:: JGRAPH_2 semantic presentation
begin
theorem Th1: :: JGRAPH_2:1
for T1, S, T2, T being non empty TopSpace
for f being Function of T1,S
for g being Function of T2,S
for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) holds
ex h being Function of T,S st
( h = f +* g & h is continuous )
proof
let T1, S, T2, T be non empty TopSpace; ::_thesis: for f being Function of T1,S
for g being Function of T2,S
for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) holds
ex h being Function of T,S st
( h = f +* g & h is continuous )
let f be Function of T1,S; ::_thesis: for g being Function of T2,S
for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) holds
ex h being Function of T,S st
( h = f +* g & h is continuous )
let g be Function of T2,S; ::_thesis: for F1, F2 being Subset of T st T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) holds
ex h being Function of T,S st
( h = f +* g & h is continuous )
let F1, F2 be Subset of T; ::_thesis: ( T1 is SubSpace of T & T2 is SubSpace of T & F1 = [#] T1 & F2 = [#] T2 & ([#] T1) \/ ([#] T2) = [#] T & F1 is closed & F2 is closed & f is continuous & g is continuous & ( for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ) implies ex h being Function of T,S st
( h = f +* g & h is continuous ) )
assume that
A1: T1 is SubSpace of T and
A2: T2 is SubSpace of T and
A3: F1 = [#] T1 and
A4: F2 = [#] T2 and
A5: ([#] T1) \/ ([#] T2) = [#] T and
A6: F1 is closed and
A7: F2 is closed and
A8: f is continuous and
A9: g is continuous and
A10: for p being set st p in ([#] T1) /\ ([#] T2) holds
f . p = g . p ; ::_thesis: ex h being Function of T,S st
( h = f +* g & h is continuous )
set h = f +* g;
A11: dom g = the carrier of T2 by FUNCT_2:def_1
.= [#] T2 ;
A12: dom f = the carrier of T1 by FUNCT_2:def_1
.= [#] T1 ;
then A13: dom (f +* g) = [#] T by A5, A11, FUNCT_4:def_1
.= the carrier of T ;
rng (f +* g) c= (rng f) \/ (rng g) by FUNCT_4:17;
then reconsider h = f +* g as Function of T,S by A13, FUNCT_2:2, XBOOLE_1:1;
take h ; ::_thesis: ( h = f +* g & h is continuous )
thus h = f +* g ; ::_thesis: h is continuous
for P being Subset of S st P is closed holds
h " P is closed
proof
let P be Subset of S; ::_thesis: ( P is closed implies h " P is closed )
set P3 = f " P;
set P4 = g " P;
[#] T1 c= [#] T by A5, XBOOLE_1:7;
then reconsider P1 = f " P as Subset of T by XBOOLE_1:1;
[#] T2 c= [#] T by A5, XBOOLE_1:7;
then reconsider P2 = g " P as Subset of T by XBOOLE_1:1;
A14: dom h = (dom f) \/ (dom g) by FUNCT_4:def_1;
A15: now__::_thesis:_for_x_being_set_holds_
(_(_x_in_(h_"_P)_/\_([#]_T2)_implies_x_in_g_"_P_)_&_(_x_in_g_"_P_implies_x_in_(h_"_P)_/\_([#]_T2)_)_)
let x be set ; ::_thesis: ( ( x in (h " P) /\ ([#] T2) implies x in g " P ) & ( x in g " P implies x in (h " P) /\ ([#] T2) ) )
thus ( x in (h " P) /\ ([#] T2) implies x in g " P ) ::_thesis: ( x in g " P implies x in (h " P) /\ ([#] T2) )
proof
assume A16: x in (h " P) /\ ([#] T2) ; ::_thesis: x in g " P
then x in h " P by XBOOLE_0:def_4;
then A17: h . x in P by FUNCT_1:def_7;
g . x = h . x by A11, A16, FUNCT_4:13;
hence x in g " P by A11, A16, A17, FUNCT_1:def_7; ::_thesis: verum
end;
assume A18: x in g " P ; ::_thesis: x in (h " P) /\ ([#] T2)
then A19: x in dom g by FUNCT_1:def_7;
g . x in P by A18, FUNCT_1:def_7;
then A20: h . x in P by A19, FUNCT_4:13;
x in dom h by A14, A19, XBOOLE_0:def_3;
then x in h " P by A20, FUNCT_1:def_7;
hence x in (h " P) /\ ([#] T2) by A18, XBOOLE_0:def_4; ::_thesis: verum
end;
A21: for x being set st x in [#] T1 holds
h . x = f . x
proof
let x be set ; ::_thesis: ( x in [#] T1 implies h . x = f . x )
assume A22: x in [#] T1 ; ::_thesis: h . x = f . x
now__::_thesis:_h_._x_=_f_._x
percases ( x in [#] T2 or not x in [#] T2 ) ;
supposeA23: x in [#] T2 ; ::_thesis: h . x = f . x
then x in ([#] T1) /\ ([#] T2) by A22, XBOOLE_0:def_4;
then f . x = g . x by A10;
hence h . x = f . x by A11, A23, FUNCT_4:13; ::_thesis: verum
end;
suppose not x in [#] T2 ; ::_thesis: h . x = f . x
hence h . x = f . x by A11, FUNCT_4:11; ::_thesis: verum
end;
end;
end;
hence h . x = f . x ; ::_thesis: verum
end;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_(h_"_P)_/\_([#]_T1)_implies_x_in_f_"_P_)_&_(_x_in_f_"_P_implies_x_in_(h_"_P)_/\_([#]_T1)_)_)
let x be set ; ::_thesis: ( ( x in (h " P) /\ ([#] T1) implies x in f " P ) & ( x in f " P implies x in (h " P) /\ ([#] T1) ) )
thus ( x in (h " P) /\ ([#] T1) implies x in f " P ) ::_thesis: ( x in f " P implies x in (h " P) /\ ([#] T1) )
proof
assume A24: x in (h " P) /\ ([#] T1) ; ::_thesis: x in f " P
then x in h " P by XBOOLE_0:def_4;
then A25: h . x in P by FUNCT_1:def_7;
f . x = h . x by A21, A24;
hence x in f " P by A12, A24, A25, FUNCT_1:def_7; ::_thesis: verum
end;
assume A26: x in f " P ; ::_thesis: x in (h " P) /\ ([#] T1)
then x in dom f by FUNCT_1:def_7;
then A27: x in dom h by A14, XBOOLE_0:def_3;
f . x in P by A26, FUNCT_1:def_7;
then h . x in P by A21, A26;
then x in h " P by A27, FUNCT_1:def_7;
hence x in (h " P) /\ ([#] T1) by A26, XBOOLE_0:def_4; ::_thesis: verum
end;
then A28: (h " P) /\ ([#] T1) = f " P by TARSKI:1;
assume A29: P is closed ; ::_thesis: h " P is closed
then f " P is closed by A8, PRE_TOPC:def_6;
then ex F01 being Subset of T st
( F01 is closed & f " P = F01 /\ ([#] T1) ) by A1, PRE_TOPC:13;
then A30: P1 is closed by A3, A6;
g " P is closed by A9, A29, PRE_TOPC:def_6;
then ex F02 being Subset of T st
( F02 is closed & g " P = F02 /\ ([#] T2) ) by A2, PRE_TOPC:13;
then A31: P2 is closed by A4, A7;
h " P = (h " P) /\ (([#] T1) \/ ([#] T2)) by A12, A11, A14, RELAT_1:132, XBOOLE_1:28
.= ((h " P) /\ ([#] T1)) \/ ((h " P) /\ ([#] T2)) by XBOOLE_1:23 ;
then h " P = (f " P) \/ (g " P) by A28, A15, TARSKI:1;
hence h " P is closed by A30, A31; ::_thesis: verum
end;
hence h is continuous by PRE_TOPC:def_6; ::_thesis: verum
end;
theorem Th2: :: JGRAPH_2:2
for n being Element of NAT
for q2 being Point of (Euclid n)
for q being Point of (TOP-REAL n)
for r being real number st q = q2 holds
Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
proof
let n be Element of NAT ; ::_thesis: for q2 being Point of (Euclid n)
for q being Point of (TOP-REAL n)
for r being real number st q = q2 holds
Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
let q2 be Point of (Euclid n); ::_thesis: for q being Point of (TOP-REAL n)
for r being real number st q = q2 holds
Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
let q be Point of (TOP-REAL n); ::_thesis: for r being real number st q = q2 holds
Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
let r be real number ; ::_thesis: ( q = q2 implies Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } )
assume A1: q = q2 ; ::_thesis: Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
A2: { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } c= { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } or x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } )
assume x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } ; ::_thesis: x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r }
then consider q4 being Element of (Euclid n) such that
A3: ( q4 = x & dist (q2,q4) < r ) ;
reconsider q44 = q4 as Point of (TOP-REAL n) by TOPREAL3:8;
dist (q2,q4) = |.(q - q44).| by A1, JGRAPH_1:28;
hence x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } by A3; ::_thesis: verum
end;
A4: { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } c= { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } or x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } )
assume x in { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } ; ::_thesis: x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r }
then consider q3 being Point of (TOP-REAL n) such that
A5: ( x = q3 & |.(q - q3).| < r ) ;
reconsider q34 = q3 as Point of (Euclid n) by TOPREAL3:8;
dist (q2,q34) = |.(q - q3).| by A1, JGRAPH_1:28;
hence x in { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } by A5; ::_thesis: verum
end;
Ball (q2,r) = { q4 where q4 is Element of (Euclid n) : dist (q2,q4) < r } by METRIC_1:17;
hence Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL n) : |.(q - q3).| < r } by A2, A4, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th3: :: JGRAPH_2:3
( (0. (TOP-REAL 2)) `1 = 0 & (0. (TOP-REAL 2)) `2 = 0 ) by EUCLID:52, EUCLID:54;
theorem Th4: :: JGRAPH_2:4
1.REAL 2 = <*1,1*> by FINSEQ_2:61;
theorem Th5: :: JGRAPH_2:5
( (1.REAL 2) `1 = 1 & (1.REAL 2) `2 = 1 ) by Th4, EUCLID:52;
theorem Th6: :: JGRAPH_2:6
( dom proj1 = the carrier of (TOP-REAL 2) & dom proj1 = REAL 2 )
proof
thus dom proj1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; ::_thesis: dom proj1 = REAL 2
hence dom proj1 = REAL 2 by EUCLID:22; ::_thesis: verum
end;
theorem Th7: :: JGRAPH_2:7
( dom proj2 = the carrier of (TOP-REAL 2) & dom proj2 = REAL 2 )
proof
thus dom proj2 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1; ::_thesis: dom proj2 = REAL 2
hence dom proj2 = REAL 2 by EUCLID:22; ::_thesis: verum
end;
theorem Th8: :: JGRAPH_2:8
for p being Point of (TOP-REAL 2) holds p = |[(proj1 . p),(proj2 . p)]|
proof
let p be Point of (TOP-REAL 2); ::_thesis: p = |[(proj1 . p),(proj2 . p)]|
( p = |[(p `1),(p `2)]| & p `1 = proj1 . p ) by EUCLID:53, PSCOMP_1:def_5;
hence p = |[(proj1 . p),(proj2 . p)]| by PSCOMP_1:def_6; ::_thesis: verum
end;
theorem Th9: :: JGRAPH_2:9
for B being Subset of (TOP-REAL 2) st B = {(0. (TOP-REAL 2))} holds
( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} )
proof
let B be Subset of (TOP-REAL 2); ::_thesis: ( B = {(0. (TOP-REAL 2))} implies ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} ) )
assume A1: B = {(0. (TOP-REAL 2))} ; ::_thesis: ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} )
now__::_thesis:_not_|[0,1]|_in_B
assume |[0,1]| in B ; ::_thesis: contradiction
then |[0,1]| `2 = 0 by A1, Th3, TARSKI:def_1;
hence contradiction by EUCLID:52; ::_thesis: verum
end;
then |[0,1]| in the carrier of (TOP-REAL 2) \ B by XBOOLE_0:def_5;
hence ( B ` <> {} & the carrier of (TOP-REAL 2) \ B <> {} ) by SUBSET_1:def_4; ::_thesis: verum
end;
theorem Th10: :: JGRAPH_2:10
for X, Y being non empty TopSpace
for f being Function of X,Y holds
( f is continuous iff for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) )
proof
let X, Y be non empty TopSpace; ::_thesis: for f being Function of X,Y holds
( f is continuous iff for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) )
let f be Function of X,Y; ::_thesis: ( f is continuous iff for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) )
A1: [#] Y <> {} ;
A2: dom f = the carrier of X by FUNCT_2:def_1;
hereby ::_thesis: ( ( for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) ) implies f is continuous )
assume A3: f is continuous ; ::_thesis: for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V )
thus for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) ::_thesis: verum
proof
let p be Point of X; ::_thesis: for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V )
let V be Subset of Y; ::_thesis: ( f . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & f .: W c= V ) )
assume ( f . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & f .: W c= V )
then A4: ( f " V is open & p in f " V ) by A2, A1, A3, FUNCT_1:def_7, TOPS_2:43;
f .: (f " V) c= V by FUNCT_1:75;
hence ex W being Subset of X st
( p in W & W is open & f .: W c= V ) by A4; ::_thesis: verum
end;
end;
assume A5: for p being Point of X
for V being Subset of Y st f . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & f .: W c= V ) ; ::_thesis: f is continuous
for G being Subset of Y st G is open holds
f " G is open
proof
let G be Subset of Y; ::_thesis: ( G is open implies f " G is open )
assume A6: G is open ; ::_thesis: f " G is open
for z being set holds
( z in f " G iff ex Q being Subset of X st
( Q is open & Q c= f " G & z in Q ) )
proof
let z be set ; ::_thesis: ( z in f " G iff ex Q being Subset of X st
( Q is open & Q c= f " G & z in Q ) )
now__::_thesis:_(_z_in_f_"_G_implies_ex_Q_being_Subset_of_X_st_
(_Q_is_open_&_Q_c=_f_"_G_&_z_in_Q_)_)
assume A7: z in f " G ; ::_thesis: ex Q being Subset of X st
( Q is open & Q c= f " G & z in Q )
then reconsider p = z as Point of X ;
f . z in G by A7, FUNCT_1:def_7;
then consider W being Subset of X such that
A8: ( p in W & W is open ) and
A9: f .: W c= G by A5, A6;
A10: W c= f " (f .: W) by A2, FUNCT_1:76;
f " (f .: W) c= f " G by A9, RELAT_1:143;
hence ex Q being Subset of X st
( Q is open & Q c= f " G & z in Q ) by A8, A10, XBOOLE_1:1; ::_thesis: verum
end;
hence ( z in f " G iff ex Q being Subset of X st
( Q is open & Q c= f " G & z in Q ) ) ; ::_thesis: verum
end;
hence f " G is open by TOPS_1:25; ::_thesis: verum
end;
hence f is continuous by A1, TOPS_2:43; ::_thesis: verum
end;
theorem Th11: :: JGRAPH_2:11
for p being Point of (TOP-REAL 2)
for G being Subset of (TOP-REAL 2) st G is open & p in G holds
ex r being real number st
( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G )
proof
let p be Point of (TOP-REAL 2); ::_thesis: for G being Subset of (TOP-REAL 2) st G is open & p in G holds
ex r being real number st
( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G )
let G be Subset of (TOP-REAL 2); ::_thesis: ( G is open & p in G implies ex r being real number st
( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) )
assume that
A1: G is open and
A2: p in G ; ::_thesis: ex r being real number st
( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G )
reconsider GG = G as Subset of TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) ;
reconsider q2 = p as Point of (Euclid 2) by TOPREAL3:8;
( TopSpaceMetr (Euclid 2) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) & GG is open ) by A1, EUCLID:def_8, PRE_TOPC:30;
then consider r being real number such that
A3: r > 0 and
A4: Ball (q2,r) c= GG by A2, TOPMETR:15;
set s = r / (sqrt 2);
A5: Ball (q2,r) = { q3 where q3 is Point of (TOP-REAL 2) : |.(p - q3).| < r } by Th2;
A6: { q where q is Point of (TOP-REAL 2) : ( (p `1) - (r / (sqrt 2)) < q `1 & q `1 < (p `1) + (r / (sqrt 2)) & (p `2) - (r / (sqrt 2)) < q `2 & q `2 < (p `2) + (r / (sqrt 2)) ) } c= Ball (q2,r)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Point of (TOP-REAL 2) : ( (p `1) - (r / (sqrt 2)) < q `1 & q `1 < (p `1) + (r / (sqrt 2)) & (p `2) - (r / (sqrt 2)) < q `2 & q `2 < (p `2) + (r / (sqrt 2)) ) } or x in Ball (q2,r) )
assume x in { q where q is Point of (TOP-REAL 2) : ( (p `1) - (r / (sqrt 2)) < q `1 & q `1 < (p `1) + (r / (sqrt 2)) & (p `2) - (r / (sqrt 2)) < q `2 & q `2 < (p `2) + (r / (sqrt 2)) ) } ; ::_thesis: x in Ball (q2,r)
then consider q being Point of (TOP-REAL 2) such that
A7: q = x and
A8: (p `1) - (r / (sqrt 2)) < q `1 and
A9: q `1 < (p `1) + (r / (sqrt 2)) and
A10: (p `2) - (r / (sqrt 2)) < q `2 and
A11: q `2 < (p `2) + (r / (sqrt 2)) ;
((p `1) + (r / (sqrt 2))) - (r / (sqrt 2)) > (q `1) - (r / (sqrt 2)) by A9, XREAL_1:14;
then A12: (p `1) - (q `1) > ((q `1) + (- (r / (sqrt 2)))) - (q `1) by XREAL_1:14;
((p `2) + (r / (sqrt 2))) - (r / (sqrt 2)) > (q `2) - (r / (sqrt 2)) by A11, XREAL_1:14;
then A13: (p `2) - (q `2) > ((q `2) + (- (r / (sqrt 2)))) - (q `2) by XREAL_1:14;
((p `2) - (r / (sqrt 2))) + (r / (sqrt 2)) < (q `2) + (r / (sqrt 2)) by A10, XREAL_1:8;
then (p `2) - (q `2) < ((q `2) + (r / (sqrt 2))) - (q `2) by XREAL_1:14;
then A14: ((p `2) - (q `2)) ^2 < (r / (sqrt 2)) ^2 by A13, SQUARE_1:50;
(r / (sqrt 2)) ^2 = (r ^2) / ((sqrt 2) ^2) by XCMPLX_1:76
.= (r ^2) / 2 by SQUARE_1:def_2 ;
then A15: ((r / (sqrt 2)) ^2) + ((r / (sqrt 2)) ^2) = r ^2 ;
((p `1) - (r / (sqrt 2))) + (r / (sqrt 2)) < (q `1) + (r / (sqrt 2)) by A8, XREAL_1:8;
then (p `1) - (q `1) < ((q `1) + (r / (sqrt 2))) - (q `1) by XREAL_1:14;
then A16: ( (p - q) `2 = (p `2) - (q `2) & ((p `1) - (q `1)) ^2 < (r / (sqrt 2)) ^2 ) by A12, SQUARE_1:50, TOPREAL3:3;
( |.(p - q).| ^2 = (((p - q) `1) ^2) + (((p - q) `2) ^2) & (p - q) `1 = (p `1) - (q `1) ) by JGRAPH_1:29, TOPREAL3:3;
then |.(p - q).| ^2 < r ^2 by A16, A14, A15, XREAL_1:8;
then |.(p - q).| < r by A3, SQUARE_1:48;
hence x in Ball (q2,r) by A5, A7; ::_thesis: verum
end;
sqrt 2 > 0 by SQUARE_1:25;
then r / (sqrt 2) > 0 by A3, XREAL_1:139;
hence ex r being real number st
( r > 0 & { q where q is Point of (TOP-REAL 2) : ( (p `1) - r < q `1 & q `1 < (p `1) + r & (p `2) - r < q `2 & q `2 < (p `2) + r ) } c= G ) by A4, A6, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th12: :: JGRAPH_2:12
for X, Y, Z being non empty TopSpace
for B being Subset of Y
for C being Subset of Z
for f being Function of X,Y
for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds
ex g being Function of X,Z st
( g is continuous & g = h * f )
proof
let X, Y, Z be non empty TopSpace; ::_thesis: for B being Subset of Y
for C being Subset of Z
for f being Function of X,Y
for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds
ex g being Function of X,Z st
( g is continuous & g = h * f )
let B be Subset of Y; ::_thesis: for C being Subset of Z
for f being Function of X,Y
for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds
ex g being Function of X,Z st
( g is continuous & g = h * f )
let C be Subset of Z; ::_thesis: for f being Function of X,Y
for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds
ex g being Function of X,Z st
( g is continuous & g = h * f )
let f be Function of X,Y; ::_thesis: for h being Function of (Y | B),(Z | C) st f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} holds
ex g being Function of X,Z st
( g is continuous & g = h * f )
let h be Function of (Y | B),(Z | C); ::_thesis: ( f is continuous & h is continuous & rng f c= B & B <> {} & C <> {} implies ex g being Function of X,Z st
( g is continuous & g = h * f ) )
assume that
A1: f is continuous and
A2: h is continuous and
A3: rng f c= B and
A4: B <> {} and
A5: C <> {} ; ::_thesis: ex g being Function of X,Z st
( g is continuous & g = h * f )
A6: the carrier of X = dom f by FUNCT_2:def_1;
the carrier of (Y | B) = [#] (Y | B)
.= B by PRE_TOPC:def_5 ;
then reconsider u = f as Function of X,(Y | B) by A3, A6, FUNCT_2:2;
reconsider V = B as non empty Subset of Y by A4;
not Y | V is empty ;
then reconsider H = Y | B as non empty TopSpace ;
reconsider F = C as non empty Subset of Z by A5;
reconsider k = u as Function of X,H ;
not Z | F is empty ;
then reconsider G = Z | C as non empty TopSpace ;
reconsider j = h as Function of H,G ;
A7: the carrier of (Z | C) = [#] (Z | C)
.= C by PRE_TOPC:def_5 ;
j * k is Function of X,G ;
then reconsider v = h * u as Function of X,Z by A7, FUNCT_2:7;
u is continuous by A1, TOPMETR:6;
then v is continuous by A2, A4, A5, PRE_TOPC:26;
hence ex g being Function of X,Z st
( g is continuous & g = h * f ) ; ::_thesis: verum
end;
definition
func Out_In_Sq -> Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) means :Def1: :: JGRAPH_2:def 1
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies it . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or it . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) );
existence
ex b1 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) )
proof
reconsider BP = NonZero (TOP-REAL 2) as non empty set by Th9;
defpred S1[ set , set ] means for p being Point of (TOP-REAL 2) st p = $1 holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies $2 = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or $2 = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) );
A1: for x being Element of BP ex y being Element of BP st S1[x,y]
proof
let x be Element of BP; ::_thesis: ex y being Element of BP st S1[x,y]
reconsider q = x as Point of (TOP-REAL 2) by TARSKI:def_3;
now__::_thesis:_(_(_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_ex_y_being_Element_of_BP_st_S1[x,y]_)_or_(_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_ex_y_being_Element_of_BP_st_S1[x,y]_)_)
percases ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) or ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ;
caseA2: ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex y being Element of BP st S1[x,y]
now__::_thesis:_not_|[(1_/_(q_`1)),(((q_`2)_/_(q_`1))_/_(q_`1))]|_in_{(0._(TOP-REAL_2))}
assume |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| in {(0. (TOP-REAL 2))} ; ::_thesis: contradiction
then 0. (TOP-REAL 2) = |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| by TARSKI:def_1;
then 0 = 1 / (q `1) by Th3, EUCLID:52;
then A3: 0 = (1 / (q `1)) * (q `1) ;
now__::_thesis:_(_(_q_`1_=_0_&_contradiction_)_or_(_q_`1_<>_0_&_contradiction_)_)
percases ( q `1 = 0 or q `1 <> 0 ) ;
caseA4: q `1 = 0 ; ::_thesis: contradiction
then q `2 = 0 by A2;
then q = 0. (TOP-REAL 2) by A4, EUCLID:53, EUCLID:54;
then q in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence contradiction by XBOOLE_0:def_5; ::_thesis: verum
end;
case q `1 <> 0 ; ::_thesis: contradiction
hence contradiction by A3, XCMPLX_1:87; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
then reconsider r = |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| as Element of BP by XBOOLE_0:def_5;
for p being Point of (TOP-REAL 2) st p = x holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies r = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or r = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) by A2;
hence ex y being Element of BP st S1[x,y] ; ::_thesis: verum
end;
caseA5: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: ex y being Element of BP st S1[x,y]
now__::_thesis:_not_|[(((q_`1)_/_(q_`2))_/_(q_`2)),(1_/_(q_`2))]|_in_{(0._(TOP-REAL_2))}
assume |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| in {(0. (TOP-REAL 2))} ; ::_thesis: contradiction
then 0. (TOP-REAL 2) = |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| by TARSKI:def_1;
then (0. (TOP-REAL 2)) `2 = 1 / (q `2) by EUCLID:52;
then A6: 0 = (1 / (q `2)) * (q `2) by Th3;
q `2 <> 0 by A5;
hence contradiction by A6, XCMPLX_1:87; ::_thesis: verum
end;
then reconsider r = |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| as Element of BP by XBOOLE_0:def_5;
for p being Point of (TOP-REAL 2) st p = x holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies r = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or r = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) by A5;
hence ex y being Element of BP st S1[x,y] ; ::_thesis: verum
end;
end;
end;
hence ex y being Element of BP st S1[x,y] ; ::_thesis: verum
end;
ex h being Function of BP,BP st
for x being Element of BP holds S1[x,h . x] from FUNCT_2:sch_3(A1);
then consider h being Function of BP,BP such that
A7: for x being Element of BP
for p being Point of (TOP-REAL 2) st p = x holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . x = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . x = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ;
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) )
assume p <> 0. (TOP-REAL 2) ; ::_thesis: ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) )
then not p in {(0. (TOP-REAL 2))} by TARSKI:def_1;
then p in NonZero (TOP-REAL 2) by XBOOLE_0:def_5;
hence ( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) by A7; ::_thesis: verum
end;
hence ex b1 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) st ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) & ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b2 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b2 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) holds
b1 = b2
proof
let h1, h2 be Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)); ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) & ( for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h2 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h2 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ) implies h1 = h2 )
assume that
A8: for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) and
A9: for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies h2 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or h2 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) ; ::_thesis: h1 = h2
for x being set st x in NonZero (TOP-REAL 2) holds
h1 . x = h2 . x
proof
let x be set ; ::_thesis: ( x in NonZero (TOP-REAL 2) implies h1 . x = h2 . x )
assume A10: x in NonZero (TOP-REAL 2) ; ::_thesis: h1 . x = h2 . x
then reconsider q = x as Point of (TOP-REAL 2) ;
not q in {(0. (TOP-REAL 2))} by A10, XBOOLE_0:def_5;
then A11: q <> 0. (TOP-REAL 2) by TARSKI:def_1;
now__::_thesis:_(_(_(_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_or_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_)_&_h1_._x_=_h2_._x_)_or_(_not_(_q_`2_<=_q_`1_&_-_(q_`1)_<=_q_`2_)_&_not_(_q_`2_>=_q_`1_&_q_`2_<=_-_(q_`1)_)_&_h1_._x_=_h2_._x_)_)
percases ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) or ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ) ;
caseA12: ( ( q `2 <= q `1 & - (q `1) <= q `2 ) or ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: h1 . x = h2 . x
then h1 . q = |[(1 / (q `1)),(((q `2) / (q `1)) / (q `1))]| by A8, A11;
hence h1 . x = h2 . x by A9, A11, A12; ::_thesis: verum
end;
caseA13: ( not ( q `2 <= q `1 & - (q `1) <= q `2 ) & not ( q `2 >= q `1 & q `2 <= - (q `1) ) ) ; ::_thesis: h1 . x = h2 . x
then h1 . q = |[(((q `1) / (q `2)) / (q `2)),(1 / (q `2))]| by A8, A11;
hence h1 . x = h2 . x by A9, A11, A13; ::_thesis: verum
end;
end;
end;
hence h1 . x = h2 . x ; ::_thesis: verum
end;
hence h1 = h2 by FUNCT_2:12; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Out_In_Sq JGRAPH_2:def_1_:_
for b1 being Function of (NonZero (TOP-REAL 2)),(NonZero (TOP-REAL 2)) holds
( b1 = Out_In_Sq iff for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) implies b1 . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or b1 . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) ) );
theorem Th13: :: JGRAPH_2:13
for p being Point of (TOP-REAL 2) holds
( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) or ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) )
A1: ( - (p `1) < p `2 implies - (- (p `1)) > - (p `2) ) by XREAL_1:24;
A2: ( - (p `1) > p `2 implies - (- (p `1)) < - (p `2) ) by XREAL_1:24;
assume ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) )
hence ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A1, A2; ::_thesis: verum
end;
theorem Th14: :: JGRAPH_2:14
for p being Point of (TOP-REAL 2) st p <> 0. (TOP-REAL 2) holds
( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p <> 0. (TOP-REAL 2) implies ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) ) )
assume A1: p <> 0. (TOP-REAL 2) ; ::_thesis: ( ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) implies Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ) & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| ) )
hereby ::_thesis: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) or Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| )
assume A2: ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
now__::_thesis:_(_(_p_`1_<=_p_`2_&_-_(p_`2)_<=_p_`1_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_or_(_p_`1_>=_p_`2_&_p_`1_<=_-_(p_`2)_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_)
percases ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by A2;
caseA3: ( p `1 <= p `2 & - (p `2) <= p `1 ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)
assume A4: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
A5: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_)
percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A4;
case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
hence ( p `1 = p `2 or p `1 = - (p `2) ) by A3, XXREAL_0:1; ::_thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
then - (p `2) >= - (- (p `1)) by XREAL_1:24;
hence ( p `1 = p `2 or p `1 = - (p `2) ) by A3, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p_`1_=_p_`2_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_or_(_p_`1_=_-_(p_`2)_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_)
percases ( p `1 = p `2 or p `1 = - (p `2) ) by A5;
caseA6: p `1 = p `2 ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
then p `1 <> 0 by A1, EUCLID:53, EUCLID:54;
then ((p `1) / (p `2)) / (p `2) = 1 / (p `1) by A6, XCMPLX_1:60;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A4, A6, Def1; ::_thesis: verum
end;
caseA7: p `1 = - (p `2) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
then A8: p `2 <> 0 by A1, EUCLID:53, EUCLID:54;
A9: ((p `1) / (p `2)) / (p `2) = (- ((p `2) / (p `2))) / (p `2) by A7
.= (- 1) / (p `2) by A8, XCMPLX_1:60
.= 1 / (p `1) by A7, XCMPLX_1:192 ;
- (p `1) = p `2 by A7;
then 1 / (p `2) = - (1 / (p `1)) by XCMPLX_1:188
.= - (((p `2) / (p `1)) / (- (p `1))) by A7, A9, XCMPLX_1:192
.= - (- (((p `2) / (p `1)) / (p `1))) by XCMPLX_1:188
.= ((p `2) / (p `1)) / (p `1) ;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A4, A9, Def1; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ; ::_thesis: verum
end;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, Def1; ::_thesis: verum
end;
caseA10: ( p `1 >= p `2 & p `1 <= - (p `2) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
now__::_thesis:_(_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_)_)_implies_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)
assume A11: ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
A12: now__::_thesis:_(_(_p_`2_<=_p_`1_&_-_(p_`1)_<=_p_`2_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_or_(_p_`2_>=_p_`1_&_p_`2_<=_-_(p_`1)_&_(_p_`1_=_p_`2_or_p_`1_=_-_(p_`2)_)_)_)
percases ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A11;
case ( p `2 <= p `1 & - (p `1) <= p `2 ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
then - (- (p `1)) >= - (p `2) by XREAL_1:24;
hence ( p `1 = p `2 or p `1 = - (p `2) ) by A10, XXREAL_0:1; ::_thesis: verum
end;
case ( p `2 >= p `1 & p `2 <= - (p `1) ) ; ::_thesis: ( p `1 = p `2 or p `1 = - (p `2) )
hence ( p `1 = p `2 or p `1 = - (p `2) ) by A10, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_p_`1_=_p_`2_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_or_(_p_`1_=_-_(p_`2)_&_Out_In_Sq_._p_=_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_)_)
percases ( p `1 = p `2 or p `1 = - (p `2) ) by A12;
caseA13: p `1 = p `2 ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
then p `1 <> 0 by A1, EUCLID:53, EUCLID:54;
then ((p `1) / (p `2)) / (p `2) = 1 / (p `1) by A13, XCMPLX_1:60;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A11, A13, Def1; ::_thesis: verum
end;
caseA14: p `1 = - (p `2) ; ::_thesis: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|
then A15: p `2 <> 0 by A1, EUCLID:53, EUCLID:54;
A16: ((p `1) / (p `2)) / (p `2) = (- ((p `2) / (p `2))) / (p `2) by A14
.= (- 1) / (p `2) by A15, XCMPLX_1:60
.= 1 / (p `1) by A14, XCMPLX_1:192 ;
- (p `1) = p `2 by A14;
then 1 / (p `2) = - (((p `1) / (p `2)) / (p `2)) by A16, XCMPLX_1:188
.= - (((p `2) / (p `1)) / (- (p `1))) by A14, XCMPLX_1:191
.= - (- (((p `2) / (p `1)) / (p `1))) by XCMPLX_1:188
.= ((p `2) / (p `1)) / (p `1) ;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, A11, A16, Def1; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ; ::_thesis: verum
end;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A1, Def1; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| ; ::_thesis: verum
end;
hereby ::_thesis: verum
A17: ( - (p `2) > p `1 implies - (- (p `2)) < - (p `1) ) by XREAL_1:24;
A18: ( - (p `2) < p `1 implies - (- (p `2)) > - (p `1) ) by XREAL_1:24;
assume ( not ( p `1 <= p `2 & - (p `2) <= p `1 ) & not ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ; ::_thesis: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]|
hence Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by A1, A18, A17, Def1; ::_thesis: verum
end;
end;
theorem Th15: :: JGRAPH_2:15
for D being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
proof
let D be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
let K0 be Subset of ((TOP-REAL 2) | D); ::_thesis: ( K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } implies rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) )
A1: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= D by PRE_TOPC:def_5 ;
then reconsider K00 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
assume A2: K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds
q `1 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `1 <> 0 )
A4: the carrier of ((TOP-REAL 2) | K00) = [#] ((TOP-REAL 2) | K00)
.= K0 by PRE_TOPC:def_5 ;
assume q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `1 <> 0
then A5: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A4;
now__::_thesis:_not_q_`1_=_0
assume A6: q `1 = 0 ; ::_thesis: contradiction
then q `2 = 0 by A5;
hence contradiction by A5, A6, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence q `1 <> 0 ; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) )
assume y in rng (Out_In_Sq | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0)
then consider x being set such that
A7: x in dom (Out_In_Sq | K0) and
A8: y = (Out_In_Sq | K0) . x by FUNCT_1:def_3;
A9: x in (dom Out_In_Sq) /\ K0 by A7, RELAT_1:61;
then A10: x in K0 by XBOOLE_0:def_4;
K0 c= the carrier of (TOP-REAL 2) by A1, XBOOLE_1:1;
then reconsider p = x as Point of (TOP-REAL 2) by A10;
A11: Out_In_Sq . p = y by A8, A10, FUNCT_1:49;
A12: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) by A2, A10;
then A13: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1;
set p9 = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]|;
K00 = [#] ((TOP-REAL 2) | K00) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K00) ;
then A14: p in the carrier of ((TOP-REAL 2) | K00) by A9, XBOOLE_0:def_4;
A15: |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) by EUCLID:52;
A16: now__::_thesis:_not_|[(1_/_(p_`1)),(((p_`2)_/_(p_`1))_/_(p_`1))]|_=_0._(TOP-REAL_2)
assume |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then 0 * (p `1) = (1 / (p `1)) * (p `1) by A15, EUCLID:52, EUCLID:54;
hence contradiction by A14, A3, XCMPLX_1:87; ::_thesis: verum
end;
A17: p `1 <> 0 by A14, A3;
now__::_thesis:_(_(_p_`1_>=_0_&_y_in_K0_)_or_(_p_`1_<_0_&_y_in_K0_)_)
percases ( p `1 >= 0 or p `1 < 0 ) ;
caseA18: p `1 >= 0 ; ::_thesis: y in K0
then ( ( (p `2) / (p `1) <= (p `1) / (p `1) & (- (1 * (p `1))) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A12, XREAL_1:72;
then A19: ( ( (p `2) / (p `1) <= 1 & ((- 1) * (p `1)) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A14, A3, XCMPLX_1:60;
then ( ( (p `2) / (p `1) <= 1 & - 1 <= (p `2) / (p `1) ) or ( (p `2) / (p `1) >= 1 & (p `2) / (p `1) <= ((- 1) * (p `1)) / (p `1) ) ) by A17, A18, XCMPLX_1:89;
then (- 1) / (p `1) <= ((p `2) / (p `1)) / (p `1) by A18, XREAL_1:72;
then A20: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A17, A18, A19, XREAL_1:72;
( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52;
hence y in K0 by A2, A11, A16, A13, A20; ::_thesis: verum
end;
caseA21: p `1 < 0 ; ::_thesis: y in K0
A22: ( not (p `2) / (p `1) >= 1 or not (p `2) / (p `1) <= - 1 ) ;
( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= (p `1) / (p `1) & (p `2) / (p `1) >= (- (1 * (p `1))) / (p `1) ) ) by A12, A21, XREAL_1:73;
then A23: ( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= ((- 1) * (p `1)) / (p `1) ) ) by A21, XCMPLX_1:60;
then ( ( (p `2) / (p `1) >= (p `1) / (p `1) & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= - 1 ) ) by A21, XCMPLX_1:89;
then (- 1) / (p `1) >= ((p `2) / (p `1)) / (p `1) by A21, A22, XCMPLX_1:60, XREAL_1:73;
then A24: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A21, A23, XREAL_1:73;
( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52;
hence y in K0 by A2, A11, A16, A13, A24; ::_thesis: verum
end;
end;
end;
then y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
hence y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum
end;
theorem Th16: :: JGRAPH_2:16
for D being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
proof
let D be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | D) st K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
let K0 be Subset of ((TOP-REAL 2) | D); ::_thesis: ( K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } implies rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0) )
A1: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= D by PRE_TOPC:def_5 ;
then reconsider K00 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
assume A2: K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
A3: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `2 <> 0 )
A4: the carrier of ((TOP-REAL 2) | K00) = [#] ((TOP-REAL 2) | K00)
.= K0 by PRE_TOPC:def_5 ;
assume q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `2 <> 0
then A5: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A4;
now__::_thesis:_not_q_`2_=_0
assume A6: q `2 = 0 ; ::_thesis: contradiction
then q `1 = 0 by A5;
hence contradiction by A5, A6, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence q `2 <> 0 ; ::_thesis: verum
end;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) )
assume y in rng (Out_In_Sq | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0)
then consider x being set such that
A7: x in dom (Out_In_Sq | K0) and
A8: y = (Out_In_Sq | K0) . x by FUNCT_1:def_3;
x in (dom Out_In_Sq) /\ K0 by A7, RELAT_1:61;
then A9: x in K0 by XBOOLE_0:def_4;
K0 c= the carrier of (TOP-REAL 2) by A1, XBOOLE_1:1;
then reconsider p = x as Point of (TOP-REAL 2) by A9;
A10: Out_In_Sq . p = y by A8, A9, FUNCT_1:49;
A11: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) by A2, A9;
then A12: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14;
A13: K00 = [#] ((TOP-REAL 2) | K00) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K00) ;
set p9 = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|;
A14: |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) by EUCLID:52;
A15: now__::_thesis:_not_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_=_0._(TOP-REAL_2)
assume |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then 0 * (p `2) = (1 / (p `2)) * (p `2) by A14, EUCLID:52, EUCLID:54;
hence contradiction by A9, A13, A3, XCMPLX_1:87; ::_thesis: verum
end;
A16: p `2 <> 0 by A9, A13, A3;
now__::_thesis:_(_(_p_`2_>=_0_&_y_in_K0_)_or_(_p_`2_<_0_&_y_in_K0_)_)
percases ( p `2 >= 0 or p `2 < 0 ) ;
caseA17: p `2 >= 0 ; ::_thesis: y in K0
then ( ( (p `1) / (p `2) <= (p `2) / (p `2) & (- (1 * (p `2))) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A11, XREAL_1:72;
then A18: ( ( (p `1) / (p `2) <= 1 & ((- 1) * (p `2)) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A9, A13, A3, XCMPLX_1:60;
then ( ( (p `1) / (p `2) <= 1 & - 1 <= (p `1) / (p `2) ) or ( (p `1) / (p `2) >= 1 & (p `1) / (p `2) <= ((- 1) * (p `2)) / (p `2) ) ) by A16, A17, XCMPLX_1:89;
then (- 1) / (p `2) <= ((p `1) / (p `2)) / (p `2) by A17, XREAL_1:72;
then A19: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A16, A17, A18, XREAL_1:72;
( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52;
hence y in K0 by A2, A10, A15, A12, A19; ::_thesis: verum
end;
caseA20: p `2 < 0 ; ::_thesis: y in K0
then ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= (p `2) / (p `2) & (p `1) / (p `2) >= (- (1 * (p `2))) / (p `2) ) ) by A11, XREAL_1:73;
then A21: ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= ((- 1) * (p `2)) / (p `2) ) ) by A20, XCMPLX_1:60;
then ( ( (p `1) / (p `2) >= 1 & ((- 1) * (p `2)) / (p `2) >= (p `1) / (p `2) ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= - 1 ) ) by A20, XCMPLX_1:89;
then (- 1) / (p `2) >= ((p `1) / (p `2)) / (p `2) by A20, XREAL_1:73;
then A22: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A20, A21, XREAL_1:73;
( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52;
hence y in K0 by A2, A10, A15, A12, A22; ::_thesis: verum
end;
end;
end;
then y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
hence y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum
end;
Lm1: 0. (TOP-REAL 2) = 0.REAL 2
by EUCLID:66;
theorem Th17: :: JGRAPH_2:17
for K0a being set
for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds
( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )
proof
A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19;
let K0a be set ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds
( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} implies ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) )
assume that
A2: K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } and
A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )
( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by Th5;
then A4: 1.REAL 2 in K0a by A2, A1;
A5: K0a c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0a or x in D )
A6: D = (D `) `
.= NonZero (TOP-REAL 2) by A3, SUBSET_1:def_4 ;
assume x in K0a ; ::_thesis: x in D
then A7: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in D by A7, A6, XBOOLE_0:def_5; ::_thesis: verum
end;
the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= D by PRE_TOPC:def_5 ;
hence K0a is non empty Subset of ((TOP-REAL 2) | D) by A4, A5; ::_thesis: K0a is non empty Subset of (TOP-REAL 2)
thus K0a is non empty Subset of (TOP-REAL 2) by A4, A5, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th18: :: JGRAPH_2:18
for K0a being set
for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds
( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )
proof
A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19;
let K0a be set ; ::_thesis: for D being non empty Subset of (TOP-REAL 2) st K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} holds
( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } & D ` = {(0. (TOP-REAL 2))} implies ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) ) )
assume that
A2: K0a = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } and
A3: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ( K0a is non empty Subset of ((TOP-REAL 2) | D) & K0a is non empty Subset of (TOP-REAL 2) )
( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by Th5;
then A4: 1.REAL 2 in K0a by A2, A1;
A5: K0a c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0a or x in D )
A6: D = (D `) `
.= NonZero (TOP-REAL 2) by A3, SUBSET_1:def_4 ;
assume x in K0a ; ::_thesis: x in D
then A7: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in D by A7, A6, XBOOLE_0:def_5; ::_thesis: verum
end;
the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= D by PRE_TOPC:def_5 ;
hence K0a is non empty Subset of ((TOP-REAL 2) | D) by A4, A5; ::_thesis: K0a is non empty Subset of (TOP-REAL 2)
thus K0a is non empty Subset of (TOP-REAL 2) by A4, A5, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th19: :: JGRAPH_2:19
for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: f2 is continuous ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous )
defpred S1[ set , set ] means for r1, r2 being real number st f1 . $1 = r1 & f2 . $1 = r2 holds
$2 = r1 + r2;
A3: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
reconsider r2 = f2 . x as Real by TOPMETR:17;
set r3 = r1 + r2;
for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds
r1 + r2 = r1 + r2 ;
hence ex y being Element of REAL st
for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds
y = r1 + r2 ; ::_thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3);
then consider f being Function of the carrier of X,REAL such that
A4: for x being Element of X
for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds
f . x = r1 + r2 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
reconsider r2 = f2 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A5: r0 > 0 and
A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
reconsider G1 = ].(r1 - (r0 / 2)),(r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17;
A7: r1 < r1 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215;
then r1 - (r0 / 2) < r1 by XREAL_1:19;
then A8: f1 . p in G1 by A7, XXREAL_1:4;
reconsider G2 = ].(r2 - (r0 / 2)),(r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17;
A9: r2 < r2 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215;
then r2 - (r0 / 2) < r2 by XREAL_1:19;
then A10: f2 . p in G2 by A9, XXREAL_1:4;
G2 is open by JORDAN6:35;
then consider W2 being Subset of X such that
A11: ( p in W2 & W2 is open ) and
A12: f2 .: W2 c= G2 by A2, A10, Th10;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A13: ( p in W1 & W1 is open ) and
A14: f1 .: W1 c= G1 by A1, A8, Th10;
set W = W1 /\ W2;
A15: g0 .: (W1 /\ W2) c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: (W1 /\ W2) or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: (W1 /\ W2) ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A16: z in dom g0 and
A17: z in W1 /\ W2 and
A18: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A16;
reconsider aa2 = f2 . pz as Real by TOPMETR:17;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A19: pz in the carrier of X ;
then A20: pz in dom f2 by FUNCT_2:def_1;
z in W2 by A17, XBOOLE_0:def_4;
then A21: f2 . pz in f2 .: W2 by A20, FUNCT_1:def_6;
then A22: r2 - (r0 / 2) < aa2 by A12, XXREAL_1:4;
A23: pz in dom f1 by A19, FUNCT_2:def_1;
z in W1 by A17, XBOOLE_0:def_4;
then A24: f1 . pz in f1 .: W1 by A23, FUNCT_1:def_6;
then r1 - (r0 / 2) < aa1 by A14, XXREAL_1:4;
then (r1 - (r0 / 2)) + (r2 - (r0 / 2)) < aa1 + aa2 by A22, XREAL_1:8;
then (r1 + r2) - ((r0 / 2) + (r0 / 2)) < aa1 + aa2 ;
then A25: r - r0 < aa1 + aa2 by A4;
A26: aa2 < r2 + (r0 / 2) by A12, A21, XXREAL_1:4;
A27: x = aa1 + aa2 by A4, A18;
then reconsider rx = x as Real ;
aa1 < r1 + (r0 / 2) by A14, A24, XXREAL_1:4;
then aa1 + aa2 < (r1 + (r0 / 2)) + (r2 + (r0 / 2)) by A26, XREAL_1:8;
then aa1 + aa2 < (r1 + r2) + ((r0 / 2) + (r0 / 2)) ;
then rx < r + r0 by A4, A27;
hence x in ].(r - r0),(r + r0).[ by A27, A25, XXREAL_1:4; ::_thesis: verum
end;
( W1 /\ W2 is open & p in W1 /\ W2 ) by A13, A11, XBOOLE_0:def_4;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A6, A15, XBOOLE_1:1; ::_thesis: verum
end;
then A28: g0 is continuous by Th10;
for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g0 . p = r1 + r2 by A4;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous ) by A28; ::_thesis: verum
end;
theorem :: JGRAPH_2:20
for X being non empty TopSpace
for a being real number ex g being Function of X,R^1 st
( ( for p being Point of X holds g . p = a ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for a being real number ex g being Function of X,R^1 st
( ( for p being Point of X holds g . p = a ) & g is continuous )
let a be real number ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X holds g . p = a ) & g is continuous )
reconsider a1 = a as Element of R^1 by TOPMETR:17, XREAL_0:def_1;
set g1 = the carrier of X --> a1;
reconsider g0 = the carrier of X --> a1 as Function of X,R^1 ;
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
set f1 = g0;
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
assume that
A1: g0 . p in V and
V is open ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set G1 = V;
g0 .: ([#] X) c= V
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in g0 .: ([#] X) or y in V )
assume y in g0 .: ([#] X) ; ::_thesis: y in V
then ex x being set st
( x in dom g0 & x in [#] X & y = g0 . x ) by FUNCT_1:def_6;
then y = a by FUNCOP_1:7;
hence y in V by A1, FUNCOP_1:7; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum
end;
then ( ( for p being Point of X holds ( the carrier of X --> a1) . p = a ) & g0 is continuous ) by Th10, FUNCOP_1:7;
hence ex g being Function of X,R^1 st
( ( for p being Point of X holds g . p = a ) & g is continuous ) ; ::_thesis: verum
end;
theorem Th21: :: JGRAPH_2:21
for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: f2 is continuous ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous )
defpred S1[ set , set ] means for r1, r2 being real number st f1 . $1 = r1 & f2 . $1 = r2 holds
$2 = r1 - r2;
A3: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
reconsider r2 = f2 . x as Real by TOPMETR:17;
set r3 = r1 - r2;
for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds
r1 - r2 = r1 - r2 ;
hence ex y being Element of REAL st
for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds
y = r1 - r2 ; ::_thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3);
then consider f being Function of the carrier of X,REAL such that
A4: for x being Element of X
for r1, r2 being real number st f1 . x = r1 & f2 . x = r2 holds
f . x = r1 - r2 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
reconsider r2 = f2 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A5: r0 > 0 and
A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
reconsider G1 = ].(r1 - (r0 / 2)),(r1 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17;
A7: r1 < r1 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215;
then r1 - (r0 / 2) < r1 by XREAL_1:19;
then A8: f1 . p in G1 by A7, XXREAL_1:4;
reconsider G2 = ].(r2 - (r0 / 2)),(r2 + (r0 / 2)).[ as Subset of R^1 by TOPMETR:17;
A9: r2 < r2 + (r0 / 2) by A5, XREAL_1:29, XREAL_1:215;
then r2 - (r0 / 2) < r2 by XREAL_1:19;
then A10: f2 . p in G2 by A9, XXREAL_1:4;
G2 is open by JORDAN6:35;
then consider W2 being Subset of X such that
A11: ( p in W2 & W2 is open ) and
A12: f2 .: W2 c= G2 by A2, A10, Th10;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A13: ( p in W1 & W1 is open ) and
A14: f1 .: W1 c= G1 by A1, A8, Th10;
set W = W1 /\ W2;
A15: g0 .: (W1 /\ W2) c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: (W1 /\ W2) or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: (W1 /\ W2) ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A16: z in dom g0 and
A17: z in W1 /\ W2 and
A18: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A16;
reconsider aa2 = f2 . pz as Real by TOPMETR:17;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A19: pz in the carrier of X ;
then A20: pz in dom f1 by FUNCT_2:def_1;
A21: pz in dom f2 by A19, FUNCT_2:def_1;
z in W2 by A17, XBOOLE_0:def_4;
then A22: f2 . pz in f2 .: W2 by A21, FUNCT_1:def_6;
then A23: r2 - (r0 / 2) < aa2 by A12, XXREAL_1:4;
A24: aa2 < r2 + (r0 / 2) by A12, A22, XXREAL_1:4;
z in W1 by A17, XBOOLE_0:def_4;
then A25: f1 . pz in f1 .: W1 by A20, FUNCT_1:def_6;
then r1 - (r0 / 2) < aa1 by A14, XXREAL_1:4;
then (r1 - (r0 / 2)) - (r2 + (r0 / 2)) < aa1 - aa2 by A24, XREAL_1:14;
then (r1 - r2) - ((r0 / 2) + (r0 / 2)) < aa1 - aa2 ;
then A26: r - r0 < aa1 - aa2 by A4;
A27: x = aa1 - aa2 by A4, A18;
then reconsider rx = x as Real ;
aa1 < r1 + (r0 / 2) by A14, A25, XXREAL_1:4;
then aa1 - aa2 < (r1 + (r0 / 2)) - (r2 - (r0 / 2)) by A23, XREAL_1:14;
then aa1 - aa2 < (r1 - r2) + ((r0 / 2) + (r0 / 2)) ;
then rx < r + r0 by A4, A27;
hence x in ].(r - r0),(r + r0).[ by A27, A26, XXREAL_1:4; ::_thesis: verum
end;
( W1 /\ W2 is open & p in W1 /\ W2 ) by A13, A11, XBOOLE_0:def_4;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A6, A15, XBOOLE_1:1; ::_thesis: verum
end;
then A28: g0 is continuous by Th10;
for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g0 . p = r1 - r2 by A4;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous ) by A28; ::_thesis: verum
end;
theorem Th22: :: JGRAPH_2:22
for X being non empty TopSpace
for f1 being Function of X,R^1 st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 * r1 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 * r1 ) & g is continuous )
let f1 be Function of X,R^1; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 * r1 ) & g is continuous ) )
defpred S1[ set , set ] means for r1 being real number st f1 . $1 = r1 holds
$2 = r1 * r1;
A1: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
set r3 = r1 * r1;
for r1 being real number st f1 . x = r1 holds
r1 * r1 = r1 * r1 ;
hence ex y being Element of REAL st
for r1 being real number st f1 . x = r1 holds
y = r1 * r1 ; ::_thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A1);
then consider f being Function of the carrier of X,REAL such that
A2: for x being Element of X
for r1 being real number st f1 . x = r1 holds
f . x = r1 * r1 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
assume A3: f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 * r1 ) & g is continuous )
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A4: r0 > 0 and
A5: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
A6: r = r1 ^2 by A2;
A7: r = r1 * r1 by A2;
then A8: 0 <= r by XREAL_1:63;
then A9: (sqrt (r + r0)) ^2 = r + r0 by A4, SQUARE_1:def_2;
now__::_thesis:_(_(_r1_>=_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_r1_<_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_)
percases ( r1 >= 0 or r1 < 0 ) ;
caseA10: r1 >= 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = (sqrt (r + r0)) - (sqrt r);
reconsider G1 = ].(r1 - ((sqrt (r + r0)) - (sqrt r))),(r1 + ((sqrt (r + r0)) - (sqrt r))).[ as Subset of R^1 by TOPMETR:17;
A11: G1 is open by JORDAN6:35;
r + r0 > r by A4, XREAL_1:29;
then sqrt (r + r0) > sqrt r by A7, SQUARE_1:27, XREAL_1:63;
then A12: (sqrt (r + r0)) - (sqrt r) > 0 by XREAL_1:50;
then A13: r1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by XREAL_1:29;
then r1 - ((sqrt (r + r0)) - (sqrt r)) < r1 by XREAL_1:19;
then f1 . p in G1 by A13, XXREAL_1:4;
then consider W1 being Subset of X such that
A14: ( p in W1 & W1 is open ) and
A15: f1 .: W1 c= G1 by A3, A11, Th10;
A16: r1 = sqrt r by A6, A10, SQUARE_1:def_2;
set W = W1;
A17: ((sqrt (r + r0)) - (sqrt r)) ^2 = (((sqrt (r + r0)) ^2) - ((2 * (sqrt (r + r0))) * (sqrt r))) + ((sqrt r) ^2)
.= ((r + r0) - ((2 * (sqrt (r + r0))) * (sqrt r))) + ((sqrt r) ^2) by A4, A8, SQUARE_1:def_2
.= (r + (r0 - ((2 * (sqrt (r + r0))) * (sqrt r)))) + r by A8, SQUARE_1:def_2
.= ((2 * r) + r0) - ((2 * (sqrt (r + r0))) * (sqrt r)) ;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A18: z in dom g0 and
A19: z in W1 and
A20: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A18;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A21: f1 . pz in f1 .: W1 by A19, FUNCT_1:def_6;
then A22: r1 - ((sqrt (r + r0)) - (sqrt r)) < aa1 by A15, XXREAL_1:4;
A23: now__::_thesis:_(_(_0_<=_r1_-_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_or_(_0_>_r1_-_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_)
percases ( 0 <= r1 - ((sqrt (r + r0)) - (sqrt r)) or 0 > r1 - ((sqrt (r + r0)) - (sqrt r)) ) ;
caseA24: 0 <= r1 - ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1
(- 2) * (((sqrt (r + r0)) - (sqrt r)) ^2) <= 0 by XREAL_1:63, XREAL_1:131;
then A25: (((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + (- (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) <= (((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + 0 by XREAL_1:7;
(r1 - ((sqrt (r + r0)) - (sqrt r))) ^2 < aa1 ^2 by A22, A24, SQUARE_1:16;
then (((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) - (aa1 ^2) < 0 by A25, XREAL_1:49;
hence r - r0 < aa1 * aa1 by A7, A16, A17, XREAL_1:48; ::_thesis: verum
end;
case 0 > r1 - ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1
then r1 < (sqrt (r + r0)) - (sqrt r) by XREAL_1:48;
then r1 ^2 < ((sqrt (r + r0)) - (sqrt r)) ^2 by A10, SQUARE_1:16;
then (r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2) < 0 by XREAL_1:49;
then ((r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2)) - ((2 * r1) * ((sqrt (r + r0)) - (sqrt r))) < 0 - 0 by A10, A12;
hence r - r0 < aa1 * aa1 by A7, A16, A17, XREAL_1:63; ::_thesis: verum
end;
end;
end;
(- r1) - ((sqrt (r + r0)) - (sqrt r)) <= r1 - ((sqrt (r + r0)) - (sqrt r)) by A10, XREAL_1:9;
then - (r1 + ((sqrt (r + r0)) - (sqrt r))) < aa1 by A22, XXREAL_0:2;
then A26: aa1 - (- (r1 + ((sqrt (r + r0)) - (sqrt r)))) > 0 by XREAL_1:50;
aa1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by A15, A21, XXREAL_1:4;
then (r1 + ((sqrt (r + r0)) - (sqrt r))) - aa1 > 0 by XREAL_1:50;
then ((r1 + ((sqrt (r + r0)) - (sqrt r))) - aa1) * ((r1 + ((sqrt (r + r0)) - (sqrt r))) + aa1) > 0 by A26, XREAL_1:129;
then ((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2) > 0 ;
then A27: aa1 ^2 < (r1 + ((sqrt (r + r0)) - (sqrt r))) ^2 by XREAL_1:47;
x = aa1 * aa1 by A2, A20;
hence x in ].(r - r0),(r + r0).[ by A7, A16, A17, A27, A23, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A5, A14, XBOOLE_1:1; ::_thesis: verum
end;
caseA28: r1 < 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = (sqrt (r + r0)) - (sqrt r);
reconsider G1 = ].(r1 - ((sqrt (r + r0)) - (sqrt r))),(r1 + ((sqrt (r + r0)) - (sqrt r))).[ as Subset of R^1 by TOPMETR:17;
A29: G1 is open by JORDAN6:35;
r + r0 > r by A4, XREAL_1:29;
then sqrt (r + r0) > sqrt r by A7, SQUARE_1:27, XREAL_1:63;
then A30: (sqrt (r + r0)) - (sqrt r) > 0 by XREAL_1:50;
then A31: r1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by XREAL_1:29;
then r1 - ((sqrt (r + r0)) - (sqrt r)) < r1 by XREAL_1:19;
then f1 . p in G1 by A31, XXREAL_1:4;
then consider W1 being Subset of X such that
A32: ( p in W1 & W1 is open ) and
A33: f1 .: W1 c= G1 by A3, A29, Th10;
A34: (- r1) ^2 = r1 ^2 ;
then A35: - r1 = sqrt r by A7, A28, SQUARE_1:22;
set W = W1;
A36: ((sqrt (r + r0)) - (sqrt r)) ^2 = ((r + r0) - ((2 * (sqrt (r + r0))) * (sqrt r))) + ((sqrt r) ^2) by A9
.= (r + (r0 - ((2 * (sqrt (r + r0))) * (sqrt r)))) + r by A7, A28, A34, SQUARE_1:22
.= ((2 * r) + r0) - ((2 * (sqrt (r + r0))) * (sqrt r)) ;
then A37: (- ((2 * r1) * ((sqrt (r + r0)) - (sqrt r)))) + (((sqrt (r + r0)) - (sqrt r)) ^2) = r0 by A7, A35;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A38: z in dom g0 and
A39: z in W1 and
A40: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A38;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A41: f1 . pz in f1 .: W1 by A39, FUNCT_1:def_6;
then A42: aa1 < r1 + ((sqrt (r + r0)) - (sqrt r)) by A33, XXREAL_1:4;
A43: now__::_thesis:_(_(_0_>=_r1_+_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_or_(_0_<_r1_+_((sqrt_(r_+_r0))_-_(sqrt_r))_&_r_-_r0_<_aa1_*_aa1_)_)
percases ( 0 >= r1 + ((sqrt (r + r0)) - (sqrt r)) or 0 < r1 + ((sqrt (r + r0)) - (sqrt r)) ) ;
caseA44: 0 >= r1 + ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1
(- 2) * (((sqrt (r + r0)) - (sqrt r)) ^2) <= 0 by XREAL_1:63, XREAL_1:131;
then A45: (((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + (- (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) <= (((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2)) + 0 by XREAL_1:7;
- aa1 > - (r1 + ((sqrt (r + r0)) - (sqrt r))) by A42, XREAL_1:24;
then (- (r1 + ((sqrt (r + r0)) - (sqrt r)))) ^2 < (- aa1) ^2 by A44, SQUARE_1:16;
then (((r1 + ((sqrt (r + r0)) - (sqrt r))) ^2) - (2 * (((sqrt (r + r0)) - (sqrt r)) ^2))) - (aa1 ^2) < 0 by A45, XREAL_1:49;
hence r - r0 < aa1 * aa1 by A7, A37, XREAL_1:48; ::_thesis: verum
end;
case 0 < r1 + ((sqrt (r + r0)) - (sqrt r)) ; ::_thesis: r - r0 < aa1 * aa1
then 0 + (- r1) < (r1 + ((sqrt (r + r0)) - (sqrt r))) + (- r1) by XREAL_1:8;
then (- r1) ^2 < ((sqrt (r + r0)) - (sqrt r)) ^2 by A28, SQUARE_1:16;
then (r1 ^2) - (r1 ^2) > (r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2) by XREAL_1:15;
then ((r1 ^2) - (((sqrt (r + r0)) - (sqrt r)) ^2)) + ((2 * r1) * ((sqrt (r + r0)) - (sqrt r))) < 0 + 0 by A28, A30;
hence r - r0 < aa1 * aa1 by A7, A35, A36, XREAL_1:63; ::_thesis: verum
end;
end;
end;
r1 - ((sqrt (r + r0)) - (sqrt r)) < aa1 by A33, A41, XXREAL_1:4;
then aa1 - (r1 - ((sqrt (r + r0)) - (sqrt r))) > 0 by XREAL_1:50;
then - ((- aa1) + (r1 - ((sqrt (r + r0)) - (sqrt r)))) > 0 ;
then A46: (r1 - ((sqrt (r + r0)) - (sqrt r))) + (- aa1) < 0 ;
(- r1) - ((sqrt (r + r0)) - (sqrt r)) >= r1 - ((sqrt (r + r0)) - (sqrt r)) by A28, XREAL_1:9;
then - ((- r1) - ((sqrt (r + r0)) - (sqrt r))) <= - (r1 - ((sqrt (r + r0)) - (sqrt r))) by XREAL_1:24;
then - (r1 - ((sqrt (r + r0)) - (sqrt r))) > aa1 by A42, XXREAL_0:2;
then (- (r1 - ((sqrt (r + r0)) - (sqrt r)))) + (r1 - ((sqrt (r + r0)) - (sqrt r))) > aa1 + (r1 - ((sqrt (r + r0)) - (sqrt r))) by XREAL_1:8;
then ((r1 - ((sqrt (r + r0)) - (sqrt r))) - aa1) * ((r1 - ((sqrt (r + r0)) - (sqrt r))) + aa1) > 0 by A46, XREAL_1:130;
then ((r1 - ((sqrt (r + r0)) - (sqrt r))) ^2) - (aa1 ^2) > 0 ;
then A47: aa1 ^2 < (r1 - ((sqrt (r + r0)) - (sqrt r))) ^2 by XREAL_1:47;
x = aa1 * aa1 by A2, A40;
hence x in ].(r - r0),(r + r0).[ by A7, A37, A47, A43, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A5, A32, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum
end;
then A48: g0 is continuous by Th10;
for p being Point of X
for r1 being real number st f1 . p = r1 holds
g0 . p = r1 * r1 by A2;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 * r1 ) & g is continuous ) by A48; ::_thesis: verum
end;
theorem Th23: :: JGRAPH_2:23
for X being non empty TopSpace
for f1 being Function of X,R^1
for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1
for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
let f1 be Function of X,R^1; ::_thesis: for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
let a be real number ; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous ) )
defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds
$2 = a * r1;
A1: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
reconsider r3 = a * r1 as Element of REAL ;
for r1 being Real st f1 . x = r1 holds
r3 = a * r1 ;
hence ex y being Element of REAL st
for r1 being Real st f1 . x = r1 holds
y = a * r1 ; ::_thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A1);
then consider f being Function of the carrier of X,REAL such that
A2: for x being Element of X
for r1 being Real st f1 . x = r1 holds
f . x = a * r1 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
A3: for p being Point of X
for r1 being real number st f1 . p = r1 holds
g0 . p = a * r1
proof
let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds
g0 . p = a * r1
let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g0 . p = a * r1 )
assume A4: f1 . p = r1 ; ::_thesis: g0 . p = a * r1
reconsider r1 = r1 as Element of REAL by XREAL_0:def_1;
g0 . p = a * r1 by A2, A4;
hence g0 . p = a * r1 ; ::_thesis: verum
end;
assume A5: f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous )
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A6: r0 > 0 and
A7: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
A8: r = a * r1 by A2;
A9: r = a * r1 by A2;
now__::_thesis:_(_(_a_>=_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_a_<_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_)
percases ( a >= 0 or a < 0 ) ;
caseA10: a >= 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
now__::_thesis:_(_(_a_>_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_a_=_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_)
percases ( a > 0 or a = 0 ) by A10;
caseA11: a > 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = r0 / a;
reconsider G1 = ].(r1 - (r0 / a)),(r1 + (r0 / a)).[ as Subset of R^1 by TOPMETR:17;
A12: r1 < r1 + (r0 / a) by A6, A11, XREAL_1:29, XREAL_1:139;
then r1 - (r0 / a) < r1 by XREAL_1:19;
then A13: f1 . p in G1 by A12, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A14: ( p in W1 & W1 is open ) and
A15: f1 .: W1 c= G1 by A5, A13, Th10;
set W = W1;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A16: z in dom g0 and
A17: z in W1 and
A18: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A16;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A19: x = a * aa1 by A2, A18;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A20: f1 . pz in f1 .: W1 by A17, FUNCT_1:def_6;
then r1 - (r0 / a) < aa1 by A15, XXREAL_1:4;
then A21: a * (r1 - (r0 / a)) < a * aa1 by A11, XREAL_1:68;
reconsider rx = x as Real by A18, XREAL_0:def_1;
A22: a * (r1 + (r0 / a)) = (a * r1) + (a * (r0 / a))
.= r + r0 by A8, A11, XCMPLX_1:87 ;
A23: a * (r1 - (r0 / a)) = (a * r1) - (a * (r0 / a))
.= r - r0 by A8, A11, XCMPLX_1:87 ;
aa1 < r1 + (r0 / a) by A15, A20, XXREAL_1:4;
then rx < r + r0 by A11, A19, A22, XREAL_1:68;
hence x in ].(r - r0),(r + r0).[ by A19, A21, A23, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A7, A14, XBOOLE_1:1; ::_thesis: verum
end;
caseA24: a = 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = r0;
reconsider G1 = ].(r1 - r0),(r1 + r0).[ as Subset of R^1 by TOPMETR:17;
A25: r1 < r1 + r0 by A6, XREAL_1:29;
then r1 - r0 < r1 by XREAL_1:19;
then A26: f1 . p in G1 by A25, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A27: ( p in W1 & W1 is open ) and
f1 .: W1 c= G1 by A5, A26, Th10;
set W = W1;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A28: z in dom g0 and
z in W1 and
A29: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A28;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
x = a * aa1 by A2, A29
.= 0 by A24 ;
hence x in ].(r - r0),(r + r0).[ by A6, A9, A24, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A7, A27, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum
end;
caseA30: a < 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = r0 / (- a);
reconsider G1 = ].(r1 - (r0 / (- a))),(r1 + (r0 / (- a))).[ as Subset of R^1 by TOPMETR:17;
- a > 0 by A30, XREAL_1:58;
then A31: r1 < r1 + (r0 / (- a)) by A6, XREAL_1:29, XREAL_1:139;
then r1 - (r0 / (- a)) < r1 by XREAL_1:19;
then A32: f1 . p in G1 by A31, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A33: ( p in W1 & W1 is open ) and
A34: f1 .: W1 c= G1 by A5, A32, Th10;
set W = W1;
- a <> 0 by A30;
then A35: (- a) * (r0 / (- a)) = r0 by XCMPLX_1:87;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A36: z in dom g0 and
A37: z in W1 and
A38: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A36;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A39: f1 . pz in f1 .: W1 by A37, FUNCT_1:def_6;
then r1 - (r0 / (- a)) < aa1 by A34, XXREAL_1:4;
then A40: a * aa1 < a * (r1 - (r0 / (- a))) by A30, XREAL_1:69;
A41: a * (r1 + (r0 / (- a))) = (a * r1) - (- (a * (r0 / (- a))))
.= r - r0 by A3, A35 ;
A42: a * (r1 - (r0 / (- a))) = (a * r1) + (- (a * (r0 / (- a))))
.= r + r0 by A3, A35 ;
aa1 < r1 + (r0 / (- a)) by A34, A39, XXREAL_1:4;
then A43: r - r0 < a * aa1 by A30, A41, XREAL_1:69;
x = a * aa1 by A2, A38;
hence x in ].(r - r0),(r + r0).[ by A40, A42, A43, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A7, A33, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum
end;
then g0 is continuous by Th10;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = a * r1 ) & g is continuous ) by A3; ::_thesis: verum
end;
theorem Th24: :: JGRAPH_2:24
for X being non empty TopSpace
for f1 being Function of X,R^1
for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 + a ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1
for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 + a ) & g is continuous )
let f1 be Function of X,R^1; ::_thesis: for a being real number st f1 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 + a ) & g is continuous )
let a be real number ; ::_thesis: ( f1 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 + a ) & g is continuous ) )
defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds
$2 = r1 + a;
A1: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
reconsider r2 = a as Element of REAL by XREAL_0:def_1;
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
set r3 = r1 + r2;
for r1 being Real st f1 . x = r1 holds
r1 + r2 = r1 + r2 ;
hence ex y being Element of REAL st
for r1 being Real st f1 . x = r1 holds
y = r1 + a ; ::_thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A1);
then consider f being Function of the carrier of X,REAL such that
A2: for x being Element of X
for r1 being Real st f1 . x = r1 holds
f . x = r1 + a ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
assume A3: f1 is continuous ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 + a ) & g is continuous )
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A4: r0 > 0 and
A5: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
set r4 = r0;
reconsider G1 = ].(r1 - r0),(r1 + r0).[ as Subset of R^1 by TOPMETR:17;
A6: r1 < r1 + r0 by A4, XREAL_1:29;
then r1 - r0 < r1 by XREAL_1:19;
then A7: f1 . p in G1 by A6, XXREAL_1:4;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A8: ( p in W1 & W1 is open ) and
A9: f1 .: W1 c= G1 by A3, A7, Th10;
set W = W1;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A10: z in dom g0 and
A11: z in W1 and
A12: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A10;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A13: f1 . pz in f1 .: W1 by A11, FUNCT_1:def_6;
then r1 - r0 < aa1 by A9, XXREAL_1:4;
then A14: (r1 - r0) + a < aa1 + a by XREAL_1:8;
A15: (r1 - r0) + a = (r1 + a) - r0
.= r - r0 by A2 ;
aa1 < r1 + r0 by A9, A13, XXREAL_1:4;
then A16: (r1 + r0) + a > aa1 + a by XREAL_1:8;
x = aa1 + a by A2, A12;
hence x in ].(r - r0),(r + r0).[ by A16, A14, A15, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A5, A8, XBOOLE_1:1; ::_thesis: verum
end;
then A17: g0 is continuous by Th10;
for p being Point of X
for r1 being real number st f1 . p = r1 holds
g0 . p = r1 + a
proof
let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds
g0 . p = r1 + a
let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g0 . p = r1 + a )
assume A18: f1 . p = r1 ; ::_thesis: g0 . p = r1 + a
reconsider r1 = r1 as Element of REAL by XREAL_0:def_1;
g0 . p = r1 + a by A2, A18;
hence g0 . p = r1 + a ; ::_thesis: verum
end;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = r1 + a ) & g is continuous ) by A17; ::_thesis: verum
end;
theorem Th25: :: JGRAPH_2:25
for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous ) )
assume A1: ( f1 is continuous & f2 is continuous ) ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous )
then consider g1 being Function of X,R^1 such that
A2: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g1 . p = r1 + r2 and
A3: g1 is continuous by Th19;
consider g3 being Function of X,R^1 such that
A4: for p being Point of X
for r1 being real number st g1 . p = r1 holds
g3 . p = r1 * r1 and
A5: g3 is continuous by A3, Th22;
consider g2 being Function of X,R^1 such that
A6: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 - r2 and
A7: g2 is continuous by A1, Th21;
consider g4 being Function of X,R^1 such that
A8: for p being Point of X
for r1 being real number st g2 . p = r1 holds
g4 . p = r1 * r1 and
A9: g4 is continuous by A7, Th22;
consider g5 being Function of X,R^1 such that
A10: for p being Point of X
for r1, r2 being real number st g3 . p = r1 & g4 . p = r2 holds
g5 . p = r1 - r2 and
A11: g5 is continuous by A5, A9, Th21;
consider g6 being Function of X,R^1 such that
A12: for p being Point of X
for r1 being real number st g5 . p = r1 holds
g6 . p = (1 / 4) * r1 and
A13: g6 is continuous by A11, Th23;
for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g6 . p = r1 * r2
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g6 . p = r1 * r2
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g6 . p = r1 * r2 )
assume A14: ( f1 . p = r1 & f2 . p = r2 ) ; ::_thesis: g6 . p = r1 * r2
then g2 . p = r1 - r2 by A6;
then A15: g4 . p = (r1 - r2) ^2 by A8;
g1 . p = r1 + r2 by A2, A14;
then g3 . p = (r1 + r2) ^2 by A4;
then g5 . p = ((r1 + r2) ^2) - ((r1 - r2) ^2) by A10, A15;
then g6 . p = (1 / 4) * ((((r1 ^2) + ((2 * r1) * r2)) + (r2 ^2)) - ((r1 - r2) ^2)) by A12
.= r1 * r2 ;
hence g6 . p = r1 * r2 ; ::_thesis: verum
end;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous ) by A13; ::_thesis: verum
end;
theorem Th26: :: JGRAPH_2:26
for X being non empty TopSpace
for f1 being Function of X,R^1 st f1 is continuous & ( for q being Point of X holds f1 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = 1 / r1 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1 being Function of X,R^1 st f1 is continuous & ( for q being Point of X holds f1 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = 1 / r1 ) & g is continuous )
let f1 be Function of X,R^1; ::_thesis: ( f1 is continuous & ( for q being Point of X holds f1 . q <> 0 ) implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = 1 / r1 ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: for q being Point of X holds f1 . q <> 0 ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = 1 / r1 ) & g is continuous )
defpred S1[ set , set ] means for r1 being Real st f1 . $1 = r1 holds
$2 = 1 / r1;
A3: for x being Element of X ex y being Element of REAL st S1[x,y]
proof
let x be Element of X; ::_thesis: ex y being Element of REAL st S1[x,y]
reconsider r1 = f1 . x as Real by TOPMETR:17;
set r3 = 1 / r1;
for r1 being Real st f1 . x = r1 holds
1 / r1 = 1 / r1 ;
hence ex y being Element of REAL st
for r1 being Real st f1 . x = r1 holds
y = 1 / r1 ; ::_thesis: verum
end;
ex f being Function of the carrier of X,REAL st
for x being Element of X holds S1[x,f . x] from FUNCT_2:sch_3(A3);
then consider f being Function of the carrier of X,REAL such that
A4: for x being Element of X
for r1 being Real st f1 . x = r1 holds
f . x = 1 / r1 ;
reconsider g0 = f as Function of X,R^1 by TOPMETR:17;
for p being Point of X
for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
proof
let p be Point of X; ::_thesis: for V being Subset of R^1 st g0 . p in V & V is open holds
ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
let V be Subset of R^1; ::_thesis: ( g0 . p in V & V is open implies ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) )
reconsider r = g0 . p as Real by TOPMETR:17;
reconsider r1 = f1 . p as Real by TOPMETR:17;
assume ( g0 . p in V & V is open ) ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
then consider r0 being Real such that
A5: r0 > 0 and
A6: ].(r - r0),(r + r0).[ c= V by FRECHET:8;
A7: r = 1 / r1 by A4;
A8: r1 <> 0 by A2;
now__::_thesis:_(_(_r1_>=_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_or_(_r1_<_0_&_ex_W_being_Subset_of_X_st_
(_p_in_W_&_W_is_open_&_g0_.:_W_c=_V_)_)_)
percases ( r1 >= 0 or r1 < 0 ) ;
caseA9: r1 >= 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = (r0 / r) / (r + r0);
reconsider G1 = ].(r1 - ((r0 / r) / (r + r0))),(r1 + ((r0 / r) / (r + r0))).[ as Subset of R^1 by TOPMETR:17;
r0 / r > 0 by A5, A8, A7, A9, XREAL_1:139;
then A10: r1 < r1 + ((r0 / r) / (r + r0)) by A5, A7, A9, XREAL_1:29, XREAL_1:139;
then r1 - ((r0 / r) / (r + r0)) < r1 by XREAL_1:19;
then A11: f1 . p in G1 by A10, XXREAL_1:4;
A12: r / (r + r0) > 0 by A5, A8, A7, A9, XREAL_1:139;
G1 is open by JORDAN6:35;
then consider W1 being Subset of X such that
A13: ( p in W1 & W1 is open ) and
A14: f1 .: W1 c= G1 by A1, A11, Th10;
set W = W1;
r1 - ((r0 / r) / (r + r0)) = (1 / r) - ((r0 / (r + r0)) / r) by A7
.= (1 - (r0 / (r + r0))) / r
.= (((r + r0) / (r + r0)) - (r0 / (r + r0))) / r by A5, A7, A9, XCMPLX_1:60
.= (((r + r0) - r0) / (r + r0)) / r
.= (r / (r + r0)) / r ;
then A15: r1 - ((r0 / r) / (r + r0)) > 0 by A8, A7, A9, A12, XREAL_1:139;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
0 < r0 ^2 by A5, SQUARE_1:12;
then r0 * r < (r0 * r) + ((r0 * r0) + (r0 * r0)) by XREAL_1:29;
then (r0 * r) - ((r0 * r0) + (r0 * r0)) < r0 * r by XREAL_1:19;
then ((r0 * r) - ((r0 * r0) + (r0 * r0))) + (r * r) < (r * r) + (r0 * r) by XREAL_1:8;
then ((r - r0) * ((r + r0) + r0)) / ((r + r0) + r0) < (r * (r + r0)) / ((r + r0) + r0) by A5, A7, A9, XREAL_1:74;
then r - r0 < (r * (r + r0)) / ((r + r0) + r0) by A5, A7, A9, XCMPLX_1:89;
then r - r0 < r / (((r + r0) + r0) / (r + r0)) by XCMPLX_1:77;
then r - r0 < r / (((r + r0) / (r + r0)) + (r0 / (r + r0))) ;
then r - r0 < (r * 1) / (1 + (r0 / (r + r0))) by A5, A7, A9, XCMPLX_1:60;
then A16: r - r0 < 1 / ((1 + (r0 / (r + r0))) / r) by XCMPLX_1:77;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A17: z in dom g0 and
A18: z in W1 and
A19: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A17;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A20: x = 1 / aa1 by A4, A19;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A21: f1 . pz in f1 .: W1 by A18, FUNCT_1:def_6;
then A22: r1 - ((r0 / r) / (r + r0)) < aa1 by A14, XXREAL_1:4;
then A23: 1 / aa1 < 1 / (r1 - ((r0 / r) / (r + r0))) by A15, XREAL_1:88;
aa1 < r1 + ((r0 / r) / (r + r0)) by A14, A21, XXREAL_1:4;
then 1 / ((1 / r) + ((r0 / r) / (r + r0))) < 1 / aa1 by A7, A15, A22, XREAL_1:76;
then A24: r - r0 < 1 / aa1 by A16, XXREAL_0:2;
1 / (r1 - ((r0 / r) / (r + r0))) = 1 / (r1 - ((r0 * (r ")) / (r + r0)))
.= 1 / (r1 - ((r0 * (1 / r)) / (r + r0)))
.= 1 / (r1 - (r0 / ((r + r0) / r1))) by A7, XCMPLX_1:77
.= 1 / ((r1 * 1) - (r1 * (r0 / (r + r0)))) by XCMPLX_1:81
.= 1 / ((1 - (r0 / (r + r0))) * r1)
.= 1 / ((((r + r0) / (r + r0)) - (r0 / (r + r0))) * r1) by A5, A7, A9, XCMPLX_1:60
.= 1 / ((((r + r0) - r0) / (r + r0)) * r1)
.= 1 / (r / ((r + r0) / r1)) by XCMPLX_1:81
.= 1 / ((r * r1) / (r + r0)) by XCMPLX_1:77
.= ((r + r0) / (r * r1)) * 1 by XCMPLX_1:80
.= (r + r0) / 1 by A8, A7, XCMPLX_0:def_7
.= r + r0 ;
hence x in ].(r - r0),(r + r0).[ by A20, A24, A23, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A6, A13, XBOOLE_1:1; ::_thesis: verum
end;
caseA25: r1 < 0 ; ::_thesis: ex W being Subset of X st
( p in W & W is open & g0 .: W c= V )
set r4 = (r0 / (- r)) / ((- r) + r0);
reconsider G1 = ].(r1 - ((r0 / (- r)) / ((- r) + r0))),(r1 + ((r0 / (- r)) / ((- r) + r0))).[ as Subset of R^1 by TOPMETR:17;
A26: G1 is open by JORDAN6:35;
A27: 0 < - r by A7, A25, XREAL_1:58;
then (- r) / ((- r) + r0) > 0 by A5, XREAL_1:139;
then - (r / ((- r) + r0)) > 0 ;
then A28: r / ((- r) + r0) < 0 ;
r0 / (- r) > 0 by A5, A27, XREAL_1:139;
then A29: r1 < r1 + ((r0 / (- r)) / ((- r) + r0)) by A5, A7, A25, XREAL_1:29, XREAL_1:139;
then r1 - ((r0 / (- r)) / ((- r) + r0)) < r1 by XREAL_1:19;
then f1 . p in G1 by A29, XXREAL_1:4;
then consider W1 being Subset of X such that
A30: ( p in W1 & W1 is open ) and
A31: f1 .: W1 c= G1 by A1, A26, Th10;
set W = W1;
r1 * ((- r) * (1 / (- r))) = r1 * 1 by A27, XCMPLX_1:87;
then (- (r * r1)) * (1 / (- r)) = r1 ;
then A32: (- 1) * (1 / (- r)) = r1 by A2, A7, XCMPLX_1:87;
then r1 + ((r0 / (- r)) / ((- r) + r0)) = (- (1 / (- r))) + ((r0 / ((- r) + r0)) / (- r))
.= ((- 1) / (- r)) + ((r0 / ((- r) + r0)) / (- r))
.= ((- 1) + (r0 / ((- r) + r0))) / (- r)
.= ((- (((- r) + r0) / ((- r) + r0))) + (r0 / ((- r) + r0))) / (- r) by A5, A7, A25, XCMPLX_1:60
.= (((- ((- r) + r0)) / ((- r) + r0)) + (r0 / ((- r) + r0))) / (- r)
.= (((r - r0) + r0) / ((- r) + r0)) / (- r)
.= (r / ((- r) + r0)) / (- r) ;
then A33: r1 + ((r0 / (- r)) / ((- r) + r0)) < 0 by A27, A28, XREAL_1:141;
g0 .: W1 c= ].(r - r0),(r + r0).[
proof
0 < r0 ^2 by A5, SQUARE_1:12;
then r0 * (- r) < (r0 * (- r)) + ((r0 * r0) + (r0 * r0)) by XREAL_1:29;
then (r0 * (- r)) - ((r0 * r0) + (r0 * r0)) < r0 * (- r) by XREAL_1:19;
then ((r0 * (- r)) - ((r0 * r0) + (r0 * r0))) + ((- r) * (- r)) < (r0 * (- r)) + ((- r) * (- r)) by XREAL_1:8;
then (((- r) - r0) * (((- r) + r0) + r0)) / (((- r) + r0) + r0) < ((- r) * ((- r) + r0)) / (((- r) + r0) + r0) by A5, A7, A25, XREAL_1:74;
then (- r) - r0 < ((- r) * ((- r) + r0)) / (((- r) + r0) + r0) by A5, A7, A25, XCMPLX_1:89;
then (- r) - r0 < (- r) / ((((- r) + r0) + r0) / ((- r) + r0)) by XCMPLX_1:77;
then (- r) - r0 < (- r) / ((((- r) + r0) / ((- r) + r0)) + (r0 / ((- r) + r0))) ;
then (- r) - r0 < ((- r) * 1) / (1 + (r0 / ((- r) + r0))) by A5, A7, A25, XCMPLX_1:60;
then (- r) - r0 < 1 / ((1 + (r0 / ((- r) + r0))) / (- r)) by XCMPLX_1:77;
then - (r + r0) < 1 / ((1 / (- r)) + ((r0 / (- r)) / ((- r) + r0))) ;
then r + r0 > - (1 / ((1 / (- r)) + ((r0 / (- r)) / ((- r) + r0)))) by XREAL_1:25;
then A34: r + r0 > 1 / (- ((1 / (- r)) + ((r0 / (- r)) / ((- r) + r0)))) by XCMPLX_1:188;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in g0 .: W1 or x in ].(r - r0),(r + r0).[ )
assume x in g0 .: W1 ; ::_thesis: x in ].(r - r0),(r + r0).[
then consider z being set such that
A35: z in dom g0 and
A36: z in W1 and
A37: g0 . z = x by FUNCT_1:def_6;
reconsider pz = z as Point of X by A35;
reconsider aa1 = f1 . pz as Real by TOPMETR:17;
A38: x = 1 / aa1 by A4, A37;
pz in the carrier of X ;
then pz in dom f1 by FUNCT_2:def_1;
then A39: f1 . pz in f1 .: W1 by A36, FUNCT_1:def_6;
then A40: aa1 < r1 + ((r0 / (- r)) / ((- r) + r0)) by A31, XXREAL_1:4;
then A41: 1 / aa1 > 1 / (r1 + ((r0 / (- r)) / ((- r) + r0))) by A33, XREAL_1:87;
r1 - ((r0 / (- r)) / ((- r) + r0)) < aa1 by A31, A39, XXREAL_1:4;
then 1 / ((- (1 / (- r))) - ((r0 / (- r)) / ((- r) + r0))) > 1 / aa1 by A32, A33, A40, XREAL_1:99;
then A42: r + r0 > 1 / aa1 by A34, XXREAL_0:2;
1 / (r1 + ((r0 / (- r)) / ((- r) + r0))) = 1 / (r1 + ((r0 * ((- r) ")) / ((- r) + r0)))
.= 1 / (r1 + ((r0 * (1 / (- r))) / ((- r) + r0)))
.= 1 / (r1 + ((- (r1 * r0)) / ((- r) + r0))) by A32
.= 1 / (r1 + (- ((r1 * r0) / ((- r) + r0))))
.= 1 / (r1 - ((r1 * r0) / ((- r) + r0)))
.= 1 / (r1 - (r0 / (((- r) + r0) / r1))) by XCMPLX_1:77
.= 1 / ((r1 * 1) - (r1 * (r0 / ((- r) + r0)))) by XCMPLX_1:81
.= 1 / (r1 * (1 - (r0 / ((- r) + r0))))
.= 1 / (((((- r) + r0) / ((- r) + r0)) - (r0 / ((- r) + r0))) * r1) by A5, A7, A25, XCMPLX_1:60
.= 1 / (((((- r) + r0) - r0) / (- (r - r0))) * r1)
.= 1 / ((- ((((- r) + r0) - r0) / (r - r0))) * r1) by XCMPLX_1:188
.= 1 / (((((- r) + r0) - r0) / (r - r0)) * (- r1))
.= 1 / ((- r) / ((r - r0) / (- r1))) by XCMPLX_1:81
.= 1 / (((- r) * (- r1)) / (r - r0)) by XCMPLX_1:77
.= ((r - r0) / ((- r) * (- r1))) * 1 by XCMPLX_1:80
.= (r - r0) / ((- r) * ((- r) ")) by A32
.= (r - r0) / 1 by A27, XCMPLX_0:def_7
.= r - r0 ;
hence x in ].(r - r0),(r + r0).[ by A38, A42, A41, XXREAL_1:4; ::_thesis: verum
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) by A6, A30, XBOOLE_1:1; ::_thesis: verum
end;
end;
end;
hence ex W being Subset of X st
( p in W & W is open & g0 .: W c= V ) ; ::_thesis: verum
end;
then A43: g0 is continuous by Th10;
for p being Point of X
for r1 being real number st f1 . p = r1 holds
g0 . p = 1 / r1
proof
let p be Point of X; ::_thesis: for r1 being real number st f1 . p = r1 holds
g0 . p = 1 / r1
let r1 be real number ; ::_thesis: ( f1 . p = r1 implies g0 . p = 1 / r1 )
assume A44: f1 . p = r1 ; ::_thesis: g0 . p = 1 / r1
reconsider r1 = r1 as Element of REAL by XREAL_0:def_1;
g0 . p = 1 / r1 by A4, A44;
hence g0 . p = 1 / r1 ; ::_thesis: verum
end;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1 being real number st f1 . p = r1 holds
g . p = 1 / r1 ) & g is continuous ) by A43; ::_thesis: verum
end;
theorem Th27: :: JGRAPH_2:27
for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: ( f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) ) ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous )
consider g1 being Function of X,R^1 such that
A3: for p being Point of X
for r2 being real number st f2 . p = r2 holds
g1 . p = 1 / r2 and
A4: g1 is continuous by A2, Th26;
consider g2 being Function of X,R^1 such that
A5: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & g1 . p = r2 holds
g2 . p = r1 * r2 and
A6: g2 is continuous by A1, A4, Th25;
for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 / r2
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 / r2
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g2 . p = r1 / r2 )
assume that
A7: f1 . p = r1 and
A8: f2 . p = r2 ; ::_thesis: g2 . p = r1 / r2
g1 . p = 1 / r2 by A3, A8;
then g2 . p = r1 * (1 / r2) by A5, A7
.= r1 / r2 ;
hence g2 . p = r1 / r2 ; ::_thesis: verum
end;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 / r2 ) & g is continuous ) by A6; ::_thesis: verum
end;
theorem Th28: :: JGRAPH_2:28
for X being non empty TopSpace
for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = (r1 / r2) / r2 ) & g is continuous )
proof
let X be non empty TopSpace; ::_thesis: for f1, f2 being Function of X,R^1 st f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) holds
ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = (r1 / r2) / r2 ) & g is continuous )
let f1, f2 be Function of X,R^1; ::_thesis: ( f1 is continuous & f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) implies ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = (r1 / r2) / r2 ) & g is continuous ) )
assume that
A1: f1 is continuous and
A2: ( f2 is continuous & ( for q being Point of X holds f2 . q <> 0 ) ) ; ::_thesis: ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = (r1 / r2) / r2 ) & g is continuous )
consider g2 being Function of X,R^1 such that
A3: for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g2 . p = r1 / r2 and
A4: g2 is continuous by A1, A2, Th27;
consider g3 being Function of X,R^1 such that
A5: for p being Point of X
for r1, r2 being real number st g2 . p = r1 & f2 . p = r2 holds
g3 . p = r1 / r2 and
A6: g3 is continuous by A2, A4, Th27;
for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = (r1 / r2) / r2
proof
let p be Point of X; ::_thesis: for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g3 . p = (r1 / r2) / r2
let r1, r2 be real number ; ::_thesis: ( f1 . p = r1 & f2 . p = r2 implies g3 . p = (r1 / r2) / r2 )
assume that
A7: f1 . p = r1 and
A8: f2 . p = r2 ; ::_thesis: g3 . p = (r1 / r2) / r2
g2 . p = r1 / r2 by A3, A7, A8;
hence g3 . p = (r1 / r2) / r2 by A5, A8; ::_thesis: verum
end;
hence ex g being Function of X,R^1 st
( ( for p being Point of X
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = (r1 / r2) / r2 ) & g is continuous ) by A6; ::_thesis: verum
end;
theorem Th29: :: JGRAPH_2:29
for K0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) holds
f is continuous
proof
reconsider g = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
let K0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),R^1; ::_thesis: ( ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ) implies f is continuous )
A1: ( dom f = the carrier of ((TOP-REAL 2) | K0) & the carrier of (TOP-REAL 2) /\ K0 = K0 ) by FUNCT_2:def_1, XBOOLE_1:28;
A2: g is continuous by JORDAN5A:27;
assume for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj1 . p ; ::_thesis: f is continuous
then A3: for x being set st x in dom f holds
f . x = proj1 . x ;
the carrier of ((TOP-REAL 2) | K0) = [#] ((TOP-REAL 2) | K0)
.= K0 by PRE_TOPC:def_5 ;
then f = g | K0 by A1, A3, Th6, FUNCT_1:46;
hence f is continuous by A2, TOPMETR:7; ::_thesis: verum
end;
theorem Th30: :: JGRAPH_2:30
for K0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) holds
f is continuous
proof
let K0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),R^1 st ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),R^1; ::_thesis: ( ( for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ) implies f is continuous )
A1: ( dom f = the carrier of ((TOP-REAL 2) | K0) & the carrier of (TOP-REAL 2) /\ K0 = K0 ) by FUNCT_2:def_1, XBOOLE_1:28;
assume for p being Point of ((TOP-REAL 2) | K0) holds f . p = proj2 . p ; ::_thesis: f is continuous
then A2: for x being set st x in dom f holds
f . x = proj2 . x ;
the carrier of ((TOP-REAL 2) | K0) = [#] ((TOP-REAL 2) | K0)
.= K0 by PRE_TOPC:def_5 ;
then f = proj2 | K0 by A1, A2, Th7, FUNCT_1:46;
hence f is continuous by JORDAN5A:27; ::_thesis: verum
end;
theorem Th31: :: JGRAPH_2:31
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) holds
f is continuous
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) implies f is continuous )
assume that
A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `1) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; ::_thesis: f is continuous
reconsider g1 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj1 . q
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj1 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def_4;
hence g1 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum
end;
A5: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0
q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by A4
.= q2 `1 by PSCOMP_1:def_5 ;
hence g1 . q <> 0 by A2; ::_thesis: verum
end;
g1 is continuous by A4, Th29;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A6: for q being Point of ((TOP-REAL 2) | K1)
for r2 being real number st g1 . q = r2 holds
g3 . q = 1 / r2 and
A7: g3 is continuous by A5, Th26;
A8: for x being set st x in dom f holds
f . x = g3 . x
proof
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A9: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
x in [#] ((TOP-REAL 2) | K1) by A9;
then x in K1 by PRE_TOPC:def_5;
then reconsider r = x as Point of (TOP-REAL 2) ;
A10: ( g1 . s = proj1 . s & proj1 . r = r `1 ) by A4, PSCOMP_1:def_5;
f . r = 1 / (r `1) by A1, A9;
hence f . x = g3 . x by A6, A10; ::_thesis: verum
end;
dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then dom f = dom g3 by FUNCT_2:def_1;
hence f is continuous by A7, A8, FUNCT_1:2; ::_thesis: verum
end;
theorem Th32: :: JGRAPH_2:32
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) holds
f is continuous
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) implies f is continuous )
assume that
A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = 1 / (p `2) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ; ::_thesis: f is continuous
reconsider g1 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj2 . q
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj2 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def_4;
hence g1 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum
end;
A5: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0
q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj2 . q by A4
.= q2 `2 by PSCOMP_1:def_6 ;
hence g1 . q <> 0 by A2; ::_thesis: verum
end;
g1 is continuous by A4, Th30;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A6: for q being Point of ((TOP-REAL 2) | K1)
for r2 being real number st g1 . q = r2 holds
g3 . q = 1 / r2 and
A7: g3 is continuous by A5, Th26;
A8: for x being set st x in dom f holds
f . x = g3 . x
proof
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A9: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
x in [#] ((TOP-REAL 2) | K1) by A9;
then x in K1 by PRE_TOPC:def_5;
then reconsider r = x as Point of (TOP-REAL 2) ;
A10: ( g1 . s = proj2 . s & proj2 . r = r `2 ) by A4, PSCOMP_1:def_6;
f . r = 1 / (r `2) by A1, A9;
hence f . x = g3 . x by A6, A10; ::_thesis: verum
end;
dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then dom f = dom g3 by FUNCT_2:def_1;
hence f is continuous by A7, A8, FUNCT_1:2; ::_thesis: verum
end;
theorem Th33: :: JGRAPH_2:33
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `2) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) holds
f is continuous
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `2) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `2) / (p `1)) / (p `1) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ) implies f is continuous )
assume that
A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `2) / (p `1)) / (p `1) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0 ; ::_thesis: f is continuous
reconsider g2 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
reconsider g1 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj1 . q
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj1 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def_4;
hence g1 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum
end;
then A5: g1 is continuous by Th29;
A6: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0
q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj1 . q by A4
.= q2 `1 by PSCOMP_1:def_5 ;
hence g1 . q <> 0 by A2; ::_thesis: verum
end;
A7: for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj2 . q
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q = proj2 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def_4;
hence g2 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum
end;
then g2 is continuous by Th30;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A8: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds
g3 . q = (r1 / r2) / r2 and
A9: g3 is continuous by A5, A6, Th28;
A10: for x being set st x in dom f holds
f . x = g3 . x
proof
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A11: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
x in [#] ((TOP-REAL 2) | K1) by A11;
then x in K1 by PRE_TOPC:def_5;
then reconsider r = x as Point of (TOP-REAL 2) ;
A12: ( proj2 . r = r `2 & proj1 . r = r `1 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A13: ( g2 . s = proj2 . s & g1 . s = proj1 . s ) by A7, A4;
f . r = ((r `2) / (r `1)) / (r `1) by A1, A11;
hence f . x = g3 . x by A8, A13, A12; ::_thesis: verum
end;
dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then dom f = dom g3 by FUNCT_2:def_1;
hence f is continuous by A9, A10, FUNCT_1:2; ::_thesis: verum
end;
theorem Th34: :: JGRAPH_2:34
for K1 being non empty Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `1) / (p `2)) / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) holds
f is continuous
proof
let K1 be non empty Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K1),R^1 st ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `1) / (p `2)) / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K1),R^1; ::_thesis: ( ( for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `1) / (p `2)) / (p `2) ) & ( for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ) implies f is continuous )
assume that
A1: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f . p = ((p `1) / (p `2)) / (p `2) and
A2: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0 ; ::_thesis: f is continuous
reconsider g2 = proj1 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
reconsider g1 = proj2 | K1 as Function of ((TOP-REAL 2) | K1),R^1 by TOPMETR:17;
A3: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
A4: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q = proj2 . q
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q = proj2 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj2 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
then q in (dom proj2) /\ K1 by A3, XBOOLE_0:def_4;
hence g1 . q = proj2 . q by FUNCT_1:48; ::_thesis: verum
end;
then A5: g1 is continuous by Th30;
A6: for q being Point of ((TOP-REAL 2) | K1) holds g1 . q <> 0
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g1 . q <> 0
q in the carrier of ((TOP-REAL 2) | K1) ;
then reconsider q2 = q as Point of (TOP-REAL 2) by A3;
g1 . q = proj2 . q by A4
.= q2 `2 by PSCOMP_1:def_6 ;
hence g1 . q <> 0 by A2; ::_thesis: verum
end;
A7: for q being Point of ((TOP-REAL 2) | K1) holds g2 . q = proj1 . q
proof
let q be Point of ((TOP-REAL 2) | K1); ::_thesis: g2 . q = proj1 . q
( q in the carrier of ((TOP-REAL 2) | K1) & dom proj1 = the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1;
then q in (dom proj1) /\ K1 by A3, XBOOLE_0:def_4;
hence g2 . q = proj1 . q by FUNCT_1:48; ::_thesis: verum
end;
then g2 is continuous by Th29;
then consider g3 being Function of ((TOP-REAL 2) | K1),R^1 such that
A8: for q being Point of ((TOP-REAL 2) | K1)
for r1, r2 being real number st g2 . q = r1 & g1 . q = r2 holds
g3 . q = (r1 / r2) / r2 and
A9: g3 is continuous by A5, A6, Th28;
A10: for x being set st x in dom f holds
f . x = g3 . x
proof
let x be set ; ::_thesis: ( x in dom f implies f . x = g3 . x )
assume A11: x in dom f ; ::_thesis: f . x = g3 . x
then reconsider s = x as Point of ((TOP-REAL 2) | K1) ;
x in [#] ((TOP-REAL 2) | K1) by A11;
then x in K1 by PRE_TOPC:def_5;
then reconsider r = x as Point of (TOP-REAL 2) ;
A12: ( proj1 . r = r `1 & proj2 . r = r `2 ) by PSCOMP_1:def_5, PSCOMP_1:def_6;
A13: ( g2 . s = proj1 . s & g1 . s = proj2 . s ) by A7, A4;
f . r = ((r `1) / (r `2)) / (r `2) by A1, A11;
hence f . x = g3 . x by A8, A13, A12; ::_thesis: verum
end;
dom g3 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1;
then dom f = dom g3 by FUNCT_2:def_1;
hence f is continuous by A9, A10, FUNCT_1:2; ::_thesis: verum
end;
theorem Th35: :: JGRAPH_2:35
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0)
for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]| ) holds
f is continuous
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0)
for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]| ) holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: for f1, f2 being Function of ((TOP-REAL 2) | K0),R^1 st f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]| ) holds
f is continuous
let f1, f2 be Function of ((TOP-REAL 2) | K0),R^1; ::_thesis: ( f1 is continuous & f2 is continuous & K0 <> {} & B0 <> {} & ( for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]| ) implies f is continuous )
assume that
A1: f1 is continuous and
A2: f2 is continuous and
A3: K0 <> {} and
A4: B0 <> {} and
A5: for x, y, r, s being real number st |[x,y]| in K0 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]| ; ::_thesis: f is continuous
reconsider B1 = B0 as non empty Subset of (TOP-REAL 2) by A4;
reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) by A3;
reconsider X = (TOP-REAL 2) | K1, Y = (TOP-REAL 2) | B1 as non empty TopSpace ;
reconsider f0 = f as Function of X,Y ;
for r being Point of X
for V being Subset of Y st f0 . r in V & V is open holds
ex W being Subset of X st
( r in W & W is open & f0 .: W c= V )
proof
let r be Point of X; ::_thesis: for V being Subset of Y st f0 . r in V & V is open holds
ex W being Subset of X st
( r in W & W is open & f0 .: W c= V )
let V be Subset of Y; ::_thesis: ( f0 . r in V & V is open implies ex W being Subset of X st
( r in W & W is open & f0 .: W c= V ) )
assume that
A6: f0 . r in V and
A7: V is open ; ::_thesis: ex W being Subset of X st
( r in W & W is open & f0 .: W c= V )
consider V2 being Subset of (TOP-REAL 2) such that
A8: V2 is open and
A9: V = V2 /\ ([#] Y) by A7, TOPS_2:24;
A10: V2 /\ ([#] Y) c= V2 by XBOOLE_1:17;
then f0 . r in V2 by A6, A9;
then reconsider p = f0 . r as Point of (TOP-REAL 2) ;
consider r2 being real number such that
A11: r2 > 0 and
A12: { q where q is Point of (TOP-REAL 2) : ( (p `1) - r2 < q `1 & q `1 < (p `1) + r2 & (p `2) - r2 < q `2 & q `2 < (p `2) + r2 ) } c= V2 by A6, A8, A9, A10, Th11;
reconsider G1 = ].((p `1) - r2),((p `1) + r2).[, G2 = ].((p `2) - r2),((p `2) + r2).[ as Subset of R^1 by TOPMETR:17;
A13: G1 is open by JORDAN6:35;
A14: r in the carrier of X ;
then r in dom f2 by FUNCT_2:def_1;
then A15: f2 . r in rng f2 by FUNCT_1:3;
r in dom f1 by A14, FUNCT_2:def_1;
then f1 . r in rng f1 by FUNCT_1:3;
then reconsider r3 = f1 . r, r4 = f2 . r as Real by A15, TOPMETR:17;
A16: the carrier of X = [#] X
.= K0 by PRE_TOPC:def_5 ;
then r in K0 ;
then reconsider pr = r as Point of (TOP-REAL 2) ;
A17: r = |[(pr `1),(pr `2)]| by EUCLID:53;
then A18: f0 . |[(pr `1),(pr `2)]| = |[r3,r4]| by A5, A16;
A19: p `2 < (p `2) + r2 by A11, XREAL_1:29;
then (p `2) - r2 < p `2 by XREAL_1:19;
then p `2 in ].((p `2) - r2),((p `2) + r2).[ by A19, XXREAL_1:4;
then ( G2 is open & f2 . r in G2 ) by A17, A18, EUCLID:52, JORDAN6:35;
then consider W2 being Subset of X such that
A20: r in W2 and
A21: W2 is open and
A22: f2 .: W2 c= G2 by A2, Th10;
A23: p `1 < (p `1) + r2 by A11, XREAL_1:29;
then (p `1) - r2 < p `1 by XREAL_1:19;
then p `1 in ].((p `1) - r2),((p `1) + r2).[ by A23, XXREAL_1:4;
then f1 . r in ].((p `1) - r2),((p `1) + r2).[ by A17, A18, EUCLID:52;
then consider W1 being Subset of X such that
A24: r in W1 and
A25: W1 is open and
A26: f1 .: W1 c= G1 by A1, A13, Th10;
reconsider W5 = W1 /\ W2 as Subset of X ;
f2 .: W5 c= f2 .: W2 by RELAT_1:123, XBOOLE_1:17;
then A27: f2 .: W5 c= G2 by A22, XBOOLE_1:1;
f1 .: W5 c= f1 .: W1 by RELAT_1:123, XBOOLE_1:17;
then A28: f1 .: W5 c= G1 by A26, XBOOLE_1:1;
A29: f0 .: W5 c= V
proof
let v be set ; :: according to TARSKI:def_3 ::_thesis: ( not v in f0 .: W5 or v in V )
assume A30: v in f0 .: W5 ; ::_thesis: v in V
then reconsider q2 = v as Point of Y ;
consider k being set such that
A31: k in dom f0 and
A32: k in W5 and
A33: q2 = f0 . k by A30, FUNCT_1:def_6;
the carrier of X = [#] X
.= K0 by PRE_TOPC:def_5 ;
then k in K0 by A31;
then reconsider r8 = k as Point of (TOP-REAL 2) ;
A34: dom f0 = the carrier of ((TOP-REAL 2) | K1) by FUNCT_2:def_1
.= [#] ((TOP-REAL 2) | K1)
.= K0 by PRE_TOPC:def_5 ;
then A35: |[(r8 `1),(r8 `2)]| in K0 by A31, EUCLID:53;
A36: dom f2 = the carrier of ((TOP-REAL 2) | K0) by FUNCT_2:def_1
.= [#] ((TOP-REAL 2) | K0)
.= K0 by PRE_TOPC:def_5 ;
then A37: f2 . |[(r8 `1),(r8 `2)]| in rng f2 by A35, FUNCT_1:def_3;
A38: dom f1 = the carrier of ((TOP-REAL 2) | K0) by FUNCT_2:def_1
.= [#] ((TOP-REAL 2) | K0)
.= K0 by PRE_TOPC:def_5 ;
then f1 . |[(r8 `1),(r8 `2)]| in rng f1 by A35, FUNCT_1:def_3;
then reconsider r7 = f1 . |[(r8 `1),(r8 `2)]|, s7 = f2 . |[(r8 `1),(r8 `2)]| as Real by A37, TOPMETR:17;
A39: |[(r8 `1),(r8 `2)]| in W5 by A32, EUCLID:53;
then f1 . |[(r8 `1),(r8 `2)]| in f1 .: W5 by A35, A38, FUNCT_1:def_6;
then A40: ( (p `1) - r2 < r7 & r7 < (p `1) + r2 ) by A28, XXREAL_1:4;
f2 . |[(r8 `1),(r8 `2)]| in f2 .: W5 by A35, A36, A39, FUNCT_1:def_6;
then A41: ( (p `2) - r2 < s7 & s7 < (p `2) + r2 ) by A27, XXREAL_1:4;
k = |[(r8 `1),(r8 `2)]| by EUCLID:53;
then A42: v = |[r7,s7]| by A5, A31, A33, A34;
( |[r7,s7]| `1 = r7 & |[r7,s7]| `2 = s7 ) by EUCLID:52;
then ( q2 in [#] Y & v in { q3 where q3 is Point of (TOP-REAL 2) : ( (p `1) - r2 < q3 `1 & q3 `1 < (p `1) + r2 & (p `2) - r2 < q3 `2 & q3 `2 < (p `2) + r2 ) } ) by A42, A40, A41;
hence v in V by A9, A12, XBOOLE_0:def_4; ::_thesis: verum
end;
r in W5 by A24, A20, XBOOLE_0:def_4;
hence ex W being Subset of X st
( r in W & W is open & f0 .: W c= V ) by A25, A21, A29; ::_thesis: verum
end;
hence f is continuous by Th10; ::_thesis: verum
end;
theorem Th36: :: JGRAPH_2:36
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } implies f is continuous )
A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19;
assume A2: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous
A3: K0 c= B0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 )
assume A4: x in K0 ; ::_thesis: x in B0
then ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in B0 by A2, A4, XBOOLE_0:def_5; ::_thesis: verum
end;
( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by Th5;
then A5: 1.REAL 2 in K0 by A2, A1;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
A6: K1 c= NonZero (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) )
assume A7: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2)
then ex p8 being Point of (TOP-REAL 2) st
( p8 = z & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence z in NonZero (TOP-REAL 2) by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
A8: dom (Out_In_Sq | K1) c= dom (proj2 * (Out_In_Sq | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj2 * (Out_In_Sq | K1)) )
assume A9: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj2 * (Out_In_Sq | K1))
then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61;
then x in dom Out_In_Sq by XBOOLE_0:def_4;
then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3;
then A10: ( dom proj2 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5;
(Out_In_Sq | K1) . x = Out_In_Sq . x by A9, FUNCT_1:47;
hence x in dom (proj2 * (Out_In_Sq | K1)) by A9, A10, FUNCT_1:11; ::_thesis: verum
end;
A11: rng (proj2 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17;
A12: NonZero (TOP-REAL 2) <> {} by Th9;
A13: dom (Out_In_Sq | K1) c= dom (proj1 * (Out_In_Sq | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj1 * (Out_In_Sq | K1)) )
assume A14: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj1 * (Out_In_Sq | K1))
then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61;
then x in dom Out_In_Sq by XBOOLE_0:def_4;
then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3;
then A15: ( dom proj1 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5;
(Out_In_Sq | K1) . x = Out_In_Sq . x by A14, FUNCT_1:47;
hence x in dom (proj1 * (Out_In_Sq | K1)) by A14, A15, FUNCT_1:11; ::_thesis: verum
end;
A16: rng (proj1 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj1 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25;
then dom (proj1 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A13, XBOOLE_0:def_10
.= (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1
.= K1 by A6, XBOOLE_1:28
.= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K1) ;
then reconsider g1 = proj1 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A16, FUNCT_2:2;
for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g1 . p = 1 / (p `1)
proof
A17: K1 c= NonZero (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) )
assume A18: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2)
then ex p8 being Point of (TOP-REAL 2) st
( p8 = z & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence z in NonZero (TOP-REAL 2) by A18, XBOOLE_0:def_5; ::_thesis: verum
end;
A19: NonZero (TOP-REAL 2) <> {} by Th9;
A20: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A19, FUNCT_2:def_1
.= K1 by A17, XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = 1 / (p `1) )
A21: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
assume A22: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = 1 / (p `1)
then ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A21;
then A23: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1;
(Out_In_Sq | K1) . p = Out_In_Sq . p by A22, A21, FUNCT_1:49;
then g1 . p = proj1 . |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by A22, A20, A21, A23, FUNCT_1:13
.= |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 by PSCOMP_1:def_5
.= 1 / (p `1) by EUCLID:52 ;
hence g1 . p = 1 / (p `1) ; ::_thesis: verum
end;
then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that
A24: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f1 . p = 1 / (p `1) ;
dom (proj2 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25;
then dom (proj2 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A8, XBOOLE_0:def_10
.= (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1
.= K1 by A6, XBOOLE_1:28
.= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K1) ;
then reconsider g2 = proj2 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A11, FUNCT_2:2;
for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g2 . p = ((p `2) / (p `1)) / (p `1)
proof
A25: NonZero (TOP-REAL 2) <> {} by Th9;
A26: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A25, FUNCT_2:def_1
.= K1 by A6, XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = ((p `2) / (p `1)) / (p `1) )
A27: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
assume A28: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = ((p `2) / (p `1)) / (p `1)
then ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A27;
then A29: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1;
(Out_In_Sq | K1) . p = Out_In_Sq . p by A28, A27, FUNCT_1:49;
then g2 . p = proj2 . |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by A28, A26, A27, A29, FUNCT_1:13
.= |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 by PSCOMP_1:def_6
.= ((p `2) / (p `1)) / (p `1) by EUCLID:52 ;
hence g2 . p = ((p `2) / (p `1)) / (p `1) ; ::_thesis: verum
end;
then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that
A30: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f2 . p = ((p `2) / (p `1)) / (p `1) ;
A31: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `1 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `1 <> 0 )
A32: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: q `1 <> 0
then A33: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A32;
now__::_thesis:_not_q_`1_=_0
assume A34: q `1 = 0 ; ::_thesis: contradiction
then q `2 = 0 by A33;
hence contradiction by A33, A34, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence q `1 <> 0 ; ::_thesis: verum
end;
then A35: f1 is continuous by A24, Th31;
A36: for x, y, r, s being real number st |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f . |[x,y]| = |[r,s]|
proof
let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in K1 & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f . |[x,y]| = |[r,s]| )
assume that
A37: |[x,y]| in K1 and
A38: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[r,s]|
set p99 = |[x,y]|;
A39: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
then A40: f1 . |[x,y]| = 1 / (|[x,y]| `1) by A24, A37;
A41: ex p3 being Point of (TOP-REAL 2) st
( |[x,y]| = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A37;
then ( ( ( |[x,y]| `2 <= |[x,y]| `1 & - (|[x,y]| `1) <= |[x,y]| `2 ) or ( |[x,y]| `2 >= |[x,y]| `1 & |[x,y]| `2 <= - (|[x,y]| `1) ) ) implies Out_In_Sq . |[x,y]| = |[(1 / (|[x,y]| `1)),(((|[x,y]| `2) / (|[x,y]| `1)) / (|[x,y]| `1))]| ) by Def1;
then (Out_In_Sq | K0) . |[x,y]| = |[(1 / (|[x,y]| `1)),(((|[x,y]| `2) / (|[x,y]| `1)) / (|[x,y]| `1))]| by A37, A41, FUNCT_1:49
.= |[r,s]| by A30, A37, A38, A39, A40 ;
hence f . |[x,y]| = |[r,s]| by A2; ::_thesis: verum
end;
f2 is continuous by A31, A30, Th33;
hence f is continuous by A5, A3, A35, A36, Th35; ::_thesis: verum
end;
theorem Th37: :: JGRAPH_2:37
for K0, B0 being Subset of (TOP-REAL 2)
for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
proof
let K0, B0 be Subset of (TOP-REAL 2); ::_thesis: for f being Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
f is continuous
let f be Function of ((TOP-REAL 2) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } implies f is continuous )
A1: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19;
assume A2: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: f is continuous
A3: K0 c= B0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 )
assume A4: x in K0 ; ::_thesis: x in B0
then ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in B0 by A2, A4, XBOOLE_0:def_5; ::_thesis: verum
end;
( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by Th5;
then A5: 1.REAL 2 in K0 by A2, A1;
then reconsider K1 = K0 as non empty Subset of (TOP-REAL 2) ;
A6: K1 c= NonZero (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) )
assume A7: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2)
then ex p8 being Point of (TOP-REAL 2) st
( p8 = z & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence z in NonZero (TOP-REAL 2) by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
A8: dom (Out_In_Sq | K1) c= dom (proj1 * (Out_In_Sq | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj1 * (Out_In_Sq | K1)) )
assume A9: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj1 * (Out_In_Sq | K1))
then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61;
then x in dom Out_In_Sq by XBOOLE_0:def_4;
then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3;
then A10: ( dom proj1 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5;
(Out_In_Sq | K1) . x = Out_In_Sq . x by A9, FUNCT_1:47;
hence x in dom (proj1 * (Out_In_Sq | K1)) by A9, A10, FUNCT_1:11; ::_thesis: verum
end;
A11: rng (proj1 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17;
A12: NonZero (TOP-REAL 2) <> {} by Th9;
A13: dom (Out_In_Sq | K1) c= dom (proj2 * (Out_In_Sq | K1))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dom (Out_In_Sq | K1) or x in dom (proj2 * (Out_In_Sq | K1)) )
assume A14: x in dom (Out_In_Sq | K1) ; ::_thesis: x in dom (proj2 * (Out_In_Sq | K1))
then x in (dom Out_In_Sq) /\ K1 by RELAT_1:61;
then x in dom Out_In_Sq by XBOOLE_0:def_4;
then Out_In_Sq . x in rng Out_In_Sq by FUNCT_1:3;
then A15: ( dom proj2 = the carrier of (TOP-REAL 2) & Out_In_Sq . x in the carrier of (TOP-REAL 2) ) by FUNCT_2:def_1, XBOOLE_0:def_5;
(Out_In_Sq | K1) . x = Out_In_Sq . x by A14, FUNCT_1:47;
hence x in dom (proj2 * (Out_In_Sq | K1)) by A14, A15, FUNCT_1:11; ::_thesis: verum
end;
A16: rng (proj2 * (Out_In_Sq | K1)) c= the carrier of R^1 by TOPMETR:17;
dom (proj2 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25;
then dom (proj2 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A13, XBOOLE_0:def_10
.= (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1
.= K1 by A6, XBOOLE_1:28
.= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K1) ;
then reconsider g1 = proj2 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A16, FUNCT_2:2;
for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g1 . p = 1 / (p `2)
proof
A17: K1 c= NonZero (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) )
assume A18: z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2)
then ex p8 being Point of (TOP-REAL 2) st
( p8 = z & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A2;
then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence z in NonZero (TOP-REAL 2) by A18, XBOOLE_0:def_5; ::_thesis: verum
end;
A19: NonZero (TOP-REAL 2) <> {} by Th9;
A20: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A19, FUNCT_2:def_1
.= K1 by A17, XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g1 . p = 1 / (p `2) )
A21: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
assume A22: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g1 . p = 1 / (p `2)
then ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A21;
then A23: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14;
(Out_In_Sq | K1) . p = Out_In_Sq . p by A22, A21, FUNCT_1:49;
then g1 . p = proj2 . |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A22, A20, A21, A23, FUNCT_1:13
.= |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 by PSCOMP_1:def_6
.= 1 / (p `2) by EUCLID:52 ;
hence g1 . p = 1 / (p `2) ; ::_thesis: verum
end;
then consider f1 being Function of ((TOP-REAL 2) | K1),R^1 such that
A24: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f1 . p = 1 / (p `2) ;
dom (proj1 * (Out_In_Sq | K1)) c= dom (Out_In_Sq | K1) by RELAT_1:25;
then dom (proj1 * (Out_In_Sq | K1)) = dom (Out_In_Sq | K1) by A8, XBOOLE_0:def_10
.= (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A12, FUNCT_2:def_1
.= K1 by A6, XBOOLE_1:28
.= [#] ((TOP-REAL 2) | K1) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K1) ;
then reconsider g2 = proj1 * (Out_In_Sq | K1) as Function of ((TOP-REAL 2) | K1),R^1 by A11, FUNCT_2:2;
for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
g2 . p = ((p `1) / (p `2)) / (p `2)
proof
A25: NonZero (TOP-REAL 2) <> {} by Th9;
A26: dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A25, FUNCT_2:def_1
.= K1 by A6, XBOOLE_1:28 ;
let p be Point of (TOP-REAL 2); ::_thesis: ( p in the carrier of ((TOP-REAL 2) | K1) implies g2 . p = ((p `1) / (p `2)) / (p `2) )
A27: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
assume A28: p in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: g2 . p = ((p `1) / (p `2)) / (p `2)
then ex p3 being Point of (TOP-REAL 2) st
( p = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A27;
then A29: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14;
(Out_In_Sq | K1) . p = Out_In_Sq . p by A28, A27, FUNCT_1:49;
then g2 . p = proj1 . |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by A28, A26, A27, A29, FUNCT_1:13
.= |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 by PSCOMP_1:def_5
.= ((p `1) / (p `2)) / (p `2) by EUCLID:52 ;
hence g2 . p = ((p `1) / (p `2)) / (p `2) ; ::_thesis: verum
end;
then consider f2 being Function of ((TOP-REAL 2) | K1),R^1 such that
A30: for p being Point of (TOP-REAL 2) st p in the carrier of ((TOP-REAL 2) | K1) holds
f2 . p = ((p `1) / (p `2)) / (p `2) ;
A31: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K1) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K1) implies q `2 <> 0 )
A32: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
assume q in the carrier of ((TOP-REAL 2) | K1) ; ::_thesis: q `2 <> 0
then A33: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A32;
now__::_thesis:_not_q_`2_=_0
assume A34: q `2 = 0 ; ::_thesis: contradiction
then q `1 = 0 by A33;
hence contradiction by A33, A34, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence q `2 <> 0 ; ::_thesis: verum
end;
then A35: f1 is continuous by A24, Th32;
A36: for x, y, s, r being real number st |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| holds
f . |[x,y]| = |[s,r]|
proof
let x, y, s, r be real number ; ::_thesis: ( |[x,y]| in K1 & s = f2 . |[x,y]| & r = f1 . |[x,y]| implies f . |[x,y]| = |[s,r]| )
assume that
A37: |[x,y]| in K1 and
A38: ( s = f2 . |[x,y]| & r = f1 . |[x,y]| ) ; ::_thesis: f . |[x,y]| = |[s,r]|
set p99 = |[x,y]|;
A39: ex p3 being Point of (TOP-REAL 2) st
( |[x,y]| = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A2, A37;
A40: the carrier of ((TOP-REAL 2) | K1) = [#] ((TOP-REAL 2) | K1)
.= K1 by PRE_TOPC:def_5 ;
then A41: f1 . |[x,y]| = 1 / (|[x,y]| `2) by A24, A37;
(Out_In_Sq | K0) . |[x,y]| = Out_In_Sq . |[x,y]| by A37, FUNCT_1:49
.= |[(((|[x,y]| `1) / (|[x,y]| `2)) / (|[x,y]| `2)),(1 / (|[x,y]| `2))]| by A39, Th14
.= |[s,r]| by A30, A37, A38, A40, A41 ;
hence f . |[x,y]| = |[s,r]| by A2; ::_thesis: verum
end;
f2 is continuous by A31, A30, Th34;
hence f is continuous by A5, A3, A35, A36, Th35; ::_thesis: verum
end;
scheme :: JGRAPH_2:sch 1
TopSubset{ P1[ set ] } :
{ p where p is Point of (TOP-REAL 2) : P1[p] } is Subset of (TOP-REAL 2)
proof
{ p where p is Point of (TOP-REAL 2) : P1[p] } c= the carrier of (TOP-REAL 2)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : P1[p] } or x in the carrier of (TOP-REAL 2) )
assume x in { p where p is Point of (TOP-REAL 2) : P1[p] } ; ::_thesis: x in the carrier of (TOP-REAL 2)
then ex p being Point of (TOP-REAL 2) st
( p = x & P1[p] ) ;
hence x in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
hence { p where p is Point of (TOP-REAL 2) : P1[p] } is Subset of (TOP-REAL 2) ; ::_thesis: verum
end;
scheme :: JGRAPH_2:sch 2
TopCompl{ P1[ set ], F1() -> Subset of (TOP-REAL 2) } :
F1() ` = { p where p is Point of (TOP-REAL 2) : P1[p] }
provided
A1: F1() = { p where p is Point of (TOP-REAL 2) : P1[p] }
proof
thus F1() ` c= { p where p is Point of (TOP-REAL 2) : P1[p] } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of (TOP-REAL 2) : P1[p] } c= F1() `
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F1() ` or x in { p where p is Point of (TOP-REAL 2) : P1[p] } )
assume A2: x in F1() ` ; ::_thesis: x in { p where p is Point of (TOP-REAL 2) : P1[p] }
then reconsider qx = x as Point of (TOP-REAL 2) ;
x in the carrier of (TOP-REAL 2) \ F1() by A2, SUBSET_1:def_4;
then not x in F1() by XBOOLE_0:def_5;
then P1[qx] by A1;
hence x in { p where p is Point of (TOP-REAL 2) : P1[p] } ; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : P1[p] } or x in F1() ` )
assume x in { p7 where p7 is Point of (TOP-REAL 2) : P1[p7] } ; ::_thesis: x in F1() `
then A3: ex p7 being Point of (TOP-REAL 2) st
( p7 = x & P1[p7] ) ;
then for q7 being Point of (TOP-REAL 2) holds
( not x = q7 or not P1[q7] ) ;
then not x in F1() by A1;
then x in the carrier of (TOP-REAL 2) \ F1() by A3, XBOOLE_0:def_5;
hence x in F1() ` by SUBSET_1:def_4; ::_thesis: verum
end;
Lm2: now__::_thesis:_for_p01,_p02,_px1,_px2_being_real_number_st_(p01_-_px1)_-_(p02_-_px2)_<=_((p01_-_p02)_/_4)_-_(-_((p01_-_p02)_/_4))_holds_
(p01_-_p02)_/_2_<=_px1_-_px2
let p01, p02, px1, px2 be real number ; ::_thesis: ( (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) implies (p01 - p02) / 2 <= px1 - px2 )
set r0 = (p01 - p02) / 4;
assume (p01 - px1) - (p02 - px2) <= ((p01 - p02) / 4) - (- ((p01 - p02) / 4)) ; ::_thesis: (p01 - p02) / 2 <= px1 - px2
then (p01 - p02) - (px1 - px2) <= ((p01 - p02) / 4) + ((p01 - p02) / 4) ;
then p01 - p02 <= (px1 - px2) + (((p01 - p02) / 4) + ((p01 - p02) / 4)) by XREAL_1:20;
then (p01 - p02) - ((p01 - p02) / 2) <= px1 - px2 by XREAL_1:20;
hence (p01 - p02) / 2 <= px1 - px2 ; ::_thesis: verum
end;
scheme :: JGRAPH_2:sch 3
ClosedSubset{ F1( Point of (TOP-REAL 2)) -> real number , F2( Point of (TOP-REAL 2)) -> real number } :
{ p where p is Point of (TOP-REAL 2) : F1(p) <= F2(p) } is closed Subset of (TOP-REAL 2)
provided
A1: for p, q being Point of (TOP-REAL 2) holds
( F1((p - q)) = F1(p) - F1(q) & F2((p - q)) = F2(p) - F2(q) ) and
A2: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (F1((p - q)) ^2) + (F2((p - q)) ^2)
proof
defpred S1[ Point of (TOP-REAL 2)] means F1($1) <= F2($1);
reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
A3: TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) = TopSpaceMetr (Euclid 2) by EUCLID:def_8;
then reconsider K21 = K2 ` as Subset of (TopSpaceMetr (Euclid 2)) ;
A4: K2 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } ;
A5: K2 ` = { p7 where p7 is Point of (TOP-REAL 2) : not S1[p7] } from JGRAPH_2:sch_2(A4);
for p being Point of (Euclid 2) st p in K21 holds
ex r being real number st
( r > 0 & Ball (p,r) c= K21 )
proof
let p be Point of (Euclid 2); ::_thesis: ( p in K21 implies ex r being real number st
( r > 0 & Ball (p,r) c= K21 ) )
assume A6: p in K21 ; ::_thesis: ex r being real number st
( r > 0 & Ball (p,r) c= K21 )
then reconsider p0 = p as Point of (TOP-REAL 2) ;
set r0 = (F1(p0) - F2(p0)) / 4;
ex p7 being Point of (TOP-REAL 2) st
( p0 = p7 & F1(p7) > F2(p7) ) by A5, A6;
then A7: F1(p0) - F2(p0) > 0 by XREAL_1:50;
then A8: (F1(p0) - F2(p0)) / 2 > 0 by XREAL_1:139;
Ball (p,((F1(p0) - F2(p0)) / 4)) c= K2 `
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Ball (p,((F1(p0) - F2(p0)) / 4)) or x in K2 ` )
A9: Ball (p,((F1(p0) - F2(p0)) / 4)) = { q where q is Element of (Euclid 2) : dist (p,q) < (F1(p0) - F2(p0)) / 4 } by METRIC_1:17;
assume A10: x in Ball (p,((F1(p0) - F2(p0)) / 4)) ; ::_thesis: x in K2 `
then reconsider px = x as Point of (TOP-REAL 2) by TOPREAL3:8;
consider q being Element of (Euclid 2) such that
A11: q = x and
A12: dist (p,q) < (F1(p0) - F2(p0)) / 4 by A10, A9;
dist (p,q) = |.(p0 - px).| by A11, JGRAPH_1:28;
then A13: |.(p0 - px).| ^2 <= ((F1(p0) - F2(p0)) / 4) ^2 by A12, SQUARE_1:15;
A14: F1((p0 - px)) = F1(p0) - F1(px) by A1;
A15: |.(p0 - px).| ^2 = (F1((p0 - px)) ^2) + (F2((p0 - px)) ^2) by A2;
F2((p0 - px)) ^2 >= 0 by XREAL_1:63;
then 0 + (F1((p0 - px)) ^2) <= (F2((p0 - px)) ^2) + (F1((p0 - px)) ^2) by XREAL_1:7;
then F1((p0 - px)) ^2 <= ((F1(p0) - F2(p0)) / 4) ^2 by A15, A13, XXREAL_0:2;
then A16: F1(p0) - F1(px) <= (F1(p0) - F2(p0)) / 4 by A7, A14, SQUARE_1:47;
A17: F2((p0 - px)) = F2(p0) - F2(px) by A1;
F1((p0 - px)) ^2 >= 0 by XREAL_1:63;
then (F2((p0 - px)) ^2) + 0 <= (F2((p0 - px)) ^2) + (F1((p0 - px)) ^2) by XREAL_1:7;
then F2((p0 - px)) ^2 <= ((F1(p0) - F2(p0)) / 4) ^2 by A15, A13, XXREAL_0:2;
then - ((F1(p0) - F2(p0)) / 4) <= F2(p0) - F2(px) by A7, A17, SQUARE_1:47;
then (F1(p0) - F1(px)) - (F2(p0) - F2(px)) <= ((F1(p0) - F2(p0)) / 4) - (- ((F1(p0) - F2(p0)) / 4)) by A16, XREAL_1:13;
then F1(px) - F2(px) > 0 by A8, Lm2;
then F1(px) > F2(px) by XREAL_1:47;
hence x in K2 ` by A5; ::_thesis: verum
end;
hence ex r being real number st
( r > 0 & Ball (p,r) c= K21 ) by A7, XREAL_1:139; ::_thesis: verum
end;
then K21 is open by TOPMETR:15;
then K2 ` is open by A3, PRE_TOPC:30;
hence { p where p is Point of (TOP-REAL 2) : F1(p) <= F2(p) } is closed Subset of (TOP-REAL 2) by TOPS_1:3; ::_thesis: verum
end;
deffunc H1( Point of (TOP-REAL 2)) -> Element of REAL = $1 `1 ;
deffunc H2( Point of (TOP-REAL 2)) -> Element of REAL = $1 `2 ;
Lm3: for p, q being Point of (TOP-REAL 2) holds
( H1(p - q) = H1(p) - H1(q) & H2(p - q) = H2(p) - H2(q) )
by TOPREAL3:3;
Lm4: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H1(p - q) ^2) + (H2(p - q) ^2)
by JGRAPH_1:29;
Lm5: { p7 where p7 is Point of (TOP-REAL 2) : H1(p7) <= H2(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch_3(Lm3, Lm4);
Lm6: for p, q being Point of (TOP-REAL 2) holds
( H2(p - q) = H2(p) - H2(q) & H1(p - q) = H1(p) - H1(q) )
by TOPREAL3:3;
Lm7: for p, q being Point of (TOP-REAL 2) holds |.(p - q).| ^2 = (H2(p - q) ^2) + (H1(p - q) ^2)
by JGRAPH_1:29;
Lm8: { p7 where p7 is Point of (TOP-REAL 2) : H2(p7) <= H1(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch_3(Lm6, Lm7);
deffunc H3( Point of (TOP-REAL 2)) -> Element of REAL = - ($1 `1);
deffunc H4( Point of (TOP-REAL 2)) -> Element of REAL = - ($1 `2);
Lm9: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_
(_H3(p_-_q)_=_H3(p)_-_H3(q)_&_H2(p_-_q)_=_H2(p)_-_H2(q)_)
let p, q be Point of (TOP-REAL 2); ::_thesis: ( H3(p - q) = H3(p) - H3(q) & H2(p - q) = H2(p) - H2(q) )
thus H3(p - q) = - ((p `1) - (q `1)) by TOPREAL3:3
.= H3(p) - H3(q) ; ::_thesis: H2(p - q) = H2(p) - H2(q)
thus H2(p - q) = H2(p) - H2(q) by TOPREAL3:3; ::_thesis: verum
end;
Lm10: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H3(p_-_q)_^2)_+_(H2(p_-_q)_^2)
let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H3(p - q) ^2) + (H2(p - q) ^2)
H3(p - q) ^2 = H1(p - q) ^2 ;
hence |.(p - q).| ^2 = (H3(p - q) ^2) + (H2(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum
end;
Lm11: { p7 where p7 is Point of (TOP-REAL 2) : H3(p7) <= H2(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch_3(Lm9, Lm10);
Lm12: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_
(_H2(p_-_q)_=_H2(p)_-_H2(q)_&_H3(p_-_q)_=_H3(p)_-_H3(q)_)
let p, q be Point of (TOP-REAL 2); ::_thesis: ( H2(p - q) = H2(p) - H2(q) & H3(p - q) = H3(p) - H3(q) )
thus H2(p - q) = H2(p) - H2(q) by TOPREAL3:3; ::_thesis: H3(p - q) = H3(p) - H3(q)
thus H3(p - q) = - ((p `1) - (q `1)) by TOPREAL3:3
.= H3(p) - H3(q) ; ::_thesis: verum
end;
Lm13: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H2(p_-_q)_^2)_+_(H3(p_-_q)_^2)
let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H2(p - q) ^2) + (H3(p - q) ^2)
(- ((p - q) `1)) ^2 = ((p - q) `1) ^2 ;
hence |.(p - q).| ^2 = (H2(p - q) ^2) + (H3(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum
end;
Lm14: { p7 where p7 is Point of (TOP-REAL 2) : H2(p7) <= H3(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch_3(Lm12, Lm13);
Lm15: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_
(_H4(p_-_q)_=_H4(p)_-_H4(q)_&_H1(p_-_q)_=_H1(p)_-_H1(q)_)
let p, q be Point of (TOP-REAL 2); ::_thesis: ( H4(p - q) = H4(p) - H4(q) & H1(p - q) = H1(p) - H1(q) )
thus H4(p - q) = - ((p `2) - (q `2)) by TOPREAL3:3
.= H4(p) - H4(q) ; ::_thesis: H1(p - q) = H1(p) - H1(q)
thus H1(p - q) = H1(p) - H1(q) by TOPREAL3:3; ::_thesis: verum
end;
Lm16: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H4(p_-_q)_^2)_+_(H1(p_-_q)_^2)
let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H4(p - q) ^2) + (H1(p - q) ^2)
(- ((p - q) `2)) ^2 = ((p - q) `2) ^2 ;
hence |.(p - q).| ^2 = (H4(p - q) ^2) + (H1(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum
end;
Lm17: { p7 where p7 is Point of (TOP-REAL 2) : H4(p7) <= H1(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch_3(Lm15, Lm16);
Lm18: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_
(_H1(p_-_q)_=_H1(p)_-_H1(q)_&_H4(p_-_q)_=_H4(p)_-_H4(q)_)
let p, q be Point of (TOP-REAL 2); ::_thesis: ( H1(p - q) = H1(p) - H1(q) & H4(p - q) = H4(p) - H4(q) )
thus H1(p - q) = H1(p) - H1(q) by TOPREAL3:3; ::_thesis: H4(p - q) = H4(p) - H4(q)
thus H4(p - q) = - ((p `2) - (q `2)) by TOPREAL3:3
.= H4(p) - H4(q) ; ::_thesis: verum
end;
Lm19: now__::_thesis:_for_p,_q_being_Point_of_(TOP-REAL_2)_holds_|.(p_-_q).|_^2_=_(H1(p_-_q)_^2)_+_(H4(p_-_q)_^2)
let p, q be Point of (TOP-REAL 2); ::_thesis: |.(p - q).| ^2 = (H1(p - q) ^2) + (H4(p - q) ^2)
H4(p - q) ^2 = H2(p - q) ^2 ;
hence |.(p - q).| ^2 = (H1(p - q) ^2) + (H4(p - q) ^2) by JGRAPH_1:29; ::_thesis: verum
end;
Lm20: { p7 where p7 is Point of (TOP-REAL 2) : H1(p7) <= H4(p7) } is closed Subset of (TOP-REAL 2)
from JGRAPH_2:sch_3(Lm18, Lm19);
theorem Th38: :: JGRAPH_2:38
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
proof
reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1) } as closed Subset of (TOP-REAL 2) by Lm14;
reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm5;
reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm11;
reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm8;
let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } implies ( f is continuous & K0 is closed ) )
defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `2 <= $1 `1 & - ($1 `1) <= $1 `2 ) or ( $1 `2 >= $1 `1 & $1 `2 <= - ($1 `1) ) );
the carrier of ((TOP-REAL 2) | B0) = [#] ((TOP-REAL 2) | B0)
.= B0 by PRE_TOPC:def_5 ;
then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
assume A1: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: ( f is continuous & K0 is closed )
K0 c= B0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 )
assume x in K0 ; ::_thesis: x in B0
then A2: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A1;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum
end;
then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7;
reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
A4: K1 /\ B0 c= K0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 /\ B0 or x in K0 )
assume A5: x in K1 /\ B0 ; ::_thesis: x in K0
then x in B0 by XBOOLE_0:def_4;
then not x in {(0. (TOP-REAL 2))} by A1, XBOOLE_0:def_5;
then A6: not x = 0. (TOP-REAL 2) by TARSKI:def_1;
x in K1 by A5, XBOOLE_0:def_4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & ( ( p7 `2 <= p7 `1 & - (p7 `1) <= p7 `2 ) or ( p7 `2 >= p7 `1 & p7 `2 <= - (p7 `1) ) ) ) ;
hence x in K0 by A1, A6; ::_thesis: verum
end;
A7: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A8: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1
now__::_thesis:_(_(_x_in_K2_/\_K3_&_x_in_K1_)_or_(_x_in_K4_/\_K5_&_x_in_K1_)_)
percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A8, XBOOLE_0:def_3;
caseA9: x in K2 /\ K3 ; ::_thesis: x in K1
then x in K3 by XBOOLE_0:def_4;
then A10: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `1) <= p8 `2 ) ;
x in K2 by A9, XBOOLE_0:def_4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 <= p7 `1 ) ;
hence x in K1 by A10; ::_thesis: verum
end;
caseA11: x in K4 /\ K5 ; ::_thesis: x in K1
then x in K5 by XBOOLE_0:def_4;
then A12: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `2 <= - (p8 `1) ) ;
x in K4 by A11, XBOOLE_0:def_4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `2 >= p7 `1 ) ;
hence x in K1 by A12; ::_thesis: verum
end;
end;
end;
hence x in K1 ; ::_thesis: verum
end;
K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) ) ;
then ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4;
hence x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum
end;
then K1 = (K2 /\ K3) \/ (K4 /\ K5) by A7, XBOOLE_0:def_10;
then A13: K1 is closed ;
K0 c= K1 /\ B0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in K1 /\ B0 )
assume x in K0 ; ::_thesis: x in K1 /\ B0
then A14: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) by A1;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
then A15: x in B0 by A1, A14, XBOOLE_0:def_5;
x in K1 by A14;
hence x in K1 /\ B0 by A15, XBOOLE_0:def_4; ::_thesis: verum
end;
then K0 = K1 /\ B0 by A4, XBOOLE_0:def_10
.= K1 /\ ([#] ((TOP-REAL 2) | B0)) by PRE_TOPC:def_5 ;
hence ( f is continuous & K0 is closed ) by A1, A3, A13, Th36, PRE_TOPC:13; ::_thesis: verum
end;
theorem Th39: :: JGRAPH_2:39
for B0 being Subset of (TOP-REAL 2)
for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
proof
reconsider K5 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2) } as closed Subset of (TOP-REAL 2) by Lm20;
reconsider K4 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm8;
reconsider K3 = { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1 } as closed Subset of (TOP-REAL 2) by Lm17;
reconsider K2 = { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } as closed Subset of (TOP-REAL 2) by Lm5;
let B0 be Subset of (TOP-REAL 2); ::_thesis: for K0 being Subset of ((TOP-REAL 2) | B0)
for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let K0 be Subset of ((TOP-REAL 2) | B0); ::_thesis: for f being Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0) st f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } holds
( f is continuous & K0 is closed )
let f be Function of (((TOP-REAL 2) | B0) | K0),((TOP-REAL 2) | B0); ::_thesis: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } implies ( f is continuous & K0 is closed ) )
defpred S1[ Point of (TOP-REAL 2)] means ( ( $1 `1 <= $1 `2 & - ($1 `2) <= $1 `1 ) or ( $1 `1 >= $1 `2 & $1 `1 <= - ($1 `2) ) );
the carrier of ((TOP-REAL 2) | B0) = [#] ((TOP-REAL 2) | B0)
.= B0 by PRE_TOPC:def_5 ;
then reconsider K1 = K0 as Subset of (TOP-REAL 2) by XBOOLE_1:1;
assume A1: ( f = Out_In_Sq | K0 & B0 = NonZero (TOP-REAL 2) & K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: ( f is continuous & K0 is closed )
K0 c= B0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in B0 )
assume x in K0 ; ::_thesis: x in B0
then A2: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A1;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in B0 by A1, A2, XBOOLE_0:def_5; ::_thesis: verum
end;
then A3: ((TOP-REAL 2) | B0) | K0 = (TOP-REAL 2) | K1 by PRE_TOPC:7;
reconsider K1 = { p7 where p7 is Point of (TOP-REAL 2) : S1[p7] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
A4: K1 /\ B0 c= K0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 /\ B0 or x in K0 )
assume A5: x in K1 /\ B0 ; ::_thesis: x in K0
then x in B0 by XBOOLE_0:def_4;
then not x in {(0. (TOP-REAL 2))} by A1, XBOOLE_0:def_5;
then A6: not x = 0. (TOP-REAL 2) by TARSKI:def_1;
x in K1 by A5, XBOOLE_0:def_4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & ( ( p7 `1 <= p7 `2 & - (p7 `2) <= p7 `1 ) or ( p7 `1 >= p7 `2 & p7 `1 <= - (p7 `2) ) ) ) ;
hence x in K0 by A1, A6; ::_thesis: verum
end;
A7: (K2 /\ K3) \/ (K4 /\ K5) c= K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (K2 /\ K3) \/ (K4 /\ K5) or x in K1 )
assume A8: x in (K2 /\ K3) \/ (K4 /\ K5) ; ::_thesis: x in K1
now__::_thesis:_(_(_x_in_K2_/\_K3_&_x_in_K1_)_or_(_x_in_K4_/\_K5_&_x_in_K1_)_)
percases ( x in K2 /\ K3 or x in K4 /\ K5 ) by A8, XBOOLE_0:def_3;
caseA9: x in K2 /\ K3 ; ::_thesis: x in K1
then x in K3 by XBOOLE_0:def_4;
then A10: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & - (p8 `2) <= p8 `1 ) ;
x in K2 by A9, XBOOLE_0:def_4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `1 <= p7 `2 ) ;
hence x in K1 by A10; ::_thesis: verum
end;
caseA11: x in K4 /\ K5 ; ::_thesis: x in K1
then x in K5 by XBOOLE_0:def_4;
then A12: ex p8 being Point of (TOP-REAL 2) st
( p8 = x & p8 `1 <= - (p8 `2) ) ;
x in K4 by A11, XBOOLE_0:def_4;
then ex p7 being Point of (TOP-REAL 2) st
( p7 = x & p7 `1 >= p7 `2 ) ;
hence x in K1 by A12; ::_thesis: verum
end;
end;
end;
hence x in K1 ; ::_thesis: verum
end;
K1 c= (K2 /\ K3) \/ (K4 /\ K5)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K1 or x in (K2 /\ K3) \/ (K4 /\ K5) )
assume x in K1 ; ::_thesis: x in (K2 /\ K3) \/ (K4 /\ K5)
then ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) ) ;
then ( ( x in K2 & x in K3 ) or ( x in K4 & x in K5 ) ) ;
then ( x in K2 /\ K3 or x in K4 /\ K5 ) by XBOOLE_0:def_4;
hence x in (K2 /\ K3) \/ (K4 /\ K5) by XBOOLE_0:def_3; ::_thesis: verum
end;
then K1 = (K2 /\ K3) \/ (K4 /\ K5) by A7, XBOOLE_0:def_10;
then A13: K1 is closed ;
K0 c= K1 /\ B0
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in K0 or x in K1 /\ B0 )
assume x in K0 ; ::_thesis: x in K1 /\ B0
then A14: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) by A1;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
then A15: x in B0 by A1, A14, XBOOLE_0:def_5;
x in K1 by A14;
hence x in K1 /\ B0 by A15, XBOOLE_0:def_4; ::_thesis: verum
end;
then K0 = K1 /\ B0 by A4, XBOOLE_0:def_10
.= K1 /\ ([#] ((TOP-REAL 2) | B0)) by PRE_TOPC:def_5 ;
hence ( f is continuous & K0 is closed ) by A1, A3, A13, Th37, PRE_TOPC:13; ::_thesis: verum
end;
theorem Th40: :: JGRAPH_2:40
for D being non empty Subset of (TOP-REAL 2) st D ` = {(0. (TOP-REAL 2))} holds
ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Out_In_Sq & h is continuous )
proof
set Y1 = |[(- 1),1]|;
reconsider B0 = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ;
let D be non empty Subset of (TOP-REAL 2); ::_thesis: ( D ` = {(0. (TOP-REAL 2))} implies ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Out_In_Sq & h is continuous ) )
assume A1: D ` = {(0. (TOP-REAL 2))} ; ::_thesis: ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Out_In_Sq & h is continuous )
then A2: D = B0 `
.= NonZero (TOP-REAL 2) by SUBSET_1:def_4 ;
A3: { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } or x in the carrier of ((TOP-REAL 2) | D) )
assume x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D)
then A4: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) ;
now__::_thesis:_x_in_D
assume not x in D ; ::_thesis: contradiction
then x in the carrier of (TOP-REAL 2) \ D by A4, XBOOLE_0:def_5;
then x in D ` by SUBSET_1:def_4;
hence contradiction by A1, A4, TARSKI:def_1; ::_thesis: verum
end;
then x in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5;
hence x in the carrier of ((TOP-REAL 2) | D) ; ::_thesis: verum
end;
A5: NonZero (TOP-REAL 2) <> {} by Th9;
A6: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19;
( ( (1.REAL 2) `2 <= (1.REAL 2) `1 & - ((1.REAL 2) `1) <= (1.REAL 2) `2 ) or ( (1.REAL 2) `2 >= (1.REAL 2) `1 & (1.REAL 2) `2 <= - ((1.REAL 2) `1) ) ) by Th5;
then 1.REAL 2 in { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } by A6;
then reconsider K0 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `2 <= p `1 & - (p `1) <= p `2 ) or ( p `2 >= p `1 & p `2 <= - (p `1) ) ) & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A3;
A7: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5
.= the carrier of (((TOP-REAL 2) | D) | K0) ;
A8: { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } c= the carrier of ((TOP-REAL 2) | D)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } or x in the carrier of ((TOP-REAL 2) | D) )
assume x in { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in the carrier of ((TOP-REAL 2) | D)
then A9: ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) ;
now__::_thesis:_x_in_D
assume not x in D ; ::_thesis: contradiction
then x in the carrier of (TOP-REAL 2) \ D by A9, XBOOLE_0:def_5;
then x in D ` by SUBSET_1:def_4;
hence contradiction by A1, A9, TARSKI:def_1; ::_thesis: verum
end;
then x in [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5;
hence x in the carrier of ((TOP-REAL 2) | D) ; ::_thesis: verum
end;
( |[(- 1),1]| `1 = - 1 & |[(- 1),1]| `2 = 1 ) by EUCLID:52;
then |[(- 1),1]| in { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } by Th3;
then reconsider K1 = { p where p is Point of (TOP-REAL 2) : ( ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) & p <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A8;
A10: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5
.= the carrier of (((TOP-REAL 2) | D) | K1) ;
A11: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= D by PRE_TOPC:def_5 ;
A12: rng (Out_In_Sq | K1) c= the carrier of (((TOP-REAL 2) | D) | K1)
proof
reconsider K10 = K1 as Subset of (TOP-REAL 2) by A11, XBOOLE_1:1;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K1) or y in the carrier of (((TOP-REAL 2) | D) | K1) )
A13: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K10) holds
q `2 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K10) implies q `2 <> 0 )
A14: the carrier of ((TOP-REAL 2) | K10) = [#] ((TOP-REAL 2) | K10)
.= K1 by PRE_TOPC:def_5 ;
assume q in the carrier of ((TOP-REAL 2) | K10) ; ::_thesis: q `2 <> 0
then A15: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `1 <= p3 `2 & - (p3 `2) <= p3 `1 ) or ( p3 `1 >= p3 `2 & p3 `1 <= - (p3 `2) ) ) & p3 <> 0. (TOP-REAL 2) ) by A14;
now__::_thesis:_not_q_`2_=_0
assume A16: q `2 = 0 ; ::_thesis: contradiction
then q `1 = 0 by A15;
hence contradiction by A15, A16, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence q `2 <> 0 ; ::_thesis: verum
end;
assume y in rng (Out_In_Sq | K1) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K1)
then consider x being set such that
A17: x in dom (Out_In_Sq | K1) and
A18: y = (Out_In_Sq | K1) . x by FUNCT_1:def_3;
A19: x in (dom Out_In_Sq) /\ K1 by A17, RELAT_1:61;
then A20: x in K1 by XBOOLE_0:def_4;
K1 c= the carrier of (TOP-REAL 2) by A11, XBOOLE_1:1;
then reconsider p = x as Point of (TOP-REAL 2) by A20;
A21: Out_In_Sq . p = y by A18, A20, FUNCT_1:49;
set p9 = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]|;
K10 = [#] ((TOP-REAL 2) | K10) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K10) ;
then A22: p in the carrier of ((TOP-REAL 2) | K10) by A19, XBOOLE_0:def_4;
A23: now__::_thesis:_not_|[(((p_`1)_/_(p_`2))_/_(p_`2)),(1_/_(p_`2))]|_=_0._(TOP-REAL_2)
assume |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 0 by EUCLID:52, EUCLID:54;
then 0 * (p `2) = (1 / (p `2)) * (p `2) by EUCLID:52;
hence contradiction by A22, A13, XCMPLX_1:87; ::_thesis: verum
end;
A24: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) by A20;
then A25: Out_In_Sq . p = |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| by Th14;
now__::_thesis:_(_(_p_`2_>=_0_&_y_in_K1_)_or_(_p_`2_<_0_&_y_in_K1_)_)
percases ( p `2 >= 0 or p `2 < 0 ) ;
caseA26: p `2 >= 0 ; ::_thesis: y in K1
then ( ( (p `1) / (p `2) <= (p `2) / (p `2) & (- (1 * (p `2))) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A24, XREAL_1:72;
then A27: ( ( (p `1) / (p `2) <= 1 & ((- 1) * (p `2)) / (p `2) <= (p `1) / (p `2) ) or ( p `1 >= p `2 & p `1 <= - (1 * (p `2)) ) ) by A22, A13, XCMPLX_1:60;
then A28: ( ( (p `1) / (p `2) <= 1 & - 1 <= (p `1) / (p `2) ) or ( (p `1) / (p `2) >= 1 & (p `1) / (p `2) <= ((- 1) * (p `2)) / (p `2) ) ) by A22, A13, A26, XCMPLX_1:89, XREAL_1:72;
A29: ( not (p `1) / (p `2) >= 1 or not (p `1) / (p `2) <= - 1 ) ;
( ( (p `1) / (p `2) <= 1 & - 1 <= (p `1) / (p `2) ) or ( (p `1) / (p `2) >= (p `2) / (p `2) & p `1 <= - (1 * (p `2)) ) ) by A22, A13, A26, A27, XCMPLX_1:89;
then (- 1) / (p `2) <= ((p `1) / (p `2)) / (p `2) by A22, A13, A26, A29, XCMPLX_1:60, XREAL_1:72;
then A30: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A26, A28, XREAL_1:72;
( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52;
hence y in K1 by A21, A23, A25, A30; ::_thesis: verum
end;
caseA31: p `2 < 0 ; ::_thesis: y in K1
then ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= (p `2) / (p `2) & (p `1) / (p `2) >= (- (1 * (p `2))) / (p `2) ) ) by A24, XREAL_1:73;
then A32: ( ( p `1 <= p `2 & - (1 * (p `2)) <= p `1 ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= ((- 1) * (p `2)) / (p `2) ) ) by A31, XCMPLX_1:60;
then ( ( (p `1) / (p `2) >= 1 & ((- 1) * (p `2)) / (p `2) >= (p `1) / (p `2) ) or ( (p `1) / (p `2) <= 1 & (p `1) / (p `2) >= - 1 ) ) by A31, XCMPLX_1:89;
then (- 1) / (p `2) >= ((p `1) / (p `2)) / (p `2) by A31, XREAL_1:73;
then A33: ( ( ((p `1) / (p `2)) / (p `2) <= 1 / (p `2) & - (1 / (p `2)) <= ((p `1) / (p `2)) / (p `2) ) or ( ((p `1) / (p `2)) / (p `2) >= 1 / (p `2) & ((p `1) / (p `2)) / (p `2) <= - (1 / (p `2)) ) ) by A31, A32, XREAL_1:73;
( |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `2 = 1 / (p `2) & |[(((p `1) / (p `2)) / (p `2)),(1 / (p `2))]| `1 = ((p `1) / (p `2)) / (p `2) ) by EUCLID:52;
hence y in K1 by A21, A23, A25, A33; ::_thesis: verum
end;
end;
end;
then y in [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5;
hence y in the carrier of (((TOP-REAL 2) | D) | K1) ; ::_thesis: verum
end;
A34: D c= K0 \/ K1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K0 \/ K1 )
assume A35: x in D ; ::_thesis: x in K0 \/ K1
then reconsider px = x as Point of (TOP-REAL 2) ;
not x in {(0. (TOP-REAL 2))} by A2, A35, XBOOLE_0:def_5;
then ( ( ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) or ( ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1, XREAL_1:26;
then ( x in K0 or x in K1 ) ;
hence x in K0 \/ K1 by XBOOLE_0:def_3; ::_thesis: verum
end;
A36: NonZero (TOP-REAL 2) <> {} by Th9;
A37: K1 c= NonZero (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K1 or z in NonZero (TOP-REAL 2) )
assume z in K1 ; ::_thesis: z in NonZero (TOP-REAL 2)
then A38: ex p8 being Point of (TOP-REAL 2) st
( p8 = z & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence z in NonZero (TOP-REAL 2) by A38, XBOOLE_0:def_5; ::_thesis: verum
end;
A39: the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= NonZero (TOP-REAL 2) by A2, PRE_TOPC:def_5 ;
A40: rng (Out_In_Sq | K0) c= the carrier of (((TOP-REAL 2) | D) | K0)
proof
reconsider K00 = K0 as Subset of (TOP-REAL 2) by A11, XBOOLE_1:1;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (Out_In_Sq | K0) or y in the carrier of (((TOP-REAL 2) | D) | K0) )
A41: for q being Point of (TOP-REAL 2) st q in the carrier of ((TOP-REAL 2) | K00) holds
q `1 <> 0
proof
let q be Point of (TOP-REAL 2); ::_thesis: ( q in the carrier of ((TOP-REAL 2) | K00) implies q `1 <> 0 )
A42: the carrier of ((TOP-REAL 2) | K00) = [#] ((TOP-REAL 2) | K00)
.= K0 by PRE_TOPC:def_5 ;
assume q in the carrier of ((TOP-REAL 2) | K00) ; ::_thesis: q `1 <> 0
then A43: ex p3 being Point of (TOP-REAL 2) st
( q = p3 & ( ( p3 `2 <= p3 `1 & - (p3 `1) <= p3 `2 ) or ( p3 `2 >= p3 `1 & p3 `2 <= - (p3 `1) ) ) & p3 <> 0. (TOP-REAL 2) ) by A42;
now__::_thesis:_not_q_`1_=_0
assume A44: q `1 = 0 ; ::_thesis: contradiction
then q `2 = 0 by A43;
hence contradiction by A43, A44, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
hence q `1 <> 0 ; ::_thesis: verum
end;
assume y in rng (Out_In_Sq | K0) ; ::_thesis: y in the carrier of (((TOP-REAL 2) | D) | K0)
then consider x being set such that
A45: x in dom (Out_In_Sq | K0) and
A46: y = (Out_In_Sq | K0) . x by FUNCT_1:def_3;
A47: x in (dom Out_In_Sq) /\ K0 by A45, RELAT_1:61;
then A48: x in K0 by XBOOLE_0:def_4;
K0 c= the carrier of (TOP-REAL 2) by A11, XBOOLE_1:1;
then reconsider p = x as Point of (TOP-REAL 2) by A48;
A49: Out_In_Sq . p = y by A46, A48, FUNCT_1:49;
set p9 = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]|;
K00 = [#] ((TOP-REAL 2) | K00) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | K00) ;
then A50: p in the carrier of ((TOP-REAL 2) | K00) by A47, XBOOLE_0:def_4;
A51: |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) by EUCLID:52;
A52: now__::_thesis:_not_|[(1_/_(p_`1)),(((p_`2)_/_(p_`1))_/_(p_`1))]|_=_0._(TOP-REAL_2)
assume |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| = 0. (TOP-REAL 2) ; ::_thesis: contradiction
then 0 * (p `1) = (1 / (p `1)) * (p `1) by A51, EUCLID:52, EUCLID:54;
hence contradiction by A50, A41, XCMPLX_1:87; ::_thesis: verum
end;
A53: ex px being Point of (TOP-REAL 2) st
( x = px & ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) by A48;
then A54: Out_In_Sq . p = |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| by Def1;
A55: p `1 <> 0 by A50, A41;
now__::_thesis:_(_(_p_`1_>=_0_&_y_in_K0_)_or_(_p_`1_<_0_&_y_in_K0_)_)
percases ( p `1 >= 0 or p `1 < 0 ) ;
caseA56: p `1 >= 0 ; ::_thesis: y in K0
A57: ( not (p `2) / (p `1) >= 1 or not (p `2) / (p `1) <= - 1 ) ;
( ( (p `2) / (p `1) <= (p `1) / (p `1) & (- (1 * (p `1))) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A53, A56, XREAL_1:72;
then A58: ( ( (p `2) / (p `1) <= 1 & ((- 1) * (p `1)) / (p `1) <= (p `2) / (p `1) ) or ( p `2 >= p `1 & p `2 <= - (1 * (p `1)) ) ) by A50, A41, XCMPLX_1:60;
then ( ( (p `2) / (p `1) <= 1 & - 1 <= (p `2) / (p `1) ) or ( (p `2) / (p `1) >= (p `1) / (p `1) & p `2 <= - (1 * (p `1)) ) ) by A50, A41, A56, XCMPLX_1:89;
then (- 1) / (p `1) <= ((p `2) / (p `1)) / (p `1) by A50, A41, A56, A57, XCMPLX_1:60, XREAL_1:72;
then A59: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A55, A56, A58, XREAL_1:72;
( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52;
hence y in K0 by A49, A52, A54, A59; ::_thesis: verum
end;
caseA60: p `1 < 0 ; ::_thesis: y in K0
A61: ( not (p `2) / (p `1) >= 1 or not (p `2) / (p `1) <= - 1 ) ;
( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= (p `1) / (p `1) & (p `2) / (p `1) >= (- (1 * (p `1))) / (p `1) ) ) by A53, A60, XREAL_1:73;
then A62: ( ( p `2 <= p `1 & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= ((- 1) * (p `1)) / (p `1) ) ) by A60, XCMPLX_1:60;
then ( ( (p `2) / (p `1) >= (p `1) / (p `1) & - (1 * (p `1)) <= p `2 ) or ( (p `2) / (p `1) <= 1 & (p `2) / (p `1) >= - 1 ) ) by A60, XCMPLX_1:89;
then (- 1) / (p `1) >= ((p `2) / (p `1)) / (p `1) by A60, A61, XCMPLX_1:60, XREAL_1:73;
then A63: ( ( ((p `2) / (p `1)) / (p `1) <= 1 / (p `1) & - (1 / (p `1)) <= ((p `2) / (p `1)) / (p `1) ) or ( ((p `2) / (p `1)) / (p `1) >= 1 / (p `1) & ((p `2) / (p `1)) / (p `1) <= - (1 / (p `1)) ) ) by A60, A62, XREAL_1:73;
( |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `1 = 1 / (p `1) & |[(1 / (p `1)),(((p `2) / (p `1)) / (p `1))]| `2 = ((p `2) / (p `1)) / (p `1) ) by EUCLID:52;
hence y in K0 by A49, A52, A54, A63; ::_thesis: verum
end;
end;
end;
then y in [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
hence y in the carrier of (((TOP-REAL 2) | D) | K0) ; ::_thesis: verum
end;
A64: K0 c= NonZero (TOP-REAL 2)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in K0 or z in NonZero (TOP-REAL 2) )
assume z in K0 ; ::_thesis: z in NonZero (TOP-REAL 2)
then A65: ex p8 being Point of (TOP-REAL 2) st
( p8 = z & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
then not z in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence z in NonZero (TOP-REAL 2) by A65, XBOOLE_0:def_5; ::_thesis: verum
end;
dom (Out_In_Sq | K0) = (dom Out_In_Sq) /\ K0 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K0 by A5, FUNCT_2:def_1
.= K0 by A64, XBOOLE_1:28 ;
then reconsider f = Out_In_Sq | K0 as Function of (((TOP-REAL 2) | D) | K0),((TOP-REAL 2) | D) by A7, A40, FUNCT_2:2, XBOOLE_1:1;
A66: K1 = [#] (((TOP-REAL 2) | D) | K1) by PRE_TOPC:def_5;
dom (Out_In_Sq | K1) = (dom Out_In_Sq) /\ K1 by RELAT_1:61
.= (NonZero (TOP-REAL 2)) /\ K1 by A36, FUNCT_2:def_1
.= K1 by A37, XBOOLE_1:28 ;
then reconsider g = Out_In_Sq | K1 as Function of (((TOP-REAL 2) | D) | K1),((TOP-REAL 2) | D) by A10, A12, FUNCT_2:2, XBOOLE_1:1;
A67: dom g = K1 by A10, FUNCT_2:def_1;
g = Out_In_Sq | K1 ;
then A68: K1 is closed by A2, Th39;
A69: K0 = [#] (((TOP-REAL 2) | D) | K0) by PRE_TOPC:def_5;
A70: for x being set st x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) holds
f . x = g . x
proof
let x be set ; ::_thesis: ( x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) implies f . x = g . x )
assume A71: x in ([#] (((TOP-REAL 2) | D) | K0)) /\ ([#] (((TOP-REAL 2) | D) | K1)) ; ::_thesis: f . x = g . x
then x in K0 by A69, XBOOLE_0:def_4;
then f . x = Out_In_Sq . x by FUNCT_1:49;
hence f . x = g . x by A66, A71, FUNCT_1:49; ::_thesis: verum
end;
f = Out_In_Sq | K0 ;
then A72: K0 is closed by A2, Th38;
A73: dom f = K0 by A7, FUNCT_2:def_1;
D = [#] ((TOP-REAL 2) | D) by PRE_TOPC:def_5;
then A74: ([#] (((TOP-REAL 2) | D) | K0)) \/ ([#] (((TOP-REAL 2) | D) | K1)) = [#] ((TOP-REAL 2) | D) by A69, A66, A34, XBOOLE_0:def_10;
A75: ( f is continuous & g is continuous ) by A2, Th38, Th39;
then consider h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) such that
A76: h = f +* g and
h is continuous by A69, A66, A74, A72, A68, A70, Th1;
( K0 = [#] (((TOP-REAL 2) | D) | K0) & K1 = [#] (((TOP-REAL 2) | D) | K1) ) by PRE_TOPC:def_5;
then A77: f tolerates g by A70, A73, A67, PARTFUN1:def_4;
A78: for x being set st x in dom h holds
h . x = Out_In_Sq . x
proof
let x be set ; ::_thesis: ( x in dom h implies h . x = Out_In_Sq . x )
assume A79: x in dom h ; ::_thesis: h . x = Out_In_Sq . x
then reconsider p = x as Point of (TOP-REAL 2) by A39, XBOOLE_0:def_5;
not x in {(0. (TOP-REAL 2))} by A39, A79, XBOOLE_0:def_5;
then A80: x <> 0. (TOP-REAL 2) by TARSKI:def_1;
now__::_thesis:_(_(_x_in_K0_&_h_._x_=_Out_In_Sq_._x_)_or_(_not_x_in_K0_&_h_._x_=_Out_In_Sq_._x_)_)
percases ( x in K0 or not x in K0 ) ;
caseA81: x in K0 ; ::_thesis: h . x = Out_In_Sq . x
h . p = (g +* f) . p by A76, A77, FUNCT_4:34
.= f . p by A73, A81, FUNCT_4:13 ;
hence h . x = Out_In_Sq . x by A81, FUNCT_1:49; ::_thesis: verum
end;
case not x in K0 ; ::_thesis: h . x = Out_In_Sq . x
then ( not ( p `2 <= p `1 & - (p `1) <= p `2 ) & not ( p `2 >= p `1 & p `2 <= - (p `1) ) ) by A80;
then ( ( p `1 <= p `2 & - (p `2) <= p `1 ) or ( p `1 >= p `2 & p `1 <= - (p `2) ) ) by XREAL_1:26;
then A82: x in K1 by A80;
then Out_In_Sq . p = g . p by FUNCT_1:49;
hence h . x = Out_In_Sq . x by A76, A67, A82, FUNCT_4:13; ::_thesis: verum
end;
end;
end;
hence h . x = Out_In_Sq . x ; ::_thesis: verum
end;
( dom h = the carrier of ((TOP-REAL 2) | D) & dom Out_In_Sq = the carrier of ((TOP-REAL 2) | D) ) by A39, FUNCT_2:def_1;
then f +* g = Out_In_Sq by A76, A78, FUNCT_1:2;
hence ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Out_In_Sq & h is continuous ) by A69, A66, A74, A72, A75, A68, A70, Th1; ::_thesis: verum
end;
theorem Th41: :: JGRAPH_2:41
for B, K0, Kb being Subset of (TOP-REAL 2) st B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } holds
ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st
( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds
f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds
f . s = s ) )
proof
set K1a = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ;
set K0a = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ;
let B, K0, Kb be Subset of (TOP-REAL 2); ::_thesis: ( B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } implies ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st
( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds
f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds
f . s = s ) ) )
assume A1: ( B = {(0. (TOP-REAL 2))} & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & Kb = { q where q is Point of (TOP-REAL 2) : ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) } ) ; ::_thesis: ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st
( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds
f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds
f . s = s ) )
then reconsider D = B ` as non empty Subset of (TOP-REAL 2) by Th9;
A2: D ` = {(0. (TOP-REAL 2))} by A1;
A3: B ` = NonZero (TOP-REAL 2) by A1, SUBSET_1:def_4;
A4: for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not Out_In_Sq . t in K0 \/ Kb
proof
let t be Point of (TOP-REAL 2); ::_thesis: ( t in K0 & t <> 0. (TOP-REAL 2) implies not Out_In_Sq . t in K0 \/ Kb )
assume that
A5: t in K0 and
A6: t <> 0. (TOP-REAL 2) ; ::_thesis: not Out_In_Sq . t in K0 \/ Kb
A7: ex p3 being Point of (TOP-REAL 2) st
( p3 = t & - 1 < p3 `1 & p3 `1 < 1 & - 1 < p3 `2 & p3 `2 < 1 ) by A1, A5;
now__::_thesis:_not_Out_In_Sq_._t_in_K0_\/_Kb
assume A8: Out_In_Sq . t in K0 \/ Kb ; ::_thesis: contradiction
now__::_thesis:_(_(_Out_In_Sq_._t_in_K0_&_contradiction_)_or_(_Out_In_Sq_._t_in_Kb_&_contradiction_)_)
percases ( Out_In_Sq . t in K0 or Out_In_Sq . t in Kb ) by A8, XBOOLE_0:def_3;
case Out_In_Sq . t in K0 ; ::_thesis: contradiction
then consider p4 being Point of (TOP-REAL 2) such that
A9: p4 = Out_In_Sq . t and
A10: - 1 < p4 `1 and
A11: p4 `1 < 1 and
A12: - 1 < p4 `2 and
A13: p4 `2 < 1 by A1;
now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_contradiction_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_contradiction_)_)
percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ;
caseA14: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction
then Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A6, Def1;
then A15: p4 `1 = 1 / (t `1) by A9, EUCLID:52;
now__::_thesis:_(_(_t_`1_>=_0_&_contradiction_)_or_(_t_`1_<_0_&_contradiction_)_)
percases ( t `1 >= 0 or t `1 < 0 ) ;
caseA16: t `1 >= 0 ; ::_thesis: contradiction
now__::_thesis:_(_(_t_`1_>_0_&_contradiction_)_or_(_t_`1_=_0_&_contradiction_)_)
percases ( t `1 > 0 or t `1 = 0 ) by A16;
caseA17: t `1 > 0 ; ::_thesis: contradiction
then (1 / (t `1)) * (t `1) < 1 * (t `1) by A11, A15, XREAL_1:68;
hence contradiction by A7, A17, XCMPLX_1:87; ::_thesis: verum
end;
caseA18: t `1 = 0 ; ::_thesis: contradiction
then t `2 = 0 by A14;
hence contradiction by A6, A18, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA19: t `1 < 0 ; ::_thesis: contradiction
then (- 1) * (t `1) > (1 / (t `1)) * (t `1) by A10, A15, XREAL_1:69;
then (- 1) * (t `1) > 1 by A19, XCMPLX_1:87;
then - (- (t `1)) <= - 1 by XREAL_1:24;
hence contradiction by A7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA20: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction
then Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A6, Def1;
then A21: p4 `2 = 1 / (t `2) by A9, EUCLID:52;
now__::_thesis:_(_(_t_`2_>=_0_&_contradiction_)_or_(_t_`2_<_0_&_contradiction_)_)
percases ( t `2 >= 0 or t `2 < 0 ) ;
caseA22: t `2 >= 0 ; ::_thesis: contradiction
now__::_thesis:_(_(_t_`2_>_0_&_contradiction_)_or_(_t_`2_=_0_&_contradiction_)_)
percases ( t `2 > 0 or t `2 = 0 ) by A22;
caseA23: t `2 > 0 ; ::_thesis: contradiction
then (1 / (t `2)) * (t `2) < 1 * (t `2) by A13, A21, XREAL_1:68;
hence contradiction by A7, A23, XCMPLX_1:87; ::_thesis: verum
end;
case t `2 = 0 ; ::_thesis: contradiction
hence contradiction by A20; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA24: t `2 < 0 ; ::_thesis: contradiction
then (- 1) * (t `2) > (1 / (t `2)) * (t `2) by A12, A21, XREAL_1:69;
then (- 1) * (t `2) > 1 by A24, XCMPLX_1:87;
then - (- (t `2)) <= - 1 by XREAL_1:24;
hence contradiction by A7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case Out_In_Sq . t in Kb ; ::_thesis: contradiction
then consider p4 being Point of (TOP-REAL 2) such that
A25: p4 = Out_In_Sq . t and
A26: ( ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ) by A1;
now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_contradiction_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_contradiction_)_)
percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ;
caseA27: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction
then A28: Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A6, Def1;
then A29: p4 `1 = 1 / (t `1) by A25, EUCLID:52;
now__::_thesis:_(_(_-_1_=_p4_`1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_or_(_p4_`1_=_1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_or_(_-_1_=_p4_`2_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_or_(_1_=_p4_`2_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_)
percases ( ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ) by A26;
case ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction
then A30: (t `1) * ((t `1) ") = - (t `1) by A29;
now__::_thesis:_(_(_t_`1_<>_0_&_contradiction_)_or_(_t_`1_=_0_&_contradiction_)_)
percases ( t `1 <> 0 or t `1 = 0 ) ;
case t `1 <> 0 ; ::_thesis: contradiction
then - (t `1) = 1 by A30, XCMPLX_0:def_7;
hence contradiction by A7; ::_thesis: verum
end;
caseA31: t `1 = 0 ; ::_thesis: contradiction
then t `2 = 0 by A27;
hence contradiction by A6, A31, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction
then A32: (t `1) * ((t `1) ") = t `1 by A29;
now__::_thesis:_(_(_t_`1_<>_0_&_contradiction_)_or_(_t_`1_=_0_&_contradiction_)_)
percases ( t `1 <> 0 or t `1 = 0 ) ;
case t `1 <> 0 ; ::_thesis: contradiction
hence contradiction by A7, A32, XCMPLX_0:def_7; ::_thesis: verum
end;
caseA33: t `1 = 0 ; ::_thesis: contradiction
then t `2 = 0 by A27;
hence contradiction by A6, A33, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA34: ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction
reconsider K01 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th17;
A35: the carrier of (((TOP-REAL 2) | D) | K01) = [#] (((TOP-REAL 2) | D) | K01)
.= K01 by PRE_TOPC:def_5 ;
A36: dom (Out_In_Sq | K01) = (dom Out_In_Sq) /\ K01 by RELAT_1:61
.= D /\ K01 by A3, FUNCT_2:def_1
.= ([#] ((TOP-REAL 2) | D)) /\ K01 by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | D) /\ K01
.= K01 by XBOOLE_1:28 ;
t in K01 by A6, A27;
then A37: (Out_In_Sq | K01) . t in rng (Out_In_Sq | K01) by A36, FUNCT_1:3;
rng (Out_In_Sq | K01) c= the carrier of (((TOP-REAL 2) | D) | K01) by Th15;
then A38: (Out_In_Sq | K01) . t in the carrier of (((TOP-REAL 2) | D) | K01) by A37;
t in K01 by A6, A27;
then Out_In_Sq . t in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } by A38, A35, FUNCT_1:49;
then A39: ex p5 being Point of (TOP-REAL 2) st
( p5 = p4 & ( ( p5 `2 <= p5 `1 & - (p5 `1) <= p5 `2 ) or ( p5 `2 >= p5 `1 & p5 `2 <= - (p5 `1) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25;
now__::_thesis:_(_(_p4_`1_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`1_&_contradiction_)_)
percases ( p4 `1 >= 1 or - 1 >= p4 `1 ) by A34, A39, XREAL_1:24;
caseA40: p4 `1 >= 1 ; ::_thesis: contradiction
then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * 1 by A29, A34, XXREAL_0:1
.= (t `2) * 1 by A29, A34, A40, XXREAL_0:1
.= t `2 ;
hence contradiction by A7, A25, A28, A34, EUCLID:52; ::_thesis: verum
end;
caseA41: - 1 >= p4 `1 ; ::_thesis: contradiction
then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * (- 1) by A29, A34, XXREAL_0:1
.= - ((t `2) / (t `1))
.= - ((t `2) * (- 1)) by A29, A34, A41, XXREAL_0:1
.= t `2 ;
hence contradiction by A7, A25, A28, A34, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA42: ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction
reconsider K01 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th17;
t in K01 by A6, A27;
then A43: Out_In_Sq . t = (Out_In_Sq | K01) . t by FUNCT_1:49;
dom (Out_In_Sq | K01) = (dom Out_In_Sq) /\ K01 by RELAT_1:61
.= D /\ K01 by A3, FUNCT_2:def_1
.= ([#] ((TOP-REAL 2) | D)) /\ K01 by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | D) /\ K01
.= K01 by XBOOLE_1:28 ;
then t in dom (Out_In_Sq | K01) by A6, A27;
then A44: (Out_In_Sq | K01) . t in rng (Out_In_Sq | K01) by FUNCT_1:3;
rng (Out_In_Sq | K01) c= the carrier of (((TOP-REAL 2) | D) | K01) by Th15;
then A45: (Out_In_Sq | K01) . t in the carrier of (((TOP-REAL 2) | D) | K01) by A44;
the carrier of (((TOP-REAL 2) | D) | K01) = [#] (((TOP-REAL 2) | D) | K01)
.= K01 by PRE_TOPC:def_5 ;
then A46: ex p5 being Point of (TOP-REAL 2) st
( p5 = p4 & ( ( p5 `2 <= p5 `1 & - (p5 `1) <= p5 `2 ) or ( p5 `2 >= p5 `1 & p5 `2 <= - (p5 `1) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25, A45, A43;
now__::_thesis:_(_(_p4_`1_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`1_&_contradiction_)_)
percases ( p4 `1 >= 1 or - 1 >= p4 `1 ) by A42, A46, XREAL_1:25;
caseA47: p4 `1 >= 1 ; ::_thesis: contradiction
then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * 1 by A29, A42, XXREAL_0:1
.= (t `2) * 1 by A29, A42, A47, XXREAL_0:1
.= t `2 ;
hence contradiction by A7, A25, A28, A42, EUCLID:52; ::_thesis: verum
end;
caseA48: - 1 >= p4 `1 ; ::_thesis: contradiction
then ((t `2) / (t `1)) / (t `1) = ((t `2) / (t `1)) * (- 1) by A29, A42, XXREAL_0:1
.= - ((t `2) / (t `1))
.= - ((t `2) * (- 1)) by A29, A42, A48, XXREAL_0:1
.= t `2 ;
hence contradiction by A7, A25, A28, A42, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA49: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: contradiction
then A50: Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A6, Def1;
then A51: p4 `2 = 1 / (t `2) by A25, EUCLID:52;
now__::_thesis:_(_(_-_1_=_p4_`2_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_or_(_p4_`2_=_1_&_-_1_<=_p4_`1_&_p4_`1_<=_1_&_contradiction_)_or_(_-_1_=_p4_`1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_or_(_1_=_p4_`1_&_-_1_<=_p4_`2_&_p4_`2_<=_1_&_contradiction_)_)
percases ( ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( p4 `2 = 1 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ) by A26;
case ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction
then A52: (t `2) * ((t `2) ") = - (t `2) by A51;
now__::_thesis:_(_(_t_`2_<>_0_&_contradiction_)_or_(_t_`2_=_0_&_contradiction_)_)
percases ( t `2 <> 0 or t `2 = 0 ) ;
case t `2 <> 0 ; ::_thesis: contradiction
then - (t `2) = 1 by A52, XCMPLX_0:def_7;
hence contradiction by A7; ::_thesis: verum
end;
case t `2 = 0 ; ::_thesis: contradiction
hence contradiction by A49; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
case ( p4 `2 = 1 & - 1 <= p4 `1 & p4 `1 <= 1 ) ; ::_thesis: contradiction
then A53: (t `2) * ((t `2) ") = t `2 by A51;
now__::_thesis:_(_(_t_`2_<>_0_&_contradiction_)_or_(_t_`2_=_0_&_contradiction_)_)
percases ( t `2 <> 0 or t `2 = 0 ) ;
case t `2 <> 0 ; ::_thesis: contradiction
hence contradiction by A7, A53, XCMPLX_0:def_7; ::_thesis: verum
end;
case t `2 = 0 ; ::_thesis: contradiction
hence contradiction by A49; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA54: ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction
reconsider K11 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th18;
A55: dom (Out_In_Sq | K11) = (dom Out_In_Sq) /\ K11 by RELAT_1:61
.= D /\ K11 by A3, FUNCT_2:def_1
.= ([#] ((TOP-REAL 2) | D)) /\ K11 by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | D) /\ K11
.= K11 by XBOOLE_1:28 ;
A56: ( ( t `1 <= t `2 & - (t `2) <= t `1 ) or ( t `1 >= t `2 & t `1 <= - (t `2) ) ) by A49, Th13;
then t in K11 by A6;
then A57: Out_In_Sq . t = (Out_In_Sq | K11) . t by FUNCT_1:49;
t in K11 by A6, A56;
then A58: (Out_In_Sq | K11) . t in rng (Out_In_Sq | K11) by A55, FUNCT_1:3;
rng (Out_In_Sq | K11) c= the carrier of (((TOP-REAL 2) | D) | K11) by Th16;
then A59: (Out_In_Sq | K11) . t in the carrier of (((TOP-REAL 2) | D) | K11) by A58;
the carrier of (((TOP-REAL 2) | D) | K11) = [#] (((TOP-REAL 2) | D) | K11)
.= K11 by PRE_TOPC:def_5 ;
then A60: ex p5 being Point of (TOP-REAL 2) st
( p5 = p4 & ( ( p5 `1 <= p5 `2 & - (p5 `2) <= p5 `1 ) or ( p5 `1 >= p5 `2 & p5 `1 <= - (p5 `2) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25, A59, A57;
now__::_thesis:_(_(_p4_`2_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`2_&_contradiction_)_)
percases ( p4 `2 >= 1 or - 1 >= p4 `2 ) by A54, A60, XREAL_1:24;
caseA61: p4 `2 >= 1 ; ::_thesis: contradiction
then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * 1 by A51, A54, XXREAL_0:1
.= (t `1) * 1 by A51, A54, A61, XXREAL_0:1
.= t `1 ;
hence contradiction by A7, A25, A50, A54, EUCLID:52; ::_thesis: verum
end;
caseA62: - 1 >= p4 `2 ; ::_thesis: contradiction
then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * (- 1) by A51, A54, XXREAL_0:1
.= - ((t `1) / (t `2))
.= - ((t `1) * (- 1)) by A51, A54, A62, XXREAL_0:1
.= t `1 ;
hence contradiction by A7, A25, A50, A54, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
caseA63: ( 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) ; ::_thesis: contradiction
reconsider K11 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th18;
A64: the carrier of (((TOP-REAL 2) | D) | K11) = [#] (((TOP-REAL 2) | D) | K11)
.= K11 by PRE_TOPC:def_5 ;
A65: dom (Out_In_Sq | K11) = (dom Out_In_Sq) /\ K11 by RELAT_1:61
.= D /\ K11 by A3, FUNCT_2:def_1
.= ([#] ((TOP-REAL 2) | D)) /\ K11 by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | D) /\ K11
.= K11 by XBOOLE_1:28 ;
A66: ( ( t `1 <= t `2 & - (t `2) <= t `1 ) or ( t `1 >= t `2 & t `1 <= - (t `2) ) ) by A49, Th13;
then t in K11 by A6;
then A67: (Out_In_Sq | K11) . t in rng (Out_In_Sq | K11) by A65, FUNCT_1:3;
rng (Out_In_Sq | K11) c= the carrier of (((TOP-REAL 2) | D) | K11) by Th16;
then A68: (Out_In_Sq | K11) . t in the carrier of (((TOP-REAL 2) | D) | K11) by A67;
t in K11 by A6, A66;
then Out_In_Sq . t in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } by A68, A64, FUNCT_1:49;
then A69: ex p5 being Point of (TOP-REAL 2) st
( p5 = p4 & ( ( p5 `1 <= p5 `2 & - (p5 `2) <= p5 `1 ) or ( p5 `1 >= p5 `2 & p5 `1 <= - (p5 `2) ) ) & p5 <> 0. (TOP-REAL 2) ) by A25;
now__::_thesis:_(_(_p4_`2_>=_1_&_contradiction_)_or_(_-_1_>=_p4_`2_&_contradiction_)_)
percases ( p4 `2 >= 1 or - 1 >= p4 `2 ) by A63, A69, XREAL_1:25;
caseA70: p4 `2 >= 1 ; ::_thesis: contradiction
then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * 1 by A51, A63, XXREAL_0:1
.= (t `1) * 1 by A51, A63, A70, XXREAL_0:1
.= t `1 ;
hence contradiction by A7, A25, A50, A63, EUCLID:52; ::_thesis: verum
end;
caseA71: - 1 >= p4 `2 ; ::_thesis: contradiction
then ((t `1) / (t `2)) / (t `2) = ((t `1) / (t `2)) * (- 1) by A51, A63, XXREAL_0:1
.= - ((t `1) / (t `2))
.= - ((t `1) * (- 1)) by A51, A63, A71, XXREAL_0:1
.= t `1 ;
hence contradiction by A7, A25, A50, A63, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence not Out_In_Sq . t in K0 \/ Kb ; ::_thesis: verum
end;
A72: for t being Point of (TOP-REAL 2) st not t in K0 \/ Kb holds
Out_In_Sq . t in K0
proof
let t be Point of (TOP-REAL 2); ::_thesis: ( not t in K0 \/ Kb implies Out_In_Sq . t in K0 )
assume A73: not t in K0 \/ Kb ; ::_thesis: Out_In_Sq . t in K0
then A74: not t in K0 by XBOOLE_0:def_3;
then A75: not t = 0. (TOP-REAL 2) by A1, Th3;
then not t in {(0. (TOP-REAL 2))} by TARSKI:def_1;
then t in NonZero (TOP-REAL 2) by XBOOLE_0:def_5;
then Out_In_Sq . t in NonZero (TOP-REAL 2) by FUNCT_2:5;
then reconsider p4 = Out_In_Sq . t as Point of (TOP-REAL 2) ;
A76: not t in Kb by A73, XBOOLE_0:def_3;
now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_Out_In_Sq_._t_in_K0_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_Out_In_Sq_._t_in_K0_)_)
percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ;
caseA77: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t in K0
A78: now__::_thesis:_(_(_t_`1_>_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_or_(_t_`1_<=_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_)
percases ( t `1 > 0 or t `1 <= 0 ) ;
caseA79: t `1 > 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
now__::_thesis:_(_(_t_`2_>_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_or_(_t_`2_<=_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_)
percases ( t `2 > 0 or t `2 <= 0 ) ;
caseA80: t `2 > 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
( - 1 >= t `1 or t `1 >= 1 or - 1 >= t `2 or t `2 >= 1 ) by A1, A74;
then A81: t `1 >= 1 by A77, A79, A80, XXREAL_0:2;
not t `1 = 1 by A1, A76, A77;
then A82: t `1 > 1 by A81, XXREAL_0:1;
then t `1 < (t `1) ^2 by SQUARE_1:14;
then t `2 < (t `1) ^2 by A77, A79, XXREAL_0:2;
then (t `2) / (t `1) < ((t `1) ^2) / (t `1) by A79, XREAL_1:74;
then (t `2) / (t `1) < t `1 by A79, XCMPLX_1:89;
then A83: ((t `2) / (t `1)) / (t `1) < (t `1) / (t `1) by A79, XREAL_1:74;
0 < (t `2) / (t `1) by A79, A80, XREAL_1:139;
then A84: ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A79, XREAL_1:74;
(t `1) / (t `1) > 1 / (t `1) by A82, XREAL_1:74;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A79, A84, A83, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum
end;
caseA85: t `2 <= 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
A86: now__::_thesis:_(_t_`1_<_1_implies_t_`1_>=_1_)
assume t `1 < 1 ; ::_thesis: t `1 >= 1
then - 1 >= t `2 by A1, A74, A79, A85;
then - (t `1) <= - 1 by A77, A79, XXREAL_0:2;
hence t `1 >= 1 by XREAL_1:24; ::_thesis: verum
end;
not t `1 = 1 by A1, A76, A77;
then A87: t `1 > 1 by A86, XXREAL_0:1;
then A88: t `1 < (t `1) ^2 by SQUARE_1:14;
- (- (t `1)) >= - (t `2) by A77, A79, XREAL_1:24;
then (t `1) ^2 > - (t `2) by A88, XXREAL_0:2;
then ((t `1) ^2) / (t `1) > (- (t `2)) / (t `1) by A79, XREAL_1:74;
then t `1 > - ((t `2) / (t `1)) by A79, XCMPLX_1:89;
then - (t `1) < - (- ((t `2) / (t `1))) by XREAL_1:24;
then A89: ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A79, XREAL_1:74;
(t `1) / (t `1) > 1 / (t `1) by A87, XREAL_1:74;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A79, A85, A89, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum
end;
end;
end;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ; ::_thesis: verum
end;
caseA90: t `1 <= 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
now__::_thesis:_(_(_t_`1_=_0_&_contradiction_)_or_(_t_`1_<_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_)
percases ( t `1 = 0 or t `1 < 0 ) by A90;
caseA91: t `1 = 0 ; ::_thesis: contradiction
then t `2 = 0 by A77;
hence contradiction by A1, A74, A91; ::_thesis: verum
end;
caseA92: t `1 < 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
now__::_thesis:_(_(_t_`2_>_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_or_(_t_`2_<=_0_&_-_1_<_1_/_(t_`1)_&_1_/_(t_`1)_<_1_&_-_1_<_((t_`2)_/_(t_`1))_/_(t_`1)_&_((t_`2)_/_(t_`1))_/_(t_`1)_<_1_)_)
percases ( t `2 > 0 or t `2 <= 0 ) ;
caseA93: t `2 > 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
( - 1 >= t `1 or t `1 >= 1 or - 1 >= t `2 or t `2 >= 1 ) by A1, A74;
then ( t `1 <= - 1 or 1 <= - (t `1) ) by A77, A92, XXREAL_0:2;
then A94: ( t `1 <= - 1 or - 1 >= - (- (t `1)) ) by XREAL_1:24;
not t `1 = - 1 by A1, A76, A77;
then A95: t `1 < - 1 by A94, XXREAL_0:1;
then (t `1) / (t `1) > (- 1) / (t `1) by XREAL_1:75;
then A96: - ((t `1) / (t `1)) < - ((- 1) / (t `1)) by XREAL_1:24;
- (t `1) < (t `1) ^2 by A95, SQUARE_1:46;
then t `2 < (t `1) ^2 by A77, A92, XXREAL_0:2;
then (t `2) / (t `1) > ((t `1) ^2) / (t `1) by A92, XREAL_1:75;
then (t `2) / (t `1) > t `1 by A92, XCMPLX_1:89;
then A97: ((t `2) / (t `1)) / (t `1) < (t `1) / (t `1) by A92, XREAL_1:75;
0 > (t `2) / (t `1) by A92, A93, XREAL_1:142;
then ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A92, XREAL_1:75;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A92, A96, A97, XCMPLX_1:60; ::_thesis: verum
end;
caseA98: t `2 <= 0 ; ::_thesis: ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 )
then ( - 1 >= t `1 or - 1 >= t `2 ) by A1, A74, A92;
then A99: t `1 <= - 1 by A77, A92, XXREAL_0:2;
not t `1 = - 1 by A1, A76, A77;
then A100: t `1 < - 1 by A99, XXREAL_0:1;
then A101: - (t `1) < (t `1) ^2 by SQUARE_1:46;
- (t `1) >= - (t `2) by A77, A92, XREAL_1:24;
then (t `1) ^2 > - (t `2) by A101, XXREAL_0:2;
then ((t `1) ^2) / (t `1) < (- (t `2)) / (t `1) by A92, XREAL_1:75;
then t `1 < - ((t `2) / (t `1)) by A92, XCMPLX_1:89;
then - (t `1) > - (- ((t `2) / (t `1))) by XREAL_1:24;
then A102: ((- 1) * (t `1)) / (t `1) < ((t `2) / (t `1)) / (t `1) by A92, XREAL_1:75;
(t `1) / (t `1) > (- 1) / (t `1) by A100, XREAL_1:75;
then 1 > (- 1) / (t `1) by A92, XCMPLX_1:60;
then - 1 < - ((- 1) / (t `1)) by XREAL_1:24;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) by A92, A98, A102, XCMPLX_1:89; ::_thesis: verum
end;
end;
end;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ; ::_thesis: verum
end;
end;
end;
hence ( - 1 < 1 / (t `1) & 1 / (t `1) < 1 & - 1 < ((t `2) / (t `1)) / (t `1) & ((t `2) / (t `1)) / (t `1) < 1 ) ; ::_thesis: verum
end;
end;
end;
Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A75, A77, Def1;
then ( p4 `1 = 1 / (t `1) & p4 `2 = ((t `2) / (t `1)) / (t `1) ) by EUCLID:52;
hence Out_In_Sq . t in K0 by A1, A78; ::_thesis: verum
end;
caseA103: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t in K0
then A104: ( ( t `1 <= t `2 & - (t `2) <= t `1 ) or ( t `1 >= t `2 & t `1 <= - (t `2) ) ) by Th13;
A105: now__::_thesis:_(_(_t_`2_>_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_or_(_t_`2_<=_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_)
percases ( t `2 > 0 or t `2 <= 0 ) ;
caseA106: t `2 > 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 )
now__::_thesis:_(_(_t_`1_>_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_or_(_t_`1_<=_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_)
percases ( t `1 > 0 or t `1 <= 0 ) ;
caseA107: t `1 > 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 )
A108: ( - 1 >= t `2 or t `2 >= 1 or - 1 >= t `1 or t `1 >= 1 ) by A1, A74;
not t `2 = 1 by A1, A76, A103, A107;
then A109: t `2 > 1 by A103, A106, A107, A108, XXREAL_0:1, XXREAL_0:2;
then t `2 < (t `2) ^2 by SQUARE_1:14;
then t `1 < (t `2) ^2 by A103, A106, XXREAL_0:2;
then (t `1) / (t `2) < ((t `2) ^2) / (t `2) by A106, XREAL_1:74;
then (t `1) / (t `2) < t `2 by A106, XCMPLX_1:89;
then A110: ((t `1) / (t `2)) / (t `2) < (t `2) / (t `2) by A106, XREAL_1:74;
0 < (t `1) / (t `2) by A106, A107, XREAL_1:139;
then A111: ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A106, XREAL_1:74;
(t `2) / (t `2) > 1 / (t `2) by A109, XREAL_1:74;
hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A106, A111, A110, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum
end;
caseA112: t `1 <= 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 )
A113: now__::_thesis:_(_t_`2_<_1_implies_t_`2_>=_1_)
assume t `2 < 1 ; ::_thesis: t `2 >= 1
then - 1 >= t `1 by A1, A74, A106, A112;
then - (t `2) <= - 1 by A104, A106, XXREAL_0:2;
hence t `2 >= 1 by XREAL_1:24; ::_thesis: verum
end;
not t `2 = 1 by A1, A76, A104;
then A114: t `2 > 1 by A113, XXREAL_0:1;
then t `2 < (t `2) ^2 by SQUARE_1:14;
then (t `2) ^2 > - (t `1) by A103, A106, XXREAL_0:2;
then ((t `2) ^2) / (t `2) > (- (t `1)) / (t `2) by A106, XREAL_1:74;
then t `2 > - ((t `1) / (t `2)) by A106, XCMPLX_1:89;
then - (t `2) < - (- ((t `1) / (t `2))) by XREAL_1:24;
then A115: ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A106, XREAL_1:74;
(t `2) / (t `2) > 1 / (t `2) by A114, XREAL_1:74;
hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A106, A112, A115, XCMPLX_1:60, XCMPLX_1:89; ::_thesis: verum
end;
end;
end;
hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) ; ::_thesis: verum
end;
caseA116: t `2 <= 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 )
then A117: t `2 < 0 by A103;
A118: ( t `1 <= t `2 or t `1 <= - (t `2) ) by A103, Th13;
now__::_thesis:_(_(_t_`1_>_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_or_(_t_`1_<=_0_&_-_1_<_1_/_(t_`2)_&_1_/_(t_`2)_<_1_&_-_1_<_((t_`1)_/_(t_`2))_/_(t_`2)_&_((t_`1)_/_(t_`2))_/_(t_`2)_<_1_)_)
percases ( t `1 > 0 or t `1 <= 0 ) ;
caseA119: t `1 > 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 )
( - 1 >= t `2 or t `2 >= 1 or - 1 >= t `1 or t `1 >= 1 ) by A1, A74;
then ( t `2 <= - 1 or 1 <= - (t `2) ) by A104, A116, XXREAL_0:2;
then A120: ( t `2 <= - 1 or - 1 >= - (- (t `2)) ) by XREAL_1:24;
not t `2 = - 1 by A1, A76, A104;
then A121: t `2 < - 1 by A120, XXREAL_0:1;
then (t `2) / (t `2) > (- 1) / (t `2) by XREAL_1:75;
then A122: - ((t `2) / (t `2)) < - ((- 1) / (t `2)) by XREAL_1:24;
- (t `2) < (t `2) ^2 by A121, SQUARE_1:46;
then t `1 < (t `2) ^2 by A116, A118, XXREAL_0:2;
then (t `1) / (t `2) > ((t `2) ^2) / (t `2) by A117, XREAL_1:75;
then (t `1) / (t `2) > t `2 by A117, XCMPLX_1:89;
then A123: ((t `1) / (t `2)) / (t `2) < (t `2) / (t `2) by A117, XREAL_1:75;
0 > (t `1) / (t `2) by A117, A119, XREAL_1:142;
then ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A117, XREAL_1:75;
hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A117, A122, A123, XCMPLX_1:60; ::_thesis: verum
end;
caseA124: t `1 <= 0 ; ::_thesis: ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 )
A125: not t `2 = - 1 by A1, A76, A104;
( - 1 >= t `2 or - 1 >= t `1 ) by A1, A74, A116, A124;
then A126: t `2 < - 1 by A103, A116, A125, XXREAL_0:1, XXREAL_0:2;
then A127: - (t `2) < (t `2) ^2 by SQUARE_1:46;
- (t `2) >= - (t `1) by A103, A116, XREAL_1:24;
then (t `2) ^2 > - (t `1) by A127, XXREAL_0:2;
then ((t `2) ^2) / (t `2) < (- (t `1)) / (t `2) by A117, XREAL_1:75;
then t `2 < - ((t `1) / (t `2)) by A117, XCMPLX_1:89;
then - (t `2) > - (- ((t `1) / (t `2))) by XREAL_1:24;
then A128: ((- 1) * (t `2)) / (t `2) < ((t `1) / (t `2)) / (t `2) by A117, XREAL_1:75;
(t `2) / (t `2) > (- 1) / (t `2) by A126, XREAL_1:75;
then 1 > (- 1) / (t `2) by A117, XCMPLX_1:60;
then - 1 < - ((- 1) / (t `2)) by XREAL_1:24;
hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) by A103, A116, A124, A128, XCMPLX_1:89; ::_thesis: verum
end;
end;
end;
hence ( - 1 < 1 / (t `2) & 1 / (t `2) < 1 & - 1 < ((t `1) / (t `2)) / (t `2) & ((t `1) / (t `2)) / (t `2) < 1 ) ; ::_thesis: verum
end;
end;
end;
Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A75, A103, Def1;
then ( p4 `2 = 1 / (t `2) & p4 `1 = ((t `1) / (t `2)) / (t `2) ) by EUCLID:52;
hence Out_In_Sq . t in K0 by A1, A105; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . t in K0 ; ::_thesis: verum
end;
A129: D = NonZero (TOP-REAL 2) by A1, SUBSET_1:def_4;
for x1, x2 being set st x1 in dom Out_In_Sq & x2 in dom Out_In_Sq & Out_In_Sq . x1 = Out_In_Sq . x2 holds
x1 = x2
proof
A130: { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } or x in D )
assume x in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x in D
then A131: ex p8 being Point of (TOP-REAL 2) st
( x = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
then not x in {(0. (TOP-REAL 2))} by TARSKI:def_1;
hence x in D by A3, A131, XBOOLE_0:def_5; ::_thesis: verum
end;
A132: 1.REAL 2 <> 0. (TOP-REAL 2) by Lm1, REVROT_1:19;
( ( (1.REAL 2) `1 <= (1.REAL 2) `2 & - ((1.REAL 2) `2) <= (1.REAL 2) `1 ) or ( (1.REAL 2) `1 >= (1.REAL 2) `2 & (1.REAL 2) `1 <= - ((1.REAL 2) `2) ) ) by Th5;
then A133: 1.REAL 2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } by A132;
the carrier of ((TOP-REAL 2) | D) = [#] ((TOP-REAL 2) | D)
.= D by PRE_TOPC:def_5 ;
then reconsider K11 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A133, A130;
reconsider K01 = { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } as non empty Subset of ((TOP-REAL 2) | D) by A2, Th17;
let x1, x2 be set ; ::_thesis: ( x1 in dom Out_In_Sq & x2 in dom Out_In_Sq & Out_In_Sq . x1 = Out_In_Sq . x2 implies x1 = x2 )
assume that
A134: x1 in dom Out_In_Sq and
A135: x2 in dom Out_In_Sq and
A136: Out_In_Sq . x1 = Out_In_Sq . x2 ; ::_thesis: x1 = x2
NonZero (TOP-REAL 2) <> {} by Th9;
then A137: dom Out_In_Sq = NonZero (TOP-REAL 2) by FUNCT_2:def_1;
then A138: x2 in D by A1, A135, SUBSET_1:def_4;
reconsider p1 = x1, p2 = x2 as Point of (TOP-REAL 2) by A134, A135, XBOOLE_0:def_5;
A139: D c= K01 \/ K11
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in K01 \/ K11 )
assume A140: x in D ; ::_thesis: x in K01 \/ K11
then reconsider px = x as Point of (TOP-REAL 2) ;
not x in {(0. (TOP-REAL 2))} by A129, A140, XBOOLE_0:def_5;
then ( ( ( ( px `2 <= px `1 & - (px `1) <= px `2 ) or ( px `2 >= px `1 & px `2 <= - (px `1) ) ) & px <> 0. (TOP-REAL 2) ) or ( ( ( px `1 <= px `2 & - (px `2) <= px `1 ) or ( px `1 >= px `2 & px `1 <= - (px `2) ) ) & px <> 0. (TOP-REAL 2) ) ) by TARSKI:def_1, XREAL_1:25;
then ( x in K01 or x in K11 ) ;
hence x in K01 \/ K11 by XBOOLE_0:def_3; ::_thesis: verum
end;
A141: x1 in D by A1, A134, A137, SUBSET_1:def_4;
now__::_thesis:_(_(_x1_in_K01_&_x1_=_x2_)_or_(_x1_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_x1_=_x2_)_)
percases ( x1 in K01 or x1 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) by A139, A141, XBOOLE_0:def_3;
case x1 in K01 ; ::_thesis: x1 = x2
then A142: ex p7 being Point of (TOP-REAL 2) st
( p1 = p7 & ( ( p7 `2 <= p7 `1 & - (p7 `1) <= p7 `2 ) or ( p7 `2 >= p7 `1 & p7 `2 <= - (p7 `1) ) ) & p7 <> 0. (TOP-REAL 2) ) ;
then A143: Out_In_Sq . p1 = |[(1 / (p1 `1)),(((p1 `2) / (p1 `1)) / (p1 `1))]| by Def1;
now__::_thesis:_(_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`2_<=_p8_`1_&_-_(p8_`1)_<=_p8_`2_)_or_(_p8_`2_>=_p8_`1_&_p8_`2_<=_-_(p8_`1)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_x1_=_x2_)_or_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_not_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`2_<=_p8_`1_&_-_(p8_`1)_<=_p8_`2_)_or_(_p8_`2_>=_p8_`1_&_p8_`2_<=_-_(p8_`1)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_contradiction_)_)
percases ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } or ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ) by A139, A138, XBOOLE_0:def_3;
case x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x1 = x2
then ex p8 being Point of (TOP-REAL 2) st
( p2 = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
then A144: |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| = |[(1 / (p1 `1)),(((p1 `2) / (p1 `1)) / (p1 `1))]| by A136, A143, Def1;
A145: p1 = |[(p1 `1),(p1 `2)]| by EUCLID:53;
set qq = |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]|;
A146: (1 / (p1 `1)) " = ((p1 `1) ") "
.= p1 `1 ;
A147: now__::_thesis:_not_p1_`1_=_0
assume A148: p1 `1 = 0 ; ::_thesis: contradiction
then p1 `2 = 0 by A142;
hence contradiction by A142, A148, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
|[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| `1 = 1 / (p2 `1) by EUCLID:52;
then A149: 1 / (p1 `1) = 1 / (p2 `1) by A144, EUCLID:52;
|[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| `2 = ((p2 `2) / (p2 `1)) / (p2 `1) by EUCLID:52;
then (p1 `2) / (p1 `1) = (p2 `2) / (p1 `1) by A144, A149, A146, A147, EUCLID:52, XCMPLX_1:53;
then p1 `2 = p2 `2 by A147, XCMPLX_1:53;
hence x1 = x2 by A149, A146, A145, EUCLID:53; ::_thesis: verum
end;
caseA150: ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: contradiction
A151: now__::_thesis:_not_p1_`1_=_0
assume A152: p1 `1 = 0 ; ::_thesis: contradiction
then p1 `2 = 0 by A142;
hence contradiction by A142, A152, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A153: now__::_thesis:_(_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_&_(p1_`2)_/_(p1_`1)_<=_1_)_or_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_&_(p1_`2)_/_(p1_`1)_<=_1_)_)
percases ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) by A142;
caseA154: ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) ; ::_thesis: (p1 `2) / (p1 `1) <= 1
then p1 `1 >= 0 ;
then (p1 `2) / (p1 `1) <= (p1 `1) / (p1 `1) by A154, XREAL_1:72;
hence (p1 `2) / (p1 `1) <= 1 by A151, XCMPLX_1:60; ::_thesis: verum
end;
caseA155: ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ; ::_thesis: (p1 `2) / (p1 `1) <= 1
then p1 `1 <= 0 ;
then (p1 `2) / (p1 `1) <= (p1 `1) / (p1 `1) by A155, XREAL_1:73;
hence (p1 `2) / (p1 `1) <= 1 by A151, XCMPLX_1:60; ::_thesis: verum
end;
end;
end;
A156: now__::_thesis:_(_(_p1_`2_<=_p1_`1_&_-_(p1_`1)_<=_p1_`2_&_-_1_<=_(p1_`2)_/_(p1_`1)_)_or_(_p1_`2_>=_p1_`1_&_p1_`2_<=_-_(p1_`1)_&_-_1_<=_(p1_`2)_/_(p1_`1)_)_)
percases ( ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) or ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ) by A142;
caseA157: ( p1 `2 <= p1 `1 & - (p1 `1) <= p1 `2 ) ; ::_thesis: - 1 <= (p1 `2) / (p1 `1)
then p1 `1 >= 0 ;
then (- (p1 `1)) / (p1 `1) <= (p1 `2) / (p1 `1) by A157, XREAL_1:72;
hence - 1 <= (p1 `2) / (p1 `1) by A151, XCMPLX_1:197; ::_thesis: verum
end;
case ( p1 `2 >= p1 `1 & p1 `2 <= - (p1 `1) ) ; ::_thesis: - 1 <= (p1 `2) / (p1 `1)
then ( - (p1 `2) >= - (- (p1 `1)) & p1 `1 <= 0 ) by XREAL_1:24;
then (- (p1 `2)) / (- (p1 `1)) >= (p1 `1) / (- (p1 `1)) by XREAL_1:72;
then (- (p1 `2)) / (- (p1 `1)) >= - 1 by A151, XCMPLX_1:198;
hence - 1 <= (p1 `2) / (p1 `1) by XCMPLX_1:191; ::_thesis: verum
end;
end;
end;
A158: ex p8 being Point of (TOP-REAL 2) st
( p2 = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) by A150;
A159: now__::_thesis:_not_p2_`2_=_0
assume A160: p2 `2 = 0 ; ::_thesis: contradiction
then p2 `1 = 0 by A158;
hence contradiction by A158, A160, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
( ( not ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) & not ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) or not p2 <> 0. (TOP-REAL 2) ) by A150;
then A161: Out_In_Sq . p2 = |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| by A158, Def1;
then ((p1 `2) / (p1 `1)) / (p1 `1) = 1 / (p2 `2) by A136, A143, SPPOL_2:1;
then A162: (p1 `2) / (p1 `1) = (1 / (p2 `2)) * (p1 `1) by A151, XCMPLX_1:87
.= (p1 `1) / (p2 `2) ;
1 / (p1 `1) = ((p2 `1) / (p2 `2)) / (p2 `2) by A136, A143, A161, SPPOL_2:1;
then A163: (p2 `1) / (p2 `2) = (1 / (p1 `1)) * (p2 `2) by A159, XCMPLX_1:87
.= (p2 `2) / (p1 `1) ;
then A164: ((p2 `1) / (p2 `2)) * ((p1 `2) / (p1 `1)) = 1 by A159, A151, A162, XCMPLX_1:112;
A165: (((p2 `1) / (p2 `2)) * ((p1 `2) / (p1 `1))) * (p1 `1) = 1 * (p1 `1) by A159, A151, A163, A162, XCMPLX_1:112;
then A166: p1 `2 <> 0 by A151;
A167: ex p9 being Point of (TOP-REAL 2) st
( p2 = p9 & ( ( p9 `1 <= p9 `2 & - (p9 `2) <= p9 `1 ) or ( p9 `1 >= p9 `2 & p9 `1 <= - (p9 `2) ) ) & p9 <> 0. (TOP-REAL 2) ) by A150;
A168: now__::_thesis:_(_(_p2_`1_<=_p2_`2_&_-_(p2_`2)_<=_p2_`1_&_-_1_<=_(p2_`1)_/_(p2_`2)_)_or_(_p2_`1_>=_p2_`2_&_p2_`1_<=_-_(p2_`2)_&_-_1_<=_(p2_`1)_/_(p2_`2)_)_)
percases ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A167;
caseA169: ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) ; ::_thesis: - 1 <= (p2 `1) / (p2 `2)
then p2 `2 >= 0 ;
then (- (p2 `2)) / (p2 `2) <= (p2 `1) / (p2 `2) by A169, XREAL_1:72;
hence - 1 <= (p2 `1) / (p2 `2) by A159, XCMPLX_1:197; ::_thesis: verum
end;
case ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ; ::_thesis: - 1 <= (p2 `1) / (p2 `2)
then ( - (p2 `1) >= - (- (p2 `2)) & p2 `2 <= 0 ) by XREAL_1:24;
then (- (p2 `1)) / (- (p2 `2)) >= (p2 `2) / (- (p2 `2)) by XREAL_1:72;
then (- (p2 `1)) / (- (p2 `2)) >= - 1 by A159, XCMPLX_1:198;
hence - 1 <= (p2 `1) / (p2 `2) by XCMPLX_1:191; ::_thesis: verum
end;
end;
end;
((p2 `1) / (p2 `2)) * (((p1 `2) / (p1 `1)) * (p1 `1)) = p1 `1 by A165;
then A170: ((p2 `1) / (p2 `2)) * (p1 `2) = p1 `1 by A151, XCMPLX_1:87;
then A171: (p2 `1) / (p2 `2) = (p1 `1) / (p1 `2) by A166, XCMPLX_1:89;
A172: now__::_thesis:_(_(_p2_`1_<=_p2_`2_&_-_(p2_`2)_<=_p2_`1_&_(p2_`1)_/_(p2_`2)_<=_1_)_or_(_p2_`1_>=_p2_`2_&_p2_`1_<=_-_(p2_`2)_&_(p2_`1)_/_(p2_`2)_<=_1_)_)
percases ( ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) or ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ) by A167;
caseA173: ( p2 `1 <= p2 `2 & - (p2 `2) <= p2 `1 ) ; ::_thesis: (p2 `1) / (p2 `2) <= 1
then p2 `2 >= 0 ;
then (p2 `1) / (p2 `2) <= (p2 `2) / (p2 `2) by A173, XREAL_1:72;
hence (p2 `1) / (p2 `2) <= 1 by A159, XCMPLX_1:60; ::_thesis: verum
end;
caseA174: ( p2 `1 >= p2 `2 & p2 `1 <= - (p2 `2) ) ; ::_thesis: (p2 `1) / (p2 `2) <= 1
then p2 `2 <= 0 ;
then (p2 `1) / (p2 `2) <= (p2 `2) / (p2 `2) by A174, XREAL_1:73;
hence (p2 `1) / (p2 `2) <= 1 by A159, XCMPLX_1:60; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_0_<=_(p2_`1)_/_(p2_`2)_&_contradiction_)_or_(_0_>_(p2_`1)_/_(p2_`2)_&_contradiction_)_)
percases ( 0 <= (p2 `1) / (p2 `2) or 0 > (p2 `1) / (p2 `2) ) ;
case 0 <= (p2 `1) / (p2 `2) ; ::_thesis: contradiction
then A175: ( ( p1 `2 > 0 & p1 `1 >= 0 ) or ( p1 `2 < 0 & p1 `1 <= 0 ) ) by A151, A170;
now__::_thesis:_not_(p1_`2)_/_(p1_`1)_<>_1
assume (p1 `2) / (p1 `1) <> 1 ; ::_thesis: contradiction
then (p1 `2) / (p1 `1) < 1 by A153, XXREAL_0:1;
hence contradiction by A164, A172, A175, XREAL_1:162; ::_thesis: verum
end;
then p1 `2 = 1 * (p1 `1) by A151, XCMPLX_1:87;
then ((p2 `1) / (p2 `2)) * (p2 `2) = 1 * (p2 `2) by A151, A171, XCMPLX_1:60
.= p2 `2 ;
then p2 `1 = p2 `2 by A159, XCMPLX_1:87;
hence contradiction by A150, A167; ::_thesis: verum
end;
case 0 > (p2 `1) / (p2 `2) ; ::_thesis: contradiction
then A176: ( ( p1 `2 < 0 & p1 `1 > 0 ) or ( p1 `2 > 0 & p1 `1 < 0 ) ) by A171, XREAL_1:143;
now__::_thesis:_not_(p1_`2)_/_(p1_`1)_<>_-_1
assume (p1 `2) / (p1 `1) <> - 1 ; ::_thesis: contradiction
then - 1 < (p1 `2) / (p1 `1) by A156, XXREAL_0:1;
hence contradiction by A164, A168, A176, XREAL_1:166; ::_thesis: verum
end;
then p1 `2 = (- 1) * (p1 `1) by A151, XCMPLX_1:87
.= - (p1 `1) ;
then - (p1 `2) = p1 `1 ;
then (p2 `1) / (p2 `2) = - 1 by A166, A171, XCMPLX_1:197;
then p2 `1 = (- 1) * (p2 `2) by A159, XCMPLX_1:87;
then - (p2 `1) = p2 `2 ;
hence contradiction by A150, A167; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
case x1 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x1 = x2
then A177: ex p7 being Point of (TOP-REAL 2) st
( p1 = p7 & ( ( p7 `1 <= p7 `2 & - (p7 `2) <= p7 `1 ) or ( p7 `1 >= p7 `2 & p7 `1 <= - (p7 `2) ) ) & p7 <> 0. (TOP-REAL 2) ) ;
then A178: Out_In_Sq . p1 = |[(((p1 `1) / (p1 `2)) / (p1 `2)),(1 / (p1 `2))]| by Th14;
now__::_thesis:_(_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_x1_=_x2_)_or_(_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`2_<=_p8_`1_&_-_(p8_`1)_<=_p8_`2_)_or_(_p8_`2_>=_p8_`1_&_p8_`2_<=_-_(p8_`1)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_not_x2_in__{__p8_where_p8_is_Point_of_(TOP-REAL_2)_:_(_(_(_p8_`1_<=_p8_`2_&_-_(p8_`2)_<=_p8_`1_)_or_(_p8_`1_>=_p8_`2_&_p8_`1_<=_-_(p8_`2)_)_)_&_p8_<>_0._(TOP-REAL_2)_)__}__&_contradiction_)_)
percases ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } or ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ) by A139, A138, XBOOLE_0:def_3;
case x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ; ::_thesis: x1 = x2
then ex p8 being Point of (TOP-REAL 2) st
( p2 = p8 & ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) ;
then A179: |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| = |[(((p1 `1) / (p1 `2)) / (p1 `2)),(1 / (p1 `2))]| by A136, A178, Th14;
A180: p1 = |[(p1 `1),(p1 `2)]| by EUCLID:53;
set qq = |[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]|;
A181: (1 / (p1 `2)) " = ((p1 `2) ") "
.= p1 `2 ;
A182: now__::_thesis:_not_p1_`2_=_0
assume A183: p1 `2 = 0 ; ::_thesis: contradiction
then p1 `1 = 0 by A177;
hence contradiction by A177, A183, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
|[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| `2 = 1 / (p2 `2) by EUCLID:52;
then A184: 1 / (p1 `2) = 1 / (p2 `2) by A179, EUCLID:52;
|[(((p2 `1) / (p2 `2)) / (p2 `2)),(1 / (p2 `2))]| `1 = ((p2 `1) / (p2 `2)) / (p2 `2) by EUCLID:52;
then (p1 `1) / (p1 `2) = (p2 `1) / (p1 `2) by A179, A184, A181, A182, EUCLID:52, XCMPLX_1:53;
then p1 `1 = p2 `1 by A182, XCMPLX_1:53;
hence x1 = x2 by A184, A181, A180, EUCLID:53; ::_thesis: verum
end;
caseA185: ( x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) } & not x2 in { p8 where p8 is Point of (TOP-REAL 2) : ( ( ( p8 `1 <= p8 `2 & - (p8 `2) <= p8 `1 ) or ( p8 `1 >= p8 `2 & p8 `1 <= - (p8 `2) ) ) & p8 <> 0. (TOP-REAL 2) ) } ) ; ::_thesis: contradiction
A186: now__::_thesis:_not_p1_`2_=_0
assume A187: p1 `2 = 0 ; ::_thesis: contradiction
then p1 `1 = 0 by A177;
hence contradiction by A177, A187, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A188: now__::_thesis:_(_(_p1_`1_<=_p1_`2_&_-_(p1_`2)_<=_p1_`1_&_(p1_`1)_/_(p1_`2)_<=_1_)_or_(_p1_`1_>=_p1_`2_&_p1_`1_<=_-_(p1_`2)_&_(p1_`1)_/_(p1_`2)_<=_1_)_)
percases ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A177;
caseA189: ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) ; ::_thesis: (p1 `1) / (p1 `2) <= 1
then p1 `2 >= 0 ;
then (p1 `1) / (p1 `2) <= (p1 `2) / (p1 `2) by A189, XREAL_1:72;
hence (p1 `1) / (p1 `2) <= 1 by A186, XCMPLX_1:60; ::_thesis: verum
end;
caseA190: ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ; ::_thesis: (p1 `1) / (p1 `2) <= 1
then p1 `2 <= 0 ;
then (p1 `1) / (p1 `2) <= (p1 `2) / (p1 `2) by A190, XREAL_1:73;
hence (p1 `1) / (p1 `2) <= 1 by A186, XCMPLX_1:60; ::_thesis: verum
end;
end;
end;
A191: now__::_thesis:_(_(_p1_`1_<=_p1_`2_&_-_(p1_`2)_<=_p1_`1_&_-_1_<=_(p1_`1)_/_(p1_`2)_)_or_(_p1_`1_>=_p1_`2_&_p1_`1_<=_-_(p1_`2)_&_-_1_<=_(p1_`1)_/_(p1_`2)_)_)
percases ( ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) or ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ) by A177;
caseA192: ( p1 `1 <= p1 `2 & - (p1 `2) <= p1 `1 ) ; ::_thesis: - 1 <= (p1 `1) / (p1 `2)
then p1 `2 >= 0 ;
then (- (p1 `2)) / (p1 `2) <= (p1 `1) / (p1 `2) by A192, XREAL_1:72;
hence - 1 <= (p1 `1) / (p1 `2) by A186, XCMPLX_1:197; ::_thesis: verum
end;
case ( p1 `1 >= p1 `2 & p1 `1 <= - (p1 `2) ) ; ::_thesis: - 1 <= (p1 `1) / (p1 `2)
then ( - (p1 `1) >= - (- (p1 `2)) & p1 `2 <= 0 ) by XREAL_1:24;
then (- (p1 `1)) / (- (p1 `2)) >= (p1 `2) / (- (p1 `2)) by XREAL_1:72;
then (- (p1 `1)) / (- (p1 `2)) >= - 1 by A186, XCMPLX_1:198;
hence - 1 <= (p1 `1) / (p1 `2) by XCMPLX_1:191; ::_thesis: verum
end;
end;
end;
A193: ex p8 being Point of (TOP-REAL 2) st
( p2 = p8 & ( ( p8 `2 <= p8 `1 & - (p8 `1) <= p8 `2 ) or ( p8 `2 >= p8 `1 & p8 `2 <= - (p8 `1) ) ) & p8 <> 0. (TOP-REAL 2) ) by A185;
A194: now__::_thesis:_not_p2_`1_=_0
assume A195: p2 `1 = 0 ; ::_thesis: contradiction
then p2 `2 = 0 by A193;
hence contradiction by A193, A195, EUCLID:53, EUCLID:54; ::_thesis: verum
end;
A196: ex p9 being Point of (TOP-REAL 2) st
( p2 = p9 & ( ( p9 `2 <= p9 `1 & - (p9 `1) <= p9 `2 ) or ( p9 `2 >= p9 `1 & p9 `2 <= - (p9 `1) ) ) & p9 <> 0. (TOP-REAL 2) ) by A185;
A197: now__::_thesis:_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_&_-_1_<=_(p2_`2)_/_(p2_`1)_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_&_-_1_<=_(p2_`2)_/_(p2_`1)_)_)
percases ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) by A196;
caseA198: ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) ; ::_thesis: - 1 <= (p2 `2) / (p2 `1)
then p2 `1 >= 0 ;
then (- (p2 `1)) / (p2 `1) <= (p2 `2) / (p2 `1) by A198, XREAL_1:72;
hence - 1 <= (p2 `2) / (p2 `1) by A194, XCMPLX_1:197; ::_thesis: verum
end;
case ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ; ::_thesis: - 1 <= (p2 `2) / (p2 `1)
then ( - (p2 `2) >= - (- (p2 `1)) & p2 `1 <= 0 ) by XREAL_1:24;
then (- (p2 `2)) / (- (p2 `1)) >= (p2 `1) / (- (p2 `1)) by XREAL_1:72;
then (- (p2 `2)) / (- (p2 `1)) >= - 1 by A194, XCMPLX_1:198;
hence - 1 <= (p2 `2) / (p2 `1) by XCMPLX_1:191; ::_thesis: verum
end;
end;
end;
A199: Out_In_Sq . p2 = |[(1 / (p2 `1)),(((p2 `2) / (p2 `1)) / (p2 `1))]| by A193, Def1;
then 1 / (p1 `2) = ((p2 `2) / (p2 `1)) / (p2 `1) by A136, A178, SPPOL_2:1;
then A200: (p2 `2) / (p2 `1) = (1 / (p1 `2)) * (p2 `1) by A194, XCMPLX_1:87
.= (p2 `1) / (p1 `2) ;
((p1 `1) / (p1 `2)) / (p1 `2) = 1 / (p2 `1) by A136, A178, A199, SPPOL_2:1;
then (p1 `1) / (p1 `2) = (1 / (p2 `1)) * (p1 `2) by A186, XCMPLX_1:87
.= (p1 `2) / (p2 `1) ;
then A201: ((p2 `2) / (p2 `1)) * ((p1 `1) / (p1 `2)) = 1 by A194, A186, A200, XCMPLX_1:112;
then A202: p1 `1 <> 0 ;
(((p2 `2) / (p2 `1)) * ((p1 `1) / (p1 `2))) * (p1 `2) = p1 `2 by A201;
then ((p2 `2) / (p2 `1)) * (((p1 `1) / (p1 `2)) * (p1 `2)) = p1 `2 ;
then ((p2 `2) / (p2 `1)) * (p1 `1) = p1 `2 by A186, XCMPLX_1:87;
then A203: (p2 `2) / (p2 `1) = (p1 `2) / (p1 `1) by A202, XCMPLX_1:89;
A204: now__::_thesis:_(_(_p2_`2_<=_p2_`1_&_-_(p2_`1)_<=_p2_`2_&_(p2_`2)_/_(p2_`1)_<=_1_)_or_(_p2_`2_>=_p2_`1_&_p2_`2_<=_-_(p2_`1)_&_(p2_`2)_/_(p2_`1)_<=_1_)_)
percases ( ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) or ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ) by A196;
caseA205: ( p2 `2 <= p2 `1 & - (p2 `1) <= p2 `2 ) ; ::_thesis: (p2 `2) / (p2 `1) <= 1
then p2 `1 >= 0 ;
then (p2 `2) / (p2 `1) <= (p2 `1) / (p2 `1) by A205, XREAL_1:72;
hence (p2 `2) / (p2 `1) <= 1 by A194, XCMPLX_1:60; ::_thesis: verum
end;
caseA206: ( p2 `2 >= p2 `1 & p2 `2 <= - (p2 `1) ) ; ::_thesis: (p2 `2) / (p2 `1) <= 1
then p2 `1 <= 0 ;
then (p2 `2) / (p2 `1) <= (p2 `1) / (p2 `1) by A206, XREAL_1:73;
hence (p2 `2) / (p2 `1) <= 1 by A194, XCMPLX_1:60; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_0_<=_(p2_`2)_/_(p2_`1)_&_contradiction_)_or_(_0_>_(p2_`2)_/_(p2_`1)_&_contradiction_)_)
percases ( 0 <= (p2 `2) / (p2 `1) or 0 > (p2 `2) / (p2 `1) ) ;
case 0 <= (p2 `2) / (p2 `1) ; ::_thesis: contradiction
then A207: ( ( p1 `1 > 0 & p1 `2 >= 0 ) or ( p1 `1 < 0 & p1 `2 <= 0 ) ) by A201, A202;
now__::_thesis:_not_(p1_`1)_/_(p1_`2)_<>_1
assume (p1 `1) / (p1 `2) <> 1 ; ::_thesis: contradiction
then (p1 `1) / (p1 `2) < 1 by A188, XXREAL_0:1;
hence contradiction by A201, A204, A207, XREAL_1:162; ::_thesis: verum
end;
then p1 `1 = 1 * (p1 `2) by A186, XCMPLX_1:87;
then ((p2 `2) / (p2 `1)) * (p2 `1) = 1 * (p2 `1) by A186, A203, XCMPLX_1:60
.= p2 `1 ;
then p2 `2 = p2 `1 by A194, XCMPLX_1:87;
hence contradiction by A185, A196; ::_thesis: verum
end;
case 0 > (p2 `2) / (p2 `1) ; ::_thesis: contradiction
then A208: ( ( p1 `1 < 0 & p1 `2 > 0 ) or ( p1 `1 > 0 & p1 `2 < 0 ) ) by A203, XREAL_1:143;
now__::_thesis:_not_(p1_`1)_/_(p1_`2)_<>_-_1
assume (p1 `1) / (p1 `2) <> - 1 ; ::_thesis: contradiction
then - 1 < (p1 `1) / (p1 `2) by A191, XXREAL_0:1;
hence contradiction by A201, A197, A208, XREAL_1:166; ::_thesis: verum
end;
then p1 `1 = (- 1) * (p1 `2) by A186, XCMPLX_1:87
.= - (p1 `2) ;
then - (p1 `1) = p1 `2 ;
then (p2 `2) / (p2 `1) = - 1 by A202, A203, XCMPLX_1:197;
then p2 `2 = (- 1) * (p2 `1) by A194, XCMPLX_1:87;
then - (p2 `2) = p2 `1 ;
hence contradiction by A185, A196; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
then A209: Out_In_Sq is one-to-one by FUNCT_1:def_4;
A210: for s being Point of (TOP-REAL 2) st s in Kb holds
Out_In_Sq . s = s
proof
let t be Point of (TOP-REAL 2); ::_thesis: ( t in Kb implies Out_In_Sq . t = t )
assume t in Kb ; ::_thesis: Out_In_Sq . t = t
then A211: ex p4 being Point of (TOP-REAL 2) st
( p4 = t & ( ( - 1 = p4 `1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( p4 `1 = 1 & - 1 <= p4 `2 & p4 `2 <= 1 ) or ( - 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) or ( 1 = p4 `2 & - 1 <= p4 `1 & p4 `1 <= 1 ) ) ) by A1;
then A212: t <> 0. (TOP-REAL 2) by EUCLID:52, EUCLID:54;
A213: not t = 0. (TOP-REAL 2) by A211, EUCLID:52, EUCLID:54;
now__::_thesis:_(_(_(_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_or_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_)_&_Out_In_Sq_._t_=_t_)_or_(_not_(_t_`2_<=_t_`1_&_-_(t_`1)_<=_t_`2_)_&_not_(_t_`2_>=_t_`1_&_t_`2_<=_-_(t_`1)_)_&_Out_In_Sq_._t_=_t_)_)
percases ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) or ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ) ;
caseA214: ( ( t `2 <= t `1 & - (t `1) <= t `2 ) or ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t = t
then A215: Out_In_Sq . t = |[(1 / (t `1)),(((t `2) / (t `1)) / (t `1))]| by A213, Def1;
A216: ( ( 1 <= t `1 & t `1 >= - 1 ) or ( 1 >= t `1 & - 1 >= - (- (t `1)) ) ) by A211, A214, XREAL_1:24;
now__::_thesis:_(_(_t_`1_=_1_&_Out_In_Sq_._t_=_t_)_or_(_t_`1_=_-_1_&_Out_In_Sq_._t_=_t_)_)
percases ( t `1 = 1 or t `1 = - 1 ) by A211, A216, XXREAL_0:1;
case t `1 = 1 ; ::_thesis: Out_In_Sq . t = t
hence Out_In_Sq . t = t by A215, EUCLID:53; ::_thesis: verum
end;
case t `1 = - 1 ; ::_thesis: Out_In_Sq . t = t
hence Out_In_Sq . t = t by A215, EUCLID:53; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . t = t ; ::_thesis: verum
end;
caseA217: ( not ( t `2 <= t `1 & - (t `1) <= t `2 ) & not ( t `2 >= t `1 & t `2 <= - (t `1) ) ) ; ::_thesis: Out_In_Sq . t = t
then A218: Out_In_Sq . t = |[(((t `1) / (t `2)) / (t `2)),(1 / (t `2))]| by A212, Def1;
now__::_thesis:_(_(_t_`2_=_1_&_Out_In_Sq_._t_=_t_)_or_(_t_`2_=_-_1_&_Out_In_Sq_._t_=_t_)_)
percases ( t `2 = 1 or t `2 = - 1 ) by A211, A217;
case t `2 = 1 ; ::_thesis: Out_In_Sq . t = t
hence Out_In_Sq . t = t by A218, EUCLID:53; ::_thesis: verum
end;
case t `2 = - 1 ; ::_thesis: Out_In_Sq . t = t
hence Out_In_Sq . t = t by A218, EUCLID:53; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . t = t ; ::_thesis: verum
end;
end;
end;
hence Out_In_Sq . t = t ; ::_thesis: verum
end;
ex h being Function of ((TOP-REAL 2) | D),((TOP-REAL 2) | D) st
( h = Out_In_Sq & h is continuous ) by A2, Th40;
hence ex f being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) st
( f is continuous & f is one-to-one & ( for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not f . t in K0 \/ Kb ) & ( for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds
f . r in K0 ) & ( for s being Point of (TOP-REAL 2) st s in Kb holds
f . s = s ) ) by A209, A4, A72, A210; ::_thesis: verum
end;
theorem Th42: :: JGRAPH_2:42
for f, g being Function of I[01],(TOP-REAL 2)
for K0 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 & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 holds
rng f meets rng g
proof
reconsider B = {(0. (TOP-REAL 2))} as Subset of (TOP-REAL 2) ;
A1: B ` <> {} by Th9;
reconsider W = B ` as non empty Subset of (TOP-REAL 2) by Th9;
defpred S1[ Point of (TOP-REAL 2)] means ( ( - 1 = $1 `1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( $1 `1 = 1 & - 1 <= $1 `2 & $1 `2 <= 1 ) or ( - 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) or ( 1 = $1 `2 & - 1 <= $1 `1 & $1 `1 <= 1 ) );
A2: the carrier of ((TOP-REAL 2) | (B `)) = [#] ((TOP-REAL 2) | (B `))
.= B ` by PRE_TOPC:def_5 ;
reconsider Kb = { q where q is Point of (TOP-REAL 2) : S1[q] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for K0 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 & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 holds
rng f meets rng g
let K0 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 & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 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 & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & rng f misses K0 & rng g misses K0 implies rng f meets rng g )
A3: dom f = the carrier of I[01] by FUNCT_2:def_1;
assume A4: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & K0 = { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } & (f . O) `1 = - 1 & (f . I) `1 = 1 & - 1 <= (f . O) `2 & (f . O) `2 <= 1 & - 1 <= (f . I) `2 & (f . I) `2 <= 1 & (g . O) `2 = - 1 & (g . I) `2 = 1 & - 1 <= (g . O) `1 & (g . O) `1 <= 1 & - 1 <= (g . I) `1 & (g . I) `1 <= 1 & (rng f) /\ K0 = {} & (rng g) /\ K0 = {} ) ; :: according to XBOOLE_0:def_7 ::_thesis: rng f meets rng g
then consider h being Function of ((TOP-REAL 2) | (B `)),((TOP-REAL 2) | (B `)) such that
A5: h is continuous and
A6: h is one-to-one and
for t being Point of (TOP-REAL 2) st t in K0 & t <> 0. (TOP-REAL 2) holds
not h . t in K0 \/ Kb and
A7: for r being Point of (TOP-REAL 2) st not r in K0 \/ Kb holds
h . r in K0 and
A8: for s being Point of (TOP-REAL 2) st s in Kb holds
h . s = s by Th41;
rng f c= B `
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in B ` )
assume A9: x in rng f ; ::_thesis: x in B `
now__::_thesis:_not_x_in_B
assume x in B ; ::_thesis: contradiction
then A10: x = 0. (TOP-REAL 2) by TARSKI:def_1;
( (0. (TOP-REAL 2)) `1 = 0 & (0. (TOP-REAL 2)) `2 = 0 ) by EUCLID:52, EUCLID:54;
then 0. (TOP-REAL 2) in K0 by A4;
hence contradiction by A4, A9, A10, XBOOLE_0:def_4; ::_thesis: verum
end;
then x in the carrier of (TOP-REAL 2) \ B by A9, XBOOLE_0:def_5;
hence x in B ` by SUBSET_1:def_4; ::_thesis: verum
end;
then A11: ex w being Function of I[01],(TOP-REAL 2) st
( w is continuous & w = h * f ) by A4, A5, A1, Th12;
then reconsider d1 = h * f as Function of I[01],(TOP-REAL 2) ;
the carrier of ((TOP-REAL 2) | W) <> {} ;
then A12: dom h = the carrier of ((TOP-REAL 2) | (B `)) by FUNCT_2:def_1;
rng g c= B `
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in B ` )
assume A13: x in rng g ; ::_thesis: x in B `
now__::_thesis:_not_x_in_B
assume x in B ; ::_thesis: contradiction
then A14: x = 0. (TOP-REAL 2) by TARSKI:def_1;
0. (TOP-REAL 2) in K0 by A4, Th3;
hence contradiction by A4, A13, A14, XBOOLE_0:def_4; ::_thesis: verum
end;
then x in the carrier of (TOP-REAL 2) \ B by A13, XBOOLE_0:def_5;
hence x in B ` by SUBSET_1:def_4; ::_thesis: verum
end;
then A15: ex w2 being Function of I[01],(TOP-REAL 2) st
( w2 is continuous & w2 = h * g ) by A4, A5, A1, Th12;
then reconsider d2 = h * g as Function of I[01],(TOP-REAL 2) ;
A16: dom g = the carrier of I[01] by FUNCT_2:def_1;
A17: for r being Point of I[01] holds
( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 )
proof
let r be Point of I[01]; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 )
A18: ( g . r in Kb implies d2 . r in K0 \/ Kb )
proof
A19: d2 . r = h . (g . r) by A16, FUNCT_1:13;
assume A20: g . r in Kb ; ::_thesis: d2 . r in K0 \/ Kb
then h . (g . r) = g . r by A8;
hence d2 . r in K0 \/ Kb by A20, A19, XBOOLE_0:def_3; ::_thesis: verum
end;
f . r in rng f by A3, FUNCT_1:3;
then A21: not f . r in K0 by A4, XBOOLE_0:def_4;
A22: ( not f . r in Kb implies d1 . r in K0 \/ Kb )
proof
assume not f . r in Kb ; ::_thesis: d1 . r in K0 \/ Kb
then not f . r in K0 \/ Kb by A21, XBOOLE_0:def_3;
then A23: h . (f . r) in K0 by A7;
d1 . r = h . (f . r) by A3, FUNCT_1:13;
hence d1 . r in K0 \/ Kb by A23, XBOOLE_0:def_3; ::_thesis: verum
end;
g . r in rng g by A16, FUNCT_1:3;
then A24: not g . r in K0 by A4, XBOOLE_0:def_4;
A25: ( not g . r in Kb implies d2 . r in K0 \/ Kb )
proof
assume not g . r in Kb ; ::_thesis: d2 . r in K0 \/ Kb
then not g . r in K0 \/ Kb by A24, XBOOLE_0:def_3;
then A26: h . (g . r) in K0 by A7;
d2 . r = h . (g . r) by A16, FUNCT_1:13;
hence d2 . r in K0 \/ Kb by A26, XBOOLE_0:def_3; ::_thesis: verum
end;
A27: ( f . r in Kb implies d1 . r in K0 \/ Kb )
proof
A28: d1 . r = h . (f . r) by A3, FUNCT_1:13;
assume A29: f . r in Kb ; ::_thesis: d1 . r in K0 \/ Kb
then h . (f . r) = f . r by A8;
hence d1 . r in K0 \/ Kb by A29, A28, XBOOLE_0:def_3; ::_thesis: verum
end;
now__::_thesis:_(_(_d1_._r_in_K0_&_d2_._r_in_K0_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_or_(_d1_._r_in_K0_&_d2_._r_in_Kb_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_or_(_d1_._r_in_Kb_&_d2_._r_in_K0_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_or_(_d1_._r_in_Kb_&_d2_._r_in_Kb_&_-_1_<=_(d1_._r)_`1_&_(d1_._r)_`1_<=_1_&_-_1_<=_(d2_._r)_`1_&_(d2_._r)_`1_<=_1_&_-_1_<=_(d1_._r)_`2_&_(d1_._r)_`2_<=_1_&_-_1_<=_(d2_._r)_`2_&_(d2_._r)_`2_<=_1_)_)
percases ( ( d1 . r in K0 & d2 . r in K0 ) or ( d1 . r in K0 & d2 . r in Kb ) or ( d1 . r in Kb & d2 . r in K0 ) or ( d1 . r in Kb & d2 . r in Kb ) ) by A22, A27, A25, A18, XBOOLE_0:def_3;
case ( d1 . r in K0 & d2 . r in K0 ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 )
then ( ex p being Point of (TOP-REAL 2) st
( p = d1 . r & - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) & ex q being Point of (TOP-REAL 2) st
( q = d2 . r & - 1 < q `1 & q `1 < 1 & - 1 < q `2 & q `2 < 1 ) ) by A4;
hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum
end;
case ( d1 . r in K0 & d2 . r in Kb ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 )
then ( ex p being Point of (TOP-REAL 2) st
( p = d1 . r & - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) & ex q being Point of (TOP-REAL 2) st
( q = d2 . r & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ) by A4;
hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum
end;
case ( d1 . r in Kb & d2 . r in K0 ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 )
then ( ex p being Point of (TOP-REAL 2) st
( p = d2 . r & - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) & ex q being Point of (TOP-REAL 2) st
( q = d1 . r & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ) by A4;
hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum
end;
case ( d1 . r in Kb & d2 . r in Kb ) ; ::_thesis: ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 )
then ( ex p being Point of (TOP-REAL 2) st
( p = d2 . r & ( ( - 1 = p `1 & - 1 <= p `2 & p `2 <= 1 ) or ( p `1 = 1 & - 1 <= p `2 & p `2 <= 1 ) or ( - 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) or ( 1 = p `2 & - 1 <= p `1 & p `1 <= 1 ) ) ) & ex q being Point of (TOP-REAL 2) st
( q = d1 . r & ( ( - 1 = q `1 & - 1 <= q `2 & q `2 <= 1 ) or ( q `1 = 1 & - 1 <= q `2 & q `2 <= 1 ) or ( - 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) or ( 1 = q `2 & - 1 <= q `1 & q `1 <= 1 ) ) ) ) ;
hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum
end;
end;
end;
hence ( - 1 <= (d1 . r) `1 & (d1 . r) `1 <= 1 & - 1 <= (d2 . r) `1 & (d2 . r) `1 <= 1 & - 1 <= (d1 . r) `2 & (d1 . r) `2 <= 1 & - 1 <= (d2 . r) `2 & (d2 . r) `2 <= 1 ) ; ::_thesis: verum
end;
f . I in Kb by A4;
then h . (f . I) = f . I by A8;
then A30: (d1 . I) `1 = 1 by A4, A3, FUNCT_1:13;
f . O in Kb by A4;
then h . (f . O) = f . O by A8;
then A31: (d1 . O) `1 = - 1 by A4, A3, FUNCT_1:13;
g . I in Kb by A4;
then h . (g . I) = g . I by A8;
then A32: (d2 . I) `2 = 1 by A4, A16, FUNCT_1:13;
g . O in Kb by A4;
then h . (g . O) = g . O by A8;
then A33: (d2 . O) `2 = - 1 by A4, A16, FUNCT_1:13;
set s = the Element of (rng d1) /\ (rng d2);
( d1 is one-to-one & d2 is one-to-one ) by A4, A6, FUNCT_1:24;
then rng d1 meets rng d2 by A4, A11, A15, A31, A30, A33, A32, A17, JGRAPH_1:47;
then A34: (rng d1) /\ (rng d2) <> {} by XBOOLE_0:def_7;
then the Element of (rng d1) /\ (rng d2) in rng d1 by XBOOLE_0:def_4;
then consider t1 being set such that
A35: t1 in dom d1 and
A36: the Element of (rng d1) /\ (rng d2) = d1 . t1 by FUNCT_1:def_3;
A37: f . t1 in rng f by A3, A35, FUNCT_1:3;
the Element of (rng d1) /\ (rng d2) in rng d2 by A34, XBOOLE_0:def_4;
then consider t2 being set such that
A38: t2 in dom d2 and
A39: the Element of (rng d1) /\ (rng d2) = d2 . t2 by FUNCT_1:def_3;
h . (f . t1) = d1 . t1 by A35, FUNCT_1:12;
then A40: h . (f . t1) = h . (g . t2) by A36, A38, A39, FUNCT_1:12;
rng g c= the carrier of (TOP-REAL 2) \ B
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in rng g or e in the carrier of (TOP-REAL 2) \ B )
assume A41: e in rng g ; ::_thesis: e in the carrier of (TOP-REAL 2) \ B
now__::_thesis:_not_e_in_B
assume e in B ; ::_thesis: contradiction
then A42: e = 0. (TOP-REAL 2) by TARSKI:def_1;
0. (TOP-REAL 2) in { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } by Th3;
hence contradiction by A4, A41, A42, XBOOLE_0:def_4; ::_thesis: verum
end;
hence e in the carrier of (TOP-REAL 2) \ B by A41, XBOOLE_0:def_5; ::_thesis: verum
end;
then A43: rng g c= the carrier of ((TOP-REAL 2) | (B `)) by A2, SUBSET_1:def_4;
dom g = the carrier of I[01] by FUNCT_2:def_1;
then A44: g . t2 in rng g by A38, FUNCT_1:3;
rng f c= the carrier of (TOP-REAL 2) \ B
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in rng f or e in the carrier of (TOP-REAL 2) \ B )
assume A45: e in rng f ; ::_thesis: e in the carrier of (TOP-REAL 2) \ B
now__::_thesis:_not_e_in_B
assume e in B ; ::_thesis: contradiction
then A46: e = 0. (TOP-REAL 2) by TARSKI:def_1;
0. (TOP-REAL 2) in { p where p is Point of (TOP-REAL 2) : ( - 1 < p `1 & p `1 < 1 & - 1 < p `2 & p `2 < 1 ) } by Th3;
hence contradiction by A4, A45, A46, XBOOLE_0:def_4; ::_thesis: verum
end;
hence e in the carrier of (TOP-REAL 2) \ B by A45, XBOOLE_0:def_5; ::_thesis: verum
end;
then rng f c= the carrier of ((TOP-REAL 2) | (B `)) by A2, SUBSET_1:def_4;
then f . t1 = g . t2 by A6, A43, A40, A12, A37, A44, FUNCT_1:def_4;
then (rng f) /\ (rng g) <> {} by A37, A44, XBOOLE_0:def_4;
hence rng f meets rng g by XBOOLE_0:def_7; ::_thesis: verum
end;
theorem Th43: :: JGRAPH_2:43
for A, B, C, D being real number
for f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds
f is continuous
proof
reconsider h11 = proj1 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
set K0 = [#] (TOP-REAL 2);
let A, B, C, D be real number ; ::_thesis: for f being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds
f is continuous
let f be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( ( for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) implies f is continuous )
A1: (TOP-REAL 2) | ([#] (TOP-REAL 2)) = TopStruct(# the carrier of (TOP-REAL 2), the topology of (TOP-REAL 2) #) by TSEP_1:93;
then reconsider h1 = h11 as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 ;
h11 is continuous by JORDAN5A:27;
then h1 is continuous by A1, PRE_TOPC:32;
then consider g1 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that
A2: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2)))
for r1 being real number st h1 . p = r1 holds
g1 . p = A * r1 and
A3: g1 is continuous by Th23;
reconsider f1 = proj1 * f as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by A1, TOPMETR:17;
consider g11 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that
A4: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2)))
for r1 being real number st g1 . p = r1 holds
g11 . p = r1 + B and
A5: g11 is continuous by A3, Th24;
reconsider f2 = proj2 * f as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by A1, TOPMETR:17;
reconsider h11 = proj2 as Function of (TOP-REAL 2),R^1 by TOPMETR:17;
reconsider h1 = h11 as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 by A1;
dom f1 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then A6: dom f1 = dom g11 by A1, FUNCT_2:def_1;
assume A7: for t being Point of (TOP-REAL 2) holds f . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ; ::_thesis: f is continuous
A8: for x being set st x in dom f1 holds
f1 . x = g11 . x
proof
let x be set ; ::_thesis: ( x in dom f1 implies f1 . x = g11 . x )
assume A9: x in dom f1 ; ::_thesis: f1 . x = g11 . x
then reconsider p = x as Point of (TOP-REAL 2) by FUNCT_2:def_1;
f1 . x = proj1 . (f . x) by A9, FUNCT_1:12;
then A10: f1 . x = proj1 . |[((A * (p `1)) + B),((C * (p `2)) + D)]| by A7
.= (A * (p `1)) + B by PSCOMP_1:65
.= (A * (proj1 . p)) + B by PSCOMP_1:def_5 ;
A * (proj1 . p) = g1 . p by A1, A2;
hence f1 . x = g11 . x by A1, A4, A10; ::_thesis: verum
end;
h11 is continuous by JORDAN5A:27;
then h1 is continuous by A1, PRE_TOPC:32;
then consider g1 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that
A11: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2)))
for r1 being real number st h1 . p = r1 holds
g1 . p = C * r1 and
A12: g1 is continuous by Th23;
consider g11 being Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),R^1 such that
A13: for p being Point of ((TOP-REAL 2) | ([#] (TOP-REAL 2)))
for r1 being real number st g1 . p = r1 holds
g11 . p = r1 + D and
A14: g11 is continuous by A12, Th24;
A15: for x being set st x in dom f2 holds
f2 . x = g11 . x
proof
let x be set ; ::_thesis: ( x in dom f2 implies f2 . x = g11 . x )
assume A16: x in dom f2 ; ::_thesis: f2 . x = g11 . x
then reconsider p = x as Point of (TOP-REAL 2) by FUNCT_2:def_1;
f2 . x = proj2 . (f . x) by A16, FUNCT_1:12;
then A17: f2 . x = proj2 . |[((A * (p `1)) + B),((C * (p `2)) + D)]| by A7
.= (C * (p `2)) + D by PSCOMP_1:65
.= (C * (proj2 . p)) + D by PSCOMP_1:def_6 ;
C * (proj2 . p) = g1 . p by A1, A11;
hence f2 . x = g11 . x by A1, A13, A17; ::_thesis: verum
end;
reconsider f0 = f as Function of ((TOP-REAL 2) | ([#] (TOP-REAL 2))),((TOP-REAL 2) | ([#] (TOP-REAL 2))) by A1;
A18: for x, y, r, s being real number st |[x,y]| in [#] (TOP-REAL 2) & r = f1 . |[x,y]| & s = f2 . |[x,y]| holds
f0 . |[x,y]| = |[r,s]|
proof
let x, y, r, s be real number ; ::_thesis: ( |[x,y]| in [#] (TOP-REAL 2) & r = f1 . |[x,y]| & s = f2 . |[x,y]| implies f0 . |[x,y]| = |[r,s]| )
assume that
|[x,y]| in [#] (TOP-REAL 2) and
A19: ( r = f1 . |[x,y]| & s = f2 . |[x,y]| ) ; ::_thesis: f0 . |[x,y]| = |[r,s]|
A20: f . |[x,y]| is Point of (TOP-REAL 2) ;
dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then ( proj1 . (f0 . |[x,y]|) = r & proj2 . (f0 . |[x,y]|) = s ) by A19, FUNCT_1:13;
hence f0 . |[x,y]| = |[r,s]| by A20, Th8; ::_thesis: verum
end;
dom f2 = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
then dom f2 = dom g11 by A1, FUNCT_2:def_1;
then A21: f2 is continuous by A14, A15, FUNCT_1:2;
f1 is continuous by A5, A6, A8, FUNCT_1:2;
then f0 is continuous by A21, A18, Th35;
hence f is continuous by A1, PRE_TOPC:34; ::_thesis: verum
end;
definition
let A, B, C, D be real number ;
func AffineMap (A,B,C,D) -> Function of (TOP-REAL 2),(TOP-REAL 2) means :Def2: :: JGRAPH_2:def 2
for t being Point of (TOP-REAL 2) holds it . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]|;
existence
ex b1 being Function of (TOP-REAL 2),(TOP-REAL 2) st
for t being Point of (TOP-REAL 2) holds b1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]|
proof
defpred S1[ set , set ] means for t being Point of (TOP-REAL 2) st t = $1 holds
$2 = |[((A * (t `1)) + B),((C * (t `2)) + D)]|;
A1: for x being set st x in the carrier of (TOP-REAL 2) holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies ex y being set st S1[x,y] )
assume x in the carrier of (TOP-REAL 2) ; ::_thesis: ex y being set st S1[x,y]
then reconsider t2 = x as Point of (TOP-REAL 2) ;
reconsider y2 = |[((A * (t2 `1)) + B),((C * (t2 `2)) + D)]| as set ;
for t being Point of (TOP-REAL 2) st t = x holds
y2 = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ;
hence ex y being set st S1[x,y] ; ::_thesis: verum
end;
ex ff being Function st
( dom ff = the carrier of (TOP-REAL 2) & ( for x being set st x in the carrier of (TOP-REAL 2) holds
S1[x,ff . x] ) ) from CLASSES1:sch_1(A1);
then consider ff being Function such that
A2: dom ff = the carrier of (TOP-REAL 2) and
A3: for x being set st x in the carrier of (TOP-REAL 2) holds
for t being Point of (TOP-REAL 2) st t = x holds
ff . x = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ;
for x being set st x in the carrier of (TOP-REAL 2) holds
ff . x in the carrier of (TOP-REAL 2)
proof
let x be set ; ::_thesis: ( x in the carrier of (TOP-REAL 2) implies ff . x in the carrier of (TOP-REAL 2) )
assume x in the carrier of (TOP-REAL 2) ; ::_thesis: ff . x in the carrier of (TOP-REAL 2)
then reconsider t = x as Point of (TOP-REAL 2) ;
ff . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| by A3;
hence ff . x in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider ff = ff as Function of (TOP-REAL 2),(TOP-REAL 2) by A2, FUNCT_2:3;
take ff ; ::_thesis: for t being Point of (TOP-REAL 2) holds ff . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]|
thus for t being Point of (TOP-REAL 2) holds ff . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| by A3; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (TOP-REAL 2),(TOP-REAL 2) st ( for t being Point of (TOP-REAL 2) holds b1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) & ( for t being Point of (TOP-REAL 2) holds b2 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) holds
b1 = b2
proof
let m1, m2 be Function of (TOP-REAL 2),(TOP-REAL 2); ::_thesis: ( ( for t being Point of (TOP-REAL 2) holds m1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) & ( for t being Point of (TOP-REAL 2) holds m2 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ) implies m1 = m2 )
assume that
A4: for t being Point of (TOP-REAL 2) holds m1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| and
A5: for t being Point of (TOP-REAL 2) holds m2 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| ; ::_thesis: m1 = m2
for x being Point of (TOP-REAL 2) holds m1 . x = m2 . x
proof
let t be Point of (TOP-REAL 2); ::_thesis: m1 . t = m2 . t
thus m1 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| by A4
.= m2 . t by A5 ; ::_thesis: verum
end;
hence m1 = m2 by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines AffineMap JGRAPH_2:def_2_:_
for A, B, C, D being real number
for b5 being Function of (TOP-REAL 2),(TOP-REAL 2) holds
( b5 = AffineMap (A,B,C,D) iff for t being Point of (TOP-REAL 2) holds b5 . t = |[((A * (t `1)) + B),((C * (t `2)) + D)]| );
registration
let a, b, c, d be real number ;
cluster AffineMap (a,b,c,d) -> continuous ;
coherence
AffineMap (a,b,c,d) is continuous
proof
for t being Point of (TOP-REAL 2) holds (AffineMap (a,b,c,d)) . t = |[((a * (t `1)) + b),((c * (t `2)) + d)]| by Def2;
hence AffineMap (a,b,c,d) is continuous by Th43; ::_thesis: verum
end;
end;
theorem Th44: :: JGRAPH_2:44
for A, B, C, D being real number st A > 0 & C > 0 holds
AffineMap (A,B,C,D) is one-to-one
proof
let A, B, C, D be real number ; ::_thesis: ( A > 0 & C > 0 implies AffineMap (A,B,C,D) is one-to-one )
assume that
A1: A > 0 and
A2: C > 0 ; ::_thesis: AffineMap (A,B,C,D) is one-to-one
set ff = AffineMap (A,B,C,D);
for x1, x2 being set st x1 in dom (AffineMap (A,B,C,D)) & x2 in dom (AffineMap (A,B,C,D)) & (AffineMap (A,B,C,D)) . x1 = (AffineMap (A,B,C,D)) . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom (AffineMap (A,B,C,D)) & x2 in dom (AffineMap (A,B,C,D)) & (AffineMap (A,B,C,D)) . x1 = (AffineMap (A,B,C,D)) . x2 implies x1 = x2 )
assume that
A3: x1 in dom (AffineMap (A,B,C,D)) and
A4: x2 in dom (AffineMap (A,B,C,D)) and
A5: (AffineMap (A,B,C,D)) . x1 = (AffineMap (A,B,C,D)) . x2 ; ::_thesis: x1 = x2
reconsider p2 = x2 as Point of (TOP-REAL 2) by A4;
reconsider p1 = x1 as Point of (TOP-REAL 2) by A3;
A6: ( (AffineMap (A,B,C,D)) . x1 = |[((A * (p1 `1)) + B),((C * (p1 `2)) + D)]| & (AffineMap (A,B,C,D)) . x2 = |[((A * (p2 `1)) + B),((C * (p2 `2)) + D)]| ) by Def2;
then (A * (p1 `1)) + B = (A * (p2 `1)) + B by A5, SPPOL_2:1;
then p1 `1 = (A * (p2 `1)) / A by A1, XCMPLX_1:89;
then A7: p1 `1 = p2 `1 by A1, XCMPLX_1:89;
(C * (p1 `2)) + D = (C * (p2 `2)) + D by A5, A6, SPPOL_2:1;
then p1 `2 = (C * (p2 `2)) / C by A2, XCMPLX_1:89;
hence x1 = x2 by A2, A7, TOPREAL3:6, XCMPLX_1:89; ::_thesis: verum
end;
hence AffineMap (A,B,C,D) is one-to-one by FUNCT_1:def_4; ::_thesis: verum
end;
theorem :: JGRAPH_2:45
for f, g being Function of I[01],(TOP-REAL 2)
for a, b, c, d being real number
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 & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds
( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds
( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) holds
rng f meets rng g
proof
defpred S1[ Point of (TOP-REAL 2)] means ( - 1 < $1 `1 & $1 `1 < 1 & - 1 < $1 `2 & $1 `2 < 1 );
reconsider K0 = { p where p is Point of (TOP-REAL 2) : S1[p] } as Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
let f, g be Function of I[01],(TOP-REAL 2); ::_thesis: for a, b, c, d being real number
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 & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds
( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds
( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) holds
rng f meets rng g
let a, b, c, d be real number ; ::_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 & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds
( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds
( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (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 & (f . O) `1 = a & (f . I) `1 = b & c <= (f . O) `2 & (f . O) `2 <= d & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & (g . I) `2 = d & a <= (g . O) `1 & (g . O) `1 <= b & a <= (g . I) `1 & (g . I) `1 <= b & a < b & c < d & ( for r being Point of I[01] holds
( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) ) & ( for r being Point of I[01] holds
( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ) implies rng f meets rng g )
assume that
A1: ( O = 0 & I = 1 ) and
A2: ( f is continuous & f is one-to-one & g is continuous & g is one-to-one ) and
A3: (f . O) `1 = a and
A4: (f . I) `1 = b and
A5: c <= (f . O) `2 and
A6: (f . O) `2 <= d and
A7: c <= (f . I) `2 and
A8: (f . I) `2 <= d and
A9: (g . O) `2 = c and
A10: (g . I) `2 = d and
A11: a <= (g . O) `1 and
A12: (g . O) `1 <= b and
A13: a <= (g . I) `1 and
A14: (g . I) `1 <= b and
A15: a < b and
A16: c < d and
A17: for r being Point of I[01] holds
( not a < (f . r) `1 or not (f . r) `1 < b or not c < (f . r) `2 or not (f . r) `2 < d ) and
A18: for r being Point of I[01] holds
( not a < (g . r) `1 or not (g . r) `1 < b or not c < (g . r) `2 or not (g . r) `2 < d ) ; ::_thesis: rng f meets rng g
set A = 2 / (b - a);
set B = 1 - ((2 * b) / (b - a));
set C = 2 / (d - c);
set D = 1 - ((2 * d) / (d - c));
set ff = AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))));
reconsider f2 = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) * f, g2 = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) * g as Function of I[01],(TOP-REAL 2) ;
A19: d - c > 0 by A16, XREAL_1:50;
then A20: 2 / (d - c) > 0 by XREAL_1:139;
A21: dom g = the carrier of I[01] by FUNCT_2:def_1;
then A22: g2 . I = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (g . I) by FUNCT_1:13
.= |[(((2 / (b - a)) * ((g . I) `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * d) + (1 - ((2 * d) / (d - c))))]| by A10, Def2 ;
then A23: (g2 . I) `2 = ((2 / (d - c)) * d) + (1 - ((2 * d) / (d - c))) by EUCLID:52
.= ((d * 2) / (d - c)) + (1 - ((2 * d) / (d - c)))
.= 1 ;
A24: g2 . O = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (g . O) by A21, FUNCT_1:13
.= |[(((2 / (b - a)) * ((g . O) `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * c) + (1 - ((2 * d) / (d - c))))]| by A9, Def2 ;
then A25: (g2 . O) `2 = ((2 / (d - c)) * c) + (1 - ((2 * d) / (d - c))) by EUCLID:52
.= ((c * 2) / (d - c)) + (1 - ((2 * d) / (d - c)))
.= ((c * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60
.= ((c * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c))
.= ((c * 2) + ((d - c) - (2 * d))) / (d - c)
.= (- (d - c)) / (d - c)
.= - ((d - c) / (d - c))
.= - 1 by A19, XCMPLX_1:60 ;
A26: b - a > 0 by A15, XREAL_1:50;
A27: ( - 1 <= (g2 . O) `1 & (g2 . O) `1 <= 1 & - 1 <= (g2 . I) `1 & (g2 . I) `1 <= 1 )
proof
reconsider s1 = (g . I) `1 as Real ;
reconsider s0 = (g . O) `1 as Real ;
A28: (a - b) / (b - a) = (- (b - a)) / (b - a)
.= - ((b - a) / (b - a))
.= - 1 by A26, XCMPLX_1:60 ;
A29: (g2 . I) `1 = ((2 / (b - a)) * s1) + (1 - ((2 * b) / (b - a))) by A22, EUCLID:52
.= ((s1 * 2) / (b - a)) + (1 - ((2 * b) / (b - a)))
.= ((s1 * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60
.= ((s1 * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a))
.= ((s1 * 2) + ((b - a) - (2 * b))) / (b - a)
.= (((s1 - b) + (s1 - b)) - (a - b)) / (b - a) ;
b - b >= s0 - b by A12, XREAL_1:9;
then (0 + (b - b)) - (a - b) >= ((s0 - b) + (s0 - b)) - (a - b) by XREAL_1:9;
then A30: (b - a) / (b - a) >= (((s0 - b) + (s0 - b)) - (a - b)) / (b - a) by A26, XREAL_1:72;
b - b >= s1 - b by A14, XREAL_1:9;
then A31: (0 + (b - b)) - (a - b) >= ((s1 - b) + (s1 - b)) - (a - b) by XREAL_1:9;
a - b <= s1 - b by A13, XREAL_1:9;
then (a - b) + (a - b) <= (s1 - b) + (s1 - b) by XREAL_1:7;
then A32: ((a - b) + (a - b)) - (a - b) <= ((s1 - b) + (s1 - b)) - (a - b) by XREAL_1:9;
a - b <= s0 - b by A11, XREAL_1:9;
then (a - b) + (a - b) <= (s0 - b) + (s0 - b) by XREAL_1:7;
then A33: ((a - b) + (a - b)) - (a - b) <= ((s0 - b) + (s0 - b)) - (a - b) by XREAL_1:9;
(g2 . O) `1 = ((2 / (b - a)) * s0) + (1 - ((2 * b) / (b - a))) by A24, EUCLID:52
.= ((s0 * 2) / (b - a)) + (1 - ((2 * b) / (b - a)))
.= ((s0 * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60
.= ((s0 * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a))
.= ((s0 * 2) + ((b - a) - (2 * b))) / (b - a)
.= (((s0 - b) + (s0 - b)) - (a - b)) / (b - a) ;
hence ( - 1 <= (g2 . O) `1 & (g2 . O) `1 <= 1 & - 1 <= (g2 . I) `1 & (g2 . I) `1 <= 1 ) by A26, A33, A28, A30, A29, A32, A31, XREAL_1:72; ::_thesis: verum
end;
A34: now__::_thesis:_not_rng_f2_meets_K0
assume rng f2 meets K0 ; ::_thesis: contradiction
then consider x being set such that
A35: x in rng f2 and
A36: x in K0 by XBOOLE_0:3;
reconsider q = x as Point of (TOP-REAL 2) by A35;
consider p being Point of (TOP-REAL 2) such that
A37: p = q and
A38: - 1 < p `1 and
A39: p `1 < 1 and
A40: - 1 < p `2 and
A41: p `2 < 1 by A36;
consider z being set such that
A42: z in dom f2 and
A43: x = f2 . z by A35, FUNCT_1:def_3;
reconsider u = z as Point of I[01] by A42;
reconsider t = f . u as Point of (TOP-REAL 2) ;
A44: ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) = (((t `1) * 2) / (b - a)) + (1 - ((2 * b) / (b - a)))
.= (((t `1) * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60
.= (((t `1) * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a))
.= (((t `1) * 2) + ((b - a) - (2 * b))) / (b - a)
.= ((2 * ((t `1) - b)) - (a - b)) / (b - a) ;
A45: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = p by A37, A42, A43, FUNCT_1:12;
A46: ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) = (((t `2) * 2) / (d - c)) + (1 - ((2 * d) / (d - c)))
.= (((t `2) * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60
.= (((t `2) * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c))
.= (((t `2) * 2) + ((d - c) - (2 * d))) / (d - c)
.= ((2 * ((t `2) - d)) - (c - d)) / (d - c) ;
A47: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = |[(((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))))]| by Def2;
then - 1 < ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) by A40, A45, EUCLID:52;
then (- 1) * (d - c) < (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A46, XREAL_1:68;
then (- 1) * (d - c) < (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87;
then ((- 1) * (d - c)) + (c - d) < ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8;
then (2 * (c - d)) / 2 < (2 * ((t `2) - d)) / 2 by XREAL_1:74;
then A48: c < t `2 by XREAL_1:9;
((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) < 1 by A41, A47, A45, EUCLID:52;
then 1 * (d - c) > (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A46, XREAL_1:68;
then 1 * (d - c) > (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87;
then (1 * (d - c)) + (c - d) > ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8;
then 0 / 2 > (((t `2) - d) * 2) / 2 ;
then A49: 0 + d > t `2 by XREAL_1:19;
((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) < 1 by A39, A47, A45, EUCLID:52;
then 1 * (b - a) > (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A44, XREAL_1:68;
then 1 * (b - a) > (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87;
then (1 * (b - a)) + (a - b) > ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8;
then 0 / 2 > (((t `1) - b) * 2) / 2 ;
then A50: 0 + b > t `1 by XREAL_1:19;
- 1 < ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) by A38, A47, A45, EUCLID:52;
then (- 1) * (b - a) < (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A44, XREAL_1:68;
then (- 1) * (b - a) < (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87;
then ((- 1) * (b - a)) + (a - b) < ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8;
then (2 * (a - b)) / 2 < (2 * ((t `1) - b)) / 2 by XREAL_1:74;
then a < t `1 by XREAL_1:9;
hence contradiction by A17, A50, A48, A49; ::_thesis: verum
end;
A51: dom f = the carrier of I[01] by FUNCT_2:def_1;
then A52: f2 . I = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (f . I) by FUNCT_1:13
.= |[(((2 / (b - a)) * b) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (1 - ((2 * d) / (d - c))))]| by A4, Def2 ;
then A53: (f2 . I) `1 = ((2 / (b - a)) * b) + (1 - ((2 * b) / (b - a))) by EUCLID:52
.= ((b * 2) / (b - a)) + (1 - ((2 * b) / (b - a)))
.= 1 ;
A54: f2 . O = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (f . O) by A51, FUNCT_1:13
.= |[(((2 / (b - a)) * a) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (1 - ((2 * d) / (d - c))))]| by A3, Def2 ;
then A55: (f2 . O) `1 = ((2 / (b - a)) * a) + (1 - ((2 * b) / (b - a))) by EUCLID:52
.= ((a * 2) / (b - a)) + (1 - ((2 * b) / (b - a)))
.= ((a * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60
.= ((a * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a))
.= ((a * 2) + ((b - a) - (2 * b))) / (b - a)
.= (- (b - a)) / (b - a)
.= - ((b - a) / (b - a))
.= - 1 by A26, XCMPLX_1:60 ;
A56: now__::_thesis:_not_rng_g2_meets_K0
assume rng g2 meets K0 ; ::_thesis: contradiction
then consider x being set such that
A57: x in rng g2 and
A58: x in K0 by XBOOLE_0:3;
reconsider q = x as Point of (TOP-REAL 2) by A57;
consider p being Point of (TOP-REAL 2) such that
A59: p = q and
A60: - 1 < p `1 and
A61: p `1 < 1 and
A62: - 1 < p `2 and
A63: p `2 < 1 by A58;
consider z being set such that
A64: z in dom g2 and
A65: x = g2 . z by A57, FUNCT_1:def_3;
reconsider u = z as Point of I[01] by A64;
reconsider t = g . u as Point of (TOP-REAL 2) ;
A66: ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) = (((t `1) * 2) / (b - a)) + (1 - ((2 * b) / (b - a)))
.= (((t `1) * 2) / (b - a)) + (((b - a) / (b - a)) - ((2 * b) / (b - a))) by A26, XCMPLX_1:60
.= (((t `1) * 2) / (b - a)) + (((b - a) - (2 * b)) / (b - a))
.= (((t `1) * 2) + ((b - a) - (2 * b))) / (b - a)
.= ((2 * ((t `1) - b)) - (a - b)) / (b - a) ;
A67: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = p by A59, A64, A65, FUNCT_1:12;
A68: ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) = (((t `2) * 2) / (d - c)) + (1 - ((2 * d) / (d - c)))
.= (((t `2) * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60
.= (((t `2) * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c))
.= (((t `2) * 2) + ((d - c) - (2 * d))) / (d - c)
.= ((2 * ((t `2) - d)) - (c - d)) / (d - c) ;
A69: (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . t = |[(((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a)))),(((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))))]| by Def2;
then - 1 < ((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) by A62, A67, EUCLID:52;
then (- 1) * (d - c) < (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A68, XREAL_1:68;
then (- 1) * (d - c) < (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87;
then ((- 1) * (d - c)) + (c - d) < ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8;
then (2 * (c - d)) / 2 < (2 * ((t `2) - d)) / 2 by XREAL_1:74;
then A70: c < t `2 by XREAL_1:9;
((2 / (d - c)) * (t `2)) + (1 - ((2 * d) / (d - c))) < 1 by A63, A69, A67, EUCLID:52;
then 1 * (d - c) > (((2 * ((t `2) - d)) - (c - d)) / (d - c)) * (d - c) by A19, A68, XREAL_1:68;
then 1 * (d - c) > (2 * ((t `2) - d)) - (c - d) by A19, XCMPLX_1:87;
then (1 * (d - c)) + (c - d) > ((2 * ((t `2) - d)) - (c - d)) + (c - d) by XREAL_1:8;
then 0 / 2 > (((t `2) - d) * 2) / 2 ;
then A71: 0 + d > t `2 by XREAL_1:19;
((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) < 1 by A61, A69, A67, EUCLID:52;
then 1 * (b - a) > (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A66, XREAL_1:68;
then 1 * (b - a) > (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87;
then (1 * (b - a)) + (a - b) > ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8;
then 0 / 2 > (((t `1) - b) * 2) / 2 ;
then A72: 0 + b > t `1 by XREAL_1:19;
- 1 < ((2 / (b - a)) * (t `1)) + (1 - ((2 * b) / (b - a))) by A60, A69, A67, EUCLID:52;
then (- 1) * (b - a) < (((2 * ((t `1) - b)) - (a - b)) / (b - a)) * (b - a) by A26, A66, XREAL_1:68;
then (- 1) * (b - a) < (2 * ((t `1) - b)) - (a - b) by A26, XCMPLX_1:87;
then ((- 1) * (b - a)) + (a - b) < ((2 * ((t `1) - b)) - (a - b)) + (a - b) by XREAL_1:8;
then (2 * (a - b)) / 2 < (2 * ((t `1) - b)) / 2 by XREAL_1:74;
then a < t `1 by XREAL_1:9;
hence contradiction by A18, A72, A70, A71; ::_thesis: verum
end;
A73: ( - 1 <= (f2 . O) `2 & (f2 . O) `2 <= 1 & - 1 <= (f2 . I) `2 & (f2 . I) `2 <= 1 )
proof
reconsider s1 = (f . I) `2 as Real ;
reconsider s0 = (f . O) `2 as Real ;
A74: (c - d) / (d - c) = (- (d - c)) / (d - c)
.= - ((d - c) / (d - c))
.= - 1 by A19, XCMPLX_1:60 ;
A75: (f2 . I) `2 = ((2 / (d - c)) * s1) + (1 - ((2 * d) / (d - c))) by A52, EUCLID:52
.= ((s1 * 2) / (d - c)) + (1 - ((2 * d) / (d - c)))
.= ((s1 * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60
.= ((s1 * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c))
.= ((s1 * 2) + ((d - c) - (2 * d))) / (d - c)
.= (((s1 - d) + (s1 - d)) - (c - d)) / (d - c) ;
d - d >= s0 - d by A6, XREAL_1:9;
then (0 + (d - d)) - (c - d) >= ((s0 - d) + (s0 - d)) - (c - d) by XREAL_1:9;
then A76: (d - c) / (d - c) >= (((s0 - d) + (s0 - d)) - (c - d)) / (d - c) by A19, XREAL_1:72;
d - d >= s1 - d by A8, XREAL_1:9;
then A77: (0 + (d - d)) - (c - d) >= ((s1 - d) + (s1 - d)) - (c - d) by XREAL_1:9;
c - d <= s1 - d by A7, XREAL_1:9;
then (c - d) + (c - d) <= (s1 - d) + (s1 - d) by XREAL_1:7;
then A78: ((c - d) + (c - d)) - (c - d) <= ((s1 - d) + (s1 - d)) - (c - d) by XREAL_1:9;
c - d <= s0 - d by A5, XREAL_1:9;
then (c - d) + (c - d) <= (s0 - d) + (s0 - d) by XREAL_1:7;
then A79: ((c - d) + (c - d)) - (c - d) <= ((s0 - d) + (s0 - d)) - (c - d) by XREAL_1:9;
(f2 . O) `2 = ((2 / (d - c)) * s0) + (1 - ((2 * d) / (d - c))) by A54, EUCLID:52
.= ((s0 * 2) / (d - c)) + (1 - ((2 * d) / (d - c)))
.= ((s0 * 2) / (d - c)) + (((d - c) / (d - c)) - ((2 * d) / (d - c))) by A19, XCMPLX_1:60
.= ((s0 * 2) / (d - c)) + (((d - c) - (2 * d)) / (d - c))
.= ((s0 * 2) + ((d - c) - (2 * d))) / (d - c)
.= (((s0 - d) + (s0 - d)) - (c - d)) / (d - c) ;
hence ( - 1 <= (f2 . O) `2 & (f2 . O) `2 <= 1 & - 1 <= (f2 . I) `2 & (f2 . I) `2 <= 1 ) by A19, A79, A74, A76, A75, A78, A77, XREAL_1:72; ::_thesis: verum
end;
set y = the Element of (rng f2) /\ (rng g2);
2 / (b - a) > 0 by A26, XREAL_1:139;
then A80: AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c)))) is one-to-one by A20, Th44;
then ( f2 is one-to-one & g2 is one-to-one ) by A2, FUNCT_1:24;
then rng f2 meets rng g2 by A1, A2, A55, A53, A25, A23, A73, A27, A34, A56, Th42;
then A81: (rng f2) /\ (rng g2) <> {} by XBOOLE_0:def_7;
then the Element of (rng f2) /\ (rng g2) in rng f2 by XBOOLE_0:def_4;
then consider x being set such that
A82: x in dom f2 and
A83: the Element of (rng f2) /\ (rng g2) = f2 . x by FUNCT_1:def_3;
dom f2 c= dom f by RELAT_1:25;
then A84: f . x in rng f by A82, FUNCT_1:3;
the Element of (rng f2) /\ (rng g2) in rng g2 by A81, XBOOLE_0:def_4;
then consider x2 being set such that
A85: x2 in dom g2 and
A86: the Element of (rng f2) /\ (rng g2) = g2 . x2 by FUNCT_1:def_3;
A87: the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (g . x2) by A85, A86, FUNCT_1:12;
dom g2 c= dom g by RELAT_1:25;
then A88: g . x2 in rng g by A85, FUNCT_1:3;
( dom (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) = the carrier of (TOP-REAL 2) & the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(1 - ((2 * b) / (b - a))),(2 / (d - c)),(1 - ((2 * d) / (d - c))))) . (f . x) ) by A82, A83, FUNCT_1:12, FUNCT_2:def_1;
then f . x = g . x2 by A80, A87, A84, A88, FUNCT_1:def_4;
then (rng f) /\ (rng g) <> {} by A84, A88, XBOOLE_0:def_4;
hence rng f meets rng g by XBOOLE_0:def_7; ::_thesis: verum
end;
theorem :: JGRAPH_2:46
( { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= p7 `1 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= p7 `2 } is closed Subset of (TOP-REAL 2) ) by Lm5, Lm8;
theorem :: JGRAPH_2:47
( { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `1) <= p7 `2 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `2 <= - (p7 `1) } is closed Subset of (TOP-REAL 2) ) by Lm11, Lm14;
theorem :: JGRAPH_2:48
( { p7 where p7 is Point of (TOP-REAL 2) : - (p7 `2) <= p7 `1 } is closed Subset of (TOP-REAL 2) & { p7 where p7 is Point of (TOP-REAL 2) : p7 `1 <= - (p7 `2) } is closed Subset of (TOP-REAL 2) ) by Lm17, Lm20;