0:def_4;
then A30: ex p being Point of (TOP-REAL 2) st
( p = x & p `1 = 0 ) ;
then (0 ^2) + ((q `2) ^2) = 1 ^2 by A28, A29, JGRAPH_3:1;
then ( q `2 = 1 or q `2 = - 1 ) by SQUARE_1:41;
then ( x = |[0,(- 1)]| or x = |[0,1]| ) by A30, A28, EUCLID:53;
hence x in {|[0,(- 1)]|,|[0,1]|} by TARSKI:def_2; ::_thesis: verum
end;
then ( First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = |[0,(- 1)]| or First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = |[0,1]| ) by A20, TARSKI:def_2;
then A31: ( (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0))) `2 = - 1 or (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0))) `2 = 1 ) by EUCLID:52;
A32: now__::_thesis:_not_r8_=_0
assume r8 = 0 ; ::_thesis: contradiction
then First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = |[(- 1),0]| by A3, A7, A22, A23, Th29;
hence contradiction by A31, EUCLID:52; ::_thesis: verum
end;
A33: rng (h2 * f) c= the carrier of R^1 ;
A34: the carrier of ((TOP-REAL 2) | R) = R by PRE_TOPC:8;
then rng f c= the carrier of (TOP-REAL 2) by XBOOLE_1:1;
then dom (h2 * f) = the carrier of I[01] by A9, RELAT_1:27;
then reconsider g0 = h2 * f as Function of I[01],R^1 by A33, FUNCT_2:2;
A35: f is one-to-one by A6, TOPS_2:def_5;
A36: f is continuous by A6, TOPS_2:def_5;
A37: ( ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f . t = p & p `2 > 0 ) implies for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
q `2 >= 0 )
proof
assume ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f . t = p & p `2 > 0 ) ; ::_thesis: for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
q `2 >= 0
then consider p being Point of (TOP-REAL 2), t being Real such that
A38: 0 < t and
A39: t < 1 and
A40: f . t = p and
A41: p `2 > 0 ;
now__::_thesis:_for_q_being_Point_of_(TOP-REAL_2)_holds_
(_not_q_in_Upper_Arc_P_or_not_q_`2_<_0_)
assume ex q being Point of (TOP-REAL 2) st
( q in Upper_Arc P & q `2 < 0 ) ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL 2) such that
A42: q in Upper_Arc P and
A43: q `2 < 0 ;
rng f = [#] ((TOP-REAL 2) | R) by A6, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
then consider x being set such that
A44: x in dom f and
A45: q = f . x by A42, FUNCT_1:def_3;
A46: dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then A47: x in { r where r is Real : ( 0 <= r & r <= 1 ) } by A44, RCOMP_1:def_1;
t in { v where v is Real : ( 0 <= v & v <= 1 ) } by A38, A39;
then A48: t in [.0,1.] by RCOMP_1:def_1;
then A49: (h2 * f) . t = h2 . p by A40, A46, FUNCT_1:13
.= p `2 by PSCOMP_1:def_6 ;
consider r being Real such that
A50: x = r and
A51: 0 <= r and
A52: r <= 1 by A47;
A53: (h2 * f) . r = h2 . q by A44, A45, A50, FUNCT_1:13
.= q `2 by PSCOMP_1:def_6 ;
now__::_thesis:_(_(_r_<_t_&_contradiction_)_or_(_t_<_r_&_contradiction_)_or_(_t_=_r_&_contradiction_)_)
percases ( r < t or t < r or t = r ) by XXREAL_0:1;
caseA54: r < t ; ::_thesis: contradiction
then reconsider B = [.r,t.] as non empty Subset of I[01] by A44, A50, A48, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g0 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A55: (q `2) * (p `2) < 0 by A41, A43, XREAL_1:132;
t in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A54;
then t in B by RCOMP_1:def_1;
then A56: p `2 = g . t by A49, FUNCT_1:49;
r in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A54;
then r in B by RCOMP_1:def_1;
then A57: q `2 = g . r by A53, FUNCT_1:49;
g0 is continuous by A36, A15, Th7, Th32;
then A58: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (r,t) = I[01] | B by A39, A51, A54, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A59: g . r1 = 0 and
A60: r < r1 and
A61: r1 < t by A54, A58, A55, A57, A56, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A60, A61;
then A62: r1 in B by RCOMP_1:def_1;
r1 < 1 by A39, A61, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A51, A60;
then A63: r1 in dom f by A46, RCOMP_1:def_1;
then f . r1 in rng f by FUNCT_1:def_3;
then f . r1 in R by A34;
then f . r1 in P by A11;
then consider q3 being Point of (TOP-REAL 2) such that
A64: q3 = f . r1 and
A65: |.q3.| = 1 by A3;
A66: q3 `2 = h2 . (f . r1) by A64, PSCOMP_1:def_6
.= g0 . r1 by A63, FUNCT_1:13
.= 0 by A59, A62, FUNCT_1:49 ;
then A67: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A65, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A67, SQUARE_1:41;
caseA68: q3 `1 = 1 ; ::_thesis: contradiction
A69: 1 in dom f by A46, XXREAL_1:1;
q3 = |[1,0]| by A66, A68, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A8, A35, A39, A61, A63, A64, A69, FUNCT_1:def_4; ::_thesis: verum
end;
caseA70: q3 `1 = - 1 ; ::_thesis: contradiction
A71: 0 in dom f by A46, XXREAL_1:1;
q3 = |[(- 1),0]| by A66, A70, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A7, A35, A51, A60, A63, A64, A71, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA72: t < r ; ::_thesis: contradiction
then reconsider B = [.t,r.] as non empty Subset of I[01] by A44, A50, A48, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g0 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A73: (q `2) * (p `2) < 0 by A41, A43, XREAL_1:132;
t in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A72;
then t in B by RCOMP_1:def_1;
then A74: p `2 = g . t by A49, FUNCT_1:49;
r in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A72;
then r in B by RCOMP_1:def_1;
then A75: q `2 = g . r by A53, FUNCT_1:49;
g0 is continuous by A36, A15, Th7, Th32;
then A76: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (t,r) = I[01] | B by A38, A52, A72, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A77: g . r1 = 0 and
A78: t < r1 and
A79: r1 < r by A72, A76, A73, A75, A74, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A78, A79;
then A80: r1 in B by RCOMP_1:def_1;
r1 < 1 by A52, A79, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A38, A78;
then A81: r1 in dom f by A46, RCOMP_1:def_1;
then f . r1 in rng f by FUNCT_1:def_3;
then f . r1 in R by A34;
then f . r1 in P by A11;
then consider q3 being Point of (TOP-REAL 2) such that
A82: q3 = f . r1 and
A83: |.q3.| = 1 by A3;
A84: q3 `2 = h2 . (f . r1) by A82, PSCOMP_1:def_6
.= (h2 * f) . r1 by A81, FUNCT_1:13
.= 0 by A77, A80, FUNCT_1:49 ;
then A85: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A83, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A85, SQUARE_1:41;
caseA86: q3 `1 = 1 ; ::_thesis: contradiction
A87: 1 in dom f by A46, XXREAL_1:1;
q3 = |[1,0]| by A84, A86, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A8, A35, A52, A79, A81, A82, A87, FUNCT_1:def_4; ::_thesis: verum
end;
caseA88: q3 `1 = - 1 ; ::_thesis: contradiction
A89: 0 in dom f by A46, XXREAL_1:1;
q3 = |[(- 1),0]| by A84, A88, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A7, A35, A38, A78, A81, A82, A89, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case t = r ; ::_thesis: contradiction
hence contradiction by A41, A43, A53, A49; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
q `2 >= 0 ; ::_thesis: verum
end;
reconsider R = Lower_Arc P as non empty Subset of (TOP-REAL 2) ;
A90: Lower_Arc P is_an_arc_of E-max P, W-min P by A4, JORDAN6:def_9;
then consider f2 being Function of I[01],((TOP-REAL 2) | R) such that
A91: f2 is being_homeomorphism and
A92: f2 . 0 = E-max P and
A93: f2 . 1 = W-min P by TOPREAL1:def_1;
A94: ( dom f2 = the carrier of I[01] & dom h2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
A95: rng (h2 * f2) c= the carrier of R^1 ;
A96: the carrier of ((TOP-REAL 2) | R) = R by PRE_TOPC:8;
then rng f2 c= the carrier of (TOP-REAL 2) by XBOOLE_1:1;
then dom (h2 * f2) = the carrier of I[01] by A94, RELAT_1:27;
then reconsider g1 = h2 * f2 as Function of I[01],R^1 by A95, FUNCT_2:2;
A97: f2 is one-to-one by A91, TOPS_2:def_5;
A98: (Upper_Arc P) \/ (Lower_Arc P) = P by A4, JORDAN6:def_9;
then A99: Lower_Arc P c= P by XBOOLE_1:7;
A100: (Lower_Arc P) /\ (Vertical_Line 0) c= {|[0,(- 1)]|,|[0,1]|}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Lower_Arc P) /\ (Vertical_Line 0) or x in {|[0,(- 1)]|,|[0,1]|} )
assume A101: x in (Lower_Arc P) /\ (Vertical_Line 0) ; ::_thesis: x in {|[0,(- 1)]|,|[0,1]|}
then x in Lower_Arc P by XBOOLE_0:def_4;
then x in P by A98, XBOOLE_0:def_3;
then consider q being Point of (TOP-REAL 2) such that
A102: q = x and
A103: |.q.| = 1 by A3;
x in Vertical_Line 0 by A101, XBOOLE_0:def_4;
then A104: ex p being Point of (TOP-REAL 2) st
( p = x & p `1 = 0 ) ;
then (0 ^2) + ((q `2) ^2) = 1 ^2 by A102, A103, JGRAPH_3:1;
then ( q `2 = 1 or q `2 = - 1 ) by SQUARE_1:41;
then ( x = |[0,(- 1)]| or x = |[0,1]| ) by A104, A102, EUCLID:53;
hence x in {|[0,(- 1)]|,|[0,1]|} by TARSKI:def_2; ::_thesis: verum
end;
A105: for p being Point of (TOP-REAL 2) holds h2 . p = proj2 . p ;
A106: f2 is continuous by A91, TOPS_2:def_5;
A107: ( ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f2 . t = p & p `2 > 0 ) implies for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
q `2 >= 0 )
proof
assume ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f2 . t = p & p `2 > 0 ) ; ::_thesis: for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
q `2 >= 0
then consider p being Point of (TOP-REAL 2), t being Real such that
A108: 0 < t and
A109: t < 1 and
A110: f2 . t = p and
A111: p `2 > 0 ;
now__::_thesis:_for_q_being_Point_of_(TOP-REAL_2)_holds_
(_not_q_in_Lower_Arc_P_or_not_q_`2_<_0_)
assume ex q being Point of (TOP-REAL 2) st
( q in Lower_Arc P & q `2 < 0 ) ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL 2) such that
A112: q in Lower_Arc P and
A113: q `2 < 0 ;
rng f2 = [#] ((TOP-REAL 2) | R) by A91, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
then consider x being set such that
A114: x in dom f2 and
A115: q = f2 . x by A112, FUNCT_1:def_3;
A116: dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then A117: x in { r where r is Real : ( 0 <= r & r <= 1 ) } by A114, RCOMP_1:def_1;
t in { v where v is Real : ( 0 <= v & v <= 1 ) } by A108, A109;
then A118: t in [.0,1.] by RCOMP_1:def_1;
then A119: (h2 * f2) . t = h2 . p by A110, A116, FUNCT_1:13
.= p `2 by PSCOMP_1:def_6 ;
consider r being Real such that
A120: x = r and
A121: 0 <= r and
A122: r <= 1 by A117;
A123: (h2 * f2) . r = h2 . q by A114, A115, A120, FUNCT_1:13
.= q `2 by PSCOMP_1:def_6 ;
now__::_thesis:_(_(_r_<_t_&_contradiction_)_or_(_t_<_r_&_contradiction_)_or_(_t_=_r_&_contradiction_)_)
percases ( r < t or t < r or t = r ) by XXREAL_0:1;
caseA124: r < t ; ::_thesis: contradiction
then reconsider B = [.r,t.] as non empty Subset of I[01] by A114, A120, A118, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g1 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A125: (q `2) * (p `2) < 0 by A111, A113, XREAL_1:132;
t in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A124;
then t in B by RCOMP_1:def_1;
then A126: p `2 = g . t by A119, FUNCT_1:49;
r in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A124;
then r in B by RCOMP_1:def_1;
then A127: q `2 = g . r by A123, FUNCT_1:49;
g1 is continuous by A106, A105, Th7, Th32;
then A128: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (r,t) = I[01] | B by A109, A121, A124, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A129: g . r1 = 0 and
A130: r < r1 and
A131: r1 < t by A124, A128, A125, A127, A126, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A130, A131;
then A132: r1 in B by RCOMP_1:def_1;
r1 < 1 by A109, A131, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A121, A130;
then A133: r1 in dom f2 by A116, RCOMP_1:def_1;
then f2 . r1 in rng f2 by FUNCT_1:def_3;
then f2 . r1 in R by A96;
then f2 . r1 in P by A99;
then consider q3 being Point of (TOP-REAL 2) such that
A134: q3 = f2 . r1 and
A135: |.q3.| = 1 by A3;
A136: q3 `2 = h2 . (f2 . r1) by A134, PSCOMP_1:def_6
.= (h2 * f2) . r1 by A133, FUNCT_1:13
.= 0 by A129, A132, FUNCT_1:49 ;
then A137: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A135, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A137, SQUARE_1:41;
caseA138: q3 `1 = 1 ; ::_thesis: contradiction
A139: 0 in dom f2 by A116, XXREAL_1:1;
q3 = |[1,0]| by A136, A138, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A92, A97, A121, A130, A133, A134, A139, FUNCT_1:def_4; ::_thesis: verum
end;
caseA140: q3 `1 = - 1 ; ::_thesis: contradiction
A141: 1 in dom f2 by A116, XXREAL_1:1;
q3 = |[(- 1),0]| by A136, A140, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A93, A97, A109, A131, A133, A134, A141, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA142: t < r ; ::_thesis: contradiction
then reconsider B = [.t,r.] as non empty Subset of I[01] by A114, A120, A118, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g1 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A143: (q `2) * (p `2) < 0 by A111, A113, XREAL_1:132;
t in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A142;
then t in B by RCOMP_1:def_1;
then A144: p `2 = g . t by A119, FUNCT_1:49;
r in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A142;
then r in B by RCOMP_1:def_1;
then A145: q `2 = g . r by A123, FUNCT_1:49;
g1 is continuous by A106, A105, Th7, Th32;
then A146: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (t,r) = I[01] | B by A108, A122, A142, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A147: g . r1 = 0 and
A148: t < r1 and
A149: r1 < r by A142, A146, A143, A145, A144, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A148, A149;
then A150: r1 in B by RCOMP_1:def_1;
r1 < 1 by A122, A149, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A108, A148;
then A151: r1 in dom f2 by A116, RCOMP_1:def_1;
then f2 . r1 in rng f2 by FUNCT_1:def_3;
then f2 . r1 in R by A96;
then f2 . r1 in P by A99;
then consider q3 being Point of (TOP-REAL 2) such that
A152: q3 = f2 . r1 and
A153: |.q3.| = 1 by A3;
A154: q3 `2 = h2 . (f2 . r1) by A152, PSCOMP_1:def_6
.= g1 . r1 by A151, FUNCT_1:13
.= 0 by A147, A150, FUNCT_1:49 ;
then A155: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A153, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A155, SQUARE_1:41;
caseA156: q3 `1 = 1 ; ::_thesis: contradiction
A157: 0 in dom f2 by A116, XXREAL_1:1;
q3 = |[1,0]| by A154, A156, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A92, A97, A108, A148, A151, A152, A157, FUNCT_1:def_4; ::_thesis: verum
end;
caseA158: q3 `1 = - 1 ; ::_thesis: contradiction
A159: 1 in dom f2 by A116, XXREAL_1:1;
q3 = |[(- 1),0]| by A154, A158, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A93, A97, A122, A149, A151, A152, A159, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case t = r ; ::_thesis: contradiction
hence contradiction by A111, A113, A123, A119; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
q `2 >= 0 ; ::_thesis: verum
end;
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A160: W-min P in Upper_Arc P by A10, XBOOLE_0:def_4;
A161: ( W-bound P = - 1 & E-bound P = 1 ) by A3, Th28;
now__::_thesis:_not_r8_=_1
assume r8 = 1 ; ::_thesis: contradiction
then First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = |[1,0]| by A3, A8, A22, A23, Th30;
hence contradiction by A31, EUCLID:52; ::_thesis: verum
end;
then A162: 1 > r8 by A25, XXREAL_0:1;
Lower_Arc P is closed by A90, JORDAN6:11;
then (Lower_Arc P) /\ (Vertical_Line 0) is closed by A26, TOPS_1:8;
then Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) in (Lower_Arc P) /\ (Vertical_Line 0) by A90, A18, JORDAN5C:def_2;
then A163: ( Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = |[0,(- 1)]| or Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = |[0,1]| ) by A100, TARSKI:def_2;
E-max P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A164: E-max P in Upper_Arc P by A10, XBOOLE_0:def_4;
A165: { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } c= Upper_Arc P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } or x in Upper_Arc P )
assume x in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } ; ::_thesis: x in Upper_Arc P
then consider p being Point of (TOP-REAL 2) such that
A166: p = x and
A167: p in P and
A168: p `2 >= 0 ;
now__::_thesis:_(_(_p_`2_=_0_&_x_in_Upper_Arc_P_)_or_(_p_`2_>_0_&_x_in_Upper_Arc_P_)_)
percases ( p `2 = 0 or p `2 > 0 ) by A168;
caseA169: p `2 = 0 ; ::_thesis: x in Upper_Arc P
ex p8 being Point of (TOP-REAL 2) st
( p8 = p & |.p8.| = 1 ) by A3, A167;
then 1 = sqrt (((p `1) ^2) + ((p `2) ^2)) by JGRAPH_3:1
.= abs (p `1) by A169, COMPLEX1:72 ;
then ( p = |[(p `1),(p `2)]| & (p `1) ^2 = 1 ^2 ) by COMPLEX1:75, EUCLID:53;
then ( p = |[1,0]| or p = |[(- 1),0]| ) by A169, SQUARE_1:41;
hence x in Upper_Arc P by A3, A164, A160, A166, Th29, Th30; ::_thesis: verum
end;
caseA170: p `2 > 0 ; ::_thesis: x in Upper_Arc P
now__::_thesis:_x_in_Upper_Arc_P
assume not x in Upper_Arc P ; ::_thesis: contradiction
then A171: x in Lower_Arc P by A98, A166, A167, XBOOLE_0:def_3;
rng f2 = [#] ((TOP-REAL 2) | R) by A91, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
then consider x2 being set such that
A172: x2 in dom f2 and
A173: p = f2 . x2 by A166, A171, FUNCT_1:def_3;
dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then x2 in { r where r is Real : ( 0 <= r & r <= 1 ) } by A172, RCOMP_1:def_1;
then consider t2 being Real such that
A174: x2 = t2 and
A175: 0 <= t2 and
A176: t2 <= 1 ;
A177: |[0,(- 1)]| `2 = - 1 by EUCLID:52;
now__::_thesis:_not_t2_=_1
assume t2 = 1 ; ::_thesis: contradiction
then p = |[(- 1),0]| by A3, A93, A173, A174, Th29;
hence contradiction by A170, EUCLID:52; ::_thesis: verum
end;
then A178: t2 < 1 by A176, XXREAL_0:1;
A179: now__::_thesis:_not_t2_=_0
assume t2 = 0 ; ::_thesis: contradiction
then p = |[1,0]| by A3, A92, A173, A174, Th30;
hence contradiction by A170, EUCLID:52; ::_thesis: verum
end;
|[0,(- 1)]| `1 = 0 by EUCLID:52;
then |.|[0,(- 1)]|.| = sqrt ((0 ^2) + ((- 1) ^2)) by A177, JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
then A180: |[0,(- 1)]| in { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ;
now__::_thesis:_(_(_|[0,(-_1)]|_in_Upper_Arc_P_&_contradiction_)_or_(_|[0,(-_1)]|_in_Lower_Arc_P_&_contradiction_)_)
percases ( |[0,(- 1)]| in Upper_Arc P or |[0,(- 1)]| in Lower_Arc P ) by A3, A98, A180, XBOOLE_0:def_3;
case |[0,(- 1)]| in Upper_Arc P ; ::_thesis: contradiction
hence contradiction by A19, A161, A31, A163, A22, A23, A24, A32, A162, A37, A177, EUCLID:52; ::_thesis: verum
end;
case |[0,(- 1)]| in Lower_Arc P ; ::_thesis: contradiction
hence contradiction by A107, A170, A173, A174, A175, A179, A178, A177; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence x in Upper_Arc P ; ::_thesis: verum
end;
end;
end;
hence x in Upper_Arc P ; ::_thesis: verum
end;
Upper_Arc P c= { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) }
proof
let x2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x2 in Upper_Arc P or x2 in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } )
assume A181: x2 in Upper_Arc P ; ::_thesis: x2 in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) }
then reconsider q3 = x2 as Point of (TOP-REAL 2) ;
q3 `2 >= 0 by A19, A161, A31, A163, A22, A23, A24, A32, A162, A37, A181, EUCLID:52;
hence x2 in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A11, A181; ::_thesis: verum
end;
hence Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A165, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th35: :: JGRAPH_5:35
for P being non empty compact Subset of (TOP-REAL 2) st P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } holds
Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) }
proof
reconsider h2 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider Q = Vertical_Line 0 as Subset of (TOP-REAL 2) ;
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } implies Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } )
set P4 = Lower_Arc P;
reconsider P1 = Lower_Arc P as Subset of (TOP-REAL 2) ;
reconsider P2 = Upper_Arc P as Subset of (TOP-REAL 2) ;
set pj = First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0));
set p8 = Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0));
A1: LSeg (|[0,(- 1)]|,|[0,1]|) c= Q
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (|[0,(- 1)]|,|[0,1]|) or x in Q )
assume x in LSeg (|[0,(- 1)]|,|[0,1]|) ; ::_thesis: x in Q
then consider l being Real such that
A2: x = ((1 - l) * |[0,(- 1)]|) + (l * |[0,1]|) and
0 <= l and
l <= 1 ;
(((1 - l) * |[0,(- 1)]|) + (l * |[0,1]|)) `1 = (((1 - l) * |[0,(- 1)]|) `1) + ((l * |[0,1]|) `1) by TOPREAL3:2
.= ((1 - l) * (|[0,(- 1)]| `1)) + ((l * |[0,1]|) `1) by TOPREAL3:4
.= ((1 - l) * (|[0,(- 1)]| `1)) + (l * (|[0,1]| `1)) by TOPREAL3:4
.= ((1 - l) * 0) + (l * (|[0,1]| `1)) by EUCLID:52
.= ((1 - l) * 0) + (l * 0) by EUCLID:52
.= 0 ;
hence x in Q by A2; ::_thesis: verum
end;
assume A3: P = { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ; ::_thesis: Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) }
then A4: P is being_simple_closed_curve by JGRAPH_3:26;
then A5: (Upper_Arc P) \/ (Lower_Arc P) = P by JORDAN6:def_9;
then A6: Lower_Arc P c= P by XBOOLE_1:7;
A7: P2 /\ Q c= {|[0,(- 1)]|,|[0,1]|}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P2 /\ Q or x in {|[0,(- 1)]|,|[0,1]|} )
assume A8: x in P2 /\ Q ; ::_thesis: x in {|[0,(- 1)]|,|[0,1]|}
then x in P2 by XBOOLE_0:def_4;
then x in P by A5, XBOOLE_0:def_3;
then consider q being Point of (TOP-REAL 2) such that
A9: q = x and
A10: |.q.| = 1 by A3;
x in Q by A8, XBOOLE_0:def_4;
then A11: ex p being Point of (TOP-REAL 2) st
( p = x & p `1 = 0 ) ;
then (0 ^2) + ((q `2) ^2) = 1 ^2 by A9, A10, JGRAPH_3:1;
then ( q `2 = 1 or q `2 = - 1 ) by SQUARE_1:41;
then ( x = |[0,(- 1)]| or x = |[0,1]| ) by A11, A9, EUCLID:53;
hence x in {|[0,(- 1)]|,|[0,1]|} by TARSKI:def_2; ::_thesis: verum
end;
A12: for p being Point of (TOP-REAL 2) holds h2 . p = proj2 . p ;
reconsider R = Lower_Arc P as non empty Subset of (TOP-REAL 2) ;
A13: Vertical_Line 0 is closed by JORDAN6:30;
A14: Vertical_Line 0 is closed by JORDAN6:30;
A15: for p being Point of (TOP-REAL 2) holds h2 . p = proj2 . p ;
A16: ( S-bound P = - 1 & N-bound P = 1 ) by A3, Th28;
A17: ( W-bound P = - 1 & E-bound P = 1 ) by A3, Th28;
then A18: P1 meets Q by A4, A16, A1, JORDAN6:70, XBOOLE_1:64;
A19: P2 meets Q by A4, A17, A16, A1, JORDAN6:69, XBOOLE_1:64;
A20: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A4, JORDAN6:def_9;
A21: Lower_Arc P is_an_arc_of E-max P, W-min P by A4, JORDAN6:def_9;
then consider f being Function of I[01],((TOP-REAL 2) | R) such that
A22: f is being_homeomorphism and
A23: f . 0 = E-max P and
A24: f . 1 = W-min P by TOPREAL1:def_1;
A25: ( dom f = the carrier of I[01] & dom h2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
A26: rng f = [#] ((TOP-REAL 2) | R) by A22, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
A27: Upper_Arc P c= P by A5, XBOOLE_1:7;
A28: rng (h2 * f) c= the carrier of R^1 ;
A29: the carrier of ((TOP-REAL 2) | R) = R by PRE_TOPC:8;
then rng f c= the carrier of (TOP-REAL 2) by XBOOLE_1:1;
then dom (h2 * f) = the carrier of I[01] by A25, RELAT_1:27;
then reconsider g0 = h2 * f as Function of I[01],R^1 by A28, FUNCT_2:2;
A30: f is one-to-one by A22, TOPS_2:def_5;
A31: f is continuous by A22, TOPS_2:def_5;
A32: ( ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f . t = p & p `2 < 0 ) implies for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
q `2 <= 0 )
proof
assume ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f . t = p & p `2 < 0 ) ; ::_thesis: for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
q `2 <= 0
then consider p being Point of (TOP-REAL 2), t being Real such that
A33: 0 < t and
A34: t < 1 and
A35: f . t = p and
A36: p `2 < 0 ;
now__::_thesis:_for_q_being_Point_of_(TOP-REAL_2)_holds_
(_not_q_in_Lower_Arc_P_or_not_q_`2_>_0_)
assume ex q being Point of (TOP-REAL 2) st
( q in Lower_Arc P & q `2 > 0 ) ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL 2) such that
A37: q in Lower_Arc P and
A38: q `2 > 0 ;
rng f = [#] ((TOP-REAL 2) | R) by A22, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
then consider x being set such that
A39: x in dom f and
A40: q = f . x by A37, FUNCT_1:def_3;
A41: dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then A42: x in { r where r is Real : ( 0 <= r & r <= 1 ) } by A39, RCOMP_1:def_1;
t in { v where v is Real : ( 0 <= v & v <= 1 ) } by A33, A34;
then A43: t in [.0,1.] by RCOMP_1:def_1;
then A44: (h2 * f) . t = h2 . p by A35, A41, FUNCT_1:13
.= p `2 by PSCOMP_1:def_6 ;
consider r being Real such that
A45: x = r and
A46: 0 <= r and
A47: r <= 1 by A42;
A48: (h2 * f) . r = h2 . q by A39, A40, A45, FUNCT_1:13
.= q `2 by PSCOMP_1:def_6 ;
now__::_thesis:_(_(_r_<_t_&_contradiction_)_or_(_t_<_r_&_contradiction_)_or_(_t_=_r_&_contradiction_)_)
percases ( r < t or t < r or t = r ) by XXREAL_0:1;
caseA49: r < t ; ::_thesis: contradiction
then reconsider B = [.r,t.] as non empty Subset of I[01] by A39, A45, A43, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g0 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A50: (q `2) * (p `2) < 0 by A36, A38, XREAL_1:132;
t in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A49;
then t in B by RCOMP_1:def_1;
then A51: p `2 = g . t by A44, FUNCT_1:49;
r in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A49;
then r in B by RCOMP_1:def_1;
then A52: q `2 = g . r by A48, FUNCT_1:49;
g0 is continuous by A31, A12, Th7, Th32;
then A53: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (r,t) = I[01] | B by A34, A46, A49, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A54: g . r1 = 0 and
A55: r < r1 and
A56: r1 < t by A49, A53, A50, A52, A51, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A55, A56;
then A57: r1 in B by RCOMP_1:def_1;
r1 < 1 by A34, A56, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A46, A55;
then A58: r1 in dom f by A41, RCOMP_1:def_1;
then f . r1 in rng f by FUNCT_1:def_3;
then f . r1 in R by A29;
then f . r1 in P by A6;
then consider q3 being Point of (TOP-REAL 2) such that
A59: q3 = f . r1 and
A60: |.q3.| = 1 by A3;
A61: q3 `2 = h2 . (f . r1) by A59, PSCOMP_1:def_6
.= (h2 * f) . r1 by A58, FUNCT_1:13
.= 0 by A54, A57, FUNCT_1:49 ;
then A62: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A60, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A62, SQUARE_1:41;
caseA63: q3 `1 = 1 ; ::_thesis: contradiction
A64: 0 in dom f by A41, XXREAL_1:1;
q3 = |[1,0]| by A61, A63, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A23, A30, A46, A55, A58, A59, A64, FUNCT_1:def_4; ::_thesis: verum
end;
caseA65: q3 `1 = - 1 ; ::_thesis: contradiction
A66: 1 in dom f by A41, XXREAL_1:1;
q3 = |[(- 1),0]| by A61, A65, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A24, A30, A34, A56, A58, A59, A66, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA67: t < r ; ::_thesis: contradiction
then reconsider B = [.t,r.] as non empty Subset of I[01] by A39, A45, A43, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g0 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A68: (q `2) * (p `2) < 0 by A36, A38, XREAL_1:132;
t in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A67;
then t in B by RCOMP_1:def_1;
then A69: p `2 = g . t by A44, FUNCT_1:49;
r in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A67;
then r in B by RCOMP_1:def_1;
then A70: q `2 = g . r by A48, FUNCT_1:49;
g0 is continuous by A31, A12, Th7, Th32;
then A71: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (t,r) = I[01] | B by A33, A47, A67, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A72: g . r1 = 0 and
A73: t < r1 and
A74: r1 < r by A67, A71, A68, A70, A69, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A73, A74;
then A75: r1 in B by RCOMP_1:def_1;
r1 < 1 by A47, A74, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A33, A73;
then A76: r1 in dom f by A41, RCOMP_1:def_1;
then f . r1 in rng f by FUNCT_1:def_3;
then f . r1 in R by A29;
then f . r1 in P by A6;
then consider q3 being Point of (TOP-REAL 2) such that
A77: q3 = f . r1 and
A78: |.q3.| = 1 by A3;
A79: q3 `2 = h2 . (f . r1) by A77, PSCOMP_1:def_6
.= (h2 * f) . r1 by A76, FUNCT_1:13
.= 0 by A72, A75, FUNCT_1:49 ;
then A80: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A78, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A80, SQUARE_1:41;
caseA81: q3 `1 = 1 ; ::_thesis: contradiction
A82: 0 in dom f by A41, XXREAL_1:1;
q3 = |[1,0]| by A79, A81, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A23, A30, A33, A73, A76, A77, A82, FUNCT_1:def_4; ::_thesis: verum
end;
caseA83: q3 `1 = - 1 ; ::_thesis: contradiction
A84: 1 in dom f by A41, XXREAL_1:1;
q3 = |[(- 1),0]| by A79, A83, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A24, A30, A47, A74, A76, A77, A84, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case t = r ; ::_thesis: contradiction
hence contradiction by A36, A38, A48, A44; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
q `2 <= 0 ; ::_thesis: verum
end;
reconsider R = Upper_Arc P as non empty Subset of (TOP-REAL 2) ;
A85: Upper_Arc P is_an_arc_of W-min P, E-max P by A4, JORDAN6:def_8;
then consider f2 being Function of I[01],((TOP-REAL 2) | R) such that
A86: f2 is being_homeomorphism and
A87: f2 . 0 = W-min P and
A88: f2 . 1 = E-max P by TOPREAL1:def_1;
A89: ( dom f2 = the carrier of I[01] & dom h2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
A90: rng (h2 * f2) c= the carrier of R^1 ;
A91: the carrier of ((TOP-REAL 2) | R) = R by PRE_TOPC:8;
then rng f2 c= the carrier of (TOP-REAL 2) by XBOOLE_1:1;
then dom (h2 * f2) = the carrier of I[01] by A89, RELAT_1:27;
then reconsider g1 = h2 * f2 as Function of I[01],R^1 by A90, FUNCT_2:2;
A92: f2 is one-to-one by A86, TOPS_2:def_5;
A93: f2 is continuous by A86, TOPS_2:def_5;
A94: ( ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f2 . t = p & p `2 < 0 ) implies for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
q `2 <= 0 )
proof
assume ex p being Point of (TOP-REAL 2) ex t being Real st
( 0 < t & t < 1 & f2 . t = p & p `2 < 0 ) ; ::_thesis: for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
q `2 <= 0
then consider p being Point of (TOP-REAL 2), t being Real such that
A95: 0 < t and
A96: t < 1 and
A97: f2 . t = p and
A98: p `2 < 0 ;
now__::_thesis:_for_q_being_Point_of_(TOP-REAL_2)_holds_
(_not_q_in_Upper_Arc_P_or_not_q_`2_>_0_)
assume ex q being Point of (TOP-REAL 2) st
( q in Upper_Arc P & q `2 > 0 ) ; ::_thesis: contradiction
then consider q being Point of (TOP-REAL 2) such that
A99: q in Upper_Arc P and
A100: q `2 > 0 ;
rng f2 = [#] ((TOP-REAL 2) | R) by A86, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
then consider x being set such that
A101: x in dom f2 and
A102: q = f2 . x by A99, FUNCT_1:def_3;
A103: dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then A104: x in { r where r is Real : ( 0 <= r & r <= 1 ) } by A101, RCOMP_1:def_1;
t in { v where v is Real : ( 0 <= v & v <= 1 ) } by A95, A96;
then A105: t in [.0,1.] by RCOMP_1:def_1;
then A106: (h2 * f2) . t = h2 . p by A97, A103, FUNCT_1:13
.= p `2 by PSCOMP_1:def_6 ;
consider r being Real such that
A107: x = r and
A108: 0 <= r and
A109: r <= 1 by A104;
A110: (h2 * f2) . r = h2 . q by A101, A102, A107, FUNCT_1:13
.= q `2 by PSCOMP_1:def_6 ;
now__::_thesis:_(_(_r_<_t_&_contradiction_)_or_(_t_<_r_&_contradiction_)_or_(_t_=_r_&_contradiction_)_)
percases ( r < t or t < r or t = r ) by XXREAL_0:1;
caseA111: r < t ; ::_thesis: contradiction
then reconsider B = [.r,t.] as non empty Subset of I[01] by A101, A107, A105, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g1 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A112: (q `2) * (p `2) < 0 by A98, A100, XREAL_1:132;
t in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A111;
then t in B by RCOMP_1:def_1;
then A113: p `2 = g . t by A106, FUNCT_1:49;
r in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A111;
then r in B by RCOMP_1:def_1;
then A114: q `2 = g . r by A110, FUNCT_1:49;
g1 is continuous by A93, A15, Th7, Th32;
then A115: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (r,t) = I[01] | B by A96, A108, A111, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A116: g . r1 = 0 and
A117: r < r1 and
A118: r1 < t by A111, A115, A112, A114, A113, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( r <= r4 & r4 <= t ) } by A117, A118;
then A119: r1 in B by RCOMP_1:def_1;
r1 < 1 by A96, A118, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A108, A117;
then A120: r1 in dom f2 by A103, RCOMP_1:def_1;
then f2 . r1 in rng f2 by FUNCT_1:def_3;
then f2 . r1 in R by A91;
then f2 . r1 in P by A27;
then consider q3 being Point of (TOP-REAL 2) such that
A121: q3 = f2 . r1 and
A122: |.q3.| = 1 by A3;
A123: q3 `2 = h2 . (f2 . r1) by A121, PSCOMP_1:def_6
.= (h2 * f2) . r1 by A120, FUNCT_1:13
.= 0 by A116, A119, FUNCT_1:49 ;
then A124: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A122, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A124, SQUARE_1:41;
caseA125: q3 `1 = 1 ; ::_thesis: contradiction
A126: 1 in dom f2 by A103, XXREAL_1:1;
q3 = |[1,0]| by A123, A125, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A88, A92, A96, A118, A120, A121, A126, FUNCT_1:def_4; ::_thesis: verum
end;
caseA127: q3 `1 = - 1 ; ::_thesis: contradiction
A128: 0 in dom f2 by A103, XXREAL_1:1;
q3 = |[(- 1),0]| by A123, A127, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A87, A92, A108, A117, A120, A121, A128, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA129: t < r ; ::_thesis: contradiction
then reconsider B = [.t,r.] as non empty Subset of I[01] by A101, A107, A105, BORSUK_1:40, XXREAL_1:1, XXREAL_2:def_12;
reconsider B0 = B as Subset of I[01] ;
reconsider g = g1 | B0 as Function of (I[01] | B0),R^1 by PRE_TOPC:9;
A130: (q `2) * (p `2) < 0 by A98, A100, XREAL_1:132;
t in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A129;
then t in B by RCOMP_1:def_1;
then A131: p `2 = g . t by A106, FUNCT_1:49;
r in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A129;
then r in B by RCOMP_1:def_1;
then A132: q `2 = g . r by A110, FUNCT_1:49;
g1 is continuous by A93, A15, Th7, Th32;
then A133: g is continuous by TOPMETR:7;
Closed-Interval-TSpace (t,r) = I[01] | B by A95, A109, A129, TOPMETR:20, TOPMETR:23;
then consider r1 being Real such that
A134: g . r1 = 0 and
A135: t < r1 and
A136: r1 < r by A129, A133, A130, A132, A131, TOPREAL5:8;
r1 in { r4 where r4 is Real : ( t <= r4 & r4 <= r ) } by A135, A136;
then A137: r1 in B by RCOMP_1:def_1;
r1 < 1 by A109, A136, XXREAL_0:2;
then r1 in { r2 where r2 is Real : ( 0 <= r2 & r2 <= 1 ) } by A95, A135;
then A138: r1 in dom f2 by A103, RCOMP_1:def_1;
then f2 . r1 in rng f2 by FUNCT_1:def_3;
then f2 . r1 in R by A91;
then f2 . r1 in P by A27;
then consider q3 being Point of (TOP-REAL 2) such that
A139: q3 = f2 . r1 and
A140: |.q3.| = 1 by A3;
A141: q3 `2 = h2 . (f2 . r1) by A139, PSCOMP_1:def_6
.= (h2 * f2) . r1 by A138, FUNCT_1:13
.= 0 by A134, A137, FUNCT_1:49 ;
then A142: 1 ^2 = ((q3 `1) ^2) + (0 ^2) by A140, JGRAPH_3:1
.= (q3 `1) ^2 ;
now__::_thesis:_(_(_q3_`1_=_1_&_contradiction_)_or_(_q3_`1_=_-_1_&_contradiction_)_)
percases ( q3 `1 = 1 or q3 `1 = - 1 ) by A142, SQUARE_1:41;
caseA143: q3 `1 = 1 ; ::_thesis: contradiction
A144: 1 in dom f2 by A103, XXREAL_1:1;
q3 = |[1,0]| by A141, A143, EUCLID:53
.= E-max P by A3, Th30 ;
hence contradiction by A88, A92, A109, A136, A138, A139, A144, FUNCT_1:def_4; ::_thesis: verum
end;
caseA145: q3 `1 = - 1 ; ::_thesis: contradiction
A146: 0 in dom f2 by A103, XXREAL_1:1;
q3 = |[(- 1),0]| by A141, A145, EUCLID:53
.= W-min P by A3, Th29 ;
hence contradiction by A87, A92, A95, A135, A138, A139, A146, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case t = r ; ::_thesis: contradiction
hence contradiction by A98, A100, A110, A106; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
q `2 <= 0 ; ::_thesis: verum
end;
A147: ( W-bound P = - 1 & E-bound P = 1 ) by A3, Th28;
Lower_Arc P is closed by A21, JORDAN6:11;
then P1 /\ Q is closed by A13, TOPS_1:8;
then A148: Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) in P1 /\ Q by A21, A18, JORDAN5C:def_2;
then Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) in P1 by XBOOLE_0:def_4;
then consider x8 being set such that
A149: x8 in dom f and
A150: Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = f . x8 by A26, FUNCT_1:def_3;
dom f = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then x8 in { r where r is Real : ( 0 <= r & r <= 1 ) } by A149, RCOMP_1:def_1;
then consider r8 being Real such that
A151: x8 = r8 and
A152: 0 <= r8 and
A153: r8 <= 1 ;
P1 /\ Q c= {|[0,(- 1)]|,|[0,1]|}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P1 /\ Q or x in {|[0,(- 1)]|,|[0,1]|} )
assume A154: x in P1 /\ Q ; ::_thesis: x in {|[0,(- 1)]|,|[0,1]|}
then x in P1 by XBOOLE_0:def_4;
then x in P by A5, XBOOLE_0:def_3;
then consider q being Point of (TOP-REAL 2) such that
A155: q = x and
A156: |.q.| = 1 by A3;
x in Q by A154, XBOOLE_0:def_4;
then A157: ex p being Point of (TOP-REAL 2) st
( p = x & p `1 = 0 ) ;
then (0 ^2) + ((q `2) ^2) = 1 ^2 by A155, A156, JGRAPH_3:1;
then ( q `2 = 1 or q `2 = - 1 ) by SQUARE_1:41;
then ( x = |[0,(- 1)]| or x = |[0,1]| ) by A157, A155, EUCLID:53;
hence x in {|[0,(- 1)]|,|[0,1]|} by TARSKI:def_2; ::_thesis: verum
end;
then ( Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = |[0,(- 1)]| or Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = |[0,1]| ) by A148, TARSKI:def_2;
then A158: ( (Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0))) `2 = - 1 or (Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0))) `2 = 1 ) by EUCLID:52;
A159: now__::_thesis:_not_r8_=_0
assume r8 = 0 ; ::_thesis: contradiction
then Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = |[1,0]| by A3, A23, A150, A151, Th30;
hence contradiction by A158, EUCLID:52; ::_thesis: verum
end;
Upper_Arc P is closed by A85, JORDAN6:11;
then P2 /\ Q is closed by A14, TOPS_1:8;
then First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) in P2 /\ Q by A85, A19, JORDAN5C:def_1;
then A160: ( First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = |[0,(- 1)]| or First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line 0)) = |[0,1]| ) by A7, TARSKI:def_2;
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A161: W-min P in Lower_Arc P by A20, XBOOLE_0:def_4;
A162: (First_Point ((Upper_Arc P),(W-min P),(E-max P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 > (Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line (((W-bound P) + (E-bound P)) / 2)))) `2 by A4, JORDAN6:def_9;
now__::_thesis:_not_r8_=_1
assume r8 = 1 ; ::_thesis: contradiction
then Last_Point ((Lower_Arc P),(E-max P),(W-min P),(Vertical_Line 0)) = |[(- 1),0]| by A3, A24, A150, A151, Th29;
hence contradiction by A158, EUCLID:52; ::_thesis: verum
end;
then A163: 1 > r8 by A153, XXREAL_0:1;
E-max P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A164: E-max P in Lower_Arc P by A20, XBOOLE_0:def_4;
A165: { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } c= Lower_Arc P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } or x in Lower_Arc P )
assume x in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } ; ::_thesis: x in Lower_Arc P
then consider p being Point of (TOP-REAL 2) such that
A166: p = x and
A167: p in P and
A168: p `2 <= 0 ;
now__::_thesis:_(_(_p_`2_=_0_&_x_in_Lower_Arc_P_)_or_(_p_`2_<_0_&_x_in_Lower_Arc_P_)_)
percases ( p `2 = 0 or p `2 < 0 ) by A168;
caseA169: p `2 = 0 ; ::_thesis: x in Lower_Arc P
ex p8 being Point of (TOP-REAL 2) st
( p8 = p & |.p8.| = 1 ) by A3, A167;
then 1 = sqrt (((p `1) ^2) + ((p `2) ^2)) by JGRAPH_3:1
.= abs (p `1) by A169, COMPLEX1:72 ;
then ( p = |[(p `1),(p `2)]| & (p `1) ^2 = 1 ^2 ) by COMPLEX1:75, EUCLID:53;
then ( p = |[1,0]| or p = |[(- 1),0]| ) by A169, SQUARE_1:41;
hence x in Lower_Arc P by A3, A164, A161, A166, Th29, Th30; ::_thesis: verum
end;
caseA170: p `2 < 0 ; ::_thesis: x in Lower_Arc P
now__::_thesis:_x_in_Lower_Arc_P
assume not x in Lower_Arc P ; ::_thesis: contradiction
then A171: x in Upper_Arc P by A5, A166, A167, XBOOLE_0:def_3;
rng f2 = [#] ((TOP-REAL 2) | R) by A86, TOPS_2:def_5
.= R by PRE_TOPC:def_5 ;
then consider x2 being set such that
A172: x2 in dom f2 and
A173: p = f2 . x2 by A166, A171, FUNCT_1:def_3;
dom f2 = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then x2 in { r where r is Real : ( 0 <= r & r <= 1 ) } by A172, RCOMP_1:def_1;
then consider t2 being Real such that
A174: x2 = t2 and
A175: 0 <= t2 and
A176: t2 <= 1 ;
A177: |[0,1]| `2 = 1 by EUCLID:52;
now__::_thesis:_not_t2_=_1
assume t2 = 1 ; ::_thesis: contradiction
then p = |[1,0]| by A3, A88, A173, A174, Th30;
hence contradiction by A170, EUCLID:52; ::_thesis: verum
end;
then A178: t2 < 1 by A176, XXREAL_0:1;
A179: now__::_thesis:_not_t2_=_0
assume t2 = 0 ; ::_thesis: contradiction
then p = |[(- 1),0]| by A3, A87, A173, A174, Th29;
hence contradiction by A170, EUCLID:52; ::_thesis: verum
end;
|[0,1]| `1 = 0 by EUCLID:52;
then |.|[0,1]|.| = sqrt ((0 ^2) + (1 ^2)) by A177, JGRAPH_3:1
.= 1 by SQUARE_1:18 ;
then A180: |[0,1]| in { q where q is Point of (TOP-REAL 2) : |.q.| = 1 } ;
now__::_thesis:_(_(_|[0,1]|_in_Lower_Arc_P_&_contradiction_)_or_(_|[0,1]|_in_Upper_Arc_P_&_contradiction_)_)
percases ( |[0,1]| in Lower_Arc P or |[0,1]| in Upper_Arc P ) by A3, A5, A180, XBOOLE_0:def_3;
case |[0,1]| in Lower_Arc P ; ::_thesis: contradiction
hence contradiction by A162, A147, A158, A160, A150, A151, A152, A159, A163, A32, A177, EUCLID:52; ::_thesis: verum
end;
case |[0,1]| in Upper_Arc P ; ::_thesis: contradiction
hence contradiction by A94, A170, A173, A174, A175, A179, A178, A177; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence x in Lower_Arc P ; ::_thesis: verum
end;
end;
end;
hence x in Lower_Arc P ; ::_thesis: verum
end;
Lower_Arc P c= { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) }
proof
let x2 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x2 in Lower_Arc P or x2 in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } )
assume A181: x2 in Lower_Arc P ; ::_thesis: x2 in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) }
then reconsider q3 = x2 as Point of (TOP-REAL 2) ;
q3 `2 <= 0 by A162, A147, A158, A160, A150, A151, A152, A159, A163, A32, A181, EUCLID:52;
hence x2 in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A6, A181; ::_thesis: verum
end;
hence Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A165, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th36: :: JGRAPH_5:36
for a, b, d, e being Real st a <= b & e > 0 holds
ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) )
proof
let a, b, d, e be Real; ::_thesis: ( a <= b & e > 0 implies ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) ) )
assume that
A1: a <= b and
A2: e > 0 ; ::_thesis: ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) )
set S = Closed-Interval-TSpace (a,b);
defpred S1[ set , set ] means for r being Real st $1 = r holds
$2 = (e * r) + d;
set X = the carrier of (Closed-Interval-TSpace (a,b));
A3: the carrier of (Closed-Interval-TSpace (a,b)) = [.a,b.] by A1, TOPMETR:18;
then reconsider B = the carrier of (Closed-Interval-TSpace (a,b)) as Subset of R^1 by TOPMETR:17;
A4: R^1 | B = Closed-Interval-TSpace (a,b) by A1, A3, TOPMETR:19;
set T = Closed-Interval-TSpace (((e * a) + d),((e * b) + d));
set Y = the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d)));
A5: e * a <= e * b by A1, A2, XREAL_1:64;
then A6: the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) = [.((e * a) + d),((e * b) + d).] by TOPMETR:18, XREAL_1:7;
then reconsider C = the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) as Subset of R^1 by TOPMETR:17;
defpred S2[ set , set ] means for r being Real st r = $1 holds
$2 = (e * r) + d;
Closed-Interval-TSpace (((e * a) + d),((e * b) + d)) = TopSpaceMetr (Closed-Interval-MSpace (((e * a) + d),((e * b) + d))) by TOPMETR:def_7;
then A7: Closed-Interval-TSpace (((e * a) + d),((e * b) + d)) is T_2 by PCOMPS_1:34;
A8: for x being set st x in the carrier of (Closed-Interval-TSpace (a,b)) holds
ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of (Closed-Interval-TSpace (a,b)) implies ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) & S1[x,y] ) )
assume A9: x in the carrier of (Closed-Interval-TSpace (a,b)) ; ::_thesis: ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) & S1[x,y] )
then reconsider r1 = x as Real by A3;
set y1 = (e * r1) + d;
r1 <= b by A3, A9, XXREAL_1:1;
then e * r1 <= e * b by A2, XREAL_1:64;
then A10: (e * r1) + d <= (e * b) + d by XREAL_1:7;
a <= r1 by A3, A9, XXREAL_1:1;
then e * a <= e * r1 by A2, XREAL_1:64;
then (e * a) + d <= (e * r1) + d by XREAL_1:7;
then ( ( for r being Real st x = r holds
(e * r1) + d = (e * r) + d ) & (e * r1) + d in the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) ) by A6, A10, XXREAL_1:1;
hence ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) & S1[x,y] ) ; ::_thesis: verum
end;
ex f being Function of the carrier of (Closed-Interval-TSpace (a,b)), the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) st
for x being set st x in the carrier of (Closed-Interval-TSpace (a,b)) holds
S1[x,f . x] from FUNCT_2:sch_1(A8);
then consider f1 being Function of the carrier of (Closed-Interval-TSpace (a,b)), the carrier of (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) such that
A11: for x being set st x in the carrier of (Closed-Interval-TSpace (a,b)) holds
S1[x,f1 . x] ;
reconsider f2 = f1 as Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) ;
A12: for r being Real st r in [.a,b.] holds
f2 . r = (e * r) + d by A3, A11;
A13: dom f2 = the carrier of (Closed-Interval-TSpace (a,b)) by FUNCT_2:def_1;
[#] (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) c= rng f2
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [#] (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) or y in rng f2 )
assume A14: y in [#] (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) ; ::_thesis: y in rng f2
then reconsider ry = y as Real by A6;
ry <= (e * b) + d by A6, A14, XXREAL_1:1;
then ((e * b) + d) - d >= ry - d by XREAL_1:9;
then (b * e) / e >= (ry - d) / e by A2, XREAL_1:72;
then A15: b >= (ry - d) / e by A2, XCMPLX_1:89;
(e * a) + d <= ry by A6, A14, XXREAL_1:1;
then ((e * a) + d) - d <= ry - d by XREAL_1:9;
then (a * e) / e <= (ry - d) / e by A2, XREAL_1:72;
then a <= (ry - d) / e by A2, XCMPLX_1:89;
then A16: (ry - d) / e in [.a,b.] by A15, XXREAL_1:1;
then f2 . ((ry - d) / e) = (e * ((ry - d) / e)) + d by A3, A11
.= (ry - d) + d by A2, XCMPLX_1:87
.= ry ;
hence y in rng f2 by A3, A13, A16, FUNCT_1:3; ::_thesis: verum
end;
then A17: rng f2 = [#] (Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) by XBOOLE_0:def_10;
then reconsider f3 = f1 as Function of (Closed-Interval-TSpace (a,b)),R^1 by A6, A13, FUNCT_2:2, TOPMETR:17;
for x1, x2 being set st x1 in dom f2 & x2 in dom f2 & f2 . x1 = f2 . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom f2 & x2 in dom f2 & f2 . x1 = f2 . x2 implies x1 = x2 )
assume that
A18: x1 in dom f2 and
A19: x2 in dom f2 and
A20: f2 . x1 = f2 . x2 ; ::_thesis: x1 = x2
reconsider r2 = x2 as Real by A3, A13, A19;
reconsider r1 = x1 as Real by A3, A13, A18;
f2 . x1 = (e * r1) + d by A11, A18;
then ((e * r1) + d) - d = ((e * r2) + d) - d by A11, A19, A20
.= e * r2 ;
then (r1 * e) / e = r2 by A2, XCMPLX_1:89;
hence x1 = x2 by A2, XCMPLX_1:89; ::_thesis: verum
end;
then A21: ( dom f2 = [#] (Closed-Interval-TSpace (a,b)) & f2 is one-to-one ) by FUNCT_1:def_4, FUNCT_2:def_1;
A22: for x being set st x in the carrier of R^1 holds
ex y being set st
( y in the carrier of R^1 & S2[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of R^1 implies ex y being set st
( y in the carrier of R^1 & S2[x,y] ) )
assume x in the carrier of R^1 ; ::_thesis: ex y being set st
( y in the carrier of R^1 & S2[x,y] )
then reconsider rx = x as Real by TOPMETR:17;
reconsider ry = (e * rx) + d as Real ;
for r being Real st r = x holds
ry = (e * r) + d ;
hence ex y being set st
( y in the carrier of R^1 & S2[x,y] ) by TOPMETR:17; ::_thesis: verum
end;
ex f4 being Function of the carrier of R^1, the carrier of R^1 st
for x being set st x in the carrier of R^1 holds
S2[x,f4 . x] from FUNCT_2:sch_1(A22);
then consider f4 being Function of the carrier of R^1, the carrier of R^1 such that
A23: for x being set st x in the carrier of R^1 holds
S2[x,f4 . x] ;
reconsider f5 = f4 as Function of R^1,R^1 ;
A24: for x being Real holds f5 . x = (e * x) + d by A23, TOPMETR:17;
A25: (dom f5) /\ B = REAL /\ B by FUNCT_2:def_1, TOPMETR:17
.= B by TOPMETR:17, XBOOLE_1:28 ;
A26: for x being set st x in dom f3 holds
f3 . x = f5 . x
proof
let x be set ; ::_thesis: ( x in dom f3 implies f3 . x = f5 . x )
assume A27: x in dom f3 ; ::_thesis: f3 . x = f5 . x
then reconsider rx = x as Real by A3, A13;
f4 . x = (e * rx) + d by A23, TOPMETR:17;
hence f3 . x = f5 . x by A11, A27; ::_thesis: verum
end;
dom f3 = B by FUNCT_2:def_1;
then f3 = f5 | B by A25, A26, FUNCT_1:46;
then A28: f3 is continuous by A24, A4, TOPMETR:7, TOPMETR:21;
A29: Closed-Interval-TSpace (a,b) is compact by A1, HEINE:4;
R^1 | C = Closed-Interval-TSpace (((e * a) + d),((e * b) + d)) by A5, A6, TOPMETR:19, XREAL_1:7;
then f2 is being_homeomorphism by A21, A17, A28, A29, A7, COMPTS_1:17, TOPMETR:6;
hence ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * a) + d),((e * b) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) ) by A12; ::_thesis: verum
end;
theorem Th37: :: JGRAPH_5:37
for a, b, d, e being Real st a <= b & e < 0 holds
ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) )
proof
let a, b, d, e be Real; ::_thesis: ( a <= b & e < 0 implies ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) ) )
assume that
A1: a <= b and
A2: e < 0 ; ::_thesis: ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) )
set S = Closed-Interval-TSpace (a,b);
defpred S1[ set , set ] means for r being Real st $1 = r holds
$2 = (e * r) + d;
set X = the carrier of (Closed-Interval-TSpace (a,b));
A3: the carrier of (Closed-Interval-TSpace (a,b)) = [.a,b.] by A1, TOPMETR:18;
then reconsider B = the carrier of (Closed-Interval-TSpace (a,b)) as Subset of R^1 by TOPMETR:17;
A4: R^1 | B = Closed-Interval-TSpace (a,b) by A1, A3, TOPMETR:19;
set T = Closed-Interval-TSpace (((e * b) + d),((e * a) + d));
set Y = the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d)));
A5: e * a >= e * b by A1, A2, XREAL_1:65;
then A6: the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) = [.((e * b) + d),((e * a) + d).] by TOPMETR:18, XREAL_1:7;
then reconsider C = the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) as Subset of R^1 by TOPMETR:17;
defpred S2[ set , set ] means for r being Real st r = $1 holds
$2 = (e * r) + d;
Closed-Interval-TSpace (((e * b) + d),((e * a) + d)) = TopSpaceMetr (Closed-Interval-MSpace (((e * b) + d),((e * a) + d))) by TOPMETR:def_7;
then A7: Closed-Interval-TSpace (((e * b) + d),((e * a) + d)) is T_2 by PCOMPS_1:34;
A8: for x being set st x in the carrier of (Closed-Interval-TSpace (a,b)) holds
ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of (Closed-Interval-TSpace (a,b)) implies ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) & S1[x,y] ) )
assume A9: x in the carrier of (Closed-Interval-TSpace (a,b)) ; ::_thesis: ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) & S1[x,y] )
then reconsider r1 = x as Real by A3;
set y1 = (e * r1) + d;
r1 <= b by A3, A9, XXREAL_1:1;
then e * r1 >= e * b by A2, XREAL_1:65;
then A10: (e * r1) + d >= (e * b) + d by XREAL_1:7;
a <= r1 by A3, A9, XXREAL_1:1;
then e * a >= e * r1 by A2, XREAL_1:65;
then (e * a) + d >= (e * r1) + d by XREAL_1:7;
then ( ( for r being Real st x = r holds
(e * r1) + d = (e * r) + d ) & (e * r1) + d in the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) ) by A6, A10, XXREAL_1:1;
hence ex y being set st
( y in the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) & S1[x,y] ) ; ::_thesis: verum
end;
ex f being Function of the carrier of (Closed-Interval-TSpace (a,b)), the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) st
for x being set st x in the carrier of (Closed-Interval-TSpace (a,b)) holds
S1[x,f . x] from FUNCT_2:sch_1(A8);
then consider f1 being Function of the carrier of (Closed-Interval-TSpace (a,b)), the carrier of (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) such that
A11: for x being set st x in the carrier of (Closed-Interval-TSpace (a,b)) holds
S1[x,f1 . x] ;
reconsider f2 = f1 as Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) ;
A12: for r being Real st r in [.a,b.] holds
f2 . r = (e * r) + d by A3, A11;
A13: dom f2 = the carrier of (Closed-Interval-TSpace (a,b)) by FUNCT_2:def_1;
[#] (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) c= rng f2
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in [#] (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) or y in rng f2 )
assume A14: y in [#] (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) ; ::_thesis: y in rng f2
then reconsider ry = y as Real by A6;
ry <= (e * a) + d by A6, A14, XXREAL_1:1;
then ((e * a) + d) - d >= ry - d by XREAL_1:9;
then (a * e) / e <= (ry - d) / e by A2, XREAL_1:73;
then A15: a <= (ry - d) / e by A2, XCMPLX_1:89;
(e * b) + d <= ry by A6, A14, XXREAL_1:1;
then ((e * b) + d) - d <= ry - d by XREAL_1:9;
then (b * e) / e >= (ry - d) / e by A2, XREAL_1:73;
then b >= (ry - d) / e by A2, XCMPLX_1:89;
then A16: (ry - d) / e in [.a,b.] by A15, XXREAL_1:1;
then f2 . ((ry - d) / e) = (e * ((ry - d) / e)) + d by A3, A11
.= (ry - d) + d by A2, XCMPLX_1:87
.= ry ;
hence y in rng f2 by A3, A13, A16, FUNCT_1:3; ::_thesis: verum
end;
then A17: rng f2 = [#] (Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) by XBOOLE_0:def_10;
then reconsider f3 = f1 as Function of (Closed-Interval-TSpace (a,b)),R^1 by A6, A13, FUNCT_2:2, TOPMETR:17;
for x1, x2 being set st x1 in dom f2 & x2 in dom f2 & f2 . x1 = f2 . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom f2 & x2 in dom f2 & f2 . x1 = f2 . x2 implies x1 = x2 )
assume that
A18: x1 in dom f2 and
A19: x2 in dom f2 and
A20: f2 . x1 = f2 . x2 ; ::_thesis: x1 = x2
reconsider r2 = x2 as Real by A3, A13, A19;
reconsider r1 = x1 as Real by A3, A13, A18;
f2 . x1 = (e * r1) + d by A11, A18;
then ((e * r1) + d) - d = ((e * r2) + d) - d by A11, A19, A20
.= e * r2 ;
then (r1 * e) / e = r2 by A2, XCMPLX_1:89;
hence x1 = x2 by A2, XCMPLX_1:89; ::_thesis: verum
end;
then A21: ( dom f2 = [#] (Closed-Interval-TSpace (a,b)) & f2 is one-to-one ) by FUNCT_1:def_4, FUNCT_2:def_1;
A22: for x being set st x in the carrier of R^1 holds
ex y being set st
( y in the carrier of R^1 & S2[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of R^1 implies ex y being set st
( y in the carrier of R^1 & S2[x,y] ) )
assume x in the carrier of R^1 ; ::_thesis: ex y being set st
( y in the carrier of R^1 & S2[x,y] )
then reconsider rx = x as Real by TOPMETR:17;
reconsider ry = (e * rx) + d as Real ;
for r being Real st r = x holds
ry = (e * r) + d ;
hence ex y being set st
( y in the carrier of R^1 & S2[x,y] ) by TOPMETR:17; ::_thesis: verum
end;
ex f4 being Function of the carrier of R^1, the carrier of R^1 st
for x being set st x in the carrier of R^1 holds
S2[x,f4 . x] from FUNCT_2:sch_1(A22);
then consider f4 being Function of the carrier of R^1, the carrier of R^1 such that
A23: for x being set st x in the carrier of R^1 holds
S2[x,f4 . x] ;
reconsider f5 = f4 as Function of R^1,R^1 ;
A24: for x being Real holds f5 . x = (e * x) + d by A23, TOPMETR:17;
A25: (dom f5) /\ B = REAL /\ B by FUNCT_2:def_1, TOPMETR:17
.= B by TOPMETR:17, XBOOLE_1:28 ;
A26: for x being set st x in dom f3 holds
f3 . x = f5 . x
proof
let x be set ; ::_thesis: ( x in dom f3 implies f3 . x = f5 . x )
assume A27: x in dom f3 ; ::_thesis: f3 . x = f5 . x
then reconsider rx = x as Real by A3, A13;
f4 . x = (e * rx) + d by A23, TOPMETR:17;
hence f3 . x = f5 . x by A11, A27; ::_thesis: verum
end;
dom f3 = B by FUNCT_2:def_1;
then f3 = f5 | B by A25, A26, FUNCT_1:46;
then A28: f3 is continuous by A24, A4, TOPMETR:7, TOPMETR:21;
A29: Closed-Interval-TSpace (a,b) is compact by A1, HEINE:4;
R^1 | C = Closed-Interval-TSpace (((e * b) + d),((e * a) + d)) by A5, A6, TOPMETR:19, XREAL_1:7;
then f2 is being_homeomorphism by A21, A17, A28, A29, A7, COMPTS_1:17, TOPMETR:6;
hence ex f being Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (((e * b) + d),((e * a) + d))) st
( f is being_homeomorphism & ( for r being Real st r in [.a,b.] holds
f . r = (e * r) + d ) ) by A12; ::_thesis: verum
end;
theorem Th38: :: JGRAPH_5:38
ex f being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) st
( f is being_homeomorphism & ( for r being Real st r in [.0,1.] holds
f . r = ((- 2) * r) + 1 ) & f . 0 = 1 & f . 1 = - 1 )
proof
consider f being Function of I[01],(Closed-Interval-TSpace ((((- 2) * 1) + 1),(((- 2) * 0) + 1))) such that
A1: f is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds
f . r = ((- 2) * r) + 1 by Th37, TOPMETR:20;
1 in [.0,1.] by XXREAL_1:1;
then A3: f . 1 = - 1 by A2;
f . 0 = ((- 2) * 0) + 1 by A2, Lm1;
hence ex f being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) st
( f is being_homeomorphism & ( for r being Real st r in [.0,1.] holds
f . r = ((- 2) * r) + 1 ) & f . 0 = 1 & f . 1 = - 1 ) by A1, A2, A3; ::_thesis: verum
end;
theorem Th39: :: JGRAPH_5:39
ex f being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) st
( f is being_homeomorphism & ( for r being Real st r in [.0,1.] holds
f . r = (2 * r) - 1 ) & f . 0 = - 1 & f . 1 = 1 )
proof
consider f being Function of I[01],(Closed-Interval-TSpace (((2 * 0) + (- 1)),((2 * 1) + (- 1)))) such that
A1: f is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds
f . r = (2 * r) + (- 1) by Th36, TOPMETR:20;
A3: for r being Real st r in [.0,1.] holds
f . r = (2 * r) - 1
proof
let r be Real; ::_thesis: ( r in [.0,1.] implies f . r = (2 * r) - 1 )
assume r in [.0,1.] ; ::_thesis: f . r = (2 * r) - 1
hence f . r = (2 * r) + (- 1) by A2
.= (2 * r) - 1 ;
::_thesis: verum
end;
1 in [.0,1.] by XXREAL_1:1;
then A4: f . 1 = (2 * 1) - 1 by A3
.= 1 ;
f . 0 = (2 * 0) - 1 by A3, Lm1
.= - 1 ;
hence ex f being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) st
( f is being_homeomorphism & ( for r being Real st r in [.0,1.] holds
f . r = (2 * r) - 1 ) & f . 0 = - 1 & f . 1 = 1 ) by A1, A3, A4; ::_thesis: verum
end;
Lm5: now__::_thesis:_for_P_being_non_empty_compact_Subset_of_(TOP-REAL_2)_st_P_=__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_|.p.|_=_1__}__holds_
(_proj1_|_(Lower_Arc_P)_is_continuous_Function_of_((TOP-REAL_2)_|_(Lower_Arc_P)),(Closed-Interval-TSpace_((-_1),1))_&_proj1_|_(Lower_Arc_P)_is_one-to-one_&_rng_(proj1_|_(Lower_Arc_P))_=_[#]_(Closed-Interval-TSpace_((-_1),1))_)
reconsider B = [.(- 1),1.] as non empty Subset of R^1 by TOPMETR:17, XXREAL_1:1;
reconsider g = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } implies ( proj1 | (Lower_Arc P) is continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) & proj1 | (Lower_Arc P) is one-to-one & rng (proj1 | (Lower_Arc P)) = [#] (Closed-Interval-TSpace ((- 1),1)) ) )
set K0 = Lower_Arc P;
reconsider g2 = g | (Lower_Arc P) as Function of ((TOP-REAL 2) | (Lower_Arc P)),R^1 by PRE_TOPC:9;
A1: for p being Point of ((TOP-REAL 2) | (Lower_Arc P)) holds g2 . p = proj1 . p
proof
let p be Point of ((TOP-REAL 2) | (Lower_Arc P)); ::_thesis: g2 . p = proj1 . p
p in the carrier of ((TOP-REAL 2) | (Lower_Arc P)) ;
then p in Lower_Arc P by PRE_TOPC:8;
hence g2 . p = proj1 . p by FUNCT_1:49; ::_thesis: verum
end;
assume A2: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } ; ::_thesis: ( proj1 | (Lower_Arc P) is continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) & proj1 | (Lower_Arc P) is one-to-one & rng (proj1 | (Lower_Arc P)) = [#] (Closed-Interval-TSpace ((- 1),1)) )
then A3: Lower_Arc P c= P by Th33;
A4: dom g2 = the carrier of ((TOP-REAL 2) | (Lower_Arc P)) by FUNCT_2:def_1;
then A5: dom g2 = Lower_Arc P by PRE_TOPC:8;
rng g2 c= the carrier of (Closed-Interval-TSpace ((- 1),1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g2 or x in the carrier of (Closed-Interval-TSpace ((- 1),1)) )
assume x in rng g2 ; ::_thesis: x in the carrier of (Closed-Interval-TSpace ((- 1),1))
then consider z being set such that
A6: z in dom g2 and
A7: x = g2 . z by FUNCT_1:def_3;
z in P by A5, A3, A6;
then consider p being Point of (TOP-REAL 2) such that
A8: p = z and
A9: |.p.| = 1 by A2;
1 ^2 = ((p `1) ^2) + ((p `2) ^2) by A9, JGRAPH_3:1;
then 1 - ((p `1) ^2) >= 0 by XREAL_1:63;
then - (1 - ((p `1) ^2)) <= 0 ;
then ((p `1) ^2) - 1 <= 0 ;
then A10: ( - 1 <= p `1 & p `1 <= 1 ) by SQUARE_1:43;
x = proj1 . p by A1, A6, A7, A8
.= p `1 by PSCOMP_1:def_5 ;
then x in [.(- 1),1.] by A10, XXREAL_1:1;
hence x in the carrier of (Closed-Interval-TSpace ((- 1),1)) by TOPMETR:18; ::_thesis: verum
end;
then reconsider g3 = g2 as Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A4, FUNCT_2:2;
dom g3 = [#] ((TOP-REAL 2) | (Lower_Arc P)) by FUNCT_2:def_1;
then A11: dom g3 = Lower_Arc P by PRE_TOPC:def_5;
A12: for x, y being set st x in dom g3 & y in dom g3 & g3 . x = g3 . y holds
x = y
proof
let x, y be set ; ::_thesis: ( x in dom g3 & y in dom g3 & g3 . x = g3 . y implies x = y )
assume that
A13: x in dom g3 and
A14: y in dom g3 and
A15: g3 . x = g3 . y ; ::_thesis: x = y
reconsider p2 = y as Point of (TOP-REAL 2) by A11, A14;
A16: g3 . y = proj1 . p2 by A1, A14
.= p2 `1 by PSCOMP_1:def_5 ;
reconsider p1 = x as Point of (TOP-REAL 2) by A11, A13;
A17: g3 . x = proj1 . p1 by A1, A13
.= p1 `1 by PSCOMP_1:def_5 ;
p2 in P by A3, A11, A14;
then ex p22 being Point of (TOP-REAL 2) st
( p2 = p22 & |.p22.| = 1 ) by A2;
then A18: 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
p2 in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 <= 0 ) } by A2, A11, A14, Th35;
then A19: ex p44 being Point of (TOP-REAL 2) st
( p2 = p44 & p44 in P & p44 `2 <= 0 ) ;
p1 in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 <= 0 ) } by A2, A11, A13, Th35;
then A20: ex p33 being Point of (TOP-REAL 2) st
( p1 = p33 & p33 in P & p33 `2 <= 0 ) ;
p1 in P by A3, A11, A13;
then ex p11 being Point of (TOP-REAL 2) st
( p1 = p11 & |.p11.| = 1 ) by A2;
then 1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
then (- (p1 `2)) ^2 = (p2 `2) ^2 by A15, A17, A16, A18;
then - (p1 `2) = sqrt ((- (p2 `2)) ^2) by A20, SQUARE_1:22;
then - (p1 `2) = - (p2 `2) by A19, SQUARE_1:22;
then p1 = |[(p2 `1),(p2 `2)]| by A15, A17, A16, EUCLID:53
.= p2 by EUCLID:53 ;
hence x = y ; ::_thesis: verum
end;
A21: [#] (Closed-Interval-TSpace ((- 1),1)) c= rng g3
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [#] (Closed-Interval-TSpace ((- 1),1)) or x in rng g3 )
assume x in [#] (Closed-Interval-TSpace ((- 1),1)) ; ::_thesis: x in rng g3
then A22: x in [.(- 1),1.] by TOPMETR:18;
then reconsider r = x as Real ;
( - 1 <= r & r <= 1 ) by A22, XXREAL_1:1;
then 1 ^2 >= r ^2 by SQUARE_1:49;
then A23: 1 - (r ^2) >= 0 by XREAL_1:48;
set q = |[r,(- (sqrt (1 - (r ^2))))]|;
A24: |[r,(- (sqrt (1 - (r ^2))))]| `2 = - (sqrt (1 - (r ^2))) by EUCLID:52;
|.|[r,(- (sqrt (1 - (r ^2))))]|.| = sqrt (((|[r,(- (sqrt (1 - (r ^2))))]| `1) ^2) + ((|[r,(- (sqrt (1 - (r ^2))))]| `2) ^2)) by JGRAPH_3:1
.= sqrt ((r ^2) + ((|[r,(- (sqrt (1 - (r ^2))))]| `2) ^2)) by EUCLID:52
.= sqrt ((r ^2) + ((- (sqrt (1 - (r ^2)))) ^2)) by EUCLID:52
.= sqrt ((r ^2) + ((sqrt (1 - (r ^2))) ^2)) ;
then |.|[r,(- (sqrt (1 - (r ^2))))]|.| = sqrt ((r ^2) + (1 - (r ^2))) by A23, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then A25: |[r,(- (sqrt (1 - (r ^2))))]| in P by A2;
0 <= sqrt (1 - (r ^2)) by A23, SQUARE_1:def_2;
then |[r,(- (sqrt (1 - (r ^2))))]| in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A25, A24;
then A26: |[r,(- (sqrt (1 - (r ^2))))]| in dom g3 by A2, A11, Th35;
then g3 . |[r,(- (sqrt (1 - (r ^2))))]| = proj1 . |[r,(- (sqrt (1 - (r ^2))))]| by A1
.= |[r,(- (sqrt (1 - (r ^2))))]| `1 by PSCOMP_1:def_5
.= r by EUCLID:52 ;
hence x in rng g3 by A26, FUNCT_1:def_3; ::_thesis: verum
end;
A27: Closed-Interval-TSpace ((- 1),1) = R^1 | B by TOPMETR:19;
g2 is continuous by A1, JGRAPH_2:29;
hence ( proj1 | (Lower_Arc P) is continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) & proj1 | (Lower_Arc P) is one-to-one & rng (proj1 | (Lower_Arc P)) = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A21, A27, A12, FUNCT_1:def_4, JGRAPH_1:45, XBOOLE_0:def_10; ::_thesis: verum
end;
Lm6: now__::_thesis:_for_P_being_non_empty_compact_Subset_of_(TOP-REAL_2)_st_P_=__{__p_where_p_is_Point_of_(TOP-REAL_2)_:_|.p.|_=_1__}__holds_
(_proj1_|_(Upper_Arc_P)_is_continuous_Function_of_((TOP-REAL_2)_|_(Upper_Arc_P)),(Closed-Interval-TSpace_((-_1),1))_&_proj1_|_(Upper_Arc_P)_is_one-to-one_&_rng_(proj1_|_(Upper_Arc_P))_=_[#]_(Closed-Interval-TSpace_((-_1),1))_)
reconsider B = [.(- 1),1.] as non empty Subset of R^1 by TOPMETR:17, XXREAL_1:1;
reconsider g = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } implies ( proj1 | (Upper_Arc P) is continuous Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) & proj1 | (Upper_Arc P) is one-to-one & rng (proj1 | (Upper_Arc P)) = [#] (Closed-Interval-TSpace ((- 1),1)) ) )
set K0 = Upper_Arc P;
reconsider g2 = g | (Upper_Arc P) as Function of ((TOP-REAL 2) | (Upper_Arc P)),R^1 by PRE_TOPC:9;
A1: for p being Point of ((TOP-REAL 2) | (Upper_Arc P)) holds g2 . p = proj1 . p
proof
let p be Point of ((TOP-REAL 2) | (Upper_Arc P)); ::_thesis: g2 . p = proj1 . p
p in the carrier of ((TOP-REAL 2) | (Upper_Arc P)) ;
then p in Upper_Arc P by PRE_TOPC:8;
hence g2 . p = proj1 . p by FUNCT_1:49; ::_thesis: verum
end;
assume A2: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } ; ::_thesis: ( proj1 | (Upper_Arc P) is continuous Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) & proj1 | (Upper_Arc P) is one-to-one & rng (proj1 | (Upper_Arc P)) = [#] (Closed-Interval-TSpace ((- 1),1)) )
then A3: Upper_Arc P c= P by Th33;
A4: dom g2 = the carrier of ((TOP-REAL 2) | (Upper_Arc P)) by FUNCT_2:def_1;
then A5: dom g2 = Upper_Arc P by PRE_TOPC:8;
rng g2 c= the carrier of (Closed-Interval-TSpace ((- 1),1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g2 or x in the carrier of (Closed-Interval-TSpace ((- 1),1)) )
assume x in rng g2 ; ::_thesis: x in the carrier of (Closed-Interval-TSpace ((- 1),1))
then consider z being set such that
A6: z in dom g2 and
A7: x = g2 . z by FUNCT_1:def_3;
z in P by A5, A3, A6;
then consider p being Point of (TOP-REAL 2) such that
A8: p = z and
A9: |.p.| = 1 by A2;
1 ^2 = ((p `1) ^2) + ((p `2) ^2) by A9, JGRAPH_3:1;
then 1 - ((p `1) ^2) >= 0 by XREAL_1:63;
then - (1 - ((p `1) ^2)) <= 0 ;
then ((p `1) ^2) - 1 <= 0 ;
then A10: ( - 1 <= p `1 & p `1 <= 1 ) by SQUARE_1:43;
x = proj1 . p by A1, A6, A7, A8
.= p `1 by PSCOMP_1:def_5 ;
then x in [.(- 1),1.] by A10, XXREAL_1:1;
hence x in the carrier of (Closed-Interval-TSpace ((- 1),1)) by TOPMETR:18; ::_thesis: verum
end;
then reconsider g3 = g2 as Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A4, FUNCT_2:2;
dom g3 = [#] ((TOP-REAL 2) | (Upper_Arc P)) by FUNCT_2:def_1;
then A11: dom g3 = Upper_Arc P by PRE_TOPC:def_5;
A12: for x, y being set st x in dom g3 & y in dom g3 & g3 . x = g3 . y holds
x = y
proof
let x, y be set ; ::_thesis: ( x in dom g3 & y in dom g3 & g3 . x = g3 . y implies x = y )
assume that
A13: x in dom g3 and
A14: y in dom g3 and
A15: g3 . x = g3 . y ; ::_thesis: x = y
reconsider p2 = y as Point of (TOP-REAL 2) by A11, A14;
A16: g3 . y = proj1 . p2 by A1, A14
.= p2 `1 by PSCOMP_1:def_5 ;
reconsider p1 = x as Point of (TOP-REAL 2) by A11, A13;
A17: g3 . x = proj1 . p1 by A1, A13
.= p1 `1 by PSCOMP_1:def_5 ;
p2 in P by A3, A11, A14;
then ex p22 being Point of (TOP-REAL 2) st
( p2 = p22 & |.p22.| = 1 ) by A2;
then A18: 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
p2 in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 >= 0 ) } by A2, A11, A14, Th34;
then A19: ex p44 being Point of (TOP-REAL 2) st
( p2 = p44 & p44 in P & p44 `2 >= 0 ) ;
p1 in P by A3, A11, A13;
then ex p11 being Point of (TOP-REAL 2) st
( p1 = p11 & |.p11.| = 1 ) by A2;
then A20: 1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
p1 in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 >= 0 ) } by A2, A11, A13, Th34;
then ex p33 being Point of (TOP-REAL 2) st
( p1 = p33 & p33 in P & p33 `2 >= 0 ) ;
then p1 `2 = sqrt ((p2 `2) ^2) by A15, A17, A16, A18, A20, SQUARE_1:22;
then p1 `2 = p2 `2 by A19, SQUARE_1:22;
then p1 = |[(p2 `1),(p2 `2)]| by A15, A17, A16, EUCLID:53
.= p2 by EUCLID:53 ;
hence x = y ; ::_thesis: verum
end;
A21: [#] (Closed-Interval-TSpace ((- 1),1)) c= rng g3
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [#] (Closed-Interval-TSpace ((- 1),1)) or x in rng g3 )
assume x in [#] (Closed-Interval-TSpace ((- 1),1)) ; ::_thesis: x in rng g3
then A22: x in [.(- 1),1.] by TOPMETR:18;
then reconsider r = x as Real ;
( - 1 <= r & r <= 1 ) by A22, XXREAL_1:1;
then 1 ^2 >= r ^2 by SQUARE_1:49;
then A23: 1 - (r ^2) >= 0 by XREAL_1:48;
set q = |[r,(sqrt (1 - (r ^2)))]|;
A24: |[r,(sqrt (1 - (r ^2)))]| `2 = sqrt (1 - (r ^2)) by EUCLID:52;
|.|[r,(sqrt (1 - (r ^2)))]|.| = sqrt (((|[r,(sqrt (1 - (r ^2)))]| `1) ^2) + ((|[r,(sqrt (1 - (r ^2)))]| `2) ^2)) by JGRAPH_3:1
.= sqrt ((r ^2) + ((|[r,(sqrt (1 - (r ^2)))]| `2) ^2)) by EUCLID:52
.= sqrt ((r ^2) + ((sqrt (1 - (r ^2))) ^2)) by EUCLID:52 ;
then |.|[r,(sqrt (1 - (r ^2)))]|.| = sqrt ((r ^2) + (1 - (r ^2))) by A23, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then A25: |[r,(sqrt (1 - (r ^2)))]| in P by A2;
0 <= sqrt (1 - (r ^2)) by A23, SQUARE_1:def_2;
then |[r,(sqrt (1 - (r ^2)))]| in { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A25, A24;
then A26: |[r,(sqrt (1 - (r ^2)))]| in dom g3 by A2, A11, Th34;
then g3 . |[r,(sqrt (1 - (r ^2)))]| = proj1 . |[r,(sqrt (1 - (r ^2)))]| by A1
.= |[r,(sqrt (1 - (r ^2)))]| `1 by PSCOMP_1:def_5
.= r by EUCLID:52 ;
hence x in rng g3 by A26, FUNCT_1:def_3; ::_thesis: verum
end;
A27: Closed-Interval-TSpace ((- 1),1) = R^1 | B by TOPMETR:19;
g2 is continuous by A1, JGRAPH_2:29;
hence ( proj1 | (Upper_Arc P) is continuous Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) & proj1 | (Upper_Arc P) is one-to-one & rng (proj1 | (Upper_Arc P)) = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A21, A27, A12, FUNCT_1:def_4, JGRAPH_1:45, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th40: :: JGRAPH_5:40
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds
ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P )
proof
reconsider g = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } implies ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P ) )
set P4 = Lower_Arc P;
set K0 = Lower_Arc P;
reconsider g2 = g | (Lower_Arc P) as Function of ((TOP-REAL 2) | (Lower_Arc P)),R^1 by PRE_TOPC:9;
A1: for p being Point of ((TOP-REAL 2) | (Lower_Arc P)) holds g2 . p = proj1 . p
proof
let p be Point of ((TOP-REAL 2) | (Lower_Arc P)); ::_thesis: g2 . p = proj1 . p
p in the carrier of ((TOP-REAL 2) | (Lower_Arc P)) ;
then p in Lower_Arc P by PRE_TOPC:8;
hence g2 . p = proj1 . p by FUNCT_1:49; ::_thesis: verum
end;
assume A2: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } ; ::_thesis: ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P )
then reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by Lm5;
A3: rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) by A2, Lm5;
A4: P is being_simple_closed_curve by A2, JGRAPH_3:26;
then A5: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
E-max P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A6: E-max P in Lower_Arc P by A5, XBOOLE_0:def_4;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A7: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
A8: g3 is one-to-one by A2, Lm5;
A9: dom g3 = [#] ((TOP-REAL 2) | (Lower_Arc P)) by FUNCT_2:def_1;
then A10: dom g3 = Lower_Arc P by PRE_TOPC:def_5;
A11: g3 is onto by A3, FUNCT_2:def_3;
A12: for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
(g3 /") . (q `1) = q
proof
reconsider g4 = g3 as Function ;
let q be Point of (TOP-REAL 2); ::_thesis: ( q in Lower_Arc P implies (g3 /") . (q `1) = q )
A13: ( q in dom g4 implies ( q = (g4 ") . (g4 . q) & q = ((g4 ") * g4) . q ) ) by A8, FUNCT_1:34;
assume A14: q in Lower_Arc P ; ::_thesis: (g3 /") . (q `1) = q
then g3 . q = proj1 . q by A1, A10
.= q `1 by PSCOMP_1:def_5 ;
hence (g3 /") . (q `1) = q by A11, A9, A8, A14, A13, PRE_TOPC:def_5, TOPS_2:def_4; ::_thesis: verum
end;
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A15: W-min P in Lower_Arc P by A5, XBOOLE_0:def_4;
A16: E-max P = |[1,0]| by A2, Th30;
A17: W-min P = |[(- 1),0]| by A2, Th29;
Lower_Arc P is_an_arc_of E-max P, W-min P by A4, JORDAN6:def_9;
then ( not Lower_Arc P is empty & Lower_Arc P is compact ) by JORDAN5A:1;
then A18: g3 /" is being_homeomorphism by A9, A3, A8, A7, COMPTS_1:17, TOPS_2:56;
A19: (g3 /") . 1 = (g3 /") . (|[1,0]| `1) by EUCLID:52
.= E-max P by A6, A12, A16 ;
(g3 /") . (- 1) = (g3 /") . (|[(- 1),0]| `1) by EUCLID:52
.= W-min P by A15, A12, A17 ;
hence ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P ) by A18, A12, A19; ::_thesis: verum
end;
theorem Th41: :: JGRAPH_5:41
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds
ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P )
proof
reconsider g = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } implies ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P ) )
set P4 = Lower_Arc P;
set K0 = Upper_Arc P;
reconsider g2 = g | (Upper_Arc P) as Function of ((TOP-REAL 2) | (Upper_Arc P)),R^1 by PRE_TOPC:9;
A1: for p being Point of ((TOP-REAL 2) | (Upper_Arc P)) holds g2 . p = proj1 . p
proof
let p be Point of ((TOP-REAL 2) | (Upper_Arc P)); ::_thesis: g2 . p = proj1 . p
p in the carrier of ((TOP-REAL 2) | (Upper_Arc P)) ;
then p in Upper_Arc P by PRE_TOPC:8;
hence g2 . p = proj1 . p by FUNCT_1:49; ::_thesis: verum
end;
assume A2: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } ; ::_thesis: ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P )
then reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by Lm6;
A3: rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) by A2, Lm6;
A4: P is being_simple_closed_curve by A2, JGRAPH_3:26;
then A5: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
E-max P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A6: E-max P in Upper_Arc P by A5, XBOOLE_0:def_4;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A7: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
A8: g3 is one-to-one by A2, Lm6;
A9: dom g3 = [#] ((TOP-REAL 2) | (Upper_Arc P)) by FUNCT_2:def_1;
then A10: dom g3 = Upper_Arc P by PRE_TOPC:def_5;
A11: g3 is onto by A3, FUNCT_2:def_3;
A12: for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
(g3 /") . (q `1) = q
proof
reconsider g4 = g3 as Function ;
let q be Point of (TOP-REAL 2); ::_thesis: ( q in Upper_Arc P implies (g3 /") . (q `1) = q )
A13: ( q in dom g4 implies ( q = (g4 ") . (g4 . q) & q = ((g4 ") * g4) . q ) ) by A8, FUNCT_1:34;
assume A14: q in Upper_Arc P ; ::_thesis: (g3 /") . (q `1) = q
then g3 . q = proj1 . q by A1, A10
.= q `1 by PSCOMP_1:def_5 ;
hence (g3 /") . (q `1) = q by A11, A9, A8, A14, A13, PRE_TOPC:def_5, TOPS_2:def_4; ::_thesis: verum
end;
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A15: W-min P in Upper_Arc P by A5, XBOOLE_0:def_4;
A16: E-max P = |[1,0]| by A2, Th30;
A17: W-min P = |[(- 1),0]| by A2, Th29;
Upper_Arc P is_an_arc_of W-min P, E-max P by A4, JORDAN6:def_8;
then ( not Upper_Arc P is empty & Upper_Arc P is compact ) by JORDAN5A:1;
then A18: g3 /" is being_homeomorphism by A9, A3, A8, A7, COMPTS_1:17, TOPS_2:56;
A19: (g3 /") . 1 = (g3 /") . (|[1,0]| `1) by EUCLID:52
.= E-max P by A6, A12, A16 ;
(g3 /") . (- 1) = (g3 /") . (|[(- 1),0]| `1) by EUCLID:52
.= W-min P by A15, A12, A17 ;
hence ex f being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
f . (q `1) = q ) & f . (- 1) = W-min P & f . 1 = E-max P ) by A18, A12, A19; ::_thesis: verum
end;
theorem Th42: :: JGRAPH_5:42
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds
ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) ) & f . 0 = E-max P & f . 1 = W-min P )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } implies ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) ) & f . 0 = E-max P & f . 1 = W-min P ) )
reconsider T = (TOP-REAL 2) | (Lower_Arc P) as non empty TopSpace ;
consider g being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) such that
A1: g is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds
g . r = ((- 2) * r) + 1 and
A3: g . 0 = 1 and
A4: g . 1 = - 1 by Th38;
assume A5: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } ; ::_thesis: ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) ) & f . 0 = E-max P & f . 1 = W-min P )
then consider f1 being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Lower_Arc P)) such that
A6: f1 is being_homeomorphism and
A7: for q being Point of (TOP-REAL 2) st q in Lower_Arc P holds
f1 . (q `1) = q and
A8: f1 . (- 1) = W-min P and
A9: f1 . 1 = E-max P by Th40;
reconsider h = f1 * g as Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) ;
A10: dom h = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then 0 in dom h by XXREAL_1:1;
then A11: h . 0 = E-max P by A9, A3, FUNCT_1:12;
A12: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st h . r1 = q1 & h . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 )
proof
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: for r1, r2 being Real st h . r1 = q1 & h . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 )
let r1, r2 be Real; ::_thesis: ( h . r1 = q1 & h . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] implies ( r1 < r2 iff q1 `1 > q2 `1 ) )
assume that
A13: h . r1 = q1 and
A14: h . r2 = q2 and
A15: r1 in [.0,1.] and
A16: r2 in [.0,1.] ; ::_thesis: ( r1 < r2 iff q1 `1 > q2 `1 )
A17: now__::_thesis:_(_r2_<_r1_implies_q2_`1_>_q1_`1_)
set s1 = ((- 2) * r2) + 1;
set s2 = ((- 2) * r1) + 1;
set p1 = |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]|;
set p2 = |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]|;
A18: |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `2 = - (sqrt (1 - ((((- 2) * r2) + 1) ^2))) by EUCLID:52;
r2 <= 1 by A16, XXREAL_1:1;
then (- 2) * r2 >= (- 2) * 1 by XREAL_1:65;
then ((- 2) * r2) + 1 >= ((- 2) * 1) + 1 by XREAL_1:7;
then A19: - 1 <= ((- 2) * r2) + 1 ;
r2 >= 0 by A16, XXREAL_1:1;
then ((- 2) * r2) + 1 <= ((- 2) * 0) + 1 by XREAL_1:7;
then (((- 2) * r2) + 1) ^2 <= 1 ^2 by A19, SQUARE_1:49;
then A20: 1 - ((((- 2) * r2) + 1) ^2) >= 0 by XREAL_1:48;
then A21: sqrt (1 - ((((- 2) * r2) + 1) ^2)) >= 0 by SQUARE_1:def_2;
|.|[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]|.| = sqrt (((|[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `1) ^2) + ((|[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `2) ^2)) by JGRAPH_3:1
.= sqrt (((((- 2) * r2) + 1) ^2) + ((sqrt (1 - ((((- 2) * r2) + 1) ^2))) ^2)) by A18, EUCLID:52
.= sqrt (((((- 2) * r2) + 1) ^2) + (1 - ((((- 2) * r2) + 1) ^2))) by A20, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| in P by A5;
then |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 <= 0 ) } by A18, A21;
then A22: ( |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `1 = ((- 2) * r2) + 1 & |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| in Lower_Arc P ) by A5, Th35, EUCLID:52;
( g . r2 = ((- 2) * r2) + 1 & dom h = [.0,1.] ) by A2, A16, BORSUK_1:40, FUNCT_2:def_1;
then h . r2 = f1 . (((- 2) * r2) + 1) by A16, FUNCT_1:12
.= |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| by A7, A22 ;
then A23: q2 `1 = ((- 2) * r2) + 1 by A14, EUCLID:52;
A24: |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `1 = ((- 2) * r1) + 1 by EUCLID:52;
r1 <= 1 by A15, XXREAL_1:1;
then (- 2) * r1 >= (- 2) * 1 by XREAL_1:65;
then ((- 2) * r1) + 1 >= ((- 2) * 1) + 1 by XREAL_1:7;
then A25: - 1 <= ((- 2) * r1) + 1 ;
r1 >= 0 by A15, XXREAL_1:1;
then ((- 2) * r1) + 1 <= ((- 2) * 0) + 1 by XREAL_1:7;
then (((- 2) * r1) + 1) ^2 <= 1 ^2 by A25, SQUARE_1:49;
then A26: 1 - ((((- 2) * r1) + 1) ^2) >= 0 by XREAL_1:48;
then A27: sqrt (1 - ((((- 2) * r1) + 1) ^2)) >= 0 by SQUARE_1:def_2;
assume r2 < r1 ; ::_thesis: q2 `1 > q1 `1
then A28: (- 2) * r2 > (- 2) * r1 by XREAL_1:69;
A29: |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `2 = - (sqrt (1 - ((((- 2) * r1) + 1) ^2))) by EUCLID:52;
|.|[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]|.| = sqrt (((|[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `1) ^2) + ((|[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `2) ^2)) by JGRAPH_3:1
.= sqrt (((((- 2) * r1) + 1) ^2) + ((sqrt (1 - ((((- 2) * r1) + 1) ^2))) ^2)) by A29, EUCLID:52
.= sqrt (((((- 2) * r1) + 1) ^2) + (1 - ((((- 2) * r1) + 1) ^2))) by A26, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| in P by A5;
then |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 <= 0 ) } by A29, A27;
then A30: |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| in Lower_Arc P by A5, Th35;
( g . r1 = ((- 2) * r1) + 1 & dom h = [.0,1.] ) by A2, A15, BORSUK_1:40, FUNCT_2:def_1;
then h . r1 = f1 . (((- 2) * r1) + 1) by A15, FUNCT_1:12
.= |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| by A7, A24, A30 ;
hence q2 `1 > q1 `1 by A13, A28, A23, A24, XREAL_1:8; ::_thesis: verum
end;
A31: now__::_thesis:_(_q1_`1_>_q2_`1_implies_r1_<_r2_)
assume A32: q1 `1 > q2 `1 ; ::_thesis: r1 < r2
now__::_thesis:_not_r1_>=_r2
assume A33: r1 >= r2 ; ::_thesis: contradiction
now__::_thesis:_(_(_r1_>_r2_&_contradiction_)_or_(_r1_=_r2_&_contradiction_)_)
percases ( r1 > r2 or r1 = r2 ) by A33, XXREAL_0:1;
case r1 > r2 ; ::_thesis: contradiction
hence contradiction by A17, A32; ::_thesis: verum
end;
case r1 = r2 ; ::_thesis: contradiction
hence contradiction by A13, A14, A32; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence r1 < r2 ; ::_thesis: verum
end;
now__::_thesis:_(_r1_<_r2_implies_q1_`1_>_q2_`1_)
assume r1 < r2 ; ::_thesis: q1 `1 > q2 `1
then (- 2) * r1 > (- 2) * r2 by XREAL_1:69;
then A34: ((- 2) * r1) + 1 > ((- 2) * r2) + 1 by XREAL_1:8;
set s1 = ((- 2) * r1) + 1;
set s2 = ((- 2) * r2) + 1;
set p1 = |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]|;
set p2 = |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]|;
A35: |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `2 = - (sqrt (1 - ((((- 2) * r1) + 1) ^2))) by EUCLID:52;
r1 <= 1 by A15, XXREAL_1:1;
then (- 2) * r1 >= (- 2) * 1 by XREAL_1:65;
then ((- 2) * r1) + 1 >= ((- 2) * 1) + 1 by XREAL_1:7;
then A36: - 1 <= ((- 2) * r1) + 1 ;
r1 >= 0 by A15, XXREAL_1:1;
then ((- 2) * r1) + 1 <= ((- 2) * 0) + 1 by XREAL_1:7;
then (((- 2) * r1) + 1) ^2 <= 1 ^2 by A36, SQUARE_1:49;
then A37: 1 - ((((- 2) * r1) + 1) ^2) >= 0 by XREAL_1:48;
then A38: sqrt (1 - ((((- 2) * r1) + 1) ^2)) >= 0 by SQUARE_1:def_2;
|.|[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]|.| = sqrt (((|[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `1) ^2) + ((|[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `2) ^2)) by JGRAPH_3:1
.= sqrt (((((- 2) * r1) + 1) ^2) + ((sqrt (1 - ((((- 2) * r1) + 1) ^2))) ^2)) by A35, EUCLID:52
.= sqrt (((((- 2) * r1) + 1) ^2) + (1 - ((((- 2) * r1) + 1) ^2))) by A37, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| in P by A5;
then |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 <= 0 ) } by A35, A38;
then A39: ( |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| `1 = ((- 2) * r1) + 1 & |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| in Lower_Arc P ) by A5, Th35, EUCLID:52;
( g . r1 = ((- 2) * r1) + 1 & dom h = [.0,1.] ) by A2, A15, BORSUK_1:40, FUNCT_2:def_1;
then h . r1 = f1 . (((- 2) * r1) + 1) by A15, FUNCT_1:12
.= |[(((- 2) * r1) + 1),(- (sqrt (1 - ((((- 2) * r1) + 1) ^2))))]| by A7, A39 ;
then A40: q1 `1 = ((- 2) * r1) + 1 by A13, EUCLID:52;
A41: |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `2 = - (sqrt (1 - ((((- 2) * r2) + 1) ^2))) by EUCLID:52;
r2 <= 1 by A16, XXREAL_1:1;
then (- 2) * r2 >= (- 2) * 1 by XREAL_1:65;
then ((- 2) * r2) + 1 >= ((- 2) * 1) + 1 by XREAL_1:7;
then A42: - 1 <= ((- 2) * r2) + 1 ;
r2 >= 0 by A16, XXREAL_1:1;
then ((- 2) * r2) + 1 <= ((- 2) * 0) + 1 by XREAL_1:7;
then (((- 2) * r2) + 1) ^2 <= 1 ^2 by A42, SQUARE_1:49;
then A43: 1 - ((((- 2) * r2) + 1) ^2) >= 0 by XREAL_1:48;
then A44: sqrt (1 - ((((- 2) * r2) + 1) ^2)) >= 0 by SQUARE_1:def_2;
|.|[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]|.| = sqrt (((|[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `1) ^2) + ((|[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `2) ^2)) by JGRAPH_3:1
.= sqrt (((((- 2) * r2) + 1) ^2) + ((sqrt (1 - ((((- 2) * r2) + 1) ^2))) ^2)) by A41, EUCLID:52
.= sqrt (((((- 2) * r2) + 1) ^2) + (1 - ((((- 2) * r2) + 1) ^2))) by A43, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| in P by A5;
then |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 <= 0 ) } by A41, A44;
then A45: ( |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| `1 = ((- 2) * r2) + 1 & |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| in Lower_Arc P ) by A5, Th35, EUCLID:52;
( g . r2 = ((- 2) * r2) + 1 & dom h = [.0,1.] ) by A2, A16, BORSUK_1:40, FUNCT_2:def_1;
then h . r2 = f1 . (((- 2) * r2) + 1) by A16, FUNCT_1:12
.= |[(((- 2) * r2) + 1),(- (sqrt (1 - ((((- 2) * r2) + 1) ^2))))]| by A7, A45 ;
hence q1 `1 > q2 `1 by A14, A34, A40, EUCLID:52; ::_thesis: verum
end;
hence ( r1 < r2 iff q1 `1 > q2 `1 ) by A31; ::_thesis: verum
end;
1 in dom h by A10, XXREAL_1:1;
then A46: h . 1 = W-min P by A8, A4, FUNCT_1:12;
reconsider f2 = f1 as Function of (Closed-Interval-TSpace ((- 1),1)),T ;
f2 * g is being_homeomorphism by A6, A1, TOPS_2:57;
hence ex f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) ) & f . 0 = E-max P & f . 1 = W-min P ) by A12, A11, A46; ::_thesis: verum
end;
theorem Th43: :: JGRAPH_5:43
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } holds
ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 ) ) & f . 0 = W-min P & f . 1 = E-max P )
proof
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } implies ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 ) ) & f . 0 = W-min P & f . 1 = E-max P ) )
reconsider T = (TOP-REAL 2) | (Upper_Arc P) as non empty TopSpace ;
consider g being Function of I[01],(Closed-Interval-TSpace ((- 1),1)) such that
A1: g is being_homeomorphism and
A2: for r being Real st r in [.0,1.] holds
g . r = (2 * r) - 1 and
A3: g . 0 = - 1 and
A4: g . 1 = 1 by Th39;
assume A5: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } ; ::_thesis: ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 ) ) & f . 0 = W-min P & f . 1 = E-max P )
then consider f1 being Function of (Closed-Interval-TSpace ((- 1),1)),((TOP-REAL 2) | (Upper_Arc P)) such that
A6: f1 is being_homeomorphism and
A7: for q being Point of (TOP-REAL 2) st q in Upper_Arc P holds
f1 . (q `1) = q and
A8: f1 . (- 1) = W-min P and
A9: f1 . 1 = E-max P by Th41;
reconsider h = f1 * g as Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) ;
A10: dom h = [.0,1.] by BORSUK_1:40, FUNCT_2:def_1;
then 0 in dom h by XXREAL_1:1;
then A11: h . 0 = W-min P by A8, A3, FUNCT_1:12;
A12: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st h . r1 = q1 & h . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 )
proof
let q1, q2 be Point of (TOP-REAL 2); ::_thesis: for r1, r2 being Real st h . r1 = q1 & h . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 )
let r1, r2 be Real; ::_thesis: ( h . r1 = q1 & h . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] implies ( r1 < r2 iff q1 `1 < q2 `1 ) )
assume that
A13: h . r1 = q1 and
A14: h . r2 = q2 and
A15: r1 in [.0,1.] and
A16: r2 in [.0,1.] ; ::_thesis: ( r1 < r2 iff q1 `1 < q2 `1 )
A17: now__::_thesis:_(_r1_>_r2_implies_q1_`1_>_q2_`1_)
r2 <= 1 by A16, XXREAL_1:1;
then 2 * r2 <= 2 * 1 by XREAL_1:64;
then A18: (2 * r2) - 1 <= (2 * 1) - 1 by XREAL_1:9;
r1 >= 0 by A15, XXREAL_1:1;
then A19: (2 * r1) - 1 >= (2 * 0) - 1 by XREAL_1:9;
set s1 = (2 * r1) - 1;
set s2 = (2 * r2) - 1;
set p1 = |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]|;
set p2 = |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]|;
A20: |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| `1 = (2 * r1) - 1 by EUCLID:52;
r2 >= 0 by A16, XXREAL_1:1;
then A21: (2 * r2) - 1 >= (2 * 0) - 1 by XREAL_1:9;
(2 * 0) - 1 = - 1 ;
then ((2 * r2) - 1) ^2 <= 1 ^2 by A18, A21, SQUARE_1:49;
then A22: 1 - (((2 * r2) - 1) ^2) >= 0 by XREAL_1:48;
then A23: sqrt (1 - (((2 * r2) - 1) ^2)) >= 0 by SQUARE_1:def_2;
r1 <= 1 by A15, XXREAL_1:1;
then 2 * r1 <= 2 * 1 by XREAL_1:64;
then A24: (2 * r1) - 1 <= (2 * 1) - 1 by XREAL_1:9;
assume r1 > r2 ; ::_thesis: q1 `1 > q2 `1
then A25: 2 * r1 > 2 * r2 by XREAL_1:68;
(2 * 0) - 1 = - 1 ;
then ((2 * r1) - 1) ^2 <= 1 ^2 by A24, A19, SQUARE_1:49;
then A26: 1 - (((2 * r1) - 1) ^2) >= 0 by XREAL_1:48;
then A27: sqrt (1 - (((2 * r1) - 1) ^2)) >= 0 by SQUARE_1:def_2;
A28: |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| `2 = sqrt (1 - (((2 * r1) - 1) ^2)) by EUCLID:52;
then |.|[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]|.| = sqrt ((((2 * r1) - 1) ^2) + ((sqrt (1 - (((2 * r1) - 1) ^2))) ^2)) by A20, JGRAPH_3:1
.= sqrt ((((2 * r1) - 1) ^2) + (1 - (((2 * r1) - 1) ^2))) by A26, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| in P by A5;
then |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 >= 0 ) } by A28, A27;
then A29: |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| in Upper_Arc P by A5, Th34;
( g . r1 = (2 * r1) - 1 & dom h = [.0,1.] ) by A2, A15, BORSUK_1:40, FUNCT_2:def_1;
then h . r1 = f1 . ((2 * r1) - 1) by A15, FUNCT_1:12
.= |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| by A7, A20, A29 ;
then A30: q1 `1 = (2 * r1) - 1 by A13, EUCLID:52;
A31: |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| `1 = (2 * r2) - 1 by EUCLID:52;
A32: |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| `2 = sqrt (1 - (((2 * r2) - 1) ^2)) by EUCLID:52;
then |.|[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]|.| = sqrt ((((2 * r2) - 1) ^2) + ((sqrt (1 - (((2 * r2) - 1) ^2))) ^2)) by A31, JGRAPH_3:1
.= sqrt ((((2 * r2) - 1) ^2) + (1 - (((2 * r2) - 1) ^2))) by A22, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| in P by A5;
then |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 >= 0 ) } by A32, A23;
then A33: |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| in Upper_Arc P by A5, Th34;
( g . r2 = (2 * r2) - 1 & dom h = [.0,1.] ) by A2, A16, BORSUK_1:40, FUNCT_2:def_1;
then h . r2 = f1 . ((2 * r2) - 1) by A16, FUNCT_1:12
.= |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| by A7, A31, A33 ;
hence q1 `1 > q2 `1 by A14, A25, A30, A31, XREAL_1:14; ::_thesis: verum
end;
A34: now__::_thesis:_(_q1_`1_<_q2_`1_implies_r1_<_r2_)
assume A35: q1 `1 < q2 `1 ; ::_thesis: r1 < r2
now__::_thesis:_not_r1_>=_r2
assume A36: r1 >= r2 ; ::_thesis: contradiction
now__::_thesis:_(_(_r1_>_r2_&_contradiction_)_or_(_r1_=_r2_&_contradiction_)_)
percases ( r1 > r2 or r1 = r2 ) by A36, XXREAL_0:1;
case r1 > r2 ; ::_thesis: contradiction
hence contradiction by A17, A35; ::_thesis: verum
end;
case r1 = r2 ; ::_thesis: contradiction
hence contradiction by A13, A14, A35; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence r1 < r2 ; ::_thesis: verum
end;
now__::_thesis:_(_r2_>_r1_implies_q2_`1_>_q1_`1_)
assume r2 > r1 ; ::_thesis: q2 `1 > q1 `1
then A37: 2 * r2 > 2 * r1 by XREAL_1:68;
set s1 = (2 * r2) - 1;
set s2 = (2 * r1) - 1;
set p1 = |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]|;
set p2 = |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]|;
A38: |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| `1 = (2 * r2) - 1 by EUCLID:52;
r2 >= 0 by A16, XXREAL_1:1;
then (2 * r2) - 1 >= (2 * 0) - 1 by XREAL_1:9;
then A39: - 1 <= (2 * r2) - 1 ;
r2 <= 1 by A16, XXREAL_1:1;
then 2 * r2 <= 2 * 1 by XREAL_1:64;
then (2 * r2) - 1 <= (2 * 1) - 1 by XREAL_1:9;
then ((2 * r2) - 1) ^2 <= 1 ^2 by A39, SQUARE_1:49;
then A40: 1 - (((2 * r2) - 1) ^2) >= 0 by XREAL_1:48;
then A41: sqrt (1 - (((2 * r2) - 1) ^2)) >= 0 by SQUARE_1:def_2;
A42: |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| `2 = sqrt (1 - (((2 * r2) - 1) ^2)) by EUCLID:52;
then |.|[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]|.| = sqrt ((((2 * r2) - 1) ^2) + ((sqrt (1 - (((2 * r2) - 1) ^2))) ^2)) by A38, JGRAPH_3:1
.= sqrt ((((2 * r2) - 1) ^2) + (1 - (((2 * r2) - 1) ^2))) by A40, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| in P by A5;
then |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 >= 0 ) } by A42, A41;
then A43: |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| in Upper_Arc P by A5, Th34;
( g . r2 = (2 * r2) - 1 & dom h = [.0,1.] ) by A2, A16, BORSUK_1:40, FUNCT_2:def_1;
then h . r2 = f1 . ((2 * r2) - 1) by A16, FUNCT_1:12
.= |[((2 * r2) - 1),(sqrt (1 - (((2 * r2) - 1) ^2)))]| by A7, A38, A43 ;
then A44: q2 `1 = (2 * r2) - 1 by A14, EUCLID:52;
A45: |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| `1 = (2 * r1) - 1 by EUCLID:52;
r1 >= 0 by A15, XXREAL_1:1;
then (2 * r1) - 1 >= (2 * 0) - 1 by XREAL_1:9;
then A46: - 1 <= (2 * r1) - 1 ;
r1 <= 1 by A15, XXREAL_1:1;
then 2 * r1 <= 2 * 1 by XREAL_1:64;
then (2 * r1) - 1 <= (2 * 1) - 1 by XREAL_1:9;
then ((2 * r1) - 1) ^2 <= 1 ^2 by A46, SQUARE_1:49;
then A47: 1 - (((2 * r1) - 1) ^2) >= 0 by XREAL_1:48;
then A48: sqrt (1 - (((2 * r1) - 1) ^2)) >= 0 by SQUARE_1:def_2;
A49: |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| `2 = sqrt (1 - (((2 * r1) - 1) ^2)) by EUCLID:52;
then |.|[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]|.| = sqrt ((((2 * r1) - 1) ^2) + ((sqrt (1 - (((2 * r1) - 1) ^2))) ^2)) by A45, JGRAPH_3:1
.= sqrt ((((2 * r1) - 1) ^2) + (1 - (((2 * r1) - 1) ^2))) by A47, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| in P by A5;
then |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| in { p3 where p3 is Point of (TOP-REAL 2) : ( p3 in P & p3 `2 >= 0 ) } by A49, A48;
then A50: |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| in Upper_Arc P by A5, Th34;
( g . r1 = (2 * r1) - 1 & dom h = [.0,1.] ) by A2, A15, BORSUK_1:40, FUNCT_2:def_1;
then h . r1 = f1 . ((2 * r1) - 1) by A15, FUNCT_1:12
.= |[((2 * r1) - 1),(sqrt (1 - (((2 * r1) - 1) ^2)))]| by A7, A45, A50 ;
hence q2 `1 > q1 `1 by A13, A37, A44, A45, XREAL_1:14; ::_thesis: verum
end;
hence ( r1 < r2 iff q1 `1 < q2 `1 ) by A34; ::_thesis: verum
end;
1 in dom h by A10, XXREAL_1:1;
then A51: h . 1 = E-max P by A9, A4, FUNCT_1:12;
reconsider f2 = f1 as Function of (Closed-Interval-TSpace ((- 1),1)),T ;
f2 * g is being_homeomorphism by A6, A1, TOPS_2:57;
hence ex f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) st
( f is being_homeomorphism & ( for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 ) ) & f . 0 = W-min P & f . 1 = E-max P ) by A12, A11, A51; ::_thesis: verum
end;
theorem Th44: :: JGRAPH_5:44
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p2 in Upper_Arc P & LE p1,p2,P holds
p1 in Upper_Arc P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p2 in Upper_Arc P & LE p1,p2,P holds
p1 in Upper_Arc P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p2 in Upper_Arc P & LE p1,p2,P implies p1 in Upper_Arc P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p2 in Upper_Arc P and
A3: LE p1,p2,P ; ::_thesis: p1 in Upper_Arc P
set P4b = Lower_Arc P;
A4: ( ( p1 in Upper_Arc P & p2 in Lower_Arc P & not p2 = W-min P ) or ( p1 in Upper_Arc P & p2 in Upper_Arc P & LE p1,p2, Upper_Arc P, W-min P, E-max P ) or ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P ) ) by A3, JORDAN6:def_10;
A5: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A6: Lower_Arc P is_an_arc_of E-max P, W-min P by JORDAN6:def_9;
A7: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A5, JORDAN6:def_9;
then E-max P in (Upper_Arc P) /\ (Lower_Arc P) by TARSKI:def_2;
then A8: E-max P in Upper_Arc P by XBOOLE_0:def_4;
now__::_thesis:_p1_in_Upper_Arc_P
assume A9: not p1 in Upper_Arc P ; ::_thesis: contradiction
then p2 in (Upper_Arc P) /\ (Lower_Arc P) by A2, A4, XBOOLE_0:def_4;
then A10: p2 = E-max P by A7, A4, A9, TARSKI:def_2;
then LE p2,p1, Lower_Arc P, E-max P, W-min P by A6, A4, A9, JORDAN5C:10;
hence contradiction by A6, A8, A4, A9, A10, JORDAN5C:12; ::_thesis: verum
end;
hence p1 in Upper_Arc P ; ::_thesis: verum
end;
theorem Th45: :: JGRAPH_5:45
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 holds
( p1 `1 > p2 `1 & p1 `2 < p2 `2 )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 holds
( p1 `1 > p2 `1 & p1 `2 < p2 `2 )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 implies ( p1 `1 > p2 `1 & p1 `2 < p2 `2 ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: p1 <> p2 and
A4: p1 `1 < 0 and
A5: p1 `2 < 0 and
A6: p2 `2 < 0 ; ::_thesis: ( p1 `1 > p2 `1 & p1 `2 < p2 `2 )
consider f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) such that
A7: f is being_homeomorphism and
A8: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) and
A9: ( f . 0 = E-max P & f . 1 = W-min P ) by A1, Th42;
A10: rng f = [#] ((TOP-REAL 2) | (Lower_Arc P)) by A7, TOPS_2:def_5
.= Lower_Arc P by PRE_TOPC:def_5 ;
A11: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A12: now__::_thesis:_not_p1_in_Upper_Arc_P
assume p1 in Upper_Arc P ; ::_thesis: contradiction
then ex p being Point of (TOP-REAL 2) st
( p1 = p & p in P & p `2 >= 0 ) by A11;
hence contradiction by A5; ::_thesis: verum
end;
then A13: LE p1,p2, Lower_Arc P, E-max P, W-min P by A2, JORDAN6:def_10;
p2 in Lower_Arc P by A2, A12, JORDAN6:def_10;
then consider x2 being set such that
A14: x2 in dom f and
A15: p2 = f . x2 by A10, FUNCT_1:def_3;
A16: dom f = [#] I[01] by A7, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then reconsider r22 = x2 as Real by A14;
A17: ( 0 <= r22 & r22 <= 1 ) by A14, A16, XXREAL_1:1;
p1 in Lower_Arc P by A2, A12, JORDAN6:def_10;
then consider x1 being set such that
A18: x1 in dom f and
A19: p1 = f . x1 by A10, FUNCT_1:def_3;
reconsider r11 = x1 as Real by A18, A16;
r11 <= 1 by A18, A16, XXREAL_1:1;
then A20: r11 <= r22 by A13, A7, A9, A19, A15, A17, JORDAN5C:def_3;
A21: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then p1 in P by A2, JORDAN7:5;
then ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1;
then 1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
then (1 ^2) - ((p1 `1) ^2) = (- (p1 `2)) ^2 ;
then - (p1 `2) = sqrt ((1 ^2) - ((- (p1 `1)) ^2)) by A5, SQUARE_1:22;
then A22: p1 `2 = - (sqrt ((1 ^2) - ((- (p1 `1)) ^2))) ;
p2 in P by A2, A21, JORDAN7:5;
then ex p4 being Point of (TOP-REAL 2) st
( p4 = p2 & |.p4.| = 1 ) by A1;
then A23: 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
then (1 ^2) - ((p2 `1) ^2) = (- (p2 `2)) ^2 ;
then - (p2 `2) = sqrt ((1 ^2) - ((- (p2 `1)) ^2)) by A6, SQUARE_1:22;
then A24: p2 `2 = - (sqrt ((1 ^2) - ((- (p2 `1)) ^2))) ;
A25: ( r11 < r22 iff p1 `1 > p2 `1 ) by A8, A18, A19, A14, A15, A16;
then - (p1 `1) < - (p2 `1) by A3, A19, A15, A20, XREAL_1:24, XXREAL_0:1;
then (- (p1 `1)) ^2 < (- (p2 `1)) ^2 by A4, SQUARE_1:16;
then (1 ^2) - ((- (p1 `1)) ^2) > (1 ^2) - ((- (p2 `1)) ^2) by XREAL_1:15;
then sqrt ((1 ^2) - ((- (p1 `1)) ^2)) > sqrt ((1 ^2) - ((- (p2 `1)) ^2)) by A23, SQUARE_1:27, XREAL_1:63;
hence ( p1 `1 > p2 `1 & p1 `2 < p2 `2 ) by A19, A15, A25, A20, A22, A24, XREAL_1:24, XXREAL_0:1; ::_thesis: verum
end;
theorem Th46: :: JGRAPH_5:46
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p2 `1 < 0 & p1 `2 >= 0 & p2 `2 >= 0 holds
( p1 `1 < p2 `1 & p1 `2 < p2 `2 )
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p2 `1 < 0 & p1 `2 >= 0 & p2 `2 >= 0 holds
( p1 `1 < p2 `1 & p1 `2 < p2 `2 )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p2 `1 < 0 & p1 `2 >= 0 & p2 `2 >= 0 implies ( p1 `1 < p2 `1 & p1 `2 < p2 `2 ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: p1 <> p2 and
A4: p2 `1 < 0 and
A5: p1 `2 >= 0 and
A6: p2 `2 >= 0 ; ::_thesis: ( p1 `1 < p2 `1 & p1 `2 < p2 `2 )
set P4 = Lower_Arc P;
A7: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A8: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
A9: p1 in P by A2, A7, JORDAN7:5;
A10: now__::_thesis:_not_p2_=_W-min_P
assume p2 = W-min P ; ::_thesis: contradiction
then LE p2,p1,P by A7, A9, JORDAN7:3;
hence contradiction by A1, A2, A3, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
A11: p2 in P by A2, A7, JORDAN7:5;
then ex p4 being Point of (TOP-REAL 2) st
( p4 = p2 & |.p4.| = 1 ) by A1;
then 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
then A12: p2 `2 = sqrt ((1 ^2) - ((- (p2 `1)) ^2)) by A6, SQUARE_1:22;
A13: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A14: now__::_thesis:_not_p2_in_Lower_Arc_P
assume A15: p2 in Lower_Arc P ; ::_thesis: contradiction
p2 in Upper_Arc P by A6, A11, A13;
then p2 in {(W-min P),(E-max P)} by A8, A15, XBOOLE_0:def_4;
then A16: ( p2 = W-min P or p2 = E-max P ) by TARSKI:def_2;
E-max P = |[1,0]| by A1, Th30;
hence contradiction by A4, A10, A16, EUCLID:52; ::_thesis: verum
end;
then A17: LE p1,p2, Upper_Arc P, W-min P, E-max P by A2, JORDAN6:def_10;
A18: ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1, A9;
then 1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
then A19: p1 `2 = sqrt ((1 ^2) - ((- (p1 `1)) ^2)) by A5, SQUARE_1:22;
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A18, JGRAPH_3:1;
then A20: (1 ^2) - ((- (p1 `1)) ^2) >= 0 by XREAL_1:63;
consider f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) such that
A21: f is being_homeomorphism and
A22: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 ) and
A23: ( f . 0 = W-min P & f . 1 = E-max P ) by A1, Th43;
A24: rng f = [#] ((TOP-REAL 2) | (Upper_Arc P)) by A21, TOPS_2:def_5
.= Upper_Arc P by PRE_TOPC:def_5 ;
p2 in Upper_Arc P by A2, A14, JORDAN6:def_10;
then consider x2 being set such that
A25: x2 in dom f and
A26: p2 = f . x2 by A24, FUNCT_1:def_3;
A27: dom f = [#] I[01] by A21, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then reconsider r22 = x2 as Real by A25;
A28: ( 0 <= r22 & r22 <= 1 ) by A25, A27, XXREAL_1:1;
p1 in Upper_Arc P by A2, A14, JORDAN6:def_10;
then consider x1 being set such that
A29: x1 in dom f and
A30: p1 = f . x1 by A24, FUNCT_1:def_3;
reconsider r11 = x1 as Real by A29, A27;
r11 <= 1 by A29, A27, XXREAL_1:1;
then A31: r11 <= r22 by A17, A21, A23, A30, A26, A28, JORDAN5C:def_3;
A32: ( r11 < r22 iff p1 `1 < p2 `1 ) by A22, A29, A30, A25, A26, A27;
then - (p1 `1) > - (p2 `1) by A3, A30, A26, A31, XREAL_1:24, XXREAL_0:1;
then (- (p1 `1)) ^2 > (- (p2 `1)) ^2 by A4, SQUARE_1:16;
then (1 ^2) - ((- (p1 `1)) ^2) < (1 ^2) - ((- (p2 `1)) ^2) by XREAL_1:15;
hence ( p1 `1 < p2 `1 & p1 `2 < p2 `2 ) by A30, A26, A32, A31, A19, A12, A20, SQUARE_1:27, XXREAL_0:1; ::_thesis: verum
end;
theorem Th47: :: JGRAPH_5:47
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p2 `2 >= 0 holds
p1 `1 < p2 `1
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p2 `2 >= 0 holds
p1 `1 < p2 `1
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p2 `2 >= 0 implies p1 `1 < p2 `1 )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: p1 <> p2 and
A4: p2 `2 >= 0 ; ::_thesis: p1 `1 < p2 `1
A5: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A6: p1 in P by A2, JORDAN7:5;
set P4 = Lower_Arc P;
A7: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A8: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A5, JORDAN6:def_9;
A9: p2 in P by A2, A5, JORDAN7:5;
A10: now__::_thesis:_(_p2_in_Lower_Arc_P_implies_p1_`1_<_p2_`1_)
A11: now__::_thesis:_not_p2_=_W-min_P
assume p2 = W-min P ; ::_thesis: contradiction
then LE p2,p1,P by A5, A6, JORDAN7:3;
hence contradiction by A1, A2, A3, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
assume A12: p2 in Lower_Arc P ; ::_thesis: p1 `1 < p2 `1
p2 in Upper_Arc P by A4, A9, A7;
then p2 in {(W-min P),(E-max P)} by A8, A12, XBOOLE_0:def_4;
then ( p2 = W-min P or p2 = E-max P ) by TARSKI:def_2;
then A13: p2 = |[1,0]| by A1, A11, Th30;
then A14: p2 `1 = 1 by EUCLID:52;
A15: ex p8 being Point of (TOP-REAL 2) st
( p8 = p1 & |.p8.| = 1 ) by A1, A6;
A16: now__::_thesis:_not_p1_`1_=_1
assume A17: p1 `1 = 1 ; ::_thesis: contradiction
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A15, JGRAPH_3:1;
then p1 `2 = 0 by A17, XCMPLX_1:6;
hence contradiction by A3, A13, A17, EUCLID:53; ::_thesis: verum
end;
p1 `1 <= 1 by A15, Th1;
hence p1 `1 < p2 `1 by A14, A16, XXREAL_0:1; ::_thesis: verum
end;
now__::_thesis:_not_p2_=_W-min_P
assume p2 = W-min P ; ::_thesis: contradiction
then LE p2,p1,P by A5, A6, JORDAN7:3;
hence contradiction by A1, A2, A3, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
then A18: ( ( p1 in Upper_Arc P & p2 in Upper_Arc P & not p2 = W-min P & LE p1,p2, Upper_Arc P, W-min P, E-max P ) or p1 `1 < p2 `1 ) by A2, A10, JORDAN6:def_10;
consider f being Function of I[01],((TOP-REAL 2) | (Upper_Arc P)) such that
A19: f is being_homeomorphism and
A20: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 < q2 `1 ) and
A21: ( f . 0 = W-min P & f . 1 = E-max P ) by A1, Th43;
A22: rng f = [#] ((TOP-REAL 2) | (Upper_Arc P)) by A19, TOPS_2:def_5
.= Upper_Arc P by PRE_TOPC:def_5 ;
now__::_thesis:_(_(_not_p1_`1_<_p2_`1_&_p1_`1_<_p2_`1_)_or_(_p1_`1_<_p2_`1_&_p1_`1_<_p2_`1_)_)
percases ( not p1 `1 < p2 `1 or p1 `1 < p2 `1 ) ;
caseA23: not p1 `1 < p2 `1 ; ::_thesis: p1 `1 < p2 `1
then consider x1 being set such that
A24: x1 in dom f and
A25: p1 = f . x1 by A18, A22, FUNCT_1:def_3;
consider x2 being set such that
A26: x2 in dom f and
A27: p2 = f . x2 by A18, A22, A23, FUNCT_1:def_3;
A28: dom f = [#] I[01] by A19, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then reconsider r22 = x2 as Real by A26;
A29: ( 0 <= r22 & r22 <= 1 ) by A26, A28, XXREAL_1:1;
reconsider r11 = x1 as Real by A24, A28;
A30: ( r11 < r22 iff p1 `1 < p2 `1 ) by A20, A24, A25, A26, A27, A28;
r11 <= 1 by A24, A28, XXREAL_1:1;
then ( r11 <= r22 or p1 `1 < p2 `1 ) by A18, A19, A21, A25, A27, A29, JORDAN5C:def_3;
hence p1 `1 < p2 `1 by A3, A25, A27, A30, XXREAL_0:1; ::_thesis: verum
end;
case p1 `1 < p2 `1 ; ::_thesis: p1 `1 < p2 `1
hence p1 `1 < p2 `1 ; ::_thesis: verum
end;
end;
end;
hence p1 `1 < p2 `1 ; ::_thesis: verum
end;
theorem Th48: :: JGRAPH_5:48
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `2 <= 0 & p1 <> W-min P holds
p1 `1 > p2 `1
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `2 <= 0 & p1 <> W-min P holds
p1 `1 > p2 `1
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `2 <= 0 & p1 <> W-min P implies p1 `1 > p2 `1 )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: p1 <> p2 and
A4: p1 `2 <= 0 and
A5: p1 <> W-min P ; ::_thesis: p1 `1 > p2 `1
A6: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A7: p2 in P by A2, JORDAN7:5;
set P4 = Lower_Arc P;
A8: Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A1, Th35;
A9: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A6, JORDAN6:def_9;
A10: p1 in P by A2, A6, JORDAN7:5;
now__::_thesis:_(_p1_in_Upper_Arc_P_implies_p1_`1_>_p2_`1_)
assume A11: p1 in Upper_Arc P ; ::_thesis: p1 `1 > p2 `1
p1 in Lower_Arc P by A4, A10, A8;
then p1 in {(W-min P),(E-max P)} by A9, A11, XBOOLE_0:def_4;
then ( p1 = W-min P or p1 = E-max P ) by TARSKI:def_2;
then A12: p1 = |[1,0]| by A1, A5, Th30;
then A13: p1 `1 = 1 by EUCLID:52;
A14: ex p9 being Point of (TOP-REAL 2) st
( p9 = p2 & |.p9.| = 1 ) by A1, A7;
A15: now__::_thesis:_not_p2_`1_=_1
assume A16: p2 `1 = 1 ; ::_thesis: contradiction
1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A14, JGRAPH_3:1;
then p2 `2 = 0 by A16, XCMPLX_1:6;
hence contradiction by A3, A12, A16, EUCLID:53; ::_thesis: verum
end;
p2 `1 <= 1 by A14, Th1;
hence p1 `1 > p2 `1 by A13, A15, XXREAL_0:1; ::_thesis: verum
end;
then A17: ( ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P ) or p1 `1 > p2 `1 ) by A2, JORDAN6:def_10;
consider f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) such that
A18: f is being_homeomorphism and
A19: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) and
A20: ( f . 0 = E-max P & f . 1 = W-min P ) by A1, Th42;
A21: rng f = [#] ((TOP-REAL 2) | (Lower_Arc P)) by A18, TOPS_2:def_5
.= Lower_Arc P by PRE_TOPC:def_5 ;
now__::_thesis:_(_(_not_p1_`1_>_p2_`1_&_p1_`1_>_p2_`1_)_or_(_p1_`1_>_p2_`1_&_p1_`1_>_p2_`1_)_)
percases ( not p1 `1 > p2 `1 or p1 `1 > p2 `1 ) ;
caseA22: not p1 `1 > p2 `1 ; ::_thesis: p1 `1 > p2 `1
then consider x1 being set such that
A23: x1 in dom f and
A24: p1 = f . x1 by A17, A21, FUNCT_1:def_3;
consider x2 being set such that
A25: x2 in dom f and
A26: p2 = f . x2 by A17, A21, A22, FUNCT_1:def_3;
A27: dom f = [#] I[01] by A18, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then reconsider r22 = x2 as Real by A25;
A28: ( 0 <= r22 & r22 <= 1 ) by A25, A27, XXREAL_1:1;
reconsider r11 = x1 as Real by A23, A27;
A29: ( r11 < r22 iff p1 `1 > p2 `1 ) by A19, A23, A24, A25, A26, A27;
r11 <= 1 by A23, A27, XXREAL_1:1;
then ( r11 <= r22 or p1 `1 > p2 `1 ) by A17, A18, A20, A24, A26, A28, JORDAN5C:def_3;
hence p1 `1 > p2 `1 by A3, A24, A26, A29, XXREAL_0:1; ::_thesis: verum
end;
case p1 `1 > p2 `1 ; ::_thesis: p1 `1 > p2 `1
hence p1 `1 > p2 `1 ; ::_thesis: verum
end;
end;
end;
hence p1 `1 > p2 `1 ; ::_thesis: verum
end;
theorem Th49: :: JGRAPH_5:49
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & ( p2 `2 >= 0 or p2 `1 >= 0 ) & LE p1,p2,P & not p1 `2 >= 0 holds
p1 `1 >= 0
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & ( p2 `2 >= 0 or p2 `1 >= 0 ) & LE p1,p2,P & not p1 `2 >= 0 holds
p1 `1 >= 0
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & ( p2 `2 >= 0 or p2 `1 >= 0 ) & LE p1,p2,P & not p1 `2 >= 0 implies p1 `1 >= 0 )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: ( p2 `2 >= 0 or p2 `1 >= 0 ) and
A3: LE p1,p2,P ; ::_thesis: ( p1 `2 >= 0 or p1 `1 >= 0 )
A4: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A5: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A6: p2 in P by A3, JORDAN7:5;
A7: Lower_Arc P is_an_arc_of E-max P, W-min P by A5, JORDAN6:def_9;
percases ( p2 `2 >= 0 or ( p2 `2 < 0 & p2 `1 >= 0 ) ) by A2;
suppose p2 `2 >= 0 ; ::_thesis: ( p1 `2 >= 0 or p1 `1 >= 0 )
then p2 in Upper_Arc P by A6, A4;
then p1 in Upper_Arc P by A1, A3, Th44;
then ex p8 being Point of (TOP-REAL 2) st
( p8 = p1 & p8 in P & p8 `2 >= 0 ) by A4;
hence ( p1 `2 >= 0 or p1 `1 >= 0 ) ; ::_thesis: verum
end;
supposeA8: ( p2 `2 < 0 & p2 `1 >= 0 ) ; ::_thesis: ( p1 `2 >= 0 or p1 `1 >= 0 )
then for p8 being Point of (TOP-REAL 2) holds
( not p8 = p2 or not p8 in P or not p8 `2 >= 0 ) ;
then A9: not p2 in Upper_Arc P by A4;
now__::_thesis:_(_(_p1_in_Upper_Arc_P_&_p2_in_Lower_Arc_P_&_not_p2_=_W-min_P_&_(_p1_`2_>=_0_or_p1_`1_>=_0_)_)_or_(_p1_in_Lower_Arc_P_&_p2_in_Lower_Arc_P_&_not_p2_=_W-min_P_&_LE_p1,p2,_Lower_Arc_P,_E-max_P,_W-min_P_&_(_p1_`2_>=_0_or_p1_`1_>=_0_)_)_)
percases ( ( p1 in Upper_Arc P & p2 in Lower_Arc P & not p2 = W-min P ) or ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P ) ) by A3, A9, JORDAN6:def_10;
case ( p1 in Upper_Arc P & p2 in Lower_Arc P & not p2 = W-min P ) ; ::_thesis: ( p1 `2 >= 0 or p1 `1 >= 0 )
then ex p8 being Point of (TOP-REAL 2) st
( p8 = p1 & p8 in P & p8 `2 >= 0 ) by A4;
hence ( p1 `2 >= 0 or p1 `1 >= 0 ) ; ::_thesis: verum
end;
caseA10: ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P ) ; ::_thesis: ( p1 `2 >= 0 or p1 `1 >= 0 )
now__::_thesis:_not_p1_=_W-min_P
assume A11: p1 = W-min P ; ::_thesis: contradiction
then LE p2,p1, Lower_Arc P, E-max P, W-min P by A7, A10, JORDAN5C:10;
hence contradiction by A7, A10, A11, JORDAN5C:12; ::_thesis: verum
end;
hence ( p1 `2 >= 0 or p1 `1 >= 0 ) by A1, A3, A8, Th48; ::_thesis: verum
end;
end;
end;
hence ( p1 `2 >= 0 or p1 `1 >= 0 ) ; ::_thesis: verum
end;
end;
end;
theorem Th50: :: JGRAPH_5:50
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `1 >= 0 & p2 `1 >= 0 holds
p1 `2 > p2 `2
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `1 >= 0 & p2 `1 >= 0 holds
p1 `2 > p2 `2
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & p1 <> p2 & p1 `1 >= 0 & p2 `1 >= 0 implies p1 `2 > p2 `2 )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: p1 <> p2 and
A4: p1 `1 >= 0 and
A5: p2 `1 >= 0 ; ::_thesis: p1 `2 > p2 `2
A6: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A7: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A8: p2 in P by A2, JORDAN7:5;
then A9: ex p3 being Point of (TOP-REAL 2) st
( p3 = p2 & |.p3.| = 1 ) by A1;
W-min P = |[(- 1),0]| by A1, Th29;
then A10: (W-min P) `2 = 0 by EUCLID:52;
A11: Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A1, Th35;
A12: p1 in P by A2, A7, JORDAN7:5;
then A13: ex p4 being Point of (TOP-REAL 2) st
( p4 = p1 & |.p4.| = 1 ) by A1;
now__::_thesis:_(_(_p1_`2_>=_0_&_p2_`2_>=_0_&_p1_`2_>_p2_`2_)_or_(_p1_`2_>=_0_&_p2_`2_<_0_&_p1_`2_>_p2_`2_)_or_(_p1_`2_<_0_&_p2_`2_>=_0_&_contradiction_)_or_(_p1_`2_<_0_&_p2_`2_<_0_&_p1_`2_>_p2_`2_)_)
percases ( ( p1 `2 >= 0 & p2 `2 >= 0 ) or ( p1 `2 >= 0 & p2 `2 < 0 ) or ( p1 `2 < 0 & p2 `2 >= 0 ) or ( p1 `2 < 0 & p2 `2 < 0 ) ) ;
caseA14: ( p1 `2 >= 0 & p2 `2 >= 0 ) ; ::_thesis: p1 `2 > p2 `2
then p1 `1 < p2 `1 by A1, A2, A3, Th47;
then (p1 `1) ^2 < (p2 `1) ^2 by A4, SQUARE_1:16;
then A15: (1 ^2) - ((p1 `1) ^2) > (1 ^2) - ((p2 `1) ^2) by XREAL_1:15;
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A13, JGRAPH_3:1;
then A16: p1 `2 = sqrt ((1 ^2) - ((p1 `1) ^2)) by A14, SQUARE_1:22;
A17: 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A9, JGRAPH_3:1;
then p2 `2 = sqrt ((1 ^2) - ((p2 `1) ^2)) by A14, SQUARE_1:22;
hence p1 `2 > p2 `2 by A15, A16, A17, SQUARE_1:27, XREAL_1:63; ::_thesis: verum
end;
case ( p1 `2 >= 0 & p2 `2 < 0 ) ; ::_thesis: p1 `2 > p2 `2
hence p1 `2 > p2 `2 ; ::_thesis: verum
end;
caseA18: ( p1 `2 < 0 & p2 `2 >= 0 ) ; ::_thesis: contradiction
then ( p1 in Lower_Arc P & p2 in Upper_Arc P ) by A12, A8, A6, A11;
then LE p2,p1,P by A10, A18, JORDAN6:def_10;
hence contradiction by A1, A2, A3, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
caseA19: ( p1 `2 < 0 & p2 `2 < 0 ) ; ::_thesis: p1 `2 > p2 `2
ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1, A12;
then A20: 1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by JGRAPH_3:1;
then (1 ^2) - ((p1 `1) ^2) = (- (p1 `2)) ^2 ;
then A21: - (p1 `2) = sqrt ((1 ^2) - ((p1 `1) ^2)) by A19, SQUARE_1:22;
for p being Point of (TOP-REAL 2) holds
( not p = p1 or not p in P or not p `2 >= 0 ) by A19;
then A22: not p1 in Upper_Arc P by A6;
then A23: LE p1,p2, Lower_Arc P, E-max P, W-min P by A2, JORDAN6:def_10;
ex p4 being Point of (TOP-REAL 2) st
( p4 = p2 & |.p4.| = 1 ) by A1, A8;
then 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by JGRAPH_3:1;
then (1 ^2) - ((p2 `1) ^2) = (- (p2 `2)) ^2 ;
then A24: - (p2 `2) = sqrt ((1 ^2) - ((p2 `1) ^2)) by A19, SQUARE_1:22;
consider f being Function of I[01],((TOP-REAL 2) | (Lower_Arc P)) such that
A25: f is being_homeomorphism and
A26: for q1, q2 being Point of (TOP-REAL 2)
for r1, r2 being Real st f . r1 = q1 & f . r2 = q2 & r1 in [.0,1.] & r2 in [.0,1.] holds
( r1 < r2 iff q1 `1 > q2 `1 ) and
A27: ( f . 0 = E-max P & f . 1 = W-min P ) by A1, Th42;
A28: rng f = [#] ((TOP-REAL 2) | (Lower_Arc P)) by A25, TOPS_2:def_5
.= Lower_Arc P by PRE_TOPC:def_5 ;
p2 in Lower_Arc P by A2, A22, JORDAN6:def_10;
then consider x2 being set such that
A29: x2 in dom f and
A30: p2 = f . x2 by A28, FUNCT_1:def_3;
A31: dom f = [#] I[01] by A25, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then reconsider r22 = x2 as Real by A29;
A32: ( 0 <= r22 & r22 <= 1 ) by A29, A31, XXREAL_1:1;
p1 in Lower_Arc P by A2, A22, JORDAN6:def_10;
then consider x1 being set such that
A33: x1 in dom f and
A34: p1 = f . x1 by A28, FUNCT_1:def_3;
reconsider r11 = x1 as Real by A33, A31;
A35: ( r11 < r22 iff p1 `1 > p2 `1 ) by A26, A33, A34, A29, A30, A31;
r11 <= 1 by A33, A31, XXREAL_1:1;
then r11 <= r22 by A23, A25, A27, A34, A30, A32, JORDAN5C:def_3;
then (p1 `1) ^2 > (p2 `1) ^2 by A3, A5, A34, A30, A35, SQUARE_1:16, XXREAL_0:1;
then (1 ^2) - ((p1 `1) ^2) < (1 ^2) - ((p2 `1) ^2) by XREAL_1:15;
then sqrt ((1 ^2) - ((p1 `1) ^2)) < sqrt ((1 ^2) - ((p2 `1) ^2)) by A20, SQUARE_1:27, XREAL_1:63;
hence p1 `2 > p2 `2 by A21, A24, XREAL_1:24; ::_thesis: verum
end;
end;
end;
hence p1 `2 > p2 `2 ; ::_thesis: verum
end;
theorem Th51: :: JGRAPH_5:51
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 < 0 & p2 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 & ( p1 `1 >= p2 `1 or p1 `2 <= p2 `2 ) holds
LE p1,p2,P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 < 0 & p2 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 & ( p1 `1 >= p2 `1 or p1 `2 <= p2 `2 ) holds
LE p1,p2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 < 0 & p2 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 & ( p1 `1 >= p2 `1 or p1 `2 <= p2 `2 ) implies LE p1,p2,P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p1 in P and
A3: p2 in P and
A4: p1 `1 < 0 and
A5: p2 `1 < 0 and
A6: p1 `2 < 0 and
A7: p2 `2 < 0 and
A8: ( p1 `1 >= p2 `1 or p1 `2 <= p2 `2 ) ; ::_thesis: LE p1,p2,P
A9: ex p3 being Point of (TOP-REAL 2) st
( p3 = p2 & |.p3.| = 1 ) by A1, A3;
set P4 = Lower_Arc P;
A10: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A11: (Upper_Arc P) \/ (Lower_Arc P) = P by JORDAN6:def_9;
A12: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A13: now__::_thesis:_p1_in_Lower_Arc_P
assume not p1 in Lower_Arc P ; ::_thesis: contradiction
then p1 in Upper_Arc P by A2, A11, XBOOLE_0:def_3;
then ex p being Point of (TOP-REAL 2) st
( p1 = p & p in P & p `2 >= 0 ) by A12;
hence contradiction by A6; ::_thesis: verum
end;
A14: now__::_thesis:_p2_in_Lower_Arc_P
assume not p2 in Lower_Arc P ; ::_thesis: contradiction
then p2 in Upper_Arc P by A3, A11, XBOOLE_0:def_3;
then ex p being Point of (TOP-REAL 2) st
( p2 = p & p in P & p `2 >= 0 ) by A12;
hence contradiction by A7; ::_thesis: verum
end;
A15: ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1, A2;
A16: now__::_thesis:_(_p1_`2_<=_p2_`2_implies_p1_`1_>=_p2_`1_)
assume p1 `2 <= p2 `2 ; ::_thesis: p1 `1 >= p2 `1
then - (p1 `2) >= - (p2 `2) by XREAL_1:24;
then (- (p1 `2)) ^2 >= (- (p2 `2)) ^2 by A7, SQUARE_1:15;
then A17: (1 ^2) - ((- (p1 `2)) ^2) <= (1 ^2) - ((- (p2 `2)) ^2) by XREAL_1:13;
A18: 1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A15, JGRAPH_3:1;
then (1 ^2) - ((- (p1 `2)) ^2) >= 0 by XREAL_1:63;
then A19: sqrt ((1 ^2) - ((- (p1 `2)) ^2)) <= sqrt ((1 ^2) - ((- (p2 `2)) ^2)) by A17, SQUARE_1:26;
1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A9, JGRAPH_3:1;
then (1 ^2) - ((- (p2 `2)) ^2) = (- (p2 `1)) ^2 ;
then A20: - (p2 `1) = sqrt ((1 ^2) - ((- (p2 `2)) ^2)) by A5, SQUARE_1:22;
(1 ^2) - ((- (p1 `2)) ^2) = (- (p1 `1)) ^2 by A18;
then - (p1 `1) = sqrt ((1 ^2) - ((- (p1 `2)) ^2)) by A4, SQUARE_1:22;
hence p1 `1 >= p2 `1 by A20, A19, XREAL_1:24; ::_thesis: verum
end;
A21: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A10, JORDAN6:def_9;
A22: Lower_Arc P is_an_arc_of E-max P, W-min P by A10, JORDAN6:def_9;
A23: W-min P = |[(- 1),0]| by A1, Th29;
for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc P))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A24: W-min P in Lower_Arc P by A21, XBOOLE_0:def_4;
set K0 = Lower_Arc P;
reconsider g0 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider g2 = g0 | (Lower_Arc P) as Function of ((TOP-REAL 2) | (Lower_Arc P)),R^1 by PRE_TOPC:9;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A25: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A1, Lm5;
let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc P)); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A26: g is being_homeomorphism and
g . 0 = E-max P and
A27: g . 1 = W-min P and
A28: g . s1 = p1 and
A29: ( 0 <= s1 & s1 <= 1 ) and
A30: g . s2 = p2 and
A31: ( 0 <= s2 & s2 <= 1 ) ; ::_thesis: s1 <= s2
A32: s2 in [.0,1.] by A31, XXREAL_1:1;
reconsider h = g3 * g as Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((- 1),1)) by TOPMETR:20;
A33: ( dom g3 = [#] ((TOP-REAL 2) | (Lower_Arc P)) & rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A1, Lm5, FUNCT_2:def_1;
( g3 is one-to-one & not Lower_Arc P is empty & Lower_Arc P is compact ) by A1, A22, Lm5, JORDAN5A:1;
then g3 is being_homeomorphism by A33, A25, COMPTS_1:17;
then A34: h is being_homeomorphism by A26, TOPMETR:20, TOPS_2:57;
A35: dom g = [#] I[01] by A26, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then A36: 1 in dom g by XXREAL_1:1;
A37: - 1 = |[(- 1),0]| `1 by EUCLID:52
.= proj1 . |[(- 1),0]| by PSCOMP_1:def_5
.= g3 . (g . 1) by A23, A27, A24, FUNCT_1:49
.= h . 1 by A36, FUNCT_1:13 ;
A38: s1 in [.0,1.] by A29, XXREAL_1:1;
A39: p2 `1 = proj1 . p2 by PSCOMP_1:def_5
.= g3 . (g . s2) by A14, A30, FUNCT_1:49
.= h . s2 by A35, A32, FUNCT_1:13 ;
p1 `1 = g0 . p1 by PSCOMP_1:def_5
.= g3 . (g . s1) by A13, A28, FUNCT_1:49
.= h . s1 by A35, A38, FUNCT_1:13 ;
hence s1 <= s2 by A8, A16, A34, A38, A32, A37, A39, Th9; ::_thesis: verum
end;
then A40: LE p1,p2, Lower_Arc P, E-max P, W-min P by A13, A14, JORDAN5C:def_3;
now__::_thesis:_not_p2_=_W-min_P
assume A41: p2 = W-min P ; ::_thesis: contradiction
W-min P = |[(- 1),0]| by A1, Th29;
hence contradiction by A7, A41, EUCLID:52; ::_thesis: verum
end;
hence LE p1,p2,P by A13, A14, A40, JORDAN6:def_10; ::_thesis: verum
end;
theorem :: JGRAPH_5:52
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 > 0 & p2 `1 > 0 & p1 `2 < 0 & p2 `2 < 0 & ( p1 `1 >= p2 `1 or p1 `2 >= p2 `2 ) holds
LE p1,p2,P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 > 0 & p2 `1 > 0 & p1 `2 < 0 & p2 `2 < 0 & ( p1 `1 >= p2 `1 or p1 `2 >= p2 `2 ) holds
LE p1,p2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 > 0 & p2 `1 > 0 & p1 `2 < 0 & p2 `2 < 0 & ( p1 `1 >= p2 `1 or p1 `2 >= p2 `2 ) implies LE p1,p2,P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p1 in P and
A3: p2 in P and
A4: p1 `1 > 0 and
A5: p2 `1 > 0 and
A6: p1 `2 < 0 and
A7: p2 `2 < 0 and
A8: ( p1 `1 >= p2 `1 or p1 `2 >= p2 `2 ) ; ::_thesis: LE p1,p2,P
A9: ex p3 being Point of (TOP-REAL 2) st
( p3 = p2 & |.p3.| = 1 ) by A1, A3;
set P4 = Lower_Arc P;
A10: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A11: (Upper_Arc P) \/ (Lower_Arc P) = P by JORDAN6:def_9;
A12: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
A13: now__::_thesis:_p1_in_Lower_Arc_P
assume not p1 in Lower_Arc P ; ::_thesis: contradiction
then p1 in Upper_Arc P by A2, A11, XBOOLE_0:def_3;
then ex p being Point of (TOP-REAL 2) st
( p1 = p & p in P & p `2 >= 0 ) by A12;
hence contradiction by A6; ::_thesis: verum
end;
A14: now__::_thesis:_p2_in_Lower_Arc_P
assume not p2 in Lower_Arc P ; ::_thesis: contradiction
then p2 in Upper_Arc P by A3, A11, XBOOLE_0:def_3;
then ex p being Point of (TOP-REAL 2) st
( p2 = p & p in P & p `2 >= 0 ) by A12;
hence contradiction by A7; ::_thesis: verum
end;
A15: ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1, A2;
A16: now__::_thesis:_(_p1_`2_>=_p2_`2_implies_p1_`1_>=_p2_`1_)
assume p1 `2 >= p2 `2 ; ::_thesis: p1 `1 >= p2 `1
then - (p1 `2) <= - (p2 `2) by XREAL_1:24;
then (- (p1 `2)) ^2 <= (- (p2 `2)) ^2 by A6, SQUARE_1:15;
then A17: (1 ^2) - ((- (p1 `2)) ^2) >= (1 ^2) - ((- (p2 `2)) ^2) by XREAL_1:13;
1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A9, JGRAPH_3:1;
then A18: ( p2 `1 = sqrt ((1 ^2) - ((- (p2 `2)) ^2)) & (1 ^2) - ((- (p2 `2)) ^2) >= 0 ) by A5, SQUARE_1:22;
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A15, JGRAPH_3:1;
then p1 `1 = sqrt ((1 ^2) - ((- (p1 `2)) ^2)) by A4, SQUARE_1:22;
hence p1 `1 >= p2 `1 by A17, A18, SQUARE_1:26; ::_thesis: verum
end;
A19: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by A10, JORDAN6:def_9;
A20: Lower_Arc P is_an_arc_of E-max P, W-min P by A10, JORDAN6:def_9;
A21: W-min P = |[(- 1),0]| by A1, Th29;
for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc P))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A22: W-min P in Lower_Arc P by A19, XBOOLE_0:def_4;
set K0 = Lower_Arc P;
reconsider g0 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider g2 = g0 | (Lower_Arc P) as Function of ((TOP-REAL 2) | (Lower_Arc P)),R^1 by PRE_TOPC:9;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A23: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A1, Lm5;
let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc P)); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A24: g is being_homeomorphism and
g . 0 = E-max P and
A25: g . 1 = W-min P and
A26: g . s1 = p1 and
A27: ( 0 <= s1 & s1 <= 1 ) and
A28: g . s2 = p2 and
A29: ( 0 <= s2 & s2 <= 1 ) ; ::_thesis: s1 <= s2
A30: s2 in [.0,1.] by A29, XXREAL_1:1;
reconsider h = g3 * g as Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((- 1),1)) by TOPMETR:20;
A31: ( dom g3 = [#] ((TOP-REAL 2) | (Lower_Arc P)) & rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A1, Lm5, FUNCT_2:def_1;
( g3 is one-to-one & not Lower_Arc P is empty & Lower_Arc P is compact ) by A1, A20, Lm5, JORDAN5A:1;
then g3 is being_homeomorphism by A31, A23, COMPTS_1:17;
then A32: h is being_homeomorphism by A24, TOPMETR:20, TOPS_2:57;
A33: dom g = [#] I[01] by A24, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then A34: 1 in dom g by XXREAL_1:1;
A35: - 1 = |[(- 1),0]| `1 by EUCLID:52
.= proj1 . |[(- 1),0]| by PSCOMP_1:def_5
.= g3 . (g . 1) by A21, A25, A22, FUNCT_1:49
.= h . 1 by A34, FUNCT_1:13 ;
A36: s1 in [.0,1.] by A27, XXREAL_1:1;
A37: p2 `1 = proj1 . p2 by PSCOMP_1:def_5
.= g3 . p2 by A14, FUNCT_1:49
.= h . s2 by A28, A33, A30, FUNCT_1:13 ;
p1 `1 = g0 . p1 by PSCOMP_1:def_5
.= g3 . (g . s1) by A13, A26, FUNCT_1:49
.= h . s1 by A33, A36, FUNCT_1:13 ;
hence s1 <= s2 by A8, A16, A32, A36, A30, A35, A37, Th9; ::_thesis: verum
end;
then A38: LE p1,p2, Lower_Arc P, E-max P, W-min P by A13, A14, JORDAN5C:def_3;
now__::_thesis:_not_p2_=_W-min_P
assume A39: p2 = W-min P ; ::_thesis: contradiction
W-min P = |[(- 1),0]| by A1, Th29;
hence contradiction by A5, A39, EUCLID:52; ::_thesis: verum
end;
hence LE p1,p2,P by A13, A14, A38, JORDAN6:def_10; ::_thesis: verum
end;
theorem Th53: :: JGRAPH_5:53
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 < 0 & p2 `1 < 0 & p1 `2 >= 0 & p2 `2 >= 0 & ( p1 `1 <= p2 `1 or p1 `2 <= p2 `2 ) holds
LE p1,p2,P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 < 0 & p2 `1 < 0 & p1 `2 >= 0 & p2 `2 >= 0 & ( p1 `1 <= p2 `1 or p1 `2 <= p2 `2 ) holds
LE p1,p2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 < 0 & p2 `1 < 0 & p1 `2 >= 0 & p2 `2 >= 0 & ( p1 `1 <= p2 `1 or p1 `2 <= p2 `2 ) implies LE p1,p2,P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p1 in P and
A3: p2 in P and
A4: p1 `1 < 0 and
A5: p2 `1 < 0 and
A6: p1 `2 >= 0 and
A7: p2 `2 >= 0 and
A8: ( p1 `1 <= p2 `1 or p1 `2 <= p2 `2 ) ; ::_thesis: LE p1,p2,P
A9: ex p3 being Point of (TOP-REAL 2) st
( p3 = p2 & |.p3.| = 1 ) by A1, A3;
set P4b = Upper_Arc P;
set P4 = Lower_Arc P;
A10: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A11: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
A12: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
then A13: p1 in Upper_Arc P by A2, A6;
A14: p2 in Upper_Arc P by A3, A7, A12;
A15: ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1, A2;
A16: now__::_thesis:_(_p1_`2_<=_p2_`2_implies_p1_`1_<=_p2_`1_)
assume p1 `2 <= p2 `2 ; ::_thesis: p1 `1 <= p2 `1
then (p1 `2) ^2 <= (p2 `2) ^2 by A6, SQUARE_1:15;
then A17: (1 ^2) - ((p1 `2) ^2) >= (1 ^2) - ((p2 `2) ^2) by XREAL_1:13;
A18: 1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A9, JGRAPH_3:1;
then (1 ^2) - ((p2 `2) ^2) >= 0 by XREAL_1:63;
then A19: sqrt ((1 ^2) - ((p1 `2) ^2)) >= sqrt ((1 ^2) - ((p2 `2) ^2)) by A17, SQUARE_1:26;
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A15, JGRAPH_3:1;
then (1 ^2) - ((p1 `2) ^2) = (- (p1 `1)) ^2 ;
then A20: - (p1 `1) = sqrt ((1 ^2) - ((p1 `2) ^2)) by A4, SQUARE_1:22;
(1 ^2) - ((p2 `2) ^2) = (- (p2 `1)) ^2 by A18;
then - (p2 `1) = sqrt ((1 ^2) - ((p2 `2) ^2)) by A5, SQUARE_1:22;
hence p1 `1 <= p2 `1 by A20, A19, XREAL_1:24; ::_thesis: verum
end;
A21: E-max P = |[1,0]| by A1, Th30;
A22: Upper_Arc P is_an_arc_of W-min P, E-max P by A10, JORDAN6:def_8;
for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc P))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min P & g . 1 = E-max P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
E-max P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A23: E-max P in Upper_Arc P by A11, XBOOLE_0:def_4;
set K0 = Upper_Arc P;
reconsider g0 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider g2 = g0 | (Upper_Arc P) as Function of ((TOP-REAL 2) | (Upper_Arc P)),R^1 by PRE_TOPC:9;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A24: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A1, Lm6;
let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc P)); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min P & g . 1 = E-max P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = W-min P & g . 1 = E-max P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A25: g is being_homeomorphism and
g . 0 = W-min P and
A26: g . 1 = E-max P and
A27: g . s1 = p1 and
A28: ( 0 <= s1 & s1 <= 1 ) and
A29: g . s2 = p2 and
A30: ( 0 <= s2 & s2 <= 1 ) ; ::_thesis: s1 <= s2
A31: s2 in [.0,1.] by A30, XXREAL_1:1;
reconsider h = g3 * g as Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((- 1),1)) by TOPMETR:20;
A32: ( dom g3 = [#] ((TOP-REAL 2) | (Upper_Arc P)) & rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A1, Lm6, FUNCT_2:def_1;
( g3 is one-to-one & not Upper_Arc P is empty & Upper_Arc P is compact ) by A1, A22, Lm6, JORDAN5A:1;
then g3 is being_homeomorphism by A32, A24, COMPTS_1:17;
then A33: h is being_homeomorphism by A25, TOPMETR:20, TOPS_2:57;
A34: dom g = [#] I[01] by A25, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then A35: 1 in dom g by XXREAL_1:1;
A36: 1 = |[1,0]| `1 by EUCLID:52
.= g0 . |[1,0]| by PSCOMP_1:def_5
.= g3 . |[1,0]| by A21, A23, FUNCT_1:49
.= h . 1 by A21, A26, A35, FUNCT_1:13 ;
A37: s1 in [.0,1.] by A28, XXREAL_1:1;
A38: p2 `1 = g0 . p2 by PSCOMP_1:def_5
.= g3 . p2 by A14, FUNCT_1:49
.= h . s2 by A29, A34, A31, FUNCT_1:13 ;
p1 `1 = g0 . p1 by PSCOMP_1:def_5
.= g3 . (g . s1) by A13, A27, FUNCT_1:49
.= h . s1 by A34, A37, FUNCT_1:13 ;
hence s1 <= s2 by A8, A16, A33, A37, A31, A36, A38, Th8; ::_thesis: verum
end;
then A39: LE p1,p2, Upper_Arc P, W-min P, E-max P by A13, A14, JORDAN5C:def_3;
p1 in Upper_Arc P by A2, A6, A12;
hence LE p1,p2,P by A14, A39, JORDAN6:def_10; ::_thesis: verum
end;
theorem Th54: :: JGRAPH_5:54
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `2 >= 0 & p2 `2 >= 0 & p1 `1 <= p2 `1 holds
LE p1,p2,P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `2 >= 0 & p2 `2 >= 0 & p1 `1 <= p2 `1 holds
LE p1,p2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `2 >= 0 & p2 `2 >= 0 & p1 `1 <= p2 `1 implies LE p1,p2,P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p1 in P and
A3: p2 in P and
A4: p1 `2 >= 0 and
A5: p2 `2 >= 0 and
A6: p1 `1 <= p2 `1 ; ::_thesis: LE p1,p2,P
A7: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
then A8: p1 in Upper_Arc P by A2, A4;
A9: p2 in Upper_Arc P by A3, A5, A7;
set P4b = Upper_Arc P;
set P4 = Lower_Arc P;
A10: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A11: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
A12: E-max P = |[1,0]| by A1, Th30;
A13: Upper_Arc P is_an_arc_of W-min P, E-max P by A10, JORDAN6:def_8;
for g being Function of I[01],((TOP-REAL 2) | (Upper_Arc P))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min P & g . 1 = E-max P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
E-max P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A14: E-max P in Upper_Arc P by A11, XBOOLE_0:def_4;
set K0 = Upper_Arc P;
reconsider g0 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider g2 = g0 | (Upper_Arc P) as Function of ((TOP-REAL 2) | (Upper_Arc P)),R^1 by PRE_TOPC:9;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A15: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Upper_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A1, Lm6;
let g be Function of I[01],((TOP-REAL 2) | (Upper_Arc P)); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = W-min P & g . 1 = E-max P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = W-min P & g . 1 = E-max P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A16: g is being_homeomorphism and
g . 0 = W-min P and
A17: g . 1 = E-max P and
A18: g . s1 = p1 and
A19: ( 0 <= s1 & s1 <= 1 ) and
A20: g . s2 = p2 and
A21: ( 0 <= s2 & s2 <= 1 ) ; ::_thesis: s1 <= s2
A22: s2 in [.0,1.] by A21, XXREAL_1:1;
reconsider h = g3 * g as Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((- 1),1)) by TOPMETR:20;
A23: ( dom g3 = [#] ((TOP-REAL 2) | (Upper_Arc P)) & rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A1, Lm6, FUNCT_2:def_1;
( g3 is one-to-one & not Upper_Arc P is empty & Upper_Arc P is compact ) by A1, A13, Lm6, JORDAN5A:1;
then g3 is being_homeomorphism by A23, A15, COMPTS_1:17;
then A24: h is being_homeomorphism by A16, TOPMETR:20, TOPS_2:57;
A25: dom g = [#] I[01] by A16, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then A26: 1 in dom g by XXREAL_1:1;
A27: 1 = |[1,0]| `1 by EUCLID:52
.= g0 . |[1,0]| by PSCOMP_1:def_5
.= g3 . |[1,0]| by A12, A14, FUNCT_1:49
.= h . 1 by A12, A17, A26, FUNCT_1:13 ;
A28: s1 in [.0,1.] by A19, XXREAL_1:1;
A29: p2 `1 = g0 . p2 by PSCOMP_1:def_5
.= g3 . p2 by A9, FUNCT_1:49
.= h . s2 by A20, A25, A22, FUNCT_1:13 ;
p1 `1 = g0 . p1 by PSCOMP_1:def_5
.= g3 . p1 by A8, FUNCT_1:49
.= h . s1 by A18, A25, A28, FUNCT_1:13 ;
hence s1 <= s2 by A6, A24, A28, A22, A27, A29, Th8; ::_thesis: verum
end;
then A30: LE p1,p2, Upper_Arc P, W-min P, E-max P by A8, A9, JORDAN5C:def_3;
p1 in Upper_Arc P by A2, A4, A7;
hence LE p1,p2,P by A9, A30, JORDAN6:def_10; ::_thesis: verum
end;
theorem Th55: :: JGRAPH_5:55
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 >= 0 & p2 `1 >= 0 & p1 `2 >= p2 `2 holds
LE p1,p2,P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 >= 0 & p2 `1 >= 0 & p1 `2 >= p2 `2 holds
LE p1,p2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `1 >= 0 & p2 `1 >= 0 & p1 `2 >= p2 `2 implies LE p1,p2,P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p1 in P and
A3: p2 in P and
A4: p1 `1 >= 0 and
A5: p2 `1 >= 0 and
A6: p1 `2 >= p2 `2 ; ::_thesis: LE p1,p2,P
A7: ex p3 being Point of (TOP-REAL 2) st
( p3 = p1 & |.p3.| = 1 ) by A1, A2;
A8: W-min P = |[(- 1),0]| by A1, Th29;
A9: ex p3 being Point of (TOP-REAL 2) st
( p3 = p2 & |.p3.| = 1 ) by A1, A3;
A10: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A1, Th34;
set P4b = Lower_Arc P;
A11: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A12: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
A13: (Upper_Arc P) \/ (Lower_Arc P) = P by A11, JORDAN6:def_9;
A14: Lower_Arc P is_an_arc_of E-max P, W-min P by A11, JORDAN6:def_9;
now__::_thesis:_(_(_p1_in_Upper_Arc_P_&_p2_in_Upper_Arc_P_&_LE_p1,p2,P_)_or_(_p1_in_Upper_Arc_P_&_not_p2_in_Upper_Arc_P_&_LE_p1,p2,P_)_or_(_not_p1_in_Upper_Arc_P_&_p2_in_Upper_Arc_P_&_contradiction_)_or_(_not_p1_in_Upper_Arc_P_&_not_p2_in_Upper_Arc_P_&_LE_p1,p2,P_)_)
percases ( ( p1 in Upper_Arc P & p2 in Upper_Arc P ) or ( p1 in Upper_Arc P & not p2 in Upper_Arc P ) or ( not p1 in Upper_Arc P & p2 in Upper_Arc P ) or ( not p1 in Upper_Arc P & not p2 in Upper_Arc P ) ) ;
caseA15: ( p1 in Upper_Arc P & p2 in Upper_Arc P ) ; ::_thesis: LE p1,p2,P
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A7, JGRAPH_3:1;
then A16: ( p1 `1 = sqrt ((1 ^2) - ((p1 `2) ^2)) & (1 ^2) - ((p1 `2) ^2) >= 0 ) by A4, SQUARE_1:22;
1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A9, JGRAPH_3:1;
then A17: p2 `1 = sqrt ((1 ^2) - ((p2 `2) ^2)) by A5, SQUARE_1:22;
A18: ex p22 being Point of (TOP-REAL 2) st
( p2 = p22 & p22 in P & p22 `2 >= 0 ) by A10, A15;
then (p1 `2) ^2 >= (p2 `2) ^2 by A6, SQUARE_1:15;
then (1 ^2) - ((p1 `2) ^2) <= (1 ^2) - ((p2 `2) ^2) by XREAL_1:13;
hence LE p1,p2,P by A1, A2, A6, A18, A17, A16, Th54, SQUARE_1:26; ::_thesis: verum
end;
caseA19: ( p1 in Upper_Arc P & not p2 in Upper_Arc P ) ; ::_thesis: LE p1,p2,P
A20: now__::_thesis:_not_p2_=_W-min_P
assume A21: p2 = W-min P ; ::_thesis: contradiction
W-min P = |[(- 1),0]| by A1, Th29;
then p2 `2 = 0 by A21, EUCLID:52;
hence contradiction by A3, A10, A19; ::_thesis: verum
end;
p2 in Lower_Arc P by A3, A13, A19, XBOOLE_0:def_3;
hence LE p1,p2,P by A19, A20, JORDAN6:def_10; ::_thesis: verum
end;
caseA22: ( not p1 in Upper_Arc P & p2 in Upper_Arc P ) ; ::_thesis: contradiction
then ex p9 being Point of (TOP-REAL 2) st
( p2 = p9 & p9 in P & p9 `2 >= 0 ) by A10;
hence contradiction by A2, A6, A10, A22; ::_thesis: verum
end;
caseA23: ( not p1 in Upper_Arc P & not p2 in Upper_Arc P ) ; ::_thesis: LE p1,p2,P
A24: - (p1 `2) <= - (p2 `2) by A6, XREAL_1:24;
p1 `2 < 0 by A2, A10, A23;
then (- (p1 `2)) ^2 <= (- (p2 `2)) ^2 by A24, SQUARE_1:15;
then A25: (1 ^2) - ((- (p1 `2)) ^2) >= (1 ^2) - ((- (p2 `2)) ^2) by XREAL_1:13;
1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A9, JGRAPH_3:1;
then A26: ( p2 `1 = sqrt ((1 ^2) - ((- (p2 `2)) ^2)) & (1 ^2) - ((- (p2 `2)) ^2) >= 0 ) by A5, SQUARE_1:22;
A27: p2 in Lower_Arc P by A3, A13, A23, XBOOLE_0:def_3;
A28: now__::_thesis:_not_p2_=_W-min_P
assume A29: p2 = W-min P ; ::_thesis: contradiction
W-min P = |[(- 1),0]| by A1, Th29;
then p2 `2 = 0 by A29, EUCLID:52;
hence contradiction by A3, A10, A23; ::_thesis: verum
end;
A30: p1 in Lower_Arc P by A2, A13, A23, XBOOLE_0:def_3;
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A7, JGRAPH_3:1;
then p1 `1 = sqrt ((1 ^2) - ((- (p1 `2)) ^2)) by A4, SQUARE_1:22;
then A31: p1 `1 >= p2 `1 by A25, A26, SQUARE_1:26;
for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc P))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A32: W-min P in Lower_Arc P by A12, XBOOLE_0:def_4;
set K0 = Lower_Arc P;
reconsider g0 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider g2 = g0 | (Lower_Arc P) as Function of ((TOP-REAL 2) | (Lower_Arc P)),R^1 by PRE_TOPC:9;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A33: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A1, Lm5;
let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc P)); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A34: g is being_homeomorphism and
g . 0 = E-max P and
A35: g . 1 = W-min P and
A36: g . s1 = p1 and
A37: ( 0 <= s1 & s1 <= 1 ) and
A38: g . s2 = p2 and
A39: ( 0 <= s2 & s2 <= 1 ) ; ::_thesis: s1 <= s2
A40: s2 in [.0,1.] by A39, XXREAL_1:1;
reconsider h = g3 * g as Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((- 1),1)) by TOPMETR:20;
A41: ( dom g3 = [#] ((TOP-REAL 2) | (Lower_Arc P)) & rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A1, Lm5, FUNCT_2:def_1;
( g3 is one-to-one & not Lower_Arc P is empty & Lower_Arc P is compact ) by A1, A14, Lm5, JORDAN5A:1;
then g3 is being_homeomorphism by A41, A33, COMPTS_1:17;
then A42: h is being_homeomorphism by A34, TOPMETR:20, TOPS_2:57;
A43: dom g = [#] I[01] by A34, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then A44: 1 in dom g by XXREAL_1:1;
A45: - 1 = |[(- 1),0]| `1 by EUCLID:52
.= proj1 . |[(- 1),0]| by PSCOMP_1:def_5
.= g3 . |[(- 1),0]| by A8, A32, FUNCT_1:49
.= h . 1 by A8, A35, A44, FUNCT_1:13 ;
A46: s1 in [.0,1.] by A37, XXREAL_1:1;
A47: p2 `1 = g0 . p2 by PSCOMP_1:def_5
.= g3 . p2 by A27, FUNCT_1:49
.= h . s2 by A38, A43, A40, FUNCT_1:13 ;
p1 `1 = g0 . p1 by PSCOMP_1:def_5
.= g3 . p1 by A30, FUNCT_1:49
.= h . s1 by A36, A43, A46, FUNCT_1:13 ;
hence s1 <= s2 by A31, A42, A46, A40, A45, A47, Th9; ::_thesis: verum
end;
then A48: LE p1,p2, Lower_Arc P, E-max P, W-min P by A30, A27, JORDAN5C:def_3;
p1 in Lower_Arc P by A2, A13, A23, XBOOLE_0:def_3;
hence LE p1,p2,P by A27, A28, A48, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
hence LE p1,p2,P ; ::_thesis: verum
end;
theorem Th56: :: JGRAPH_5:56
for p1, p2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `2 <= 0 & p2 `2 <= 0 & p2 <> W-min P & p1 `1 >= p2 `1 holds
LE p1,p2,P
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `2 <= 0 & p2 `2 <= 0 & p2 <> W-min P & p1 `1 >= p2 `1 holds
LE p1,p2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p1 in P & p2 in P & p1 `2 <= 0 & p2 `2 <= 0 & p2 <> W-min P & p1 `1 >= p2 `1 implies LE p1,p2,P )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p1 in P and
A3: p2 in P and
A4: p1 `2 <= 0 and
A5: p2 `2 <= 0 and
A6: p2 <> W-min P and
A7: p1 `1 >= p2 `1 ; ::_thesis: LE p1,p2,P
A8: Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A1, Th35;
then A9: p1 in Lower_Arc P by A2, A4;
set P4 = Lower_Arc P;
A10: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A11: (Upper_Arc P) /\ (Lower_Arc P) = {(W-min P),(E-max P)} by JORDAN6:def_9;
A12: W-min P = |[(- 1),0]| by A1, Th29;
A13: p2 in Lower_Arc P by A3, A5, A8;
A14: Lower_Arc P is_an_arc_of E-max P, W-min P by A10, JORDAN6:def_9;
for g being Function of I[01],((TOP-REAL 2) | (Lower_Arc P))
for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
proof
W-min P in {(W-min P),(E-max P)} by TARSKI:def_2;
then A15: W-min P in Lower_Arc P by A11, XBOOLE_0:def_4;
set K0 = Lower_Arc P;
reconsider g0 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider g2 = g0 | (Lower_Arc P) as Function of ((TOP-REAL 2) | (Lower_Arc P)),R^1 by PRE_TOPC:9;
Closed-Interval-TSpace ((- 1),1) = TopSpaceMetr (Closed-Interval-MSpace ((- 1),1)) by TOPMETR:def_7;
then A16: Closed-Interval-TSpace ((- 1),1) is T_2 by PCOMPS_1:34;
reconsider g3 = g2 as continuous Function of ((TOP-REAL 2) | (Lower_Arc P)),(Closed-Interval-TSpace ((- 1),1)) by A1, Lm5;
let g be Function of I[01],((TOP-REAL 2) | (Lower_Arc P)); ::_thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
let s1, s2 be Real; ::_thesis: ( g is being_homeomorphism & g . 0 = E-max P & g . 1 = W-min P & g . s1 = p1 & 0 <= s1 & s1 <= 1 & g . s2 = p2 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A17: g is being_homeomorphism and
g . 0 = E-max P and
A18: g . 1 = W-min P and
A19: g . s1 = p1 and
A20: ( 0 <= s1 & s1 <= 1 ) and
A21: g . s2 = p2 and
A22: ( 0 <= s2 & s2 <= 1 ) ; ::_thesis: s1 <= s2
A23: s2 in [.0,1.] by A22, XXREAL_1:1;
reconsider h = g3 * g as Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace ((- 1),1)) by TOPMETR:20;
A24: ( dom g3 = [#] ((TOP-REAL 2) | (Lower_Arc P)) & rng g3 = [#] (Closed-Interval-TSpace ((- 1),1)) ) by A1, Lm5, FUNCT_2:def_1;
( g3 is one-to-one & not Lower_Arc P is empty & Lower_Arc P is compact ) by A1, A14, Lm5, JORDAN5A:1;
then g3 is being_homeomorphism by A24, A16, COMPTS_1:17;
then A25: h is being_homeomorphism by A17, TOPMETR:20, TOPS_2:57;
A26: dom g = [#] I[01] by A17, TOPS_2:def_5
.= [.0,1.] by BORSUK_1:40 ;
then A27: 1 in dom g by XXREAL_1:1;
A28: - 1 = |[(- 1),0]| `1 by EUCLID:52
.= proj1 . |[(- 1),0]| by PSCOMP_1:def_5
.= g3 . |[(- 1),0]| by A12, A15, FUNCT_1:49
.= h . 1 by A12, A18, A27, FUNCT_1:13 ;
A29: s1 in [.0,1.] by A20, XXREAL_1:1;
A30: p2 `1 = g0 . p2 by PSCOMP_1:def_5
.= g3 . p2 by A13, FUNCT_1:49
.= h . s2 by A21, A26, A23, FUNCT_1:13 ;
p1 `1 = g0 . p1 by PSCOMP_1:def_5
.= g3 . p1 by A9, FUNCT_1:49
.= h . s1 by A19, A26, A29, FUNCT_1:13 ;
hence s1 <= s2 by A7, A25, A29, A23, A28, A30, Th9; ::_thesis: verum
end;
then A31: LE p1,p2, Lower_Arc P, E-max P, W-min P by A9, A13, JORDAN5C:def_3;
( p1 in Lower_Arc P & p2 in Lower_Arc P ) by A2, A3, A4, A5, A8;
hence LE p1,p2,P by A6, A31, JORDAN6:def_10; ::_thesis: verum
end;
theorem Th57: :: JGRAPH_5:57
for cn being Real
for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 <= 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 <= 0
proof
let cn be Real; ::_thesis: for q being Point of (TOP-REAL 2) st - 1 < cn & cn < 1 & q `2 <= 0 holds
for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 <= 0
let q be Point of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & q `2 <= 0 implies for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 <= 0 )
assume that
A1: - 1 < cn and
A2: cn < 1 and
A3: q `2 <= 0 ; ::_thesis: for p being Point of (TOP-REAL 2) st p = (cn -FanMorphS) . q holds
p `2 <= 0
let p be Point of (TOP-REAL 2); ::_thesis: ( p = (cn -FanMorphS) . q implies p `2 <= 0 )
assume A4: p = (cn -FanMorphS) . q ; ::_thesis: p `2 <= 0
percases ( q `2 < 0 or q `2 = 0 ) by A3;
supposeA5: q `2 < 0 ; ::_thesis: p `2 <= 0
now__::_thesis:_(_(_(q_`1)_/_|.q.|_<_cn_&_p_`2_<=_0_)_or_(_(q_`1)_/_|.q.|_>=_cn_&_p_`2_<=_0_)_)
percases ( (q `1) / |.q.| < cn or (q `1) / |.q.| >= cn ) ;
case (q `1) / |.q.| < cn ; ::_thesis: p `2 <= 0
hence p `2 <= 0 by A1, A4, A5, JGRAPH_4:138; ::_thesis: verum
end;
case (q `1) / |.q.| >= cn ; ::_thesis: p `2 <= 0
hence p `2 <= 0 by A2, A4, A5, JGRAPH_4:137; ::_thesis: verum
end;
end;
end;
hence p `2 <= 0 ; ::_thesis: verum
end;
suppose q `2 = 0 ; ::_thesis: p `2 <= 0
hence p `2 <= 0 by A4, JGRAPH_4:113; ::_thesis: verum
end;
end;
end;
theorem Th58: :: JGRAPH_5:58
for cn being Real
for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & q1 = (cn -FanMorphS) . p1 & q2 = (cn -FanMorphS) . p2 holds
LE q1,q2,P
proof
let cn be Real; ::_thesis: for p1, p2, q1, q2 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & q1 = (cn -FanMorphS) . p1 & q2 = (cn -FanMorphS) . p2 holds
LE q1,q2,P
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st - 1 < cn & cn < 1 & P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & q1 = (cn -FanMorphS) . p1 & q2 = (cn -FanMorphS) . p2 holds
LE q1,q2,P
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( - 1 < cn & cn < 1 & P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & q1 = (cn -FanMorphS) . p1 & q2 = (cn -FanMorphS) . p2 implies LE q1,q2,P )
assume that
A1: ( - 1 < cn & cn < 1 ) and
A2: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A3: LE p1,p2,P and
A4: q1 = (cn -FanMorphS) . p1 and
A5: q2 = (cn -FanMorphS) . p2 ; ::_thesis: LE q1,q2,P
A6: P is being_simple_closed_curve by A2, JGRAPH_3:26;
W-min P = |[(- 1),0]| by A2, Th29;
then A7: (W-min P) `2 = 0 by EUCLID:52;
then A8: (cn -FanMorphS) . (W-min P) = W-min P by JGRAPH_4:113;
p2 in the carrier of (TOP-REAL 2) ;
then A9: p2 in dom (cn -FanMorphS) by FUNCT_2:def_1;
W-min P in the carrier of (TOP-REAL 2) ;
then A10: W-min P in dom (cn -FanMorphS) by FUNCT_2:def_1;
A11: Lower_Arc P c= P by A2, Th33;
A12: cn -FanMorphS is one-to-one by A1, JGRAPH_4:133;
A13: Upper_Arc P c= P by A2, Th33;
A14: now__::_thesis:_(_(_p1_in_Upper_Arc_P_&_p1_in_P_)_or_(_p1_in_Lower_Arc_P_&_p1_in_P_)_)
percases ( p1 in Upper_Arc P or p1 in Lower_Arc P ) by A3, JORDAN6:def_10;
case p1 in Upper_Arc P ; ::_thesis: p1 in P
hence p1 in P by A13; ::_thesis: verum
end;
case p1 in Lower_Arc P ; ::_thesis: p1 in P
hence p1 in P by A11; ::_thesis: verum
end;
end;
end;
A15: now__::_thesis:_(_not_q2_=_W-min_P_or_(_q1_in_Upper_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_)_or_(_q1_in_Upper_Arc_P_&_q2_in_Upper_Arc_P_&_LE_q1,q2,_Upper_Arc_P,_W-min_P,_E-max_P_)_or_(_q1_in_Lower_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_&_LE_q1,q2,_Lower_Arc_P,_E-max_P,_W-min_P_)_)
assume A16: q2 = W-min P ; ::_thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) )
then p2 = W-min P by A5, A8, A10, A9, A12, FUNCT_1:def_4;
then LE p2,p1,P by A6, A14, JORDAN7:3;
then A17: q1 = q2 by A2, A3, A4, A5, JGRAPH_3:26, JORDAN6:57;
W-min P in Lower_Arc P by A6, JORDAN7:1;
then LE q1,q2,P by A2, A11, A16, A17, JGRAPH_3:26, JORDAN6:56;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) by JORDAN6:def_10; ::_thesis: verum
end;
A18: Upper_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 >= 0 ) } by A2, Th34;
A19: Lower_Arc P = { p where p is Point of (TOP-REAL 2) : ( p in P & p `2 <= 0 ) } by A2, Th35;
percases ( ( p1 in Upper_Arc P & p2 in Lower_Arc P & not p2 = W-min P ) or ( p1 in Upper_Arc P & p2 in Upper_Arc P & LE p1,p2, Upper_Arc P, W-min P, E-max P ) or ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P & not p1 in Upper_Arc P ) ) by A3, JORDAN6:def_10;
supposeA20: ( p1 in Upper_Arc P & p2 in Lower_Arc P & not p2 = W-min P ) ; ::_thesis: LE q1,q2,P
A21: |.q2.| = |.p2.| by A5, JGRAPH_4:128;
A22: ex p9 being Point of (TOP-REAL 2) st
( p9 = p2 & p9 in P & p9 `2 <= 0 ) by A19, A20;
then ex p10 being Point of (TOP-REAL 2) st
( p10 = p2 & |.p10.| = 1 ) by A2;
then A23: q2 in P by A2, A21;
A24: ex p8 being Point of (TOP-REAL 2) st
( p8 = p1 & p8 in P & p8 `2 >= 0 ) by A18, A20;
q2 `2 <= 0 by A1, A5, A22, Th57;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) by A4, A19, A15, A20, A24, A23, JGRAPH_4:113; :: according to JORDAN6:def_10 ::_thesis: verum
end;
supposeA25: ( p1 in Upper_Arc P & p2 in Upper_Arc P & LE p1,p2, Upper_Arc P, W-min P, E-max P ) ; ::_thesis: LE q1,q2,P
then ex p8 being Point of (TOP-REAL 2) st
( p8 = p1 & p8 in P & p8 `2 >= 0 ) by A18;
then A26: p1 = (cn -FanMorphS) . p1 by JGRAPH_4:113;
ex p9 being Point of (TOP-REAL 2) st
( p9 = p2 & p9 in P & p9 `2 >= 0 ) by A18, A25;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) by A4, A5, A25, A26, JGRAPH_4:113; :: according to JORDAN6:def_10 ::_thesis: verum
end;
supposeA27: ( p1 in Lower_Arc P & p2 in Lower_Arc P & not p2 = W-min P & LE p1,p2, Lower_Arc P, E-max P, W-min P & not p1 in Upper_Arc P ) ; ::_thesis: LE q1,q2,P
then A28: ex p8 being Point of (TOP-REAL 2) st
( p8 = p1 & p8 in P & p8 `2 <= 0 ) by A19;
then A29: ex p10 being Point of (TOP-REAL 2) st
( p10 = p1 & |.p10.| = 1 ) by A2;
A30: ex p9 being Point of (TOP-REAL 2) st
( p9 = p2 & p9 in P & p9 `2 <= 0 ) by A19, A27;
then A31: ex p11 being Point of (TOP-REAL 2) st
( p11 = p2 & |.p11.| = 1 ) by A2;
A32: q2 `2 <= 0 by A1, A5, A30, Th57;
A33: |.q2.| = |.p2.| by A5, JGRAPH_4:128;
then A34: q2 in P by A2, A31;
A35: q1 `2 <= 0 by A1, A4, A28, Th57;
A36: |.q1.| = |.p1.| by A4, JGRAPH_4:128;
then A37: q1 in P by A2, A29;
now__::_thesis:_(_(_p1_=_W-min_P_&_(_(_q1_in_Upper_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_)_or_(_q1_in_Upper_Arc_P_&_q2_in_Upper_Arc_P_&_LE_q1,q2,_Upper_Arc_P,_W-min_P,_E-max_P_)_or_(_q1_in_Lower_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_&_LE_q1,q2,_Lower_Arc_P,_E-max_P,_W-min_P_)_)_)_or_(_p1_<>_W-min_P_&_(_(_q1_in_Upper_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_)_or_(_q1_in_Upper_Arc_P_&_q2_in_Upper_Arc_P_&_LE_q1,q2,_Upper_Arc_P,_W-min_P,_E-max_P_)_or_(_q1_in_Lower_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_&_LE_q1,q2,_Lower_Arc_P,_E-max_P,_W-min_P_)_)_)_)
percases ( p1 = W-min P or p1 <> W-min P ) ;
caseA38: p1 = W-min P ; ::_thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) )
then p1 = (cn -FanMorphS) . p1 by A7, JGRAPH_4:113;
then LE q1,q2,P by A4, A6, A34, A38, JORDAN7:3;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) by JORDAN6:def_10; ::_thesis: verum
end;
caseA39: p1 <> W-min P ; ::_thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) )
now__::_thesis:_(_(_p1_`1_=_p2_`1_&_(_(_q1_in_Upper_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_)_or_(_q1_in_Upper_Arc_P_&_q2_in_Upper_Arc_P_&_LE_q1,q2,_Upper_Arc_P,_W-min_P,_E-max_P_)_or_(_q1_in_Lower_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_&_LE_q1,q2,_Lower_Arc_P,_E-max_P,_W-min_P_)_)_)_or_(_p1_`1_>_p2_`1_&_(_(_q1_in_Upper_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_)_or_(_q1_in_Upper_Arc_P_&_q2_in_Upper_Arc_P_&_LE_q1,q2,_Upper_Arc_P,_W-min_P,_E-max_P_)_or_(_q1_in_Lower_Arc_P_&_q2_in_Lower_Arc_P_&_not_q2_=_W-min_P_&_LE_q1,q2,_Lower_Arc_P,_E-max_P,_W-min_P_)_)_)_)
percases ( p1 `1 = p2 `1 or p1 `1 > p2 `1 ) by A2, A3, A28, A39, Th48;
caseA40: p1 `1 = p2 `1 ; ::_thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) )
A41: p2 = |[(p2 `1),(p2 `2)]| by EUCLID:53;
A42: now__::_thesis:_(_p1_`2_=_-_(p2_`2)_implies_p1_=_p2_)
assume A43: p1 `2 = - (p2 `2) ; ::_thesis: p1 = p2
then p2 `2 = 0 by A28, A30, XREAL_1:58;
hence p1 = p2 by A40, A41, A43, EUCLID:53; ::_thesis: verum
end;
((p1 `1) ^2) + ((p1 `2) ^2) = 1 ^2 by A29, JGRAPH_3:1
.= ((p1 `1) ^2) + ((p2 `2) ^2) by A31, A40, JGRAPH_3:1 ;
then A44: ( p1 `2 = p2 `2 or p1 `2 = - (p2 `2) ) by SQUARE_1:40;
p1 = |[(p1 `1),(p1 `2)]| by EUCLID:53;
then LE q1,q2,P by A2, A4, A5, A34, A40, A44, A41, A42, JGRAPH_3:26, JORDAN6:56;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) by JORDAN6:def_10; ::_thesis: verum
end;
case p1 `1 > p2 `1 ; ::_thesis: ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) )
then (p1 `1) / |.p1.| > (p2 `1) / |.p2.| by A29, A31;
then A45: (q1 `1) / |.q1.| >= (q2 `1) / |.q2.| by A1, A4, A5, A28, A30, A29, A31, Th27;
q2 <> W-min P by A5, A8, A10, A9, A12, A27, FUNCT_1:def_4;
then LE q1,q2,P by A2, A36, A33, A35, A32, A29, A31, A37, A34, A45, Th56;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) by JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) ; ::_thesis: verum
end;
end;
end;
hence ( ( q1 in Upper_Arc P & q2 in Lower_Arc P & not q2 = W-min P ) or ( q1 in Upper_Arc P & q2 in Upper_Arc P & LE q1,q2, Upper_Arc P, W-min P, E-max P ) or ( q1 in Lower_Arc P & q2 in Lower_Arc P & not q2 = W-min P & LE q1,q2, Lower_Arc P, E-max P, W-min P ) ) ; :: according to JORDAN6:def_10 ::_thesis: verum
end;
end;
end;
theorem Th59: :: JGRAPH_5:59
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `1 < 0 & p1 `2 >= 0 & p2 `1 < 0 & p2 `2 >= 0 & p3 `1 < 0 & p3 `2 >= 0 & p4 `1 < 0 & p4 `2 >= 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `1 < 0 & p1 `2 >= 0 & p2 `1 < 0 & p2 `2 >= 0 & p3 `1 < 0 & p3 `2 >= 0 & p4 `1 < 0 & p4 `2 >= 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `1 < 0 & p1 `2 >= 0 & p2 `1 < 0 & p2 `2 >= 0 & p3 `1 < 0 & p3 `2 >= 0 & p4 `1 < 0 & p4 `2 >= 0 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1 `1 < 0 and
A6: p1 `2 >= 0 and
A7: p2 `1 < 0 and
A8: p2 `2 >= 0 and
A9: p3 `1 < 0 and
A10: p3 `2 >= 0 and
A11: p4 `1 < 0 and
A12: p4 `2 >= 0 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
consider r being real number such that
A13: p4 `1 < r and
A14: r < 0 by A11, XREAL_1:5;
reconsider r1 = r as Real by XREAL_0:def_1;
set s = sqrt (1 - (r1 ^2));
A15: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then p4 in P by A4, JORDAN7:5;
then A16: ex p being Point of (TOP-REAL 2) st
( p = p4 & |.p.| = 1 ) by A1;
then - 1 <= p4 `1 by Th1;
then - 1 <= r1 by A13, XXREAL_0:2;
then r1 ^2 <= 1 ^2 by A14, SQUARE_1:49;
then A17: 1 - (r1 ^2) >= 0 by XREAL_1:48;
then A18: (sqrt (1 - (r1 ^2))) ^2 = 1 - (r1 ^2) by SQUARE_1:def_2;
then A19: (1 - ((sqrt (1 - (r1 ^2))) ^2)) + ((sqrt (1 - (r1 ^2))) ^2) > 0 + ((sqrt (1 - (r1 ^2))) ^2) by A14, SQUARE_1:12, XREAL_1:8;
then A20: - 1 < sqrt (1 - (r1 ^2)) by SQUARE_1:52;
A21: sqrt (1 - (r1 ^2)) < 1 by A19, SQUARE_1:52;
then consider f1 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A22: f1 = (sqrt (1 - (r1 ^2))) -FanMorphW and
A23: f1 is being_homeomorphism by A20, JGRAPH_4:41;
set q11 = f1 . p1;
set q22 = f1 . p2;
set q33 = f1 . p3;
set q44 = f1 . p4;
A24: sqrt (1 - (r1 ^2)) >= 0 by A17, SQUARE_1:def_2;
p3 in P by A3, A15, JORDAN7:5;
then A25: ex p33 being Point of (TOP-REAL 2) st
( p33 = p3 & |.p33.| = 1 ) by A1;
then ( (p3 `2) / |.p3.| < (p4 `2) / |.p4.| or p3 = p4 ) by A1, A4, A11, A12, A16, Th46;
then A26: ( ((f1 . p3) `2) / |.(f1 . p3).| < ((f1 . p4) `2) / |.(f1 . p4).| or p3 = p4 ) by A9, A11, A20, A21, A22, JGRAPH_4:46;
(p4 `1) ^2 > r1 ^2 by A13, A14, SQUARE_1:44;
then A27: 1 - ((p4 `1) ^2) < 1 - (r1 ^2) by XREAL_1:15;
A28: ( p3 `1 < p4 `1 or p3 = p4 ) by A1, A4, A9, A10, A12, Th46;
then - (p3 `1) >= - (p4 `1) by XREAL_1:24;
then (- (p3 `1)) ^2 >= (- (p4 `1)) ^2 by A11, SQUARE_1:15;
then 1 - ((p3 `1) ^2) <= 1 - ((p4 `1) ^2) by XREAL_1:10;
then A29: 1 - ((p3 `1) ^2) < (sqrt (1 - (r1 ^2))) ^2 by A27, A18, XXREAL_0:2;
( p2 `1 < p3 `1 or p2 = p3 ) by A1, A3, A7, A8, A10, Th46;
then A30: p2 `1 <= p4 `1 by A28, XXREAL_0:2;
then - (p2 `1) >= - (p4 `1) by XREAL_1:24;
then (- (p2 `1)) ^2 >= (- (p4 `1)) ^2 by A11, SQUARE_1:15;
then 1 - ((p2 `1) ^2) <= 1 - ((p4 `1) ^2) by XREAL_1:10;
then A31: 1 - ((p2 `1) ^2) < (sqrt (1 - (r1 ^2))) ^2 by A27, A18, XXREAL_0:2;
( p1 `1 < p2 `1 or p1 = p2 ) by A1, A2, A5, A6, A8, Th46;
then p1 `1 <= p4 `1 by A30, XXREAL_0:2;
then - (p1 `1) >= - (p4 `1) by XREAL_1:24;
then (- (p1 `1)) ^2 >= (- (p4 `1)) ^2 by A11, SQUARE_1:15;
then 1 - ((p1 `1) ^2) <= 1 - ((p4 `1) ^2) by XREAL_1:10;
then A32: 1 - ((p1 `1) ^2) < (sqrt (1 - (r1 ^2))) ^2 by A27, A18, XXREAL_0:2;
1 ^2 = ((p3 `1) ^2) + ((p3 `2) ^2) by A25, JGRAPH_3:1;
then A33: (p3 `2) / |.p3.| < sqrt (1 - (r1 ^2)) by A25, A24, A29, SQUARE_1:48;
then A34: (f1 . p3) `1 < 0 by A9, A20, A22, JGRAPH_4:43;
p2 in P by A2, A15, JORDAN7:5;
then A35: ex p22 being Point of (TOP-REAL 2) st
( p22 = p2 & |.p22.| = 1 ) by A1;
then A36: |.(f1 . p2).| = 1 by A22, JGRAPH_4:33;
then A37: f1 . p2 in P by A1;
( (p2 `2) / |.p2.| < (p3 `2) / |.p3.| or p2 = p3 ) by A1, A3, A9, A10, A35, A25, Th46;
then A38: ( ((f1 . p2) `2) / |.(f1 . p2).| < ((f1 . p3) `2) / |.(f1 . p3).| or p2 = p3 ) by A7, A9, A20, A21, A22, JGRAPH_4:46;
A39: |.(f1 . p3).| = 1 by A25, A22, JGRAPH_4:33;
then A40: f1 . p3 in P by A1;
1 ^2 = ((p2 `1) ^2) + ((p2 `2) ^2) by A35, JGRAPH_3:1;
then A41: (p2 `2) / |.p2.| < sqrt (1 - (r1 ^2)) by A35, A24, A31, SQUARE_1:48;
then A42: (f1 . p2) `2 < 0 by A7, A20, A22, JGRAPH_4:43;
A43: (f1 . p2) `1 < 0 by A7, A20, A22, A41, JGRAPH_4:43;
1 ^2 = ((p4 `1) ^2) + ((p4 `2) ^2) by A16, JGRAPH_3:1;
then (p4 `2) / |.p4.| < sqrt (1 - (r1 ^2)) by A27, A16, A18, A24, SQUARE_1:48;
then A44: ( (f1 . p4) `1 < 0 & (f1 . p4) `2 < 0 ) by A11, A20, A22, JGRAPH_4:43;
p1 in P by A2, A15, JORDAN7:5;
then A45: ex p11 being Point of (TOP-REAL 2) st
( p11 = p1 & |.p11.| = 1 ) by A1;
then ( (p1 `2) / |.p1.| < (p2 `2) / |.p2.| or p1 = p2 ) by A1, A2, A7, A8, A35, Th46;
then A46: ( ((f1 . p1) `2) / |.(f1 . p1).| < ((f1 . p2) `2) / |.(f1 . p2).| or p1 = p2 ) by A5, A7, A20, A21, A22, JGRAPH_4:46;
1 ^2 = ((p1 `1) ^2) + ((p1 `2) ^2) by A45, JGRAPH_3:1;
then A47: (p1 `2) / |.p1.| < sqrt (1 - (r1 ^2)) by A45, A24, A32, SQUARE_1:48;
then A48: (f1 . p1) `1 < 0 by A5, A20, A22, JGRAPH_4:43;
A49: |.(f1 . p1).| = 1 by A45, A22, JGRAPH_4:33;
then f1 . p1 in P by A1;
then A50: LE f1 . p1,f1 . p2,P by A1, A49, A36, A37, A48, A43, A42, A46, Th51;
A51: ( (f1 . p2) `1 < 0 & (f1 . p2) `2 < 0 ) by A7, A20, A22, A41, JGRAPH_4:43;
A52: ( (f1 . p1) `1 < 0 & (f1 . p1) `2 < 0 ) by A5, A20, A22, A47, JGRAPH_4:43;
A53: for q being Point of (TOP-REAL 2) holds |.(f1 . q).| = |.q.| by A22, JGRAPH_4:33;
( (f1 . p3) `1 < 0 & (f1 . p3) `2 < 0 ) by A9, A20, A22, A33, JGRAPH_4:43;
then A54: LE f1 . p2,f1 . p3,P by A1, A36, A37, A39, A40, A43, A38, Th51;
A55: (f1 . p3) `2 < 0 by A9, A20, A22, A33, JGRAPH_4:43;
A56: |.(f1 . p4).| = 1 by A16, A22, JGRAPH_4:33;
then f1 . p4 in P by A1;
then LE f1 . p3,f1 . p4,P by A1, A39, A40, A56, A34, A44, A26, Th51;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A23, A53, A52, A51, A34, A55, A44, A50, A54; ::_thesis: verum
end;
theorem Th60: :: JGRAPH_5:60
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 >= 0 & q2 `1 < 0 & q2 `2 >= 0 & q3 `1 < 0 & q3 `2 >= 0 & q4 `1 < 0 & q4 `2 >= 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 >= 0 & q2 `1 < 0 & q2 `2 >= 0 & q3 `1 < 0 & q3 `2 >= 0 & q4 `1 < 0 & q4 `2 >= 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 >= 0 & q2 `1 < 0 & q2 `2 >= 0 & q3 `1 < 0 & q3 `2 >= 0 & q4 `1 < 0 & q4 `2 >= 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1 `2 >= 0 and
A6: p2 `2 >= 0 and
A7: p3 `2 >= 0 and
A8: p4 `2 > 0 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 >= 0 & q2 `1 < 0 & q2 `2 >= 0 & q3 `1 < 0 & q3 `2 >= 0 & q4 `1 < 0 & q4 `2 >= 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A9: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then p4 in P by A4, JORDAN7:5;
then A10: ex p being Point of (TOP-REAL 2) st
( p = p4 & |.p.| = 1 ) by A1;
A11: now__::_thesis:_not_p4_`1_=_1
assume p4 `1 = 1 ; ::_thesis: contradiction
then 1 ^2 = 1 + ((p4 `2) ^2) by A10, JGRAPH_3:1;
hence contradiction by A8, XCMPLX_1:6; ::_thesis: verum
end;
p4 `1 <= 1 by A10, Th1;
then p4 `1 < 1 by A11, XXREAL_0:1;
then consider r being real number such that
A12: p4 `1 < r and
A13: r < 1 by XREAL_1:5;
reconsider r1 = r as Real by XREAL_0:def_1;
- 1 <= p4 `1 by A10, Th1;
then A14: - 1 < r1 by A12, XXREAL_0:2;
then consider f1 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A15: f1 = r1 -FanMorphN and
A16: f1 is being_homeomorphism by A13, JGRAPH_4:74;
set q11 = f1 . p1;
set q22 = f1 . p2;
set q33 = f1 . p3;
set q44 = f1 . p4;
A17: for q being Point of (TOP-REAL 2) holds |.(f1 . q).| = |.q.| by A15, JGRAPH_4:66;
A18: ( p3 `1 < p4 `1 or p3 = p4 ) by A1, A4, A8, Th47;
then A19: p3 `1 < r1 by A12, XXREAL_0:2;
p3 in P by A3, A9, JORDAN7:5;
then A20: ex p33 being Point of (TOP-REAL 2) st
( p33 = p3 & |.p33.| = 1 ) by A1;
then ( (p3 `1) / |.p3.| < (p4 `1) / |.p4.| or p3 = p4 ) by A1, A4, A8, A10, Th47;
then A21: ( ((f1 . p3) `1) / |.(f1 . p3).| < ((f1 . p4) `1) / |.(f1 . p4).| or p3 = p4 ) by A7, A8, A10, A20, A13, A14, A15, Th21;
A22: (p3 `1) / |.p3.| < r1 by A20, A12, A18, XXREAL_0:2;
then A23: (f1 . p3) `2 >= 0 by A7, A20, A13, A14, A15, Th20;
A24: ( p1 `1 < p2 `1 or p1 = p2 ) by A1, A2, A6, Th47;
(p4 `1) / |.p4.| < r1 by A10, A12;
then A25: ( (f1 . p4) `1 < 0 & (f1 . p4) `2 > 0 ) by A8, A14, A15, JGRAPH_4:76;
p2 in P by A2, A9, JORDAN7:5;
then A26: ex p22 being Point of (TOP-REAL 2) st
( p22 = p2 & |.p22.| = 1 ) by A1;
then A27: |.(f1 . p2).| = 1 by A15, JGRAPH_4:66;
then A28: f1 . p2 in P by A1;
A29: ( p2 `1 < p3 `1 or p2 = p3 ) by A1, A3, A7, Th47;
then A30: (p2 `1) / |.p2.| < r1 by A26, A19, XXREAL_0:2;
then A31: (f1 . p2) `2 >= 0 by A6, A26, A13, A14, A15, Th20;
p1 in P by A2, A9, JORDAN7:5;
then A32: ex p11 being Point of (TOP-REAL 2) st
( p11 = p1 & |.p11.| = 1 ) by A1;
then ( (p1 `1) / |.p1.| < (p2 `1) / |.p2.| or p1 = p2 ) by A1, A2, A6, A26, Th47;
then A33: ( ((f1 . p1) `1) / |.(f1 . p1).| < ((f1 . p2) `1) / |.(f1 . p2).| or p1 = p2 ) by A5, A6, A32, A26, A13, A14, A15, Th21;
p2 `1 < r1 by A29, A19, XXREAL_0:2;
then A34: (p1 `1) / |.p1.| < r1 by A32, A24, XXREAL_0:2;
then A35: (f1 . p1) `2 >= 0 by A5, A32, A13, A14, A15, Th20;
A36: (f1 . p2) `1 < 0 by A6, A26, A13, A14, A15, A30, Th20;
A37: |.(f1 . p1).| = 1 by A32, A15, JGRAPH_4:66;
then f1 . p1 in P by A1;
then A38: LE f1 . p1,f1 . p2,P by A1, A37, A27, A28, A31, A36, A35, A33, Th53;
A39: |.(f1 . p3).| = 1 by A20, A15, JGRAPH_4:66;
then A40: f1 . p3 in P by A1;
A41: (f1 . p3) `1 < 0 by A7, A20, A13, A14, A15, A22, Th20;
A42: ( (f1 . p2) `1 < 0 & (f1 . p2) `2 >= 0 ) by A6, A26, A13, A14, A15, A30, Th20;
A43: ( ( (f1 . p1) `1 < 0 & (f1 . p1) `2 >= 0 ) or ( (f1 . p1) `1 < 0 & (f1 . p1) `2 = 0 ) ) by A5, A32, A13, A14, A15, A34, Th20;
A44: |.(f1 . p4).| = 1 by A10, A15, JGRAPH_4:66;
then f1 . p4 in P by A1;
then A45: LE f1 . p3,f1 . p4,P by A1, A39, A40, A44, A25, A23, A21, Th53;
( (p2 `1) / |.p2.| < (p3 `1) / |.p3.| or p2 = p3 ) by A1, A3, A7, A26, A20, Th47;
then ( ((f1 . p2) `1) / |.(f1 . p2).| < ((f1 . p3) `1) / |.(f1 . p3).| or p2 = p3 ) by A6, A7, A26, A20, A13, A14, A15, Th21;
then LE f1 . p2,f1 . p3,P by A1, A27, A28, A39, A40, A31, A23, A41, Th53;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 >= 0 & q2 `1 < 0 & q2 `2 >= 0 & q3 `1 < 0 & q3 `2 >= 0 & q4 `1 < 0 & q4 `2 >= 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A16, A17, A25, A43, A42, A38, A23, A41, A45; ::_thesis: verum
end;
theorem Th61: :: JGRAPH_5:61
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: ( LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 `2 >= 0 & p2 `2 >= 0 & p3 `2 >= 0 & p4 `2 > 0 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
consider f being Function of (TOP-REAL 2),(TOP-REAL 2), q1, q2, q3, q4 being Point of (TOP-REAL 2) such that
A3: f is being_homeomorphism and
A4: for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| and
A5: ( q1 = f . p1 & q2 = f . p2 ) and
A6: ( q3 = f . p3 & q4 = f . p4 ) and
A7: ( q1 `1 < 0 & q1 `2 >= 0 & q2 `1 < 0 & q2 `2 >= 0 & q3 `1 < 0 & q3 `2 >= 0 & q4 `1 < 0 & q4 `2 >= 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A1, A2, Th60;
consider f2 being Function of (TOP-REAL 2),(TOP-REAL 2), q1b, q2b, q3b, q4b being Point of (TOP-REAL 2) such that
A8: f2 is being_homeomorphism and
A9: for q being Point of (TOP-REAL 2) holds |.(f2 . q).| = |.q.| and
A10: ( q1b = f2 . q1 & q2b = f2 . q2 ) and
A11: ( q3b = f2 . q3 & q4b = f2 . q4 ) and
A12: ( q1b `1 < 0 & q1b `2 < 0 & q2b `1 < 0 & q2b `2 < 0 & q3b `1 < 0 & q3b `2 < 0 & q4b `1 < 0 & q4b `2 < 0 & LE q1b,q2b,P & LE q2b,q3b,P & LE q3b,q4b,P ) by A1, A7, Th59;
reconsider f3 = f2 * f as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A13: f3 is being_homeomorphism by A3, A8, TOPS_2:57;
A14: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A15: ( f3 . p3 = q3b & f3 . p4 = q4b ) by A6, A11, FUNCT_1:13;
A16: for q being Point of (TOP-REAL 2) holds |.(f3 . q).| = |.q.|
proof
let q be Point of (TOP-REAL 2); ::_thesis: |.(f3 . q).| = |.q.|
dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then f3 . q = f2 . (f . q) by FUNCT_1:13;
hence |.(f3 . q).| = |.(f . q).| by A9
.= |.q.| by A4 ;
::_thesis: verum
end;
( f3 . p1 = q1b & f3 . p2 = q2b ) by A5, A10, A14, FUNCT_1:13;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A12, A13, A16, A15; ::_thesis: verum
end;
theorem Th62: :: JGRAPH_5:62
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: ( p1 `2 >= 0 or p1 `1 >= 0 ) and
A6: ( p2 `2 >= 0 or p2 `1 >= 0 ) and
A7: ( p3 `2 >= 0 or p3 `1 >= 0 ) and
A8: ( p4 `2 > 0 or p4 `1 > 0 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A9: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A10: p4 in P by A4, JORDAN7:5;
then A11: ex p44 being Point of (TOP-REAL 2) st
( p44 = p4 & |.p44.| = 1 ) by A1;
then A12: - 1 <= p4 `2 by Th1;
now__::_thesis:_not_p4_`2_=_-_1
assume A13: p4 `2 = - 1 ; ::_thesis: contradiction
1 ^2 = ((p4 `1) ^2) + ((p4 `2) ^2) by A11, JGRAPH_3:1
.= ((p4 `1) ^2) + 1 by A13 ;
hence contradiction by A8, A13, XCMPLX_1:6; ::_thesis: verum
end;
then p4 `2 > - 1 by A12, XXREAL_0:1;
then consider r being real number such that
A14: - 1 < r and
A15: r < p4 `2 by XREAL_1:5;
reconsider r1 = r as Real by XREAL_0:def_1;
p4 `2 <= 1 by A11, Th1;
then A16: r1 < 1 by A15, XXREAL_0:2;
then consider f1 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A17: f1 = r1 -FanMorphE and
A18: f1 is being_homeomorphism by A14, JGRAPH_4:105;
set q11 = f1 . p1;
set q22 = f1 . p2;
set q33 = f1 . p3;
set q44 = f1 . p4;
A19: |.(f1 . p4).| = 1 by A11, A17, JGRAPH_4:97;
then A20: f1 . p4 in P by A1;
A21: p1 in P by A2, A9, JORDAN7:5;
then A22: ex p11 being Point of (TOP-REAL 2) st
( p11 = p1 & |.p11.| = 1 ) by A1;
then A23: |.(f1 . p1).| = 1 by A17, JGRAPH_4:97;
then A24: f1 . p1 in P by A1;
A25: p3 in P by A3, A9, JORDAN7:5;
then A26: ex p33 being Point of (TOP-REAL 2) st
( p33 = p3 & |.p33.| = 1 ) by A1;
then A27: |.(f1 . p3).| = 1 by A17, JGRAPH_4:97;
then A28: f1 . p3 in P by A1;
A29: p2 in P by A2, A9, JORDAN7:5;
then A30: ex p22 being Point of (TOP-REAL 2) st
( p22 = p2 & |.p22.| = 1 ) by A1;
then A31: |.(f1 . p2).| = 1 by A17, JGRAPH_4:97;
then A32: f1 . p2 in P by A1;
now__::_thesis:_(_(_p4_`2_<=_0_&_ex_f_being_Function_of_(TOP-REAL_2),(TOP-REAL_2)_ex_q1,_q2,_q3,_q4_being_Point_of_(TOP-REAL_2)_st_
(_f_is_being_homeomorphism_&_(_for_q_being_Point_of_(TOP-REAL_2)_holds_|.(f_._q).|_=_|.q.|_)_&_q1_=_f_._p1_&_q2_=_f_._p2_&_q3_=_f_._p3_&_q4_=_f_._p4_&_q1_`2_>=_0_&_q2_`2_>=_0_&_q3_`2_>=_0_&_q4_`2_>_0_&_LE_q1,q2,P_&_LE_q2,q3,P_&_LE_q3,q4,P_)_)_or_(_p4_`2_>_0_&_ex_f_being_Function_of_(TOP-REAL_2),(TOP-REAL_2)_ex_q1,_q2,_q3,_q4_being_Point_of_(TOP-REAL_2)_st_
(_f_is_being_homeomorphism_&_(_for_q_being_Point_of_(TOP-REAL_2)_holds_|.(f_._q).|_=_|.q.|_)_&_q1_=_f_._p1_&_q2_=_f_._p2_&_q3_=_f_._p3_&_q4_=_f_._p4_&_q1_`2_>=_0_&_q2_`2_>=_0_&_q3_`2_>=_0_&_q4_`2_>_0_&_LE_q1,q2,P_&_LE_q2,q3,P_&_LE_q3,q4,P_)_)_)
percases ( p4 `2 <= 0 or p4 `2 > 0 ) ;
caseA33: p4 `2 <= 0 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A34: Upper_Arc P = { p7 where p7 is Point of (TOP-REAL 2) : ( p7 in P & p7 `2 >= 0 ) } by A1, Th34;
A35: (p4 `2) / |.p4.| > r1 by A11, A15;
then A36: (f1 . p4) `1 > 0 by A8, A15, A17, A33, JGRAPH_4:106;
A37: now__::_thesis:_not_(f1_._p4)_`2_=_0
set q8 = |[(sqrt (1 - (r1 ^2))),r1]|;
assume A38: (f1 . p4) `2 = 0 ; ::_thesis: contradiction
1 ^2 = (((f1 . p4) `1) ^2) + (((f1 . p4) `2) ^2) by A19, JGRAPH_3:1
.= ((f1 . p4) `1) ^2 by A38 ;
then ( (f1 . p4) `1 = - 1 or (f1 . p4) `1 = 1 ) by SQUARE_1:41;
then A39: f1 . p4 = |[1,0]| by A8, A15, A17, A33, A35, A38, EUCLID:53, JGRAPH_4:106;
set r8 = f1 . |[(sqrt (1 - (r1 ^2))),r1]|;
1 ^2 > r1 ^2 by A14, A16, SQUARE_1:50;
then A40: 1 - (r1 ^2) > 0 by XREAL_1:50;
A41: |[(sqrt (1 - (r1 ^2))),r1]| `1 = sqrt (1 - (r1 ^2)) by EUCLID:52;
then A42: |[(sqrt (1 - (r1 ^2))),r1]| `1 > 0 by A40, SQUARE_1:25;
|[(sqrt (1 - (r1 ^2))),r1]| `2 = r1 by EUCLID:52;
then |.|[(sqrt (1 - (r1 ^2))),r1]|.| = sqrt (((sqrt (1 - (r1 ^2))) ^2) + (r1 ^2)) by A41, JGRAPH_3:1;
then A43: |.|[(sqrt (1 - (r1 ^2))),r1]|.| = sqrt ((1 - (r1 ^2)) + (r1 ^2)) by A40, SQUARE_1:def_2
.= 1 by SQUARE_1:18 ;
then A44: (|[(sqrt (1 - (r1 ^2))),r1]| `2) / |.|[(sqrt (1 - (r1 ^2))),r1]|.| = r1 by EUCLID:52;
then A45: (f1 . |[(sqrt (1 - (r1 ^2))),r1]|) `2 = 0 by A17, A42, JGRAPH_4:111;
|.(f1 . |[(sqrt (1 - (r1 ^2))),r1]|).| = 1 by A17, A43, JGRAPH_4:97;
then 1 ^2 = (((f1 . |[(sqrt (1 - (r1 ^2))),r1]|) `1) ^2) + (((f1 . |[(sqrt (1 - (r1 ^2))),r1]|) `2) ^2) by JGRAPH_3:1
.= ((f1 . |[(sqrt (1 - (r1 ^2))),r1]|) `1) ^2 by A45 ;
then ( (f1 . |[(sqrt (1 - (r1 ^2))),r1]|) `1 = - 1 or (f1 . |[(sqrt (1 - (r1 ^2))),r1]|) `1 = 1 ) by SQUARE_1:41;
then A46: f1 . |[(sqrt (1 - (r1 ^2))),r1]| = |[1,0]| by A17, A44, A42, A45, EUCLID:53, JGRAPH_4:111;
( f1 is one-to-one & dom f1 = the carrier of (TOP-REAL 2) ) by A14, A16, A17, FUNCT_2:def_1, JGRAPH_4:102;
then p4 = |[(sqrt (1 - (r1 ^2))),r1]| by A39, A46, FUNCT_1:def_4;
hence contradiction by A15, EUCLID:52; ::_thesis: verum
end;
A47: (f1 . p4) `2 >= 0 by A8, A15, A17, A33, A35, JGRAPH_4:106;
A48: Lower_Arc P = { p7 where p7 is Point of (TOP-REAL 2) : ( p7 in P & p7 `2 <= 0 ) } by A1, Th35;
A49: now__::_thesis:_(_(_p3_`1_<=_0_&_(f1_._p1)_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_(f1_._p3)_`2_>=_0_&_(f1_._p4)_`2_>_0_&_LE_f1_._p1,f1_._p2,P_&_LE_f1_._p2,f1_._p3,P_&_LE_f1_._p3,f1_._p4,P_)_or_(_p3_`1_>_0_&_(f1_._p1)_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_(f1_._p3)_`2_>=_0_&_(f1_._p4)_`2_>_0_&_LE_f1_._p1,f1_._p2,P_&_LE_f1_._p2,f1_._p3,P_&_LE_f1_._p3,f1_._p4,P_)_)
percases ( p3 `1 <= 0 or p3 `1 > 0 ) ;
caseA50: p3 `1 <= 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P )
then A51: f1 . p3 = p3 by A17, JGRAPH_4:82;
A52: now__::_thesis:_(_(_p3_`1_=_0_&_(f1_._p3)_`2_>=_0_)_or_(_p3_`1_<_0_&_(f1_._p3)_`2_>=_0_)_)
percases ( p3 `1 = 0 or p3 `1 < 0 ) by A50;
caseA53: p3 `1 = 0 ; ::_thesis: (f1 . p3) `2 >= 0
A54: now__::_thesis:_not_(f1_._p3)_`2_=_-_1
assume (f1 . p3) `2 = - 1 ; ::_thesis: contradiction
then - 1 >= p4 `2 by A1, A4, A7, A8, A33, A51, Th50;
hence contradiction by A14, A15, XXREAL_0:2; ::_thesis: verum
end;
1 ^2 = (0 ^2) + (((f1 . p3) `2) ^2) by A26, A51, A53, JGRAPH_3:1
.= ((f1 . p3) `2) ^2 ;
hence (f1 . p3) `2 >= 0 by A54, SQUARE_1:41; ::_thesis: verum
end;
case p3 `1 < 0 ; ::_thesis: (f1 . p3) `2 >= 0
hence (f1 . p3) `2 >= 0 by A7, A17, JGRAPH_4:82; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p2_<>_W-min_P_&_(f1_._p1)_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_(f1_._p3)_`2_>=_0_&_(f1_._p4)_`2_>_0_&_LE_f1_._p1,f1_._p2,P_&_LE_f1_._p2,f1_._p3,P_&_LE_f1_._p3,f1_._p4,P_)_or_(_p2_=_W-min_P_&_(f1_._p1)_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_(f1_._p3)_`2_>=_0_&_(f1_._p4)_`2_>_0_&_LE_f1_._p1,f1_._p2,P_&_LE_f1_._p2,f1_._p3,P_&_LE_f1_._p3,f1_._p4,P_)_)
percases ( p2 <> W-min P or p2 = W-min P ) ;
caseA55: p2 <> W-min P ; ::_thesis: ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P )
A56: now__::_thesis:_not_p2_`2_<_0
A57: p3 in Upper_Arc P by A25, A34, A51, A52;
assume A58: p2 `2 < 0 ; ::_thesis: contradiction
then p2 in Lower_Arc P by A29, A48;
then LE p3,p2,P by A55, A57, JORDAN6:def_10;
hence contradiction by A1, A3, A51, A52, A58, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
A59: p2 `1 <= p3 `1 by A1, A3, A51, A52, Th47;
then A60: f1 . p2 = p2 by A17, A50, JGRAPH_4:82;
now__::_thesis:_(_(_p1_<>_W-min_P_&_(f1_._p1)_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_(f1_._p3)_`2_>=_0_&_(f1_._p4)_`2_>_0_&_LE_f1_._p1,f1_._p2,P_&_LE_f1_._p2,f1_._p3,P_&_LE_f1_._p3,f1_._p4,P_)_or_(_p1_=_W-min_P_&_(f1_._p1)_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_(f1_._p3)_`2_>=_0_&_(f1_._p4)_`2_>_0_&_LE_f1_._p1,f1_._p2,P_&_LE_f1_._p2,f1_._p3,P_&_LE_f1_._p3,f1_._p4,P_)_)
percases ( p1 <> W-min P or p1 = W-min P ) ;
caseA61: p1 <> W-min P ; ::_thesis: ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P )
A62: now__::_thesis:_not_p1_`2_<_0
A63: p2 in Upper_Arc P by A29, A34, A56;
assume A64: p1 `2 < 0 ; ::_thesis: contradiction
then p1 in Lower_Arc P by A21, A48;
then LE p2,p1,P by A61, A63, JORDAN6:def_10;
hence contradiction by A1, A2, A56, A64, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
p1 `1 <= p2 `1 by A1, A2, A56, Th47;
hence ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P ) by A1, A2, A3, A17, A28, A20, A36, A47, A37, A51, A52, A56, A59, A60, A62, Th54, JGRAPH_4:82; ::_thesis: verum
end;
caseA65: p1 = W-min P ; ::_thesis: ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P )
A66: W-min P = |[(- 1),0]| by A1, Th29;
then p1 `1 = - 1 by A65, EUCLID:52;
then p1 = f1 . p1 by A17, JGRAPH_4:82;
hence ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P ) by A1, A2, A3, A25, A17, A20, A36, A47, A37, A51, A52, A56, A59, A65, A66, Th54, EUCLID:52, JGRAPH_4:82; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P ) ; ::_thesis: verum
end;
caseA67: p2 = W-min P ; ::_thesis: ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P )
W-min P = |[(- 1),0]| by A1, Th29;
then A68: p2 `1 = - 1 by A67, EUCLID:52;
then ( p2 = f1 . p2 & p1 `1 <= p2 `1 ) by A1, A2, A6, A17, Th47, JGRAPH_4:82;
hence ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P ) by A1, A2, A3, A5, A6, A14, A15, A17, A28, A20, A33, A36, A47, A37, A51, A52, A68, Th54, JGRAPH_4:82; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P ) ; ::_thesis: verum
end;
caseA69: p3 `1 > 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P )
A70: now__::_thesis:_(_(_p3_<>_p4_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_&_(f1_._p3)_`2_>=_0_&_LE_f1_._p3,f1_._p4,P_)_or_(_p3_=_p4_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_&_(f1_._p3)_`2_>=_0_&_LE_f1_._p3,f1_._p4,P_)_)
percases ( p3 <> p4 or p3 = p4 ) ;
caseA71: p3 <> p4 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P & (f1 . p3) `2 >= 0 & LE f1 . p3,f1 . p4,P )
A72: now__::_thesis:_(_p2_`1_=_0_implies_not_p2_`2_=_-_1_)
A73: LE p2,p4,P by A1, A3, A4, JGRAPH_3:26, JORDAN6:58;
assume that
A74: p2 `1 = 0 and
A75: p2 `2 = - 1 ; ::_thesis: contradiction
p2 `2 <= p4 `2 by A11, A75, Th1;
then LE p4,p2,P by A1, A8, A29, A10, A33, A74, Th55;
hence contradiction by A1, A8, A74, A75, A73, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
p3 `2 > p4 `2 by A1, A4, A8, A33, A69, A71, Th50;
then A76: (p3 `2) / |.p3.| >= r1 by A26, A15, XXREAL_0:2;
then A77: (f1 . p3) `1 > 0 by A16, A17, A69, JGRAPH_4:106;
A78: (f1 . p3) `2 >= 0 by A16, A17, A69, A76, JGRAPH_4:106;
A79: now__::_thesis:_(_not_p2_`1_=_0_or_p2_`2_=_1_or_p2_`2_=_-_1_)
assume p2 `1 = 0 ; ::_thesis: ( p2 `2 = 1 or p2 `2 = - 1 )
then 1 ^2 = (0 ^2) + ((p2 `2) ^2) by A30, JGRAPH_3:1;
hence ( p2 `2 = 1 or p2 `2 = - 1 ) by SQUARE_1:40; ::_thesis: verum
end;
A80: now__::_thesis:_(_(_p2_`1_<=_0_&_p2_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_or_(_p2_`1_>_0_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_)
percases ( ( p2 `1 <= 0 & p2 `2 >= 0 ) or p2 `1 > 0 ) by A6, A79, A72;
caseA81: ( p2 `1 <= 0 & p2 `2 >= 0 ) ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
then f1 . p2 = p2 by A17, JGRAPH_4:82;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) by A1, A29, A28, A77, A78, A81, Th54; ::_thesis: verum
end;
caseA82: p2 `1 > 0 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
then A83: (f1 . p2) `1 > 0 by A14, A16, A17, Th22;
now__::_thesis:_(_(_p2_=_p3_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_or_(_p2_<>_p3_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_)
percases ( p2 = p3 or p2 <> p3 ) ;
case p2 = p3 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) by A9, A16, A17, A28, A69, A76, JGRAPH_4:106, JORDAN6:56; ::_thesis: verum
end;
case p2 <> p3 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
then (p2 `2) / |.p2.| > (p3 `2) / |.p3.| by A1, A3, A30, A26, A69, A82, Th50;
then ((f1 . p2) `2) / |.(f1 . p2).| > ((f1 . p3) `2) / |.(f1 . p3).| by A30, A26, A14, A16, A17, A69, A82, Th24;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) by A1, A16, A17, A31, A32, A27, A28, A69, A76, A77, A83, Th55, JGRAPH_4:106; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) ; ::_thesis: verum
end;
end;
end;
(p3 `2) / |.p3.| > (p4 `2) / |.p4.| by A1, A4, A8, A11, A26, A33, A69, A71, Th50;
then ((f1 . p3) `2) / |.(f1 . p3).| > ((f1 . p4) `2) / |.(f1 . p4).| by A8, A11, A26, A14, A15, A17, A33, A69, Th24;
then ((f1 . p3) `2) ^2 > ((f1 . p4) `2) ^2 by A27, A19, A47, SQUARE_1:16;
then A84: (1 ^2) - (((f1 . p3) `2) ^2) < (1 ^2) - (((f1 . p4) `2) ^2) by XREAL_1:15;
1 ^2 = (((f1 . p4) `1) ^2) + (((f1 . p4) `2) ^2) by A19, JGRAPH_3:1;
then A85: (f1 . p4) `1 = sqrt ((1 ^2) - (((f1 . p4) `2) ^2)) by A36, SQUARE_1:22;
A86: 1 ^2 = (((f1 . p3) `1) ^2) + (((f1 . p3) `2) ^2) by A27, JGRAPH_3:1;
then (f1 . p3) `1 = sqrt ((1 ^2) - (((f1 . p3) `2) ^2)) by A77, SQUARE_1:22;
then (f1 . p3) `1 < (f1 . p4) `1 by A86, A85, A84, SQUARE_1:27, XREAL_1:63;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P & (f1 . p3) `2 >= 0 & LE f1 . p3,f1 . p4,P ) by A1, A28, A20, A47, A78, A80, Th54; ::_thesis: verum
end;
caseA87: p3 = p4 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P & (f1 . p3) `2 >= 0 & LE f1 . p3,f1 . p4,P )
A88: now__::_thesis:_(_p2_`1_=_0_implies_not_p2_`2_=_-_1_)
A89: LE p2,p4,P by A1, A3, A4, JGRAPH_3:26, JORDAN6:58;
assume A90: ( p2 `1 = 0 & p2 `2 = - 1 ) ; ::_thesis: contradiction
then LE p4,p2,P by A1, A8, A29, A10, A12, A33, Th55;
hence contradiction by A1, A8, A90, A89, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
A91: now__::_thesis:_(_not_p2_`1_=_0_or_p2_`2_=_1_or_p2_`2_=_-_1_)
assume p2 `1 = 0 ; ::_thesis: ( p2 `2 = 1 or p2 `2 = - 1 )
then 1 ^2 = (0 ^2) + ((p2 `2) ^2) by A30, JGRAPH_3:1;
hence ( p2 `2 = 1 or p2 `2 = - 1 ) by SQUARE_1:40; ::_thesis: verum
end;
A92: (p3 `2) / |.p3.| >= r1 by A26, A15, A87;
then A93: (f1 . p3) `1 > 0 by A16, A17, A69, JGRAPH_4:106;
A94: (f1 . p3) `2 >= 0 by A16, A17, A69, A92, JGRAPH_4:106;
now__::_thesis:_(_(_p2_`1_<=_0_&_p2_`2_>=_0_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_or_(_p2_`1_>_0_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_)
percases ( ( p2 `1 <= 0 & p2 `2 >= 0 ) or p2 `1 > 0 ) by A6, A91, A88;
caseA95: ( p2 `1 <= 0 & p2 `2 >= 0 ) ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
then f1 . p2 = p2 by A17, JGRAPH_4:82;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) by A1, A29, A28, A93, A94, A95, Th54; ::_thesis: verum
end;
caseA96: p2 `1 > 0 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
then A97: (f1 . p2) `1 > 0 by A14, A16, A17, Th22;
now__::_thesis:_(_(_p2_=_p3_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_or_(_p2_<>_p3_&_(f1_._p2)_`2_>=_0_&_LE_f1_._p2,f1_._p3,P_)_)
percases ( p2 = p3 or p2 <> p3 ) ;
case p2 = p3 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) by A9, A16, A17, A28, A69, A92, JGRAPH_4:106, JORDAN6:56; ::_thesis: verum
end;
case p2 <> p3 ; ::_thesis: ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P )
then (p2 `2) / |.p2.| > (p3 `2) / |.p3.| by A1, A3, A30, A26, A69, A96, Th50;
then ((f1 . p2) `2) / |.(f1 . p2).| > ((f1 . p3) `2) / |.(f1 . p3).| by A30, A26, A14, A16, A17, A69, A96, Th24;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) by A1, A16, A17, A31, A32, A27, A28, A69, A92, A93, A97, Th55, JGRAPH_4:106; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P ) ; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p2) `2 >= 0 & LE f1 . p2,f1 . p3,P & (f1 . p3) `2 >= 0 & LE f1 . p3,f1 . p4,P ) by A1, A28, A36, A47, A87, Th54; ::_thesis: verum
end;
end;
end;
A98: now__::_thesis:_(_p1_`1_=_0_implies_not_p1_`2_=_-_1_)
LE p1,p3,P by A1, A2, A3, JGRAPH_3:26, JORDAN6:58;
then A99: LE p1,p4,P by A1, A4, JGRAPH_3:26, JORDAN6:58;
assume A100: ( p1 `1 = 0 & p1 `2 = - 1 ) ; ::_thesis: contradiction
then LE p4,p1,P by A1, A8, A21, A10, A12, A33, Th55;
hence contradiction by A1, A8, A100, A99, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
A101: now__::_thesis:_(_not_p2_`1_=_0_or_p2_`2_=_1_or_p2_`2_=_-_1_)
assume p2 `1 = 0 ; ::_thesis: ( p2 `2 = 1 or p2 `2 = - 1 )
then 1 ^2 = (0 ^2) + ((p2 `2) ^2) by A30, JGRAPH_3:1;
hence ( p2 `2 = 1 or p2 `2 = - 1 ) by SQUARE_1:40; ::_thesis: verum
end;
A102: now__::_thesis:_(_p2_`1_=_0_implies_not_p2_`2_=_-_1_)
A103: LE p2,p4,P by A1, A3, A4, JGRAPH_3:26, JORDAN6:58;
assume that
A104: p2 `1 = 0 and
A105: p2 `2 = - 1 ; ::_thesis: contradiction
p2 `2 <= p4 `2 by A11, A105, Th1;
then LE p4,p2,P by A1, A8, A29, A10, A33, A104, Th55;
hence contradiction by A1, A8, A104, A105, A103, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
A106: now__::_thesis:_(_not_p1_`1_=_0_or_p1_`2_=_1_or_p1_`2_=_-_1_)
assume p1 `1 = 0 ; ::_thesis: ( p1 `2 = 1 or p1 `2 = - 1 )
then 1 ^2 = (0 ^2) + ((p1 `2) ^2) by A22, JGRAPH_3:1;
hence ( p1 `2 = 1 or p1 `2 = - 1 ) by SQUARE_1:40; ::_thesis: verum
end;
now__::_thesis:_(_(_p1_`1_<=_0_&_p1_`2_>=_0_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_or_(_p1_`1_>_0_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_)
percases ( ( p1 `1 <= 0 & p1 `2 >= 0 ) or p1 `1 > 0 ) by A5, A106, A98;
caseA107: ( p1 `1 <= 0 & p1 `2 >= 0 ) ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
then A108: p1 = f1 . p1 by A17, JGRAPH_4:82;
A109: (f1 . p1) `2 >= 0 by A17, A107, JGRAPH_4:82;
now__::_thesis:_(_(_p2_`1_<=_0_&_p2_`2_>=_0_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_or_(_p2_`1_>_0_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_)
percases ( ( p2 `1 <= 0 & p2 `2 >= 0 ) or p2 `1 > 0 ) by A6, A101, A102;
case ( p2 `1 <= 0 & p2 `2 >= 0 ) ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) by A2, A17, A107, A108, JGRAPH_4:82; ::_thesis: verum
end;
case p2 `1 > 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
then (f1 . p1) `1 < (f1 . p2) `1 by A14, A16, A17, A107, A108, Th22;
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) by A1, A24, A32, A70, A109, Th54; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) ; ::_thesis: verum
end;
caseA110: p1 `1 > 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
then A111: (f1 . p1) `1 > 0 by A14, A16, A17, Th22;
now__::_thesis:_(_(_p2_`1_<=_0_&_p2_`2_>=_0_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_or_(_p2_`1_>_0_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_)
percases ( ( p2 `1 <= 0 & p2 `2 >= 0 ) or p2 `1 > 0 ) by A6, A101, A102;
caseA112: ( p2 `1 <= 0 & p2 `2 >= 0 ) ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
now__::_thesis:_not_p1_`2_<_0
A113: p2 in Upper_Arc P by A29, A34, A112;
assume A114: p1 `2 < 0 ; ::_thesis: contradiction
W-min P = |[(- 1),0]| by A1, Th29;
then A115: p1 <> W-min P by A114, EUCLID:52;
p1 in Lower_Arc P by A21, A48, A114;
then LE p2,p1,P by A113, A115, JORDAN6:def_10;
hence contradiction by A1, A2, A110, A112, JGRAPH_3:26, JORDAN6:57; ::_thesis: verum
end;
then LE p2,p1,P by A1, A21, A29, A110, A112, Th54;
then f1 . p1 = f1 . p2 by A1, A2, JGRAPH_3:26, JORDAN6:57;
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) by A9, A17, A24, A112, JGRAPH_4:82, JORDAN6:56; ::_thesis: verum
end;
caseA116: p2 `1 > 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
then A117: (f1 . p2) `1 > 0 by A14, A16, A17, Th22;
now__::_thesis:_(_(_p1_=_p2_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_or_(_p1_<>_p2_&_(f1_._p1)_`2_>=_0_&_LE_f1_._p1,f1_._p2,P_)_)
percases ( p1 = p2 or p1 <> p2 ) ;
case p1 = p2 ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) by A1, A24, A70, JGRAPH_3:26, JORDAN6:56; ::_thesis: verum
end;
case p1 <> p2 ; ::_thesis: ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P )
then (p1 `2) / |.p1.| > (p2 `2) / |.p2.| by A1, A2, A22, A30, A110, A116, Th50;
then ((f1 . p1) `2) / |.(f1 . p1).| > ((f1 . p2) `2) / |.(f1 . p2).| by A22, A30, A14, A16, A17, A110, A116, Th24;
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) by A1, A23, A24, A31, A32, A70, A111, A117, Th55; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) ; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p1) `2 >= 0 & LE f1 . p1,f1 . p2,P ) ; ::_thesis: verum
end;
end;
end;
hence ( (f1 . p1) `2 >= 0 & (f1 . p2) `2 >= 0 & (f1 . p3) `2 >= 0 & (f1 . p4) `2 > 0 & LE f1 . p1,f1 . p2,P & LE f1 . p2,f1 . p3,P & LE f1 . p3,f1 . p4,P ) by A8, A15, A17, A33, A35, A37, A70, JGRAPH_4:106; ::_thesis: verum
end;
end;
end;
for q being Point of (TOP-REAL 2) holds |.(f1 . q).| = |.q.| by A17, JGRAPH_4:97;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A18, A49; ::_thesis: verum
end;
caseA118: p4 `2 > 0 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A119: Lower_Arc P = { p8 where p8 is Point of (TOP-REAL 2) : ( p8 in P & p8 `2 <= 0 ) } by A1, Th35;
A120: now__::_thesis:_not_p4_in_Lower_Arc_P
assume p4 in Lower_Arc P ; ::_thesis: contradiction
then ex p9 being Point of (TOP-REAL 2) st
( p9 = p4 & p9 in P & p9 `2 <= 0 ) by A119;
hence contradiction by A118; ::_thesis: verum
end;
A121: Upper_Arc P = { p7 where p7 is Point of (TOP-REAL 2) : ( p7 in P & p7 `2 >= 0 ) } by A1, Th34;
( ( p3 in Upper_Arc P & p4 in Lower_Arc P & not p4 = W-min P ) or ( p3 in Upper_Arc P & p4 in Upper_Arc P & LE p3,p4, Upper_Arc P, W-min P, E-max P ) or ( p3 in Lower_Arc P & p4 in Lower_Arc P & not p4 = W-min P & LE p3,p4, Lower_Arc P, E-max P, W-min P ) ) by A4, JORDAN6:def_10;
then A122: ex p33 being Point of (TOP-REAL 2) st
( p33 = p3 & p33 in P & p33 `2 >= 0 ) by A121, A120;
set f4 = id (TOP-REAL 2);
A123: ( (id (TOP-REAL 2)) . p3 = p3 & (id (TOP-REAL 2)) . p4 = p4 ) by FUNCT_1:18;
A124: for q being Point of (TOP-REAL 2) holds |.((id (TOP-REAL 2)) . q).| = |.q.| by FUNCT_1:18;
A125: LE p2,p4,P by A1, A3, A4, JGRAPH_3:26, JORDAN6:58;
then ( ( p2 in Upper_Arc P & p4 in Lower_Arc P & not p4 = W-min P ) or ( p2 in Upper_Arc P & p4 in Upper_Arc P & LE p2,p4, Upper_Arc P, W-min P, E-max P ) or ( p2 in Lower_Arc P & p4 in Lower_Arc P & not p4 = W-min P & LE p2,p4, Lower_Arc P, E-max P, W-min P ) ) by JORDAN6:def_10;
then A126: ex p22 being Point of (TOP-REAL 2) st
( p22 = p2 & p22 in P & p22 `2 >= 0 ) by A121, A120;
LE p1,p4,P by A1, A2, A125, JGRAPH_3:26, JORDAN6:58;
then ( ( p1 in Upper_Arc P & p4 in Lower_Arc P & not p4 = W-min P ) or ( p1 in Upper_Arc P & p4 in Upper_Arc P & LE p1,p4, Upper_Arc P, W-min P, E-max P ) or ( p1 in Lower_Arc P & p4 in Lower_Arc P & not p4 = W-min P & LE p1,p4, Lower_Arc P, E-max P, W-min P ) ) by JORDAN6:def_10;
then A127: ex p11 being Point of (TOP-REAL 2) st
( p11 = p1 & p11 in P & p11 `2 >= 0 ) by A121, A120;
( (id (TOP-REAL 2)) . p1 = p1 & (id (TOP-REAL 2)) . p2 = p2 ) by FUNCT_1:18;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A2, A3, A4, A118, A122, A126, A127, A123, A124; ::_thesis: verum
end;
end;
end;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) ; ::_thesis: verum
end;
theorem Th63: :: JGRAPH_5:63
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: ( LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & ( p1 `2 >= 0 or p1 `1 >= 0 ) & ( p2 `2 >= 0 or p2 `1 >= 0 ) & ( p3 `2 >= 0 or p3 `1 >= 0 ) & ( p4 `2 > 0 or p4 `1 > 0 ) ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
consider f being Function of (TOP-REAL 2),(TOP-REAL 2), q1, q2, q3, q4 being Point of (TOP-REAL 2) such that
A3: f is being_homeomorphism and
A4: for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| and
A5: ( q1 = f . p1 & q2 = f . p2 ) and
A6: ( q3 = f . p3 & q4 = f . p4 ) and
A7: ( q1 `2 >= 0 & q2 `2 >= 0 & q3 `2 >= 0 & q4 `2 > 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A1, A2, Th62;
consider f2 being Function of (TOP-REAL 2),(TOP-REAL 2), q1b, q2b, q3b, q4b being Point of (TOP-REAL 2) such that
A8: f2 is being_homeomorphism and
A9: for q being Point of (TOP-REAL 2) holds |.(f2 . q).| = |.q.| and
A10: ( q1b = f2 . q1 & q2b = f2 . q2 ) and
A11: ( q3b = f2 . q3 & q4b = f2 . q4 ) and
A12: ( q1b `1 < 0 & q1b `2 < 0 & q2b `1 < 0 & q2b `2 < 0 & q3b `1 < 0 & q3b `2 < 0 & q4b `1 < 0 & q4b `2 < 0 & LE q1b,q2b,P & LE q2b,q3b,P & LE q3b,q4b,P ) by A1, A7, Th61;
reconsider f3 = f2 * f as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A13: f3 is being_homeomorphism by A3, A8, TOPS_2:57;
A14: dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A15: ( f3 . p3 = q3b & f3 . p4 = q4b ) by A6, A11, FUNCT_1:13;
A16: for q being Point of (TOP-REAL 2) holds |.(f3 . q).| = |.q.|
proof
let q be Point of (TOP-REAL 2); ::_thesis: |.(f3 . q).| = |.q.|
dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then f3 . q = f2 . (f . q) by FUNCT_1:13;
hence |.(f3 . q).| = |.(f . q).| by A9
.= |.q.| by A4 ;
::_thesis: verum
end;
( f3 . p1 = q1b & f3 . p2 = q2b ) by A5, A10, A14, FUNCT_1:13;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A12, A13, A16, A15; ::_thesis: verum
end;
theorem :: JGRAPH_5:64
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p4 = W-min P & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p4 = W-min P & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & p4 = W-min P & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: p4 = W-min P and
A3: LE p1,p2,P and
A4: LE p2,p3,P and
A5: LE p3,p4,P ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A6: Upper_Arc P = { p7 where p7 is Point of (TOP-REAL 2) : ( p7 in P & p7 `2 >= 0 ) } by A1, Th34;
A7: W-min P = |[(- 1),0]| by A1, Th29;
then A8: (W-min P) `2 = 0 by EUCLID:52;
A9: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then p4 in P by A5, JORDAN7:5;
then A10: p4 in Upper_Arc P by A2, A6, A8;
A11: Upper_Arc P is_an_arc_of W-min P, E-max P by A9, JORDAN6:def_8;
A12: p3 in Upper_Arc P by A1, A5, A10, Th44;
then LE p4,p3, Upper_Arc P, W-min P, E-max P by A2, A11, JORDAN5C:10;
then LE p4,p3,P by A10, A12, JORDAN6:def_10;
then A13: p3 = p4 by A1, A5, JGRAPH_3:26, JORDAN6:57;
A14: LE p2,p4,P by A1, A4, A5, JGRAPH_3:26, JORDAN6:58;
A15: p2 in Upper_Arc P by A1, A4, A12, Th44;
then LE p4,p2, Upper_Arc P, W-min P, E-max P by A2, A11, JORDAN5C:10;
then LE p4,p2,P by A10, A15, JORDAN6:def_10;
then A16: p2 = p4 by A1, A14, JGRAPH_3:26, JORDAN6:57;
A17: (W-min P) `1 = - 1 by A7, EUCLID:52;
A18: p1 in Upper_Arc P by A1, A3, A15, Th44;
then LE p4,p1, Upper_Arc P, W-min P, E-max P by A2, A11, JORDAN5C:10;
then LE p4,p1,P by A10, A18, JORDAN6:def_10;
then p1 = p4 by A1, A3, A16, JGRAPH_3:26, JORDAN6:57;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A1, A2, A3, A17, A8, A13, A16, Th59; ::_thesis: verum
end;
theorem Th65: :: JGRAPH_5:65
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A5: Lower_Arc P = { p7 where p7 is Point of (TOP-REAL 2) : ( p7 in P & p7 `2 <= 0 ) } by A1, Th35;
A6: W-min P = |[(- 1),0]| by A1, Th29;
then A7: (W-min P) `2 = 0 by EUCLID:52;
A8: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then A9: p1 in P by A2, JORDAN7:5;
A10: Upper_Arc P is_an_arc_of W-min P, E-max P by A8, JORDAN6:def_8;
A11: Upper_Arc P = { p7 where p7 is Point of (TOP-REAL 2) : ( p7 in P & p7 `2 >= 0 ) } by A1, Th34;
A12: p4 in P by A4, A8, JORDAN7:5;
then A13: ex p44 being Point of (TOP-REAL 2) st
( p44 = p4 & |.p44.| = 1 ) by A1;
then A14: p4 `1 <= 1 by Th1;
A15: - 1 <= p4 `1 by A13, Th1;
now__::_thesis:_(_(_p4_`1_=_-_1_&_ex_f_being_Function_of_(TOP-REAL_2),(TOP-REAL_2)_ex_q1,_q2,_q3,_q4_being_Point_of_(TOP-REAL_2)_st_
(_f_is_being_homeomorphism_&_(_for_q_being_Point_of_(TOP-REAL_2)_holds_|.(f_._q).|_=_|.q.|_)_&_q1_=_f_._p1_&_q2_=_f_._p2_&_q3_=_f_._p3_&_q4_=_f_._p4_&_q1_`1_<_0_&_q1_`2_<_0_&_q2_`1_<_0_&_q2_`2_<_0_&_q3_`1_<_0_&_q3_`2_<_0_&_q4_`1_<_0_&_q4_`2_<_0_&_LE_q1,q2,P_&_LE_q2,q3,P_&_LE_q3,q4,P_)_)_or_(_p4_`1_<>_-_1_&_ex_f_being_Function_of_(TOP-REAL_2),(TOP-REAL_2)_ex_q1,_q2,_q3,_q4_being_Point_of_(TOP-REAL_2)_st_
(_f_is_being_homeomorphism_&_(_for_q_being_Point_of_(TOP-REAL_2)_holds_|.(f_._q).|_=_|.q.|_)_&_q1_=_f_._p1_&_q2_=_f_._p2_&_q3_=_f_._p3_&_q4_=_f_._p4_&_q1_`1_<_0_&_q1_`2_<_0_&_q2_`1_<_0_&_q2_`2_<_0_&_q3_`1_<_0_&_q3_`2_<_0_&_q4_`1_<_0_&_q4_`2_<_0_&_LE_q1,q2,P_&_LE_q2,q3,P_&_LE_q3,q4,P_)_)_)
percases ( p4 `1 = - 1 or p4 `1 <> - 1 ) ;
caseA16: p4 `1 = - 1 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
1 ^2 = ((p4 `1) ^2) + ((p4 `2) ^2) by A13, JGRAPH_3:1
.= ((p4 `2) ^2) + 1 by A16 ;
then A17: p4 `2 = 0 by XCMPLX_1:6;
then A18: p4 in Upper_Arc P by A12, A11;
A19: p4 = W-min P by A6, A16, A17, EUCLID:53;
A20: now__::_thesis:_(_(_p1_in_Upper_Arc_P_&_LE_p4,p1,P_)_or_(_not_p1_in_Upper_Arc_P_&_LE_p4,p1,P_)_)
percases ( p1 in Upper_Arc P or not p1 in Upper_Arc P ) ;
caseA21: p1 in Upper_Arc P ; ::_thesis: LE p4,p1,P
then LE p4,p1, Upper_Arc P, W-min P, E-max P by A10, A19, JORDAN5C:10;
hence LE p4,p1,P by A18, A21, JORDAN6:def_10; ::_thesis: verum
end;
case not p1 in Upper_Arc P ; ::_thesis: LE p4,p1,P
then A22: p1 `2 < 0 by A9, A11;
then p1 in Lower_Arc P by A9, A5;
hence LE p4,p1,P by A7, A18, A22, JORDAN6:def_10; ::_thesis: verum
end;
end;
end;
then A23: LE p4,p2,P by A1, A2, JGRAPH_3:26, JORDAN6:58;
then LE p4,p3,P by A1, A3, JGRAPH_3:26, JORDAN6:58;
then A24: p3 = p4 by A1, A4, JGRAPH_3:26, JORDAN6:57;
LE p2,p4,P by A1, A3, A4, JGRAPH_3:26, JORDAN6:58;
then A25: p2 = p4 by A1, A23, JGRAPH_3:26, JORDAN6:57;
LE p1,p3,P by A1, A2, A3, JGRAPH_3:26, JORDAN6:58;
then LE p1,p4,P by A1, A4, JGRAPH_3:26, JORDAN6:58;
then p4 = p1 by A1, A20, JGRAPH_3:26, JORDAN6:57;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A1, A2, A16, A17, A25, A24, Th59; ::_thesis: verum
end;
caseA26: p4 `1 <> - 1 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
then p4 `1 > - 1 by A15, XXREAL_0:1;
then consider r being real number such that
A27: - 1 < r and
A28: r < p4 `1 by XREAL_1:5;
reconsider r1 = r as Real by XREAL_0:def_1;
A29: r1 < 1 by A14, A28, XXREAL_0:2;
then consider f1 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A30: f1 = r1 -FanMorphS and
A31: f1 is being_homeomorphism by A27, JGRAPH_4:136;
set q11 = f1 . p1;
set q22 = f1 . p2;
set q33 = f1 . p3;
set q44 = f1 . p4;
now__::_thesis:_(_(_(_p4_`1_>_0_or_p4_`2_>=_0_)_&_ex_f_being_Function_of_(TOP-REAL_2),(TOP-REAL_2)_ex_q1,_q2,_q3,_q4_being_Point_of_(TOP-REAL_2)_st_
(_f_is_being_homeomorphism_&_(_for_q_being_Point_of_(TOP-REAL_2)_holds_|.(f_._q).|_=_|.q.|_)_&_q1_=_f_._p1_&_q2_=_f_._p2_&_q3_=_f_._p3_&_q4_=_f_._p4_&_q1_`1_<_0_&_q1_`2_<_0_&_q2_`1_<_0_&_q2_`2_<_0_&_q3_`1_<_0_&_q3_`2_<_0_&_q4_`1_<_0_&_q4_`2_<_0_&_LE_q1,q2,P_&_LE_q2,q3,P_&_LE_q3,q4,P_)_)_or_(_p4_`1_<=_0_&_p4_`2_<_0_&_ex_f_being_Function_of_(TOP-REAL_2),(TOP-REAL_2)_ex_q1,_q2,_q3,_q4_being_Point_of_(TOP-REAL_2)_st_
(_f_is_being_homeomorphism_&_(_for_q_being_Point_of_(TOP-REAL_2)_holds_|.(f_._q).|_=_|.q.|_)_&_q1_=_f_._p1_&_q2_=_f_._p2_&_q3_=_f_._p3_&_q4_=_f_._p4_&_q1_`1_<_0_&_q1_`2_<_0_&_q2_`1_<_0_&_q2_`2_<_0_&_q3_`1_<_0_&_q3_`2_<_0_&_q4_`1_<_0_&_q4_`2_<_0_&_LE_q1,q2,P_&_LE_q2,q3,P_&_LE_q3,q4,P_)_)_)
percases ( p4 `1 > 0 or p4 `2 >= 0 or ( p4 `1 <= 0 & p4 `2 < 0 ) ) ;
caseA32: ( p4 `1 > 0 or p4 `2 >= 0 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
A33: now__::_thesis:_(_p4_`2_=_0_implies_not_p4_`1_<=_0_)
assume that
A34: p4 `2 = 0 and
A35: p4 `1 <= 0 ; ::_thesis: contradiction
1 ^2 = ((p4 `1) ^2) + ((p4 `2) ^2) by A13, JGRAPH_3:1
.= (p4 `1) ^2 by A34 ;
hence contradiction by A26, A35, SQUARE_1:40; ::_thesis: verum
end;
A36: ( p3 `1 >= 0 or p3 `2 >= 0 ) by A1, A4, A32, Th49;
then A37: ( p2 `1 >= 0 or p2 `2 >= 0 ) by A1, A3, Th49;
then ( p1 `1 >= 0 or p1 `2 >= 0 ) by A1, A2, Th49;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A1, A2, A3, A4, A32, A36, A37, A33, Th63; ::_thesis: verum
end;
caseA38: ( p4 `1 <= 0 & p4 `2 < 0 ) ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P )
(p4 `1) / |.p4.| > r1 by A13, A28;
then A39: (f1 . p4) `1 > 0 by A27, A28, A30, A38, Th26;
A40: LE f1 . p3,f1 . p4,P by A1, A4, A27, A29, A30, Th58;
W-min P = |[(- 1),0]| by A1, Th29;
then A41: (W-min P) `2 = 0 by EUCLID:52;
A42: now__::_thesis:_(_(_(f1_._p3)_`2_>=_0_&_(_(f1_._p3)_`2_>=_0_or_(f1_._p3)_`1_>=_0_)_)_or_(_(f1_._p3)_`2_<_0_&_(_(f1_._p3)_`2_>=_0_or_(f1_._p3)_`1_>=_0_)_)_)
percases ( (f1 . p3) `2 >= 0 or (f1 . p3) `2 < 0 ) ;
case (f1 . p3) `2 >= 0 ; ::_thesis: ( (f1 . p3) `2 >= 0 or (f1 . p3) `1 >= 0 )
hence ( (f1 . p3) `2 >= 0 or (f1 . p3) `1 >= 0 ) ; ::_thesis: verum
end;
case (f1 . p3) `2 < 0 ; ::_thesis: ( (f1 . p3) `2 >= 0 or (f1 . p3) `1 >= 0 )
thus ( (f1 . p3) `2 >= 0 or (f1 . p3) `1 >= 0 ) by A1, A39, A40, A41, Th48; ::_thesis: verum
end;
end;
end;
A43: LE f1 . p2,f1 . p3,P by A1, A3, A27, A29, A30, Th58;
A44: now__::_thesis:_(_(_(f1_._p2)_`2_>=_0_&_(_(f1_._p2)_`2_>=_0_or_(f1_._p2)_`1_>=_0_)_)_or_(_(f1_._p2)_`2_<_0_&_(_(f1_._p2)_`2_>=_0_or_(f1_._p2)_`1_>=_0_)_)_)
percases ( (f1 . p2) `2 >= 0 or (f1 . p2) `2 < 0 ) ;
case (f1 . p2) `2 >= 0 ; ::_thesis: ( (f1 . p2) `2 >= 0 or (f1 . p2) `1 >= 0 )
hence ( (f1 . p2) `2 >= 0 or (f1 . p2) `1 >= 0 ) ; ::_thesis: verum
end;
case (f1 . p2) `2 < 0 ; ::_thesis: ( (f1 . p2) `2 >= 0 or (f1 . p2) `1 >= 0 )
thus ( (f1 . p2) `2 >= 0 or (f1 . p2) `1 >= 0 ) by A1, A8, A39, A40, A43, A41, Th48, JORDAN6:58; ::_thesis: verum
end;
end;
end;
A45: LE f1 . p1,f1 . p2,P by A1, A2, A27, A29, A30, Th58;
A46: LE f1 . p2,f1 . p4,P by A1, A40, A43, JGRAPH_3:26, JORDAN6:58;
now__::_thesis:_(_(_(f1_._p1)_`2_>=_0_&_(_(f1_._p1)_`2_>=_0_or_(f1_._p1)_`1_>=_0_)_)_or_(_(f1_._p1)_`2_<_0_&_(_(f1_._p1)_`2_>=_0_or_(f1_._p1)_`1_>=_0_)_)_)
percases ( (f1 . p1) `2 >= 0 or (f1 . p1) `2 < 0 ) ;
case (f1 . p1) `2 >= 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 or (f1 . p1) `1 >= 0 )
hence ( (f1 . p1) `2 >= 0 or (f1 . p1) `1 >= 0 ) ; ::_thesis: verum
end;
case (f1 . p1) `2 < 0 ; ::_thesis: ( (f1 . p1) `2 >= 0 or (f1 . p1) `1 >= 0 )
thus ( (f1 . p1) `2 >= 0 or (f1 . p1) `1 >= 0 ) by A1, A8, A39, A46, A45, A41, Th48, JORDAN6:58; ::_thesis: verum
end;
end;
end;
then consider f2 being Function of (TOP-REAL 2),(TOP-REAL 2), q81, q82, q83, q84 being Point of (TOP-REAL 2) such that
A47: f2 is being_homeomorphism and
A48: for q being Point of (TOP-REAL 2) holds |.(f2 . q).| = |.q.| and
A49: ( q81 = f2 . (f1 . p1) & q82 = f2 . (f1 . p2) ) and
A50: ( q83 = f2 . (f1 . p3) & q84 = f2 . (f1 . p4) ) and
A51: ( q81 `1 < 0 & q81 `2 < 0 & q82 `1 < 0 & q82 `2 < 0 & q83 `1 < 0 & q83 `2 < 0 & q84 `1 < 0 & q84 `2 < 0 & LE q81,q82,P & LE q82,q83,P & LE q83,q84,P ) by A1, A39, A40, A43, A45, A42, A44, Th63;
reconsider f3 = f2 * f1 as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A52: dom f1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A53: ( f3 . p1 = q81 & f3 . p2 = q82 ) by A49, FUNCT_1:13;
A54: for q being Point of (TOP-REAL 2) holds |.(f3 . q).| = |.q.|
proof
let q be Point of (TOP-REAL 2); ::_thesis: |.(f3 . q).| = |.q.|
dom f1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then f3 . q = f2 . (f1 . q) by FUNCT_1:13;
hence |.(f3 . q).| = |.(f1 . q).| by A48
.= |.q.| by A30, JGRAPH_4:128 ;
::_thesis: verum
end;
A55: ( f3 . p3 = q83 & f3 . p4 = q84 ) by A50, A52, FUNCT_1:13;
f3 is being_homeomorphism by A31, A47, TOPS_2:57;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A51, A54, A53, A55; ::_thesis: verum
end;
end;
end;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) ; ::_thesis: verum
end;
end;
end;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) ex q1, q2, q3, q4 being Point of (TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & q1 = f . p1 & q2 = f . p2 & q3 = f . p3 & q4 = f . p4 & q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 & q4 `2 < 0 & LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) ; ::_thesis: verum
end;
begin
theorem Th66: :: JGRAPH_5:66
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 <> p2 & p2 <> p3 & p3 <> p4 & p1 `1 < 0 & p2 `1 < 0 & p3 `1 < 0 & p4 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 & p3 `2 < 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 <> p2 & p2 <> p3 & p3 <> p4 & p1 `1 < 0 & p2 `1 < 0 & p3 `1 < 0 & p4 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 & p3 `2 < 0 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 <> p2 & p2 <> p3 & p3 <> p4 & p1 `1 < 0 & p2 `1 < 0 & p3 `1 < 0 & p4 `1 < 0 & p1 `2 < 0 & p2 `2 < 0 & p3 `2 < 0 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: LE p1,p2,P and
A3: LE p2,p3,P and
A4: LE p3,p4,P and
A5: p1 <> p2 and
A6: p2 <> p3 and
A7: p3 <> p4 and
A8: p1 `1 < 0 and
A9: p2 `1 < 0 and
A10: p3 `1 < 0 and
A11: p4 `1 < 0 and
A12: p1 `2 < 0 and
A13: p2 `2 < 0 and
A14: p3 `2 < 0 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 )
set q2 = ((p1 `2) -FanMorphW) . p2;
set q3 = ((p1 `2) -FanMorphW) . p3;
A15: p1 `2 < p2 `2 by A1, A2, A5, A8, A12, Th45;
set q1 = ((p1 `2) -FanMorphW) . p1;
A16: P is being_simple_closed_curve by A1, JGRAPH_3:26;
then p1 in P by A2, JORDAN7:5;
then A17: ex p11 being Point of (TOP-REAL 2) st
( p11 = p1 & |.p11.| = 1 ) by A1;
then A18: (p1 `2) / |.p1.| = p1 `2 ;
then A19: (((p1 `2) -FanMorphW) . p1) `2 = 0 by A8, JGRAPH_4:47;
A20: |.(((p1 `2) -FanMorphW) . p1).| = 1 by A17, JGRAPH_4:33;
then A21: ((((p1 `2) -FanMorphW) . p1) `2) / |.(((p1 `2) -FanMorphW) . p1).| = (((p1 `2) -FanMorphW) . p1) `2 ;
p2 in P by A2, A16, JORDAN7:5;
then A22: ex p22 being Point of (TOP-REAL 2) st
( p22 = p2 & |.p22.| = 1 ) by A1;
then A23: (p2 `2) / |.p2.| = p2 `2 ;
then A24: (((p1 `2) -FanMorphW) . p2) `1 < 0 by A9, A12, A15, JGRAPH_4:42;
A25: |.(((p1 `2) -FanMorphW) . p2).| = 1 by A22, JGRAPH_4:33;
then A26: ((((p1 `2) -FanMorphW) . p2) `2) / |.(((p1 `2) -FanMorphW) . p2).| = (((p1 `2) -FanMorphW) . p2) `2 ;
then A27: (((p1 `2) -FanMorphW) . p1) `2 < (((p1 `2) -FanMorphW) . p2) `2 by A8, A9, A12, A15, A18, A23, A21, JGRAPH_4:44;
then A28: (((p1 `2) -FanMorphW) . p2) `1 < 1 by A25, A19, Th2;
p3 in P by A3, A16, JORDAN7:5;
then A29: ex p33 being Point of (TOP-REAL 2) st
( p33 = p3 & |.p33.| = 1 ) by A1;
then A30: |.(((p1 `2) -FanMorphW) . p3).| = 1 by JGRAPH_4:33;
then A31: ((((p1 `2) -FanMorphW) . p3) `2) / |.(((p1 `2) -FanMorphW) . p3).| = (((p1 `2) -FanMorphW) . p3) `2 ;
set r3 = (((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3);
set r2 = (((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2);
A32: ((((p1 `2) -FanMorphW) . p3) `1) / |.(((p1 `2) -FanMorphW) . p3).| = (((p1 `2) -FanMorphW) . p3) `1 by A30;
A33: p2 `2 < p3 `2 by A1, A3, A6, A9, A13, Th45;
then A34: (p3 `2) / |.p3.| > p1 `2 by A15, A29, XXREAL_0:2;
then A35: (((p1 `2) -FanMorphW) . p3) `1 < 0 by A10, A12, JGRAPH_4:42;
A36: 1 ^2 = (((((p1 `2) -FanMorphW) . p3) `1) ^2) + (((((p1 `2) -FanMorphW) . p3) `2) ^2) by A30, JGRAPH_3:1;
A37: 1 ^2 = (((((p1 `2) -FanMorphW) . p2) `1) ^2) + (((((p1 `2) -FanMorphW) . p2) `2) ^2) by A25, JGRAPH_3:1;
(p3 `2) / |.p3.| > p2 `2 by A1, A3, A6, A9, A13, A29, Th45;
then A38: (((p1 `2) -FanMorphW) . p2) `2 < (((p1 `2) -FanMorphW) . p3) `2 by A9, A10, A12, A15, A23, A26, A31, A34, JGRAPH_4:44;
then ((((p1 `2) -FanMorphW) . p3) `2) ^2 > ((((p1 `2) -FanMorphW) . p2) `2) ^2 by A19, A27, SQUARE_1:16;
then (- ((((p1 `2) -FanMorphW) . p2) `1)) ^2 > ((((p1 `2) -FanMorphW) . p3) `1) ^2 by A37, A36, XREAL_1:8;
then A39: - (- ((((p1 `2) -FanMorphW) . p2) `1)) < (((p1 `2) -FanMorphW) . p3) `1 by A24, SQUARE_1:48;
A40: 0 < (((p1 `2) -FanMorphW) . p3) `2 by A8, A10, A12, A18, A21, A31, A19, A34, JGRAPH_4:44;
then A41: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2 > 0 by A39, A28, A32, JGRAPH_4:75;
A42: |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)).| = 1 by A30, JGRAPH_4:66;
then A43: (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1) / |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)).| = ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1 ;
A44: - 1 < p1 `2 by A8, A12, A17, Th2;
then consider f1 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A45: f1 = (p1 `2) -FanMorphW and
A46: f1 is being_homeomorphism by A12, JGRAPH_4:41;
A47: - 1 < (((p1 `2) -FanMorphW) . p2) `1 by A12, A44, A25, A19, A27, Th2;
then consider f2 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A48: f2 = ((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN and
A49: f2 is being_homeomorphism by A28, JGRAPH_4:74;
A50: ((((p1 `2) -FanMorphW) . p2) `1) / |.(((p1 `2) -FanMorphW) . p2).| = (((p1 `2) -FanMorphW) . p2) `1 by A25;
then A51: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `1 = 0 by A19, A27, JGRAPH_4:80;
A52: |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)).| = 1 by A25, JGRAPH_4:66;
then A53: (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `1) / |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)).| = ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `1 ;
then A54: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `1 < ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1 by A19, A27, A38, A39, A47, A28, A50, A32, A43, JGRAPH_4:79;
then A55: - 1 < ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2 by A12, A44, A42, A51, Th2;
(((p1 `2) -FanMorphW) . p1) `2 < (((p1 `2) -FanMorphW) . p2) `2 by A8, A9, A12, A15, A18, A23, A21, A26, JGRAPH_4:44;
then A56: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `1 < ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1 by A19, A40, A39, A47, A28, A50, A32, A53, A43, JGRAPH_4:79;
set q4 = ((p1 `2) -FanMorphW) . p4;
p4 in P by A4, A16, JORDAN7:5;
then A57: ex p44 being Point of (TOP-REAL 2) st
( p44 = p4 & |.p44.| = 1 ) by A1;
then A58: |.(((p1 `2) -FanMorphW) . p4).| = 1 by JGRAPH_4:33;
then A59: ((((p1 `2) -FanMorphW) . p4) `2) / |.(((p1 `2) -FanMorphW) . p4).| = (((p1 `2) -FanMorphW) . p4) `2 ;
p3 `2 < p4 `2 by A1, A4, A7, A10, A14, Th45;
then (p4 `2) / |.p4.| > p2 `2 by A33, A57, XXREAL_0:2;
then A60: (p4 `2) / |.p4.| > p1 `2 by A15, XXREAL_0:2;
(p4 `2) / |.p4.| > p3 `2 by A1, A4, A7, A10, A14, A57, Th45;
then (((p1 `2) -FanMorphW) . p3) `2 < (((p1 `2) -FanMorphW) . p4) `2 by A10, A11, A12, A29, A31, A59, A34, A60, JGRAPH_4:44;
then A61: ((((p1 `2) -FanMorphW) . p4) `2) ^2 > ((((p1 `2) -FanMorphW) . p3) `2) ^2 by A19, A27, A38, SQUARE_1:16;
1 ^2 = (((((p1 `2) -FanMorphW) . p4) `1) ^2) + (((((p1 `2) -FanMorphW) . p4) `2) ^2) by A58, JGRAPH_3:1;
then (- ((((p1 `2) -FanMorphW) . p3) `1)) ^2 > ((((p1 `2) -FanMorphW) . p4) `1) ^2 by A36, A61, XREAL_1:8;
then - (- ((((p1 `2) -FanMorphW) . p3) `1)) < (((p1 `2) -FanMorphW) . p4) `1 by A35, SQUARE_1:48;
then A62: ((((p1 `2) -FanMorphW) . p4) `1) / |.(((p1 `2) -FanMorphW) . p4).| > (((p1 `2) -FanMorphW) . p3) `1 by A58;
set r4 = (((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4);
A63: 1 ^2 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1) ^2) + ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) ^2) by A42, JGRAPH_3:1;
A64: |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)).| = 1 by A58, JGRAPH_4:66;
then A65: (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `1) / |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)).| = ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `1 ;
set r1 = (((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1);
|.(((p1 `2) -FanMorphW) . p1).| ^2 = (((((p1 `2) -FanMorphW) . p1) `1) ^2) + (((((p1 `2) -FanMorphW) . p1) `2) ^2) by JGRAPH_3:1;
then A66: ( (((p1 `2) -FanMorphW) . p1) `1 = - 1 or (((p1 `2) -FanMorphW) . p1) `1 = 1 ) by A20, A19, SQUARE_1:40;
then A67: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1)) `1 = - 1 by A8, A18, A19, JGRAPH_4:47, JGRAPH_4:49;
A68: 1 ^2 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `1) ^2) + ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2) ^2) by A64, JGRAPH_3:1;
0 < (((p1 `2) -FanMorphW) . p4) `2 by A8, A11, A12, A18, A21, A59, A19, A60, JGRAPH_4:44;
then A69: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1 < ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `1 by A40, A47, A28, A32, A43, A65, A62, JGRAPH_4:79;
then (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `1) ^2 > (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `1) ^2 by A51, A56, SQUARE_1:16;
then (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) ^2) - ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2) ^2)) + ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2) ^2) > 0 + ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2) ^2) by A63, A68, XREAL_1:8;
then A70: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2 > ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2 by A41, SQUARE_1:48;
set s4 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4));
set s1 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1));
((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1)) `2 = 0 by A19, JGRAPH_4:49;
then A71: (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1))) `2 = 0 by A67, JGRAPH_4:82;
set t4 = (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)));
set s3 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3));
set s2 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2));
A72: |.(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))).| ^2 = (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) `1) ^2) + (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) `2) ^2) by JGRAPH_3:1;
A73: (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) / |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)).| = ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2 by A42;
then A74: (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) `2 = 0 by A51, A56, JGRAPH_4:111;
|.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)).| ^2 = ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `1) ^2) + ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `2) ^2) by JGRAPH_3:1;
then A75: ( ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `2 = - 1 or ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) `2 = 1 ) by A52, A51, SQUARE_1:40;
then (((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2) = |[0,1]| by A19, A27, A50, A51, EUCLID:53, JGRAPH_4:80;
then A76: ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2)) = |[0,1]| by A51, JGRAPH_4:82;
(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2))) `2 = 1 by A19, A27, A50, A51, A75, JGRAPH_4:80, JGRAPH_4:82;
then A77: (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p2))) = |[0,1]| by A76, JGRAPH_4:113;
A78: ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2 < 1 by A42, A51, A54, Th2;
then consider f3 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A79: f3 = (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE and
A80: f3 is being_homeomorphism by A55, JGRAPH_4:105;
A81: dom (f2 * f1) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
A82: (((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2) / |.((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)).| = ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)) `2 by A64;
then A83: ((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) `2) / |.(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))).| > ((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `2) / |.(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))).| by A51, A56, A69, A70, A55, A78, A73, JGRAPH_4:110;
A84: |.(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))).| = 1 by A64, JGRAPH_4:97;
then A85: ((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) / |.(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))).| = (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1 ;
then A86: ((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))) `1 = 0 by A84, A74, A83, JGRAPH_4:142;
(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `2 < 0 by A51, A56, A69, A70, A55, A82, JGRAPH_4:107;
then A87: (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1 < 1 by A84, Th2;
- 1 < (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1 by A51, A56, A69, A70, A55, A82, JGRAPH_4:107;
then consider f4 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A88: f4 = ((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS and
A89: f4 is being_homeomorphism by A87, JGRAPH_4:136;
reconsider g = f4 * (f3 * (f2 * f1)) as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A90: dom (f3 * (f2 * f1)) = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
f2 * f1 is being_homeomorphism by A46, A49, TOPS_2:57;
then f3 * (f2 * f1) is being_homeomorphism by A80, TOPS_2:57;
then A91: g is being_homeomorphism by A89, TOPS_2:57;
A92: dom g = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A93: g . p2 = f4 . ((f3 * (f2 * f1)) . p2) by FUNCT_1:12
.= f4 . (f3 . ((f2 * f1) . p2)) by A90, FUNCT_1:12
.= |[0,1]| by A45, A48, A79, A88, A77, A81, FUNCT_1:12 ;
|.(((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))).| = 1 by A42, JGRAPH_4:97;
then ( (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) `1 = - 1 or (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) `1 = 1 ) by A74, A72, SQUARE_1:40;
then ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) = |[1,0]| by A51, A56, A73, A74, EUCLID:53, JGRAPH_4:111;
then A94: (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3))) = |[1,0]| by A74, JGRAPH_4:113;
((p1 `2) -FanMorphW) . p1 = |[(- 1),0]| by A8, A18, A19, A66, EUCLID:53, JGRAPH_4:47;
then (((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1) = |[(- 1),0]| by A19, JGRAPH_4:49;
then ((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1)) = |[(- 1),0]| by A67, JGRAPH_4:82;
then A95: (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p1))) = |[(- 1),0]| by A71, JGRAPH_4:113;
A96: |.((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))).| ^2 = ((((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))) `1) ^2) + ((((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))) `2) ^2) by JGRAPH_3:1;
|.((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))).| = 1 by A84, JGRAPH_4:128;
then ( ((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))) `2 = - 1 or ((((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4)))) `2 = 1 ) by A86, A96, SQUARE_1:40;
then A97: (((((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) `1) -FanMorphS) . (((((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p3)) `2) -FanMorphE) . ((((((p1 `2) -FanMorphW) . p2) `1) -FanMorphN) . (((p1 `2) -FanMorphW) . p4))) = |[0,(- 1)]| by A84, A74, A83, A85, A86, EUCLID:53, JGRAPH_4:142;
A98: for q being Point of (TOP-REAL 2) holds |.(g . q).| = |.q.|
proof
let q be Point of (TOP-REAL 2); ::_thesis: |.(g . q).| = |.q.|
A99: |.((f2 * f1) . q).| = |.(f2 . (f1 . q)).| by A81, FUNCT_1:12
.= |.(f1 . q).| by A48, JGRAPH_4:66
.= |.q.| by A45, JGRAPH_4:33 ;
A100: |.((f3 * (f2 * f1)) . q).| = |.(f3 . ((f2 * f1) . q)).| by A90, FUNCT_1:12
.= |.q.| by A79, A99, JGRAPH_4:97 ;
thus |.(g . q).| = |.(f4 . ((f3 * (f2 * f1)) . q)).| by A92, FUNCT_1:12
.= |.q.| by A88, A100, JGRAPH_4:128 ; ::_thesis: verum
end;
A101: g . p3 = f4 . ((f3 * (f2 * f1)) . p3) by A92, FUNCT_1:12
.= f4 . (f3 . ((f2 * f1) . p3)) by A90, FUNCT_1:12
.= |[1,0]| by A45, A48, A79, A88, A94, A81, FUNCT_1:12 ;
A102: g . p4 = f4 . ((f3 * (f2 * f1)) . p4) by A92, FUNCT_1:12
.= f4 . (f3 . ((f2 * f1) . p4)) by A90, FUNCT_1:12
.= |[0,(- 1)]| by A45, A48, A79, A88, A97, A81, FUNCT_1:12 ;
g . p1 = f4 . ((f3 * (f2 * f1)) . p1) by A92, FUNCT_1:12
.= f4 . (f3 . ((f2 * f1) . p1)) by A90, FUNCT_1:12
.= |[(- 1),0]| by A45, A48, A79, A88, A95, A81, FUNCT_1:12 ;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 ) by A91, A98, A93, A101, A102; ::_thesis: verum
end;
theorem Th67: :: JGRAPH_5:67
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 <> p2 & p2 <> p3 & p3 <> p4 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 )
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 <> p2 & p2 <> p3 & p3 <> p4 holds
ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 )
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P & p1 <> p2 & p2 <> p3 & p3 <> p4 implies ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 ) )
assume that
A1: P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } and
A2: ( LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) and
A3: ( p1 <> p2 & p2 <> p3 ) and
A4: p3 <> p4 ; ::_thesis: ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 )
consider f being Function of (TOP-REAL 2),(TOP-REAL 2), q1, q2, q3, q4 being Point of (TOP-REAL 2) such that
A5: f is being_homeomorphism and
A6: for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| and
A7: ( q1 = f . p1 & q2 = f . p2 ) and
A8: q3 = f . p3 and
A9: q4 = f . p4 and
A10: ( q1 `1 < 0 & q1 `2 < 0 & q2 `1 < 0 & q2 `2 < 0 & q3 `1 < 0 & q3 `2 < 0 & q4 `1 < 0 ) and
q4 `2 < 0 and
A11: ( LE q1,q2,P & LE q2,q3,P & LE q3,q4,P ) by A1, A2, Th65;
A12: ( dom f = the carrier of (TOP-REAL 2) & f is one-to-one ) by A5, FUNCT_2:def_1, TOPS_2:def_5;
then A13: q3 <> q4 by A4, A8, A9, FUNCT_1:def_4;
( q1 <> q2 & q2 <> q3 ) by A3, A7, A8, A12, FUNCT_1:def_4;
then consider f2 being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A14: f2 is being_homeomorphism and
A15: for q being Point of (TOP-REAL 2) holds |.(f2 . q).| = |.q.| and
A16: ( |[(- 1),0]| = f2 . q1 & |[0,1]| = f2 . q2 ) and
A17: ( |[1,0]| = f2 . q3 & |[0,(- 1)]| = f2 . q4 ) by A1, A10, A11, A13, Th66;
reconsider f3 = f2 * f as Function of (TOP-REAL 2),(TOP-REAL 2) ;
A18: f3 is being_homeomorphism by A5, A14, TOPS_2:57;
A19: dom f3 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A20: ( f3 . p1 = |[(- 1),0]| & f3 . p2 = |[0,1]| ) by A7, A16, FUNCT_1:12;
A21: for q being Point of (TOP-REAL 2) holds |.(f3 . q).| = |.q.|
proof
let q be Point of (TOP-REAL 2); ::_thesis: |.(f3 . q).| = |.q.|
|.(f3 . q).| = |.(f2 . (f . q)).| by A19, FUNCT_1:12
.= |.(f . q).| by A15
.= |.q.| by A6 ;
hence |.(f3 . q).| = |.q.| ; ::_thesis: verum
end;
( f3 . p3 = |[1,0]| & f3 . p4 = |[0,(- 1)]| ) by A8, A9, A17, A19, FUNCT_1:12;
hence ex f being Function of (TOP-REAL 2),(TOP-REAL 2) st
( f is being_homeomorphism & ( for q being Point of (TOP-REAL 2) holds |.(f . q).| = |.q.| ) & |[(- 1),0]| = f . p1 & |[0,1]| = f . p2 & |[1,0]| = f . p3 & |[0,(- 1)]| = f . p4 ) by A18, A21, A20; ::_thesis: verum
end;
Lm7: |[(- 1),0]| `1 = - 1
by EUCLID:52;
Lm8: |[(- 1),0]| `2 = 0
by EUCLID:52;
Lm9: ( |[1,0]| `1 = 1 & |[1,0]| `2 = 0 )
by EUCLID:52;
Lm10: |[0,(- 1)]| `1 = 0
by EUCLID:52;
Lm11: |[0,(- 1)]| `2 = - 1
by EUCLID:52;
Lm12: |[0,1]| `1 = 0
by EUCLID:52;
Lm13: |[0,1]| `2 = 1
by EUCLID:52;
Lm14: now__::_thesis:_(_|.|[(-_1),0]|.|_=_1_&_|.|[1,0]|.|_=_1_&_|.|[0,(-_1)]|.|_=_1_&_|.|[0,1]|.|_=_1_)
thus |.|[(- 1),0]|.| = sqrt (((- 1) ^2) + (0 ^2)) by Lm7, Lm8, JGRAPH_3:1
.= 1 by SQUARE_1:18 ; ::_thesis: ( |.|[1,0]|.| = 1 & |.|[0,(- 1)]|.| = 1 & |.|[0,1]|.| = 1 )
thus |.|[1,0]|.| = sqrt ((1 ^2) + (0 ^2)) by Lm9, JGRAPH_3:1
.= 1 by SQUARE_1:18 ; ::_thesis: ( |.|[0,(- 1)]|.| = 1 & |.|[0,1]|.| = 1 )
thus |.|[0,(- 1)]|.| = sqrt ((0 ^2) + ((- 1) ^2)) by Lm10, Lm11, JGRAPH_3:1
.= 1 by SQUARE_1:18 ; ::_thesis: |.|[0,1]|.| = 1
thus |.|[0,1]|.| = sqrt ((0 ^2) + (1 ^2)) by Lm12, Lm13, JGRAPH_3:1
.= 1 by SQUARE_1:18 ; ::_thesis: verum
end;
Lm15: 0 in [.0,1.]
by XXREAL_1:1;
Lm16: 1 in [.0,1.]
by XXREAL_1:1;
theorem :: JGRAPH_5:68
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g )
assume A1: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A2: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
A3: dom g = the carrier of I[01] by FUNCT_2:def_1;
A4: dom f = the carrier of I[01] by FUNCT_2:def_1;
percases ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 or ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ) ;
supposeA5: ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 ) ; ::_thesis: rng f meets rng g
now__::_thesis:_(_(_p1_=_p2_&_rng_f_meets_rng_g_)_or_(_p2_=_p3_&_rng_f_meets_rng_g_)_or_(_p3_=_p4_&_rng_f_meets_rng_g_)_)
percases ( p1 = p2 or p2 = p3 or p3 = p4 ) by A5;
caseA6: p1 = p2 ; ::_thesis: rng f meets rng g
( p1 in rng f & p2 in rng g ) by A2, A4, A3, Lm15, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A6, XBOOLE_0:3; ::_thesis: verum
end;
caseA7: p2 = p3 ; ::_thesis: rng f meets rng g
( p3 in rng f & p2 in rng g ) by A2, A4, A3, Lm15, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A7, XBOOLE_0:3; ::_thesis: verum
end;
caseA8: p3 = p4 ; ::_thesis: rng f meets rng g
( p3 in rng f & p4 in rng g ) by A2, A4, A3, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A8, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence rng f meets rng g ; ::_thesis: verum
end;
suppose ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ; ::_thesis: rng f meets rng g
then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A9: h is being_homeomorphism and
A10: for q being Point of (TOP-REAL 2) holds |.(h . q).| = |.q.| and
A11: |[(- 1),0]| = h . p1 and
A12: |[0,1]| = h . p2 and
A13: |[1,0]| = h . p3 and
A14: |[0,(- 1)]| = h . p4 by A1, Th67;
A15: h is one-to-one by A9, TOPS_2:def_5;
reconsider h1 = h as Function ;
reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) );
{ q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) );
A16: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
{ q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) );
{ q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) );
{ q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ;
A17: - (|[0,1]| `1) = 0 by Lm12;
reconsider g2 = h * g as Function of I[01],(TOP-REAL 2) ;
A18: - (|[0,(- 1)]| `1) = 0 by Lm10;
A19: dom g2 = the carrier of I[01] by FUNCT_2:def_1;
then g2 . 0 = |[0,1]| by A2, A12, Lm15, BORSUK_1:40, FUNCT_1:12;
then A20: g2 . O in KYP by A17, Lm13, Lm14;
A21: rng g2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 )
assume y in rng g2 ; ::_thesis: y in C0
then consider x being set such that
A22: x in dom g2 and
A23: y = g2 . x by FUNCT_1:def_3;
A24: g . x in rng g by A3, A22, FUNCT_1:def_3;
then reconsider qg = g . x as Point of (TOP-REAL 2) ;
g . x in C0 by A2, A24;
then A25: ex q5 being Point of (TOP-REAL 2) st
( q5 = g . x & |.q5.| <= 1 ) by A2;
A26: |.(h . qg).| = |.qg.| by A10;
g2 . x = h . (g . x) by A22, FUNCT_1:12;
hence y in C0 by A2, A23, A25, A26; ::_thesis: verum
end;
reconsider f2 = h * f as Function of I[01],(TOP-REAL 2) ;
A27: - (|[(- 1),0]| `1) = 1 by Lm7;
A28: dom f2 = the carrier of I[01] by FUNCT_2:def_1;
then f2 . 1 = |[1,0]| by A2, A13, Lm16, BORSUK_1:40, FUNCT_1:12;
then A29: f2 . I in KXP by Lm9, Lm14;
A30: rng f2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 )
assume y in rng f2 ; ::_thesis: y in C0
then consider x being set such that
A31: x in dom f2 and
A32: y = f2 . x by FUNCT_1:def_3;
A33: f . x in rng f by A4, A31, FUNCT_1:def_3;
then reconsider qf = f . x as Point of (TOP-REAL 2) ;
f . x in C0 by A2, A33;
then A34: ex q5 being Point of (TOP-REAL 2) st
( q5 = f . x & |.q5.| <= 1 ) by A2;
A35: |.(h . qf).| = |.qf.| by A10;
f2 . x = h . (f . x) by A31, FUNCT_1:12;
hence y in C0 by A2, A32, A34, A35; ::_thesis: verum
end;
g2 . 1 = |[0,(- 1)]| by A2, A14, A19, Lm16, BORSUK_1:40, FUNCT_1:12;
then A36: g2 . I in KYN by A18, Lm11, Lm14;
f2 . 0 = |[(- 1),0]| by A2, A11, A28, Lm15, BORSUK_1:40, FUNCT_1:12;
then A37: f2 . O in KXN by A27, Lm8, Lm14;
( f2 is continuous & f2 is one-to-one & g2 is continuous & g2 is one-to-one ) by A2, A9, Th5, Th6;
then rng f2 meets rng g2 by A2, A30, A21, A37, A29, A36, A20, Th13;
then consider x2 being set such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being set such that
A40: z3 in dom g2 and
A41: x2 = g2 . z3 by A39, FUNCT_1:def_3;
A42: g . z3 in rng g by A3, A40, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (g . z3)) by A40, A41, FUNCT_1:12
.= g . z3 by A15, A16, A42, FUNCT_1:34 ;
then A43: (h1 ") . x2 in rng g by A3, A40, FUNCT_1:def_3;
consider z2 being set such that
A44: z2 in dom f2 and
A45: x2 = f2 . z2 by A38, FUNCT_1:def_3;
A46: f . z2 in rng f by A4, A44, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (f . z2)) by A44, A45, FUNCT_1:12
.= f . z2 by A15, A16, A46, FUNCT_1:34 ;
then (h1 ") . x2 in rng f by A4, A44, FUNCT_1:def_3;
hence rng f meets rng g by A43, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
theorem :: JGRAPH_5:69
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g )
assume A1: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A2: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| <= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
A3: dom g = the carrier of I[01] by FUNCT_2:def_1;
A4: dom f = the carrier of I[01] by FUNCT_2:def_1;
percases ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 or ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ) ;
supposeA5: ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 ) ; ::_thesis: rng f meets rng g
now__::_thesis:_(_(_p1_=_p2_&_rng_f_meets_rng_g_)_or_(_p2_=_p3_&_rng_f_meets_rng_g_)_or_(_p3_=_p4_&_rng_f_meets_rng_g_)_)
percases ( p1 = p2 or p2 = p3 or p3 = p4 ) by A5;
caseA6: p1 = p2 ; ::_thesis: rng f meets rng g
( p1 in rng f & p2 in rng g ) by A2, A4, A3, Lm15, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A6, XBOOLE_0:3; ::_thesis: verum
end;
caseA7: p2 = p3 ; ::_thesis: rng f meets rng g
( p3 in rng f & p2 in rng g ) by A2, A4, A3, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A7, XBOOLE_0:3; ::_thesis: verum
end;
caseA8: p3 = p4 ; ::_thesis: rng f meets rng g
( p3 in rng f & p4 in rng g ) by A2, A4, A3, Lm15, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A8, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence rng f meets rng g ; ::_thesis: verum
end;
suppose ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ; ::_thesis: rng f meets rng g
then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A9: h is being_homeomorphism and
A10: for q being Point of (TOP-REAL 2) holds |.(h . q).| = |.q.| and
A11: |[(- 1),0]| = h . p1 and
A12: |[0,1]| = h . p2 and
A13: |[1,0]| = h . p3 and
A14: |[0,(- 1)]| = h . p4 by A1, Th67;
A15: h is one-to-one by A9, TOPS_2:def_5;
reconsider h1 = h as Function ;
reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) );
{ q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) );
A16: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
{ q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) );
{ q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) );
{ q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ;
A17: - (|[0,1]| `1) = 0 by Lm12;
reconsider g2 = h * g as Function of I[01],(TOP-REAL 2) ;
A18: - (|[0,(- 1)]| `1) = 0 by Lm10;
A19: dom g2 = the carrier of I[01] by FUNCT_2:def_1;
then g2 . 0 = |[0,(- 1)]| by A2, A14, Lm15, BORSUK_1:40, FUNCT_1:12;
then A20: g2 . O in KYN by A18, Lm11, Lm14;
A21: rng g2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 )
assume y in rng g2 ; ::_thesis: y in C0
then consider x being set such that
A22: x in dom g2 and
A23: y = g2 . x by FUNCT_1:def_3;
A24: g . x in rng g by A3, A22, FUNCT_1:def_3;
then reconsider qg = g . x as Point of (TOP-REAL 2) ;
g . x in C0 by A2, A24;
then A25: ex q5 being Point of (TOP-REAL 2) st
( q5 = g . x & |.q5.| <= 1 ) by A2;
A26: |.(h . qg).| = |.qg.| by A10;
g2 . x = h . (g . x) by A22, FUNCT_1:12;
hence y in C0 by A2, A23, A25, A26; ::_thesis: verum
end;
reconsider f2 = h * f as Function of I[01],(TOP-REAL 2) ;
A27: - (|[(- 1),0]| `1) = 1 by Lm7;
A28: dom f2 = the carrier of I[01] by FUNCT_2:def_1;
then f2 . 1 = |[1,0]| by A2, A13, Lm16, BORSUK_1:40, FUNCT_1:12;
then A29: f2 . I in KXP by Lm9, Lm14;
A30: rng f2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 )
assume y in rng f2 ; ::_thesis: y in C0
then consider x being set such that
A31: x in dom f2 and
A32: y = f2 . x by FUNCT_1:def_3;
A33: f . x in rng f by A4, A31, FUNCT_1:def_3;
then reconsider qf = f . x as Point of (TOP-REAL 2) ;
f . x in C0 by A2, A33;
then A34: ex q5 being Point of (TOP-REAL 2) st
( q5 = f . x & |.q5.| <= 1 ) by A2;
A35: |.(h . qf).| = |.qf.| by A10;
f2 . x = h . (f . x) by A31, FUNCT_1:12;
hence y in C0 by A2, A32, A34, A35; ::_thesis: verum
end;
g2 . 1 = |[0,1]| by A2, A12, A19, Lm16, BORSUK_1:40, FUNCT_1:12;
then A36: g2 . I in KYP by A17, Lm13, Lm14;
f2 . 0 = |[(- 1),0]| by A2, A11, A28, Lm15, BORSUK_1:40, FUNCT_1:12;
then A37: f2 . O in KXN by A27, Lm8, Lm14;
( f2 is continuous & f2 is one-to-one & g2 is continuous & g2 is one-to-one ) by A2, A9, Th5, Th6;
then rng f2 meets rng g2 by A2, A30, A21, A37, A29, A20, A36, JGRAPH_3:44;
then consider x2 being set such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being set such that
A40: z3 in dom g2 and
A41: x2 = g2 . z3 by A39, FUNCT_1:def_3;
A42: g . z3 in rng g by A3, A40, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (g . z3)) by A40, A41, FUNCT_1:12
.= g . z3 by A15, A16, A42, FUNCT_1:34 ;
then A43: (h1 ") . x2 in rng g by A3, A40, FUNCT_1:def_3;
consider z2 being set such that
A44: z2 in dom f2 and
A45: x2 = f2 . z2 by A38, FUNCT_1:def_3;
A46: f . z2 in rng f by A4, A44, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (f . z2)) by A44, A45, FUNCT_1:12
.= f . z2 by A15, A16, A46, FUNCT_1:34 ;
then (h1 ") . x2 in rng f by A4, A44, FUNCT_1:def_3;
hence rng f meets rng g by A43, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
theorem :: JGRAPH_5:70
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g )
assume A1: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A2: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p4 & g . 1 = p2 & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
A3: dom g = the carrier of I[01] by FUNCT_2:def_1;
A4: dom f = the carrier of I[01] by FUNCT_2:def_1;
percases ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 or ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ) ;
supposeA5: ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 ) ; ::_thesis: rng f meets rng g
now__::_thesis:_(_(_p1_=_p2_&_rng_f_meets_rng_g_)_or_(_p2_=_p3_&_rng_f_meets_rng_g_)_or_(_p3_=_p4_&_rng_f_meets_rng_g_)_)
percases ( p1 = p2 or p2 = p3 or p3 = p4 ) by A5;
caseA6: p1 = p2 ; ::_thesis: rng f meets rng g
( p1 in rng f & p2 in rng g ) by A2, A4, A3, Lm15, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A6, XBOOLE_0:3; ::_thesis: verum
end;
caseA7: p2 = p3 ; ::_thesis: rng f meets rng g
( p3 in rng f & p2 in rng g ) by A2, A4, A3, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A7, XBOOLE_0:3; ::_thesis: verum
end;
caseA8: p3 = p4 ; ::_thesis: rng f meets rng g
( p3 in rng f & p4 in rng g ) by A2, A4, A3, Lm15, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A8, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence rng f meets rng g ; ::_thesis: verum
end;
suppose ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ; ::_thesis: rng f meets rng g
then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A9: h is being_homeomorphism and
A10: for q being Point of (TOP-REAL 2) holds |.(h . q).| = |.q.| and
A11: |[(- 1),0]| = h . p1 and
A12: |[0,1]| = h . p2 and
A13: |[1,0]| = h . p3 and
A14: |[0,(- 1)]| = h . p4 by A1, Th67;
A15: h is one-to-one by A9, TOPS_2:def_5;
reconsider h1 = h as Function ;
reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) );
{ q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) );
A16: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
{ q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) );
{ q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) );
{ q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ;
A17: - (|[0,1]| `1) = 0 by Lm12;
reconsider g2 = h * g as Function of I[01],(TOP-REAL 2) ;
A18: - (|[0,(- 1)]| `1) = 0 by Lm10;
A19: dom g2 = the carrier of I[01] by FUNCT_2:def_1;
then g2 . 0 = |[0,(- 1)]| by A2, A14, Lm15, BORSUK_1:40, FUNCT_1:12;
then A20: g2 . O in KYN by A18, Lm11, Lm14;
A21: rng g2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 )
assume y in rng g2 ; ::_thesis: y in C0
then consider x being set such that
A22: x in dom g2 and
A23: y = g2 . x by FUNCT_1:def_3;
A24: g . x in rng g by A3, A22, FUNCT_1:def_3;
then reconsider qg = g . x as Point of (TOP-REAL 2) ;
g . x in C0 by A2, A24;
then A25: ex q5 being Point of (TOP-REAL 2) st
( q5 = g . x & |.q5.| >= 1 ) by A2;
A26: |.(h . qg).| = |.qg.| by A10;
g2 . x = h . (g . x) by A22, FUNCT_1:12;
hence y in C0 by A2, A23, A25, A26; ::_thesis: verum
end;
reconsider f2 = h * f as Function of I[01],(TOP-REAL 2) ;
A27: - (|[(- 1),0]| `1) = 1 by Lm7;
A28: dom f2 = the carrier of I[01] by FUNCT_2:def_1;
then f2 . 1 = |[1,0]| by A2, A13, Lm16, BORSUK_1:40, FUNCT_1:12;
then A29: f2 . I in KXP by Lm9, Lm14;
A30: rng f2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 )
assume y in rng f2 ; ::_thesis: y in C0
then consider x being set such that
A31: x in dom f2 and
A32: y = f2 . x by FUNCT_1:def_3;
A33: f . x in rng f by A4, A31, FUNCT_1:def_3;
then reconsider qf = f . x as Point of (TOP-REAL 2) ;
f . x in C0 by A2, A33;
then A34: ex q5 being Point of (TOP-REAL 2) st
( q5 = f . x & |.q5.| >= 1 ) by A2;
A35: |.(h . qf).| = |.qf.| by A10;
f2 . x = h . (f . x) by A31, FUNCT_1:12;
hence y in C0 by A2, A32, A34, A35; ::_thesis: verum
end;
g2 . 1 = |[0,1]| by A2, A12, A19, Lm16, BORSUK_1:40, FUNCT_1:12;
then A36: g2 . I in KYP by A17, Lm13, Lm14;
f2 . 0 = |[(- 1),0]| by A2, A11, A28, Lm15, BORSUK_1:40, FUNCT_1:12;
then A37: f2 . O in KXN by A27, Lm8, Lm14;
( f2 is continuous & f2 is one-to-one & g2 is continuous & g2 is one-to-one ) by A2, A9, Th5, Th6;
then rng f2 meets rng g2 by A2, A30, A21, A37, A29, A20, A36, Th14;
then consider x2 being set such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being set such that
A40: z3 in dom g2 and
A41: x2 = g2 . z3 by A39, FUNCT_1:def_3;
A42: g . z3 in rng g by A3, A40, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (g . z3)) by A40, A41, FUNCT_1:12
.= g . z3 by A15, A16, A42, FUNCT_1:34 ;
then A43: (h1 ") . x2 in rng g by A3, A40, FUNCT_1:def_3;
consider z2 being set such that
A44: z2 in dom f2 and
A45: x2 = f2 . z2 by A38, FUNCT_1:def_3;
A46: f . z2 in rng f by A4, A44, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (f . z2)) by A44, A45, FUNCT_1:12
.= f . z2 by A15, A16, A46, FUNCT_1:34 ;
then (h1 ") . x2 in rng f by A4, A44, FUNCT_1:def_3;
hence rng f meets rng g by A43, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
theorem :: JGRAPH_5:71
for p1, p2, p3, p4 being Point of (TOP-REAL 2)
for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
proof
let p1, p2, p3, p4 be Point of (TOP-REAL 2); ::_thesis: for P being non empty compact Subset of (TOP-REAL 2)
for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let P be non empty compact Subset of (TOP-REAL 2); ::_thesis: for C0 being Subset of (TOP-REAL 2) st P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P holds
for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let C0 be Subset of (TOP-REAL 2); ::_thesis: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P implies for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g )
assume A1: ( P = { p where p is Point of (TOP-REAL 2) : |.p.| = 1 } & LE p1,p2,P & LE p2,p3,P & LE p3,p4,P ) ; ::_thesis: for f, g being Function of I[01],(TOP-REAL 2) st f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 holds
rng f meets rng g
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 implies rng f meets rng g )
assume A2: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one & C0 = { p where p is Point of (TOP-REAL 2) : |.p.| >= 1 } & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & rng f c= C0 & rng g c= C0 ) ; ::_thesis: rng f meets rng g
A3: dom g = the carrier of I[01] by FUNCT_2:def_1;
A4: dom f = the carrier of I[01] by FUNCT_2:def_1;
percases ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 or ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ) ;
supposeA5: ( not p1 <> p2 or not p2 <> p3 or not p3 <> p4 ) ; ::_thesis: rng f meets rng g
now__::_thesis:_(_(_p1_=_p2_&_rng_f_meets_rng_g_)_or_(_p2_=_p3_&_rng_f_meets_rng_g_)_or_(_p3_=_p4_&_rng_f_meets_rng_g_)_)
percases ( p1 = p2 or p2 = p3 or p3 = p4 ) by A5;
caseA6: p1 = p2 ; ::_thesis: rng f meets rng g
( p1 in rng f & p2 in rng g ) by A2, A4, A3, Lm15, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A6, XBOOLE_0:3; ::_thesis: verum
end;
caseA7: p2 = p3 ; ::_thesis: rng f meets rng g
( p3 in rng f & p2 in rng g ) by A2, A4, A3, Lm15, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A7, XBOOLE_0:3; ::_thesis: verum
end;
caseA8: p3 = p4 ; ::_thesis: rng f meets rng g
( p3 in rng f & p4 in rng g ) by A2, A4, A3, Lm16, BORSUK_1:40, FUNCT_1:def_3;
hence rng f meets rng g by A8, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence rng f meets rng g ; ::_thesis: verum
end;
suppose ( p1 <> p2 & p2 <> p3 & p3 <> p4 ) ; ::_thesis: rng f meets rng g
then consider h being Function of (TOP-REAL 2),(TOP-REAL 2) such that
A9: h is being_homeomorphism and
A10: for q being Point of (TOP-REAL 2) holds |.(h . q).| = |.q.| and
A11: |[(- 1),0]| = h . p1 and
A12: |[0,1]| = h . p2 and
A13: |[1,0]| = h . p3 and
A14: |[0,(- 1)]| = h . p4 by A1, Th67;
reconsider f2 = h * f, g2 = h * g as Function of I[01],(TOP-REAL 2) ;
A15: - (|[0,(- 1)]| `1) = 0 by Lm10;
A16: rng g2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng g2 or y in C0 )
assume y in rng g2 ; ::_thesis: y in C0
then consider x being set such that
A17: x in dom g2 and
A18: y = g2 . x by FUNCT_1:def_3;
A19: g . x in rng g by A3, A17, FUNCT_1:def_3;
then reconsider qg = g . x as Point of (TOP-REAL 2) ;
g . x in C0 by A2, A19;
then A20: ex q5 being Point of (TOP-REAL 2) st
( q5 = g . x & |.q5.| >= 1 ) by A2;
A21: |.(h . qg).| = |.qg.| by A10;
g2 . x = h . (g . x) by A17, FUNCT_1:12;
hence y in C0 by A2, A18, A20, A21; ::_thesis: verum
end;
A22: rng f2 c= C0
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f2 or y in C0 )
assume y in rng f2 ; ::_thesis: y in C0
then consider x being set such that
A23: x in dom f2 and
A24: y = f2 . x by FUNCT_1:def_3;
A25: f . x in rng f by A4, A23, FUNCT_1:def_3;
then reconsider qf = f . x as Point of (TOP-REAL 2) ;
f . x in C0 by A2, A25;
then A26: ex q5 being Point of (TOP-REAL 2) st
( q5 = f . x & |.q5.| >= 1 ) by A2;
A27: |.(h . qf).| = |.qf.| by A10;
f2 . x = h . (f . x) by A23, FUNCT_1:12;
hence y in C0 by A2, A24, A26, A27; ::_thesis: verum
end;
reconsider h1 = h as Function ;
reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
defpred S1[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 >= - ($1 `1) );
{ q1 where q1 is Point of (TOP-REAL 2) : S1[q1] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXP = { q1 where q1 is Point of (TOP-REAL 2) : ( |.q1.| = 1 & q1 `2 <= q1 `1 & q1 `2 >= - (q1 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S2[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) );
A28: dom h = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
{ q2 where q2 is Point of (TOP-REAL 2) : S2[q2] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KXN = { q2 where q2 is Point of (TOP-REAL 2) : ( |.q2.| = 1 & q2 `2 >= q2 `1 & q2 `2 <= - (q2 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S3[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 >= $1 `1 & $1 `2 >= - ($1 `1) );
{ q3 where q3 is Point of (TOP-REAL 2) : S3[q3] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYP = { q3 where q3 is Point of (TOP-REAL 2) : ( |.q3.| = 1 & q3 `2 >= q3 `1 & q3 `2 >= - (q3 `1) ) } as Subset of (TOP-REAL 2) ;
defpred S4[ Point of (TOP-REAL 2)] means ( |.$1.| = 1 & $1 `2 <= $1 `1 & $1 `2 <= - ($1 `1) );
{ q4 where q4 is Point of (TOP-REAL 2) : S4[q4] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider KYN = { q4 where q4 is Point of (TOP-REAL 2) : ( |.q4.| = 1 & q4 `2 <= q4 `1 & q4 `2 <= - (q4 `1) ) } as Subset of (TOP-REAL 2) ;
A29: - (|[(- 1),0]| `1) = 1 by Lm7;
A30: - (|[0,1]| `1) = 0 by Lm12;
A31: dom g2 = the carrier of I[01] by FUNCT_2:def_1;
then g2 . 0 = |[0,1]| by A2, A12, Lm15, BORSUK_1:40, FUNCT_1:12;
then A32: g2 . O in KYP by A30, Lm13, Lm14;
g2 . 1 = |[0,(- 1)]| by A2, A14, A31, Lm16, BORSUK_1:40, FUNCT_1:12;
then A33: g2 . I in KYN by A15, Lm11, Lm14;
A34: dom f2 = the carrier of I[01] by FUNCT_2:def_1;
then f2 . 1 = |[1,0]| by A2, A13, Lm16, BORSUK_1:40, FUNCT_1:12;
then A35: f2 . I in KXP by Lm9, Lm14;
f2 . 0 = |[(- 1),0]| by A2, A11, A34, Lm15, BORSUK_1:40, FUNCT_1:12;
then A36: f2 . O in KXN by A29, Lm8, Lm14;
A37: h is one-to-one by A9, TOPS_2:def_5;
( f2 is continuous & f2 is one-to-one & g2 is continuous & g2 is one-to-one ) by A2, A9, Th5, Th6;
then rng f2 meets rng g2 by A2, A22, A16, A36, A35, A33, A32, Th15;
then consider x2 being set such that
A38: x2 in rng f2 and
A39: x2 in rng g2 by XBOOLE_0:3;
consider z3 being set such that
A40: z3 in dom g2 and
A41: x2 = g2 . z3 by A39, FUNCT_1:def_3;
A42: g . z3 in rng g by A3, A40, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (g . z3)) by A40, A41, FUNCT_1:12
.= g . z3 by A37, A28, A42, FUNCT_1:34 ;
then A43: (h1 ") . x2 in rng g by A3, A40, FUNCT_1:def_3;
consider z2 being set such that
A44: z2 in dom f2 and
A45: x2 = f2 . z2 by A38, FUNCT_1:def_3;
A46: f . z2 in rng f by A4, A44, FUNCT_1:def_3;
(h1 ") . x2 = (h1 ") . (h . (f . z2)) by A44, A45, FUNCT_1:12
.= f . z2 by A37, A28, A46, FUNCT_1:34 ;
then (h1 ") . x2 in rng f by A4, A44, FUNCT_1:def_3;
hence rng f meets rng g by A43, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;