:: JORDAN semantic presentation
begin
set T2 = TOP-REAL 2;
Lm1: for A, B, C, Z being set st A c= Z & B c= Z & C c= Z holds
(A \/ B) \/ C c= Z
proof
let A, B, C, Z be set ; ::_thesis: ( A c= Z & B c= Z & C c= Z implies (A \/ B) \/ C c= Z )
assume that
A1: A c= Z and
A2: B c= Z ; ::_thesis: ( not C c= Z or (A \/ B) \/ C c= Z )
A \/ B c= Z by A1, A2, XBOOLE_1:8;
hence ( not C c= Z or (A \/ B) \/ C c= Z ) by XBOOLE_1:8; ::_thesis: verum
end;
Lm2: for A, B, C, D, Z being set st A c= Z & B c= Z & C c= Z & D c= Z holds
((A \/ B) \/ C) \/ D c= Z
proof
let A, B, C, D, Z be set ; ::_thesis: ( A c= Z & B c= Z & C c= Z & D c= Z implies ((A \/ B) \/ C) \/ D c= Z )
assume that
A1: A c= Z and
A2: B c= Z and
A3: C c= Z ; ::_thesis: ( not D c= Z or ((A \/ B) \/ C) \/ D c= Z )
(A \/ B) \/ C c= Z by A1, A2, A3, Lm1;
hence ( not D c= Z or ((A \/ B) \/ C) \/ D c= Z ) by XBOOLE_1:8; ::_thesis: verum
end;
Lm3: for A, B, C, D, Z being set st A misses Z & B misses Z & C misses Z & D misses Z holds
((A \/ B) \/ C) \/ D misses Z
proof
let A, B, C, D, Z be set ; ::_thesis: ( A misses Z & B misses Z & C misses Z & D misses Z implies ((A \/ B) \/ C) \/ D misses Z )
assume that
A1: A misses Z and
A2: B misses Z and
A3: C misses Z ; ::_thesis: ( not D misses Z or ((A \/ B) \/ C) \/ D misses Z )
(A \/ B) \/ C misses Z by A1, A2, A3, XBOOLE_1:114;
hence ( not D misses Z or ((A \/ B) \/ C) \/ D misses Z ) by XBOOLE_1:70; ::_thesis: verum
end;
registration
let M be Reflexive symmetric triangle MetrStruct ;
let x, y be Point of M;
cluster dist (x,y) -> non negative ;
coherence
not dist (x,y) is negative by METRIC_1:5;
end;
registration
let n be Element of NAT ;
let x, y be Point of (TOP-REAL n);
cluster dist (x,y) -> non negative ;
coherence
not dist (x,y) is negative
proof
ex p, q being Point of (Euclid n) st
( p = x & q = y & dist (x,y) = dist (p,q) ) by TOPREAL6:def_1;
hence 0 <= dist (x,y) ; :: according to XXREAL_0:def_7 ::_thesis: verum
end;
end;
theorem Th1: :: JORDAN:1
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n) st p1 <> p2 holds
(1 / 2) * (p1 + p2) <> p1
proof
let n be Element of NAT ; ::_thesis: for p1, p2 being Point of (TOP-REAL n) st p1 <> p2 holds
(1 / 2) * (p1 + p2) <> p1
let p1, p2 be Point of (TOP-REAL n); ::_thesis: ( p1 <> p2 implies (1 / 2) * (p1 + p2) <> p1 )
set r = 1 / 2;
assume that
A1: p1 <> p2 and
A2: (1 / 2) * (p1 + p2) = p1 ; ::_thesis: contradiction
(1 / 2) * (p1 + p2) = ((1 / 2) * p1) + ((1 / 2) * p2) by EUCLID:32;
then 0. (TOP-REAL n) = p1 - (((1 / 2) * p1) + ((1 / 2) * p2)) by A2, EUCLID:42
.= (p1 - ((1 / 2) * p1)) - ((1 / 2) * p2) by EUCLID:46
.= ((1 * p1) - ((1 / 2) * p1)) - ((1 / 2) * p2) by EUCLID:29
.= ((1 - (1 / 2)) * p1) - ((1 / 2) * p2) by EUCLID:50
.= (1 / 2) * (p1 - p2) by EUCLID:49 ;
then p1 - p2 = 0. (TOP-REAL n) by EUCLID:31;
hence contradiction by A1, EUCLID:43; ::_thesis: verum
end;
theorem Th2: :: JORDAN:2
for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
p1 `2 < ((1 / 2) * (p1 + p2)) `2
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 `2 < p2 `2 implies p1 `2 < ((1 / 2) * (p1 + p2)) `2 )
assume A1: p1 `2 < p2 `2 ; ::_thesis: p1 `2 < ((1 / 2) * (p1 + p2)) `2
((1 / 2) * (p1 + p2)) `2 = (1 / 2) * ((p1 + p2) `2) by TOPREAL3:4
.= (1 / 2) * ((p1 `2) + (p2 `2)) by TOPREAL3:2
.= ((p1 `2) + (p2 `2)) / 2 ;
hence p1 `2 < ((1 / 2) * (p1 + p2)) `2 by A1, XREAL_1:226; ::_thesis: verum
end;
theorem Th3: :: JORDAN:3
for p1, p2 being Point of (TOP-REAL 2) st p1 `2 < p2 `2 holds
((1 / 2) * (p1 + p2)) `2 < p2 `2
proof
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 `2 < p2 `2 implies ((1 / 2) * (p1 + p2)) `2 < p2 `2 )
assume A1: p1 `2 < p2 `2 ; ::_thesis: ((1 / 2) * (p1 + p2)) `2 < p2 `2
((1 / 2) * (p1 + p2)) `2 = (1 / 2) * ((p1 + p2) `2) by TOPREAL3:4
.= (1 / 2) * ((p1 `2) + (p2 `2)) by TOPREAL3:2
.= ((p1 `2) + (p2 `2)) / 2 ;
hence ((1 / 2) * (p1 + p2)) `2 < p2 `2 by A1, XREAL_1:226; ::_thesis: verum
end;
theorem Th4: :: JORDAN:4
for B being Subset of (TOP-REAL 2)
for A being vertical Subset of (TOP-REAL 2) holds A /\ B is vertical
proof
let B be Subset of (TOP-REAL 2); ::_thesis: for A being vertical Subset of (TOP-REAL 2) holds A /\ B is vertical
let A be vertical Subset of (TOP-REAL 2); ::_thesis: A /\ B is vertical
let p, q be Point of (TOP-REAL 2); :: according to SPPOL_1:def_3 ::_thesis: ( not p in A /\ B or not q in A /\ B or p `1 = q `1 )
assume that
A1: p in A /\ B and
A2: q in A /\ B ; ::_thesis: p `1 = q `1
A3: p in A by A1, XBOOLE_0:def_4;
q in A by A2, XBOOLE_0:def_4;
hence p `1 = q `1 by A3, SPPOL_1:def_3; ::_thesis: verum
end;
theorem :: JORDAN:5
for B being Subset of (TOP-REAL 2)
for A being horizontal Subset of (TOP-REAL 2) holds A /\ B is horizontal
proof
let B be Subset of (TOP-REAL 2); ::_thesis: for A being horizontal Subset of (TOP-REAL 2) holds A /\ B is horizontal
let A be horizontal Subset of (TOP-REAL 2); ::_thesis: A /\ B is horizontal
let p, q be Point of (TOP-REAL 2); :: according to SPPOL_1:def_2 ::_thesis: ( not p in A /\ B or not q in A /\ B or p `2 = q `2 )
assume that
A1: p in A /\ B and
A2: q in A /\ B ; ::_thesis: p `2 = q `2
A3: p in A by A1, XBOOLE_0:def_4;
q in A by A2, XBOOLE_0:def_4;
hence p `2 = q `2 by A3, SPPOL_1:def_2; ::_thesis: verum
end;
theorem :: JORDAN:6
for p, p1, p2 being Point of (TOP-REAL 2) st p in LSeg (p1,p2) & LSeg (p1,p2) is vertical holds
LSeg (p,p2) is vertical
proof
let p, p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (p1,p2) & LSeg (p1,p2) is vertical implies LSeg (p,p2) is vertical )
assume A1: p in LSeg (p1,p2) ; ::_thesis: ( not LSeg (p1,p2) is vertical or LSeg (p,p2) is vertical )
assume A2: LSeg (p1,p2) is vertical ; ::_thesis: LSeg (p,p2) is vertical
then A3: p1 `1 = p2 `1 by SPPOL_1:16;
p1 in LSeg (p1,p2) by RLTOPSP1:68;
then p `1 = p1 `1 by A1, A2, SPPOL_1:def_3;
hence LSeg (p,p2) is vertical by A3, SPPOL_1:16; ::_thesis: verum
end;
theorem :: JORDAN:7
for p, p1, p2 being Point of (TOP-REAL 2) st p in LSeg (p1,p2) & LSeg (p1,p2) is horizontal holds
LSeg (p,p2) is horizontal
proof
let p, p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (p1,p2) & LSeg (p1,p2) is horizontal implies LSeg (p,p2) is horizontal )
assume A1: p in LSeg (p1,p2) ; ::_thesis: ( not LSeg (p1,p2) is horizontal or LSeg (p,p2) is horizontal )
assume A2: LSeg (p1,p2) is horizontal ; ::_thesis: LSeg (p,p2) is horizontal
then A3: p1 `2 = p2 `2 by SPPOL_1:15;
p1 in LSeg (p1,p2) by RLTOPSP1:68;
then p `2 = p1 `2 by A1, A2, SPPOL_1:def_2;
hence LSeg (p,p2) is horizontal by A3, SPPOL_1:15; ::_thesis: verum
end;
registration
let P be Subset of (TOP-REAL 2);
cluster LSeg ((SW-corner P),(SE-corner P)) -> horizontal ;
coherence
LSeg ((SW-corner P),(SE-corner P)) is horizontal
proof
(SW-corner P) `2 = S-bound P by EUCLID:52
.= (SE-corner P) `2 by EUCLID:52 ;
hence LSeg ((SW-corner P),(SE-corner P)) is horizontal by SPPOL_1:15; ::_thesis: verum
end;
cluster LSeg ((NW-corner P),(SW-corner P)) -> vertical ;
coherence
LSeg ((NW-corner P),(SW-corner P)) is vertical
proof
(NW-corner P) `1 = W-bound P by EUCLID:52
.= (SW-corner P) `1 by EUCLID:52 ;
hence LSeg ((NW-corner P),(SW-corner P)) is vertical by SPPOL_1:16; ::_thesis: verum
end;
cluster LSeg ((NE-corner P),(SE-corner P)) -> vertical ;
coherence
LSeg ((NE-corner P),(SE-corner P)) is vertical
proof
(NE-corner P) `1 = E-bound P by EUCLID:52
.= (SE-corner P) `1 by EUCLID:52 ;
hence LSeg ((NE-corner P),(SE-corner P)) is vertical by SPPOL_1:16; ::_thesis: verum
end;
end;
registration
let P be Subset of (TOP-REAL 2);
cluster LSeg ((SE-corner P),(SW-corner P)) -> horizontal ;
coherence
LSeg ((SE-corner P),(SW-corner P)) is horizontal ;
cluster LSeg ((SW-corner P),(NW-corner P)) -> vertical ;
coherence
LSeg ((SW-corner P),(NW-corner P)) is vertical ;
cluster LSeg ((SE-corner P),(NE-corner P)) -> vertical ;
coherence
LSeg ((SE-corner P),(NE-corner P)) is vertical ;
end;
registration
cluster non empty compact vertical -> with_the_max_arc for Element of bool the carrier of (TOP-REAL 2);
coherence
for b1 being Subset of (TOP-REAL 2) st b1 is vertical & not b1 is empty & b1 is compact holds
b1 is with_the_max_arc
proof
let A be Subset of (TOP-REAL 2); ::_thesis: ( A is vertical & not A is empty & A is compact implies A is with_the_max_arc )
assume A1: ( A is vertical & not A is empty & A is compact ) ; ::_thesis: A is with_the_max_arc
then A2: W-bound A = E-bound A by SPRECT_1:15;
A3: E-min A in A by A1, SPRECT_1:14;
(E-min A) `1 = E-bound A by EUCLID:52;
then E-min A in Vertical_Line (((W-bound A) + (E-bound A)) / 2) by A2, JORDAN6:31;
hence A meets Vertical_Line (((W-bound A) + (E-bound A)) / 2) by A3, XBOOLE_0:3; :: according to JORDAN21:def_1 ::_thesis: verum
end;
end;
theorem Th8: :: JORDAN:8
for r being real number
for p1, p2 being Point of (TOP-REAL 2) st p1 `1 <= r & r <= p2 `1 holds
LSeg (p1,p2) meets Vertical_Line r
proof
let r be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 `1 <= r & r <= p2 `1 holds
LSeg (p1,p2) meets Vertical_Line r
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 `1 <= r & r <= p2 `1 implies LSeg (p1,p2) meets Vertical_Line r )
assume that
A1: p1 `1 <= r and
A2: r <= p2 `1 ; ::_thesis: LSeg (p1,p2) meets Vertical_Line r
set a = p1 `1 ;
set b = p2 `1 ;
set l = (r - (p1 `1)) / ((p2 `1) - (p1 `1));
set k = ((1 - ((r - (p1 `1)) / ((p2 `1) - (p1 `1)))) * p1) + (((r - (p1 `1)) / ((p2 `1) - (p1 `1))) * p2);
A3: (p1 `1) - (p1 `1) <= r - (p1 `1) by A1, XREAL_1:9;
A4: r - (p1 `1) <= (p2 `1) - (p1 `1) by A2, XREAL_1:9;
then (r - (p1 `1)) / ((p2 `1) - (p1 `1)) <= 1 by A3, XREAL_1:183;
then A5: ((1 - ((r - (p1 `1)) / ((p2 `1) - (p1 `1)))) * p1) + (((r - (p1 `1)) / ((p2 `1) - (p1 `1))) * p2) in LSeg (p1,p2) by A3, A4;
percases ( p1 `1 <> p2 `1 or p1 `1 = p2 `1 ) ;
suppose p1 `1 <> p2 `1 ; ::_thesis: LSeg (p1,p2) meets Vertical_Line r
then A6: (p2 `1) - (p1 `1) <> 0 ;
(((1 - ((r - (p1 `1)) / ((p2 `1) - (p1 `1)))) * p1) + (((r - (p1 `1)) / ((p2 `1) - (p1 `1))) * p2)) `1 = ((1 - ((r - (p1 `1)) / ((p2 `1) - (p1 `1)))) * (p1 `1)) + (((r - (p1 `1)) / ((p2 `1) - (p1 `1))) * (p2 `1)) by TOPREAL9:41
.= (p1 `1) + (((r - (p1 `1)) / ((p2 `1) - (p1 `1))) * ((p2 `1) - (p1 `1)))
.= (p1 `1) + (r - (p1 `1)) by A6, XCMPLX_1:87 ;
then ((1 - ((r - (p1 `1)) / ((p2 `1) - (p1 `1)))) * p1) + (((r - (p1 `1)) / ((p2 `1) - (p1 `1))) * p2) in Vertical_Line r by JORDAN6:31;
hence LSeg (p1,p2) meets Vertical_Line r by A5, XBOOLE_0:3; ::_thesis: verum
end;
supposeA7: p1 `1 = p2 `1 ; ::_thesis: LSeg (p1,p2) meets Vertical_Line r
A8: p1 in LSeg (p1,p2) by RLTOPSP1:68;
p1 `1 = r by A1, A2, A7, XXREAL_0:1;
then p1 in Vertical_Line r by JORDAN6:31;
hence LSeg (p1,p2) meets Vertical_Line r by A8, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN:9
for r being real number
for p1, p2 being Point of (TOP-REAL 2) st p1 `2 <= r & r <= p2 `2 holds
LSeg (p1,p2) meets Horizontal_Line r
proof
let r be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st p1 `2 <= r & r <= p2 `2 holds
LSeg (p1,p2) meets Horizontal_Line r
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( p1 `2 <= r & r <= p2 `2 implies LSeg (p1,p2) meets Horizontal_Line r )
assume that
A1: p1 `2 <= r and
A2: r <= p2 `2 ; ::_thesis: LSeg (p1,p2) meets Horizontal_Line r
set a = p1 `2 ;
set b = p2 `2 ;
set l = (r - (p1 `2)) / ((p2 `2) - (p1 `2));
set k = ((1 - ((r - (p1 `2)) / ((p2 `2) - (p1 `2)))) * p1) + (((r - (p1 `2)) / ((p2 `2) - (p1 `2))) * p2);
A3: (p1 `2) - (p1 `2) <= r - (p1 `2) by A1, XREAL_1:9;
A4: r - (p1 `2) <= (p2 `2) - (p1 `2) by A2, XREAL_1:9;
then (r - (p1 `2)) / ((p2 `2) - (p1 `2)) <= 1 by A3, XREAL_1:183;
then A5: ((1 - ((r - (p1 `2)) / ((p2 `2) - (p1 `2)))) * p1) + (((r - (p1 `2)) / ((p2 `2) - (p1 `2))) * p2) in LSeg (p1,p2) by A3, A4;
percases ( p1 `2 <> p2 `2 or p1 `2 = p2 `2 ) ;
suppose p1 `2 <> p2 `2 ; ::_thesis: LSeg (p1,p2) meets Horizontal_Line r
then A6: (p2 `2) - (p1 `2) <> 0 ;
(((1 - ((r - (p1 `2)) / ((p2 `2) - (p1 `2)))) * p1) + (((r - (p1 `2)) / ((p2 `2) - (p1 `2))) * p2)) `2 = ((1 - ((r - (p1 `2)) / ((p2 `2) - (p1 `2)))) * (p1 `2)) + (((r - (p1 `2)) / ((p2 `2) - (p1 `2))) * (p2 `2)) by TOPREAL9:42
.= (p1 `2) + (((r - (p1 `2)) / ((p2 `2) - (p1 `2))) * ((p2 `2) - (p1 `2)))
.= (p1 `2) + (r - (p1 `2)) by A6, XCMPLX_1:87 ;
then ((1 - ((r - (p1 `2)) / ((p2 `2) - (p1 `2)))) * p1) + (((r - (p1 `2)) / ((p2 `2) - (p1 `2))) * p2) in Horizontal_Line r by JORDAN6:32;
hence LSeg (p1,p2) meets Horizontal_Line r by A5, XBOOLE_0:3; ::_thesis: verum
end;
supposeA7: p1 `2 = p2 `2 ; ::_thesis: LSeg (p1,p2) meets Horizontal_Line r
A8: p1 in LSeg (p1,p2) by RLTOPSP1:68;
p1 `2 = r by A1, A2, A7, XXREAL_0:1;
then p1 in Horizontal_Line r by JORDAN6:32;
hence LSeg (p1,p2) meets Horizontal_Line r by A8, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
registration
let n be Element of NAT ;
cluster empty -> bounded for Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st b1 is empty holds
b1 is bounded ;
cluster non bounded -> non empty for Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st not b1 is bounded holds
not b1 is empty ;
end;
registration
let n be non empty Nat;
cluster functional open closed non bounded convex for Element of bool the carrier of (TOP-REAL n);
existence
ex b1 being Subset of (TOP-REAL n) st
( b1 is open & b1 is closed & not b1 is bounded & b1 is convex )
proof
take [#] (TOP-REAL n) ; ::_thesis: ( [#] (TOP-REAL n) is open & [#] (TOP-REAL n) is closed & not [#] (TOP-REAL n) is bounded & [#] (TOP-REAL n) is convex )
reconsider n = n as Element of NAT by ORDINAL1:def_12;
n >= 1 by NAT_1:14;
then not [#] (TOP-REAL n) is bounded by JORDAN2C:35;
hence ( [#] (TOP-REAL n) is open & [#] (TOP-REAL n) is closed & not [#] (TOP-REAL n) is bounded & [#] (TOP-REAL n) is convex ) ; ::_thesis: verum
end;
end;
theorem Th10: :: JORDAN:10
for C being compact Subset of (TOP-REAL 2) holds (north_halfline (UMP C)) \ {(UMP C)} misses C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: (north_halfline (UMP C)) \ {(UMP C)} misses C
set p = UMP C;
set L = north_halfline (UMP C);
set w = ((W-bound C) + (E-bound C)) / 2;
assume (north_halfline (UMP C)) \ {(UMP C)} meets C ; ::_thesis: contradiction
then consider x being set such that
A1: x in (north_halfline (UMP C)) \ {(UMP C)} and
A2: x in C by XBOOLE_0:3;
A3: x in north_halfline (UMP C) by A1, ZFMISC_1:56;
A4: x <> UMP C by A1, ZFMISC_1:56;
reconsider x = x as Point of (TOP-REAL 2) by A1;
A5: x `1 = (UMP C) `1 by A3, TOPREAL1:def_10;
A6: x `2 >= (UMP C) `2 by A3, TOPREAL1:def_10;
x `2 <> (UMP C) `2 by A4, A5, TOPREAL3:6;
then A7: x `2 > (UMP C) `2 by A6, XXREAL_0:1;
x `1 = ((W-bound C) + (E-bound C)) / 2 by A5, EUCLID:52;
then x in Vertical_Line (((W-bound C) + (E-bound C)) / 2) by JORDAN6:31;
then x in C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) by A2, XBOOLE_0:def_4;
hence contradiction by A7, JORDAN21:28; ::_thesis: verum
end;
theorem Th11: :: JORDAN:11
for C being compact Subset of (TOP-REAL 2) holds (south_halfline (LMP C)) \ {(LMP C)} misses C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: (south_halfline (LMP C)) \ {(LMP C)} misses C
set p = LMP C;
set L = south_halfline (LMP C);
set w = ((W-bound C) + (E-bound C)) / 2;
assume (south_halfline (LMP C)) \ {(LMP C)} meets C ; ::_thesis: contradiction
then consider x being set such that
A1: x in (south_halfline (LMP C)) \ {(LMP C)} and
A2: x in C by XBOOLE_0:3;
A3: x in south_halfline (LMP C) by A1, ZFMISC_1:56;
A4: x <> LMP C by A1, ZFMISC_1:56;
reconsider x = x as Point of (TOP-REAL 2) by A1;
A5: x `1 = (LMP C) `1 by A3, TOPREAL1:def_12;
A6: x `2 <= (LMP C) `2 by A3, TOPREAL1:def_12;
x `2 <> (LMP C) `2 by A4, A5, TOPREAL3:6;
then A7: x `2 < (LMP C) `2 by A6, XXREAL_0:1;
x `1 = ((W-bound C) + (E-bound C)) / 2 by A5, EUCLID:52;
then x in Vertical_Line (((W-bound C) + (E-bound C)) / 2) by JORDAN6:31;
then x in C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) by A2, XBOOLE_0:def_4;
hence contradiction by A7, JORDAN21:29; ::_thesis: verum
end;
theorem Th12: :: JORDAN:12
for C being compact Subset of (TOP-REAL 2) holds (north_halfline (UMP C)) \ {(UMP C)} c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: (north_halfline (UMP C)) \ {(UMP C)} c= UBD C
set A = (north_halfline (UMP C)) \ {(UMP C)};
reconsider A = (north_halfline (UMP C)) \ {(UMP C)} as non bounded Subset of (TOP-REAL 2) by JORDAN2C:122, TOPREAL6:90;
A is convex by JORDAN21:6;
hence (north_halfline (UMP C)) \ {(UMP C)} c= UBD C by Th10, JORDAN2C:125; ::_thesis: verum
end;
theorem Th13: :: JORDAN:13
for C being compact Subset of (TOP-REAL 2) holds (south_halfline (LMP C)) \ {(LMP C)} c= UBD C
proof
let C be compact Subset of (TOP-REAL 2); ::_thesis: (south_halfline (LMP C)) \ {(LMP C)} c= UBD C
set A = (south_halfline (LMP C)) \ {(LMP C)};
reconsider A = (south_halfline (LMP C)) \ {(LMP C)} as non bounded Subset of (TOP-REAL 2) by JORDAN2C:123, TOPREAL6:90;
A is convex by JORDAN21:7;
hence (south_halfline (LMP C)) \ {(LMP C)} c= UBD C by Th11, JORDAN2C:125; ::_thesis: verum
end;
theorem Th14: :: JORDAN:14
for A, B being Subset of (TOP-REAL 2) st A is_inside_component_of B holds
UBD B misses A
proof
let A, B be Subset of (TOP-REAL 2); ::_thesis: ( A is_inside_component_of B implies UBD B misses A )
assume A is_inside_component_of B ; ::_thesis: UBD B misses A
then A c= BDD B by JORDAN2C:22;
hence UBD B misses A by JORDAN2C:24, XBOOLE_1:63; ::_thesis: verum
end;
theorem :: JORDAN:15
for A, B being Subset of (TOP-REAL 2) st A is_outside_component_of B holds
BDD B misses A
proof
let A, B be Subset of (TOP-REAL 2); ::_thesis: ( A is_outside_component_of B implies BDD B misses A )
assume A1: A is_outside_component_of B ; ::_thesis: BDD B misses A
BDD B misses UBD B by JORDAN2C:24;
hence BDD B misses A by A1, JORDAN2C:23, XBOOLE_1:63; ::_thesis: verum
end;
Lm4: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for U being Subset of ((TOP-REAL 2) | (C `)) st p in C holds
{p} misses U
proof
let p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve
for U being Subset of ((TOP-REAL 2) | (C `)) st p in C holds
{p} misses U
let C be Simple_closed_curve; ::_thesis: for U being Subset of ((TOP-REAL 2) | (C `)) st p in C holds
{p} misses U
let U be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: ( p in C implies {p} misses U )
assume A1: p in C ; ::_thesis: {p} misses U
A2: U is Subset of (TOP-REAL 2) by PRE_TOPC:11;
the carrier of ((TOP-REAL 2) | (C `)) = C ` by PRE_TOPC:8;
then U misses C by A2, SUBSET_1:23;
then not p in U by A1, XBOOLE_0:3;
hence {p} misses U by ZFMISC_1:50; ::_thesis: verum
end;
set C0 = Closed-Interval-TSpace (0,1);
set C1 = Closed-Interval-TSpace ((- 1),1);
set l0 = (#) ((- 1),1);
set l1 = ((- 1),1) (#) ;
set h1 = L[01] (((#) ((- 1),1)),(((- 1),1) (#)));
Lm5: the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] = [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]
by BORSUK_1:def_2;
Lm6: now__::_thesis:_for_T_being_non_empty_TopSpace
for_a_being_Real_holds_the_carrier_of_T_-->_a_is_continuous
let T be non empty TopSpace; ::_thesis: for a being Real holds the carrier of T --> a is continuous
let a be Real; ::_thesis: the carrier of T --> a is continuous
set c = the carrier of T;
set f = the carrier of T --> a;
thus the carrier of T --> a is continuous ::_thesis: verum
proof
A1: dom ( the carrier of T --> a) = the carrier of T by FUNCT_2:def_1;
A2: rng ( the carrier of T --> a) = {a} by FUNCOP_1:8;
let Y be Subset of REAL; :: according to PSCOMP_1:def_3 ::_thesis: ( not Y is closed or ( the carrier of T --> a) " Y is closed )
assume Y is closed ; ::_thesis: ( the carrier of T --> a) " Y is closed
percases ( a in Y or not a in Y ) ;
suppose a in Y ; ::_thesis: ( the carrier of T --> a) " Y is closed
then A3: rng ( the carrier of T --> a) c= Y by A2, ZFMISC_1:31;
( the carrier of T --> a) " Y = ( the carrier of T --> a) " ((rng ( the carrier of T --> a)) /\ Y) by RELAT_1:133
.= ( the carrier of T --> a) " (rng ( the carrier of T --> a)) by A3, XBOOLE_1:28
.= [#] T by A1, RELAT_1:134 ;
hence ( the carrier of T --> a) " Y is closed ; ::_thesis: verum
end;
suppose not a in Y ; ::_thesis: ( the carrier of T --> a) " Y is closed
then A4: rng ( the carrier of T --> a) misses Y by A2, ZFMISC_1:50;
( the carrier of T --> a) " Y = ( the carrier of T --> a) " ((rng ( the carrier of T --> a)) /\ Y) by RELAT_1:133
.= ( the carrier of T --> a) " {} by A4, XBOOLE_0:def_7
.= {} T ;
hence ( the carrier of T --> a) " Y is closed ; ::_thesis: verum
end;
end;
end;
end;
theorem Th16: :: JORDAN:16
for n being Element of NAT
for r being positive real number
for a being Point of (TOP-REAL n) holds a in Ball (a,r)
proof
let n be Element of NAT ; ::_thesis: for r being positive real number
for a being Point of (TOP-REAL n) holds a in Ball (a,r)
let r be positive real number ; ::_thesis: for a being Point of (TOP-REAL n) holds a in Ball (a,r)
let a be Point of (TOP-REAL n); ::_thesis: a in Ball (a,r)
|.(a - a).| = 0 by TOPRNS_1:28;
hence a in Ball (a,r) by TOPREAL9:7; ::_thesis: verum
end;
theorem Th17: :: JORDAN:17
for n being Element of NAT
for r being non negative real number
for p being Point of (TOP-REAL n) holds p is Point of (Tdisk (p,r))
proof
let n be Element of NAT ; ::_thesis: for r being non negative real number
for p being Point of (TOP-REAL n) holds p is Point of (Tdisk (p,r))
let r be non negative real number ; ::_thesis: for p being Point of (TOP-REAL n) holds p is Point of (Tdisk (p,r))
let p be Point of (TOP-REAL n); ::_thesis: p is Point of (Tdisk (p,r))
A1: the carrier of (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
|.(p - p).| = 0 by TOPRNS_1:28;
hence p is Point of (Tdisk (p,r)) by A1, TOPREAL9:8; ::_thesis: verum
end;
registration
let r be positive real number ;
let n be non empty Element of NAT ;
let p, q be Point of (TOP-REAL n);
cluster(cl_Ball (p,r)) \ {q} -> non empty ;
coherence
not (cl_Ball (p,r)) \ {q} is empty
proof
A1: the carrier of (Tcircle (p,r)) = Sphere (p,r) by TOPREALB:9;
A2: the carrier of (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
A3: Sphere (p,r) c= cl_Ball (p,r) by TOPREAL9:17;
set a = the Point of (Tcircle (p,r));
A4: the Point of (Tcircle (p,r)) in Sphere (p,r) by A1;
percases ( the Point of (Tcircle (p,r)) = q or the Point of (Tcircle (p,r)) <> q ) ;
supposeA5: the Point of (Tcircle (p,r)) = q ; ::_thesis: not (cl_Ball (p,r)) \ {q} is empty
A6: p is Point of (Tdisk (p,r)) by Th17;
|.(p - p).| <> r by TOPRNS_1:28;
then p <> q by A1, A5, TOPREAL9:9;
hence not (cl_Ball (p,r)) \ {q} is empty by A2, A6, ZFMISC_1:56; ::_thesis: verum
end;
suppose the Point of (Tcircle (p,r)) <> q ; ::_thesis: not (cl_Ball (p,r)) \ {q} is empty
hence not (cl_Ball (p,r)) \ {q} is empty by A3, A4, ZFMISC_1:56; ::_thesis: verum
end;
end;
end;
end;
theorem Th18: :: JORDAN:18
for r, s being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) st r <= s holds
Ball (x,r) c= Ball (x,s)
proof
let r, s be real number ; ::_thesis: for n being Element of NAT
for x being Point of (TOP-REAL n) st r <= s holds
Ball (x,r) c= Ball (x,s)
let n be Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n) st r <= s holds
Ball (x,r) c= Ball (x,s)
let x be Point of (TOP-REAL n); ::_thesis: ( r <= s implies Ball (x,r) c= Ball (x,s) )
reconsider xe = x as Point of (Euclid n) by TOPREAL3:8;
A1: Ball (x,r) = Ball (xe,r) by TOPREAL9:13;
Ball (x,s) = Ball (xe,s) by TOPREAL9:13;
hence ( r <= s implies Ball (x,r) c= Ball (x,s) ) by A1, PCOMPS_1:1; ::_thesis: verum
end;
theorem Th19: :: JORDAN:19
for r being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) holds (cl_Ball (x,r)) \ (Ball (x,r)) = Sphere (x,r)
proof
let r be real number ; ::_thesis: for n being Element of NAT
for x being Point of (TOP-REAL n) holds (cl_Ball (x,r)) \ (Ball (x,r)) = Sphere (x,r)
let n be Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n) holds (cl_Ball (x,r)) \ (Ball (x,r)) = Sphere (x,r)
let x be Point of (TOP-REAL n); ::_thesis: (cl_Ball (x,r)) \ (Ball (x,r)) = Sphere (x,r)
thus (cl_Ball (x,r)) \ (Ball (x,r)) c= Sphere (x,r) :: according to XBOOLE_0:def_10 ::_thesis: Sphere (x,r) c= (cl_Ball (x,r)) \ (Ball (x,r))
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in (cl_Ball (x,r)) \ (Ball (x,r)) or a in Sphere (x,r) )
assume A1: a in (cl_Ball (x,r)) \ (Ball (x,r)) ; ::_thesis: a in Sphere (x,r)
then reconsider a = a as Point of (TOP-REAL n) ;
A2: a in cl_Ball (x,r) by A1, XBOOLE_0:def_5;
A3: not a in Ball (x,r) by A1, XBOOLE_0:def_5;
A4: |.(a - x).| <= r by A2, TOPREAL9:8;
|.(a - x).| >= r by A3, TOPREAL9:7;
then |.(a - x).| = r by A4, XXREAL_0:1;
hence a in Sphere (x,r) by TOPREAL9:9; ::_thesis: verum
end;
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Sphere (x,r) or a in (cl_Ball (x,r)) \ (Ball (x,r)) )
assume A5: a in Sphere (x,r) ; ::_thesis: a in (cl_Ball (x,r)) \ (Ball (x,r))
then reconsider a = a as Point of (TOP-REAL n) ;
A6: |.(a - x).| = r by A5, TOPREAL9:9;
then A7: a in cl_Ball (x,r) by TOPREAL9:8;
not a in Ball (x,r) by A6, TOPREAL9:7;
hence a in (cl_Ball (x,r)) \ (Ball (x,r)) by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th20: :: JORDAN:20
for r being real number
for n being Element of NAT
for y, x being Point of (TOP-REAL n) st y in Sphere (x,r) holds
(LSeg (x,y)) \ {x,y} c= Ball (x,r)
proof
let r be real number ; ::_thesis: for n being Element of NAT
for y, x being Point of (TOP-REAL n) st y in Sphere (x,r) holds
(LSeg (x,y)) \ {x,y} c= Ball (x,r)
let n be Element of NAT ; ::_thesis: for y, x being Point of (TOP-REAL n) st y in Sphere (x,r) holds
(LSeg (x,y)) \ {x,y} c= Ball (x,r)
let y, x be Point of (TOP-REAL n); ::_thesis: ( y in Sphere (x,r) implies (LSeg (x,y)) \ {x,y} c= Ball (x,r) )
assume A1: y in Sphere (x,r) ; ::_thesis: (LSeg (x,y)) \ {x,y} c= Ball (x,r)
percases ( r = 0 or r <> 0 ) ;
supposeA2: r = 0 ; ::_thesis: (LSeg (x,y)) \ {x,y} c= Ball (x,r)
reconsider xe = x as Point of (Euclid n) by TOPREAL3:8;
Sphere (x,r) = Sphere (xe,r) by TOPREAL9:15;
then Sphere (x,r) = {x} by A2, TOPREAL6:54;
then A3: x = y by A1, TARSKI:def_1;
A4: LSeg (x,x) = {x} by RLTOPSP1:70;
A5: {x,x} = {x} by ENUMSET1:29;
{x} \ {x} = {} by XBOOLE_1:37;
hence (LSeg (x,y)) \ {x,y} c= Ball (x,r) by A3, A4, A5, XBOOLE_1:2; ::_thesis: verum
end;
supposeA6: r <> 0 ; ::_thesis: (LSeg (x,y)) \ {x,y} c= Ball (x,r)
let k be set ; :: according to TARSKI:def_3 ::_thesis: ( not k in (LSeg (x,y)) \ {x,y} or k in Ball (x,r) )
assume A7: k in (LSeg (x,y)) \ {x,y} ; ::_thesis: k in Ball (x,r)
then k in LSeg (x,y) by XBOOLE_0:def_5;
then consider l being Real such that
A8: k = ((1 - l) * x) + (l * y) and
A9: 0 <= l and
A10: l <= 1 ;
reconsider k = k as Point of (TOP-REAL n) by A8;
not k in {x,y} by A7, XBOOLE_0:def_5;
then k <> y by TARSKI:def_2;
then l <> 1 by A8, TOPREAL9:4;
then A11: l < 1 by A10, XXREAL_0:1;
k - x = (((1 - l) * x) - x) + (l * y) by A8, EUCLID:26
.= (((1 * x) - (l * x)) - x) + (l * y) by EUCLID:50
.= ((x - (l * x)) - x) + (l * y) by EUCLID:29
.= ((x + (- (l * x))) + (- x)) + (l * y)
.= ((x + (- x)) + (- (l * x))) + (l * y) by EUCLID:26
.= ((x - x) - (l * x)) + (l * y)
.= ((0. (TOP-REAL n)) - (l * x)) + (l * y) by EUCLID:42
.= (l * y) - (l * x) by EUCLID:27
.= l * (y - x) by EUCLID:49 ;
then A12: |.(k - x).| = (abs l) * |.(y - x).| by TOPRNS_1:7
.= l * |.(y - x).| by A9, ABSVALUE:def_1
.= l * r by A1, TOPREAL9:9 ;
0 <= r by A1;
then l * r < 1 * r by A6, A11, XREAL_1:68;
hence k in Ball (x,r) by A12, TOPREAL9:7; ::_thesis: verum
end;
end;
end;
theorem Th21: :: JORDAN:21
for r, s being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) st r < s holds
cl_Ball (x,r) c= Ball (x,s)
proof
let r, s be real number ; ::_thesis: for n being Element of NAT
for x being Point of (TOP-REAL n) st r < s holds
cl_Ball (x,r) c= Ball (x,s)
let n be Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n) st r < s holds
cl_Ball (x,r) c= Ball (x,s)
let x be Point of (TOP-REAL n); ::_thesis: ( r < s implies cl_Ball (x,r) c= Ball (x,s) )
assume A1: r < s ; ::_thesis: cl_Ball (x,r) c= Ball (x,s)
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in cl_Ball (x,r) or a in Ball (x,s) )
assume A2: a in cl_Ball (x,r) ; ::_thesis: a in Ball (x,s)
then reconsider a = a as Point of (TOP-REAL n) ;
|.(a - x).| <= r by A2, TOPREAL9:8;
then |.(a - x).| < s by A1, XXREAL_0:2;
hence a in Ball (x,s) by TOPREAL9:7; ::_thesis: verum
end;
theorem Th22: :: JORDAN:22
for r, s being real number
for n being Element of NAT
for x being Point of (TOP-REAL n) st r < s holds
Sphere (x,r) c= Ball (x,s)
proof
let r, s be real number ; ::_thesis: for n being Element of NAT
for x being Point of (TOP-REAL n) st r < s holds
Sphere (x,r) c= Ball (x,s)
let n be Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n) st r < s holds
Sphere (x,r) c= Ball (x,s)
let x be Point of (TOP-REAL n); ::_thesis: ( r < s implies Sphere (x,r) c= Ball (x,s) )
assume r < s ; ::_thesis: Sphere (x,r) c= Ball (x,s)
then A1: cl_Ball (x,r) c= Ball (x,s) by Th21;
Sphere (x,r) c= cl_Ball (x,r) by TOPREAL9:17;
hence Sphere (x,r) c= Ball (x,s) by A1, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th23: :: JORDAN:23
for n being Element of NAT
for x being Point of (TOP-REAL n)
for r being non zero real number holds Cl (Ball (x,r)) = cl_Ball (x,r)
proof
let n be Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n)
for r being non zero real number holds Cl (Ball (x,r)) = cl_Ball (x,r)
let x be Point of (TOP-REAL n); ::_thesis: for r being non zero real number holds Cl (Ball (x,r)) = cl_Ball (x,r)
let r be non zero real number ; ::_thesis: Cl (Ball (x,r)) = cl_Ball (x,r)
thus Cl (Ball (x,r)) c= cl_Ball (x,r) by TOPREAL9:16, TOPS_1:5; :: according to XBOOLE_0:def_10 ::_thesis: cl_Ball (x,r) c= Cl (Ball (x,r))
percases ( Ball (x,r) is empty or not Ball (x,r) is empty ) ;
suppose Ball (x,r) is empty ; ::_thesis: cl_Ball (x,r) c= Cl (Ball (x,r))
then r < 0 ;
hence cl_Ball (x,r) c= Cl (Ball (x,r)) ; ::_thesis: verum
end;
supposeA1: not Ball (x,r) is empty ; ::_thesis: cl_Ball (x,r) c= Cl (Ball (x,r))
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in cl_Ball (x,r) or a in Cl (Ball (x,r)) )
assume A2: a in cl_Ball (x,r) ; ::_thesis: a in Cl (Ball (x,r))
then reconsider a = a as Point of (TOP-REAL n) ;
reconsider ae = a as Point of (Euclid n) by TOPREAL3:8;
A3: 0 < r by A1;
for s being real number st 0 < s & s < r holds
Ball (ae,s) meets Ball (x,r)
proof
let s be real number ; ::_thesis: ( 0 < s & s < r implies Ball (ae,s) meets Ball (x,r) )
assume that
A4: 0 < s and
A5: s < r ; ::_thesis: Ball (ae,s) meets Ball (x,r)
now__::_thesis:_Ball_(a,s)_meets_Ball_(x,r)
A6: (Ball (x,r)) \/ (Sphere (x,r)) = cl_Ball (x,r) by TOPREAL9:18;
percases ( a in Ball (x,r) or a in Sphere (x,r) ) by A2, A6, XBOOLE_0:def_3;
supposeA7: a in Ball (x,r) ; ::_thesis: Ball (a,s) meets Ball (x,r)
|.(a - a).| = 0 by TOPRNS_1:28;
then a in Ball (a,s) by A4, TOPREAL9:7;
hence Ball (a,s) meets Ball (x,r) by A7, XBOOLE_0:3; ::_thesis: verum
end;
supposeA8: a in Sphere (x,r) ; ::_thesis: Ball (a,s) meets Ball (x,r)
then A9: |.(a - x).| = r by TOPREAL9:9;
|.(x - x).| = 0 by TOPRNS_1:28;
then A10: x in Ball (x,r) by A3, TOPREAL9:7;
set z = s / (2 * r);
set q = ((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x);
1 * r < 2 * r by A3, XREAL_1:68;
then s < 2 * r by A5, XXREAL_0:2;
then A11: s / (2 * r) < 1 by A4, XREAL_1:189;
0 < 2 * r by A3, XREAL_1:129;
then A12: 0 < s / (2 * r) by A4, XREAL_1:139;
A13: ((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x) in LSeg (a,x) by A3, A4, A11;
Ball (x,r) misses Sphere (x,r) by TOPREAL9:19;
then A14: a <> x by A8, A10, XBOOLE_0:3;
then A15: ((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x) <> a by A12, TOPREAL9:4;
((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x) <> x by A11, A14, TOPREAL9:4;
then not ((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x) in {a,x} by A15, TARSKI:def_2;
then A16: ((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x) in (LSeg (a,x)) \ {a,x} by A13, XBOOLE_0:def_5;
A17: (LSeg (a,x)) \ {a,x} c= Ball (x,r) by A8, Th20;
(((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x)) - a = (((1 - (s / (2 * r))) * a) - a) + ((s / (2 * r)) * x) by EUCLID:26
.= (((1 * a) - ((s / (2 * r)) * a)) - a) + ((s / (2 * r)) * x) by EUCLID:50
.= ((a - ((s / (2 * r)) * a)) - a) + ((s / (2 * r)) * x) by EUCLID:29
.= ((a + (- ((s / (2 * r)) * a))) + (- a)) + ((s / (2 * r)) * x)
.= ((a + (- a)) + (- ((s / (2 * r)) * a))) + ((s / (2 * r)) * x) by EUCLID:26
.= ((a - a) - ((s / (2 * r)) * a)) + ((s / (2 * r)) * x)
.= ((0. (TOP-REAL n)) - ((s / (2 * r)) * a)) + ((s / (2 * r)) * x) by EUCLID:42
.= ((s / (2 * r)) * x) - ((s / (2 * r)) * a) by EUCLID:27
.= (s / (2 * r)) * (x - a) by EUCLID:49 ;
then |.((((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x)) - a).| = (abs (s / (2 * r))) * |.(x - a).| by TOPRNS_1:7
.= (s / (2 * r)) * |.(x - a).| by A3, A4, ABSVALUE:def_1
.= (s / (2 * r)) * |.(a - x).| by TOPRNS_1:27
.= s / 2 by A9, XCMPLX_1:92 ;
then A18: ((1 - (s / (2 * r))) * a) + ((s / (2 * r)) * x) in Sphere (a,(s / 2)) by TOPREAL9:9;
s / 2 < s / 1 by A4, XREAL_1:76;
then Sphere (a,(s / 2)) c= Ball (a,s) by Th22;
hence Ball (a,s) meets Ball (x,r) by A16, A17, A18, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence Ball (ae,s) meets Ball (x,r) by TOPREAL9:13; ::_thesis: verum
end;
hence a in Cl (Ball (x,r)) by A3, GOBOARD6:93; ::_thesis: verum
end;
end;
end;
theorem Th24: :: JORDAN:24
for n being Element of NAT
for x being Point of (TOP-REAL n)
for r being non zero real number holds Fr (Ball (x,r)) = Sphere (x,r)
proof
let n be Element of NAT ; ::_thesis: for x being Point of (TOP-REAL n)
for r being non zero real number holds Fr (Ball (x,r)) = Sphere (x,r)
let x be Point of (TOP-REAL n); ::_thesis: for r being non zero real number holds Fr (Ball (x,r)) = Sphere (x,r)
let r be non zero real number ; ::_thesis: Fr (Ball (x,r)) = Sphere (x,r)
set P = Ball (x,r);
thus Fr (Ball (x,r)) = (Cl (Ball (x,r))) \ (Ball (x,r)) by TOPS_1:42
.= (cl_Ball (x,r)) \ (Ball (x,r)) by Th23
.= Sphere (x,r) by Th19 ; ::_thesis: verum
end;
registration
let n be non empty Element of NAT ;
cluster bounded -> proper for Element of bool the carrier of (TOP-REAL n);
coherence
for b1 being Subset of (TOP-REAL n) st b1 is bounded holds
b1 is proper
proof
not [#] (TOP-REAL n) is bounded by JORDAN2C:35, NAT_1:14;
hence for b1 being Subset of (TOP-REAL n) st b1 is bounded holds
b1 is proper by SUBSET_1:def_6; ::_thesis: verum
end;
end;
registration
let n be Element of NAT ;
cluster functional non empty closed bounded convex for Element of bool the carrier of (TOP-REAL n);
existence
ex b1 being Subset of (TOP-REAL n) st
( not b1 is empty & b1 is closed & b1 is convex & b1 is bounded )
proof
take cl_Ball ((0. (TOP-REAL n)),1) ; ::_thesis: ( not cl_Ball ((0. (TOP-REAL n)),1) is empty & cl_Ball ((0. (TOP-REAL n)),1) is closed & cl_Ball ((0. (TOP-REAL n)),1) is convex & cl_Ball ((0. (TOP-REAL n)),1) is bounded )
thus ( not cl_Ball ((0. (TOP-REAL n)),1) is empty & cl_Ball ((0. (TOP-REAL n)),1) is closed & cl_Ball ((0. (TOP-REAL n)),1) is convex & cl_Ball ((0. (TOP-REAL n)),1) is bounded ) ; ::_thesis: verum
end;
cluster functional non empty open bounded convex for Element of bool the carrier of (TOP-REAL n);
existence
ex b1 being Subset of (TOP-REAL n) st
( not b1 is empty & b1 is open & b1 is convex & b1 is bounded )
proof
take Ball ((0. (TOP-REAL n)),1) ; ::_thesis: ( not Ball ((0. (TOP-REAL n)),1) is empty & Ball ((0. (TOP-REAL n)),1) is open & Ball ((0. (TOP-REAL n)),1) is convex & Ball ((0. (TOP-REAL n)),1) is bounded )
thus ( not Ball ((0. (TOP-REAL n)),1) is empty & Ball ((0. (TOP-REAL n)),1) is open & Ball ((0. (TOP-REAL n)),1) is convex & Ball ((0. (TOP-REAL n)),1) is bounded ) ; ::_thesis: verum
end;
end;
registration
let n be Element of NAT ;
let A be bounded Subset of (TOP-REAL n);
cluster Cl A -> bounded ;
coherence
Cl A is bounded by TOPREAL6:63;
end;
registration
let n be Element of NAT ;
let A be bounded Subset of (TOP-REAL n);
cluster Fr A -> bounded ;
coherence
Fr A is bounded by TOPREAL6:89;
end;
theorem Th25: :: JORDAN:25
for n being Element of NAT
for A being closed Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) st not p in A holds
ex r being positive real number st Ball (p,r) misses A
proof
let n be Element of NAT ; ::_thesis: for A being closed Subset of (TOP-REAL n)
for p being Point of (TOP-REAL n) st not p in A holds
ex r being positive real number st Ball (p,r) misses A
let A be closed Subset of (TOP-REAL n); ::_thesis: for p being Point of (TOP-REAL n) st not p in A holds
ex r being positive real number st Ball (p,r) misses A
let p be Point of (TOP-REAL n); ::_thesis: ( not p in A implies ex r being positive real number st Ball (p,r) misses A )
assume not p in A ; ::_thesis: ex r being positive real number st Ball (p,r) misses A
then A1: p in A ` by SUBSET_1:29;
reconsider e = p as Point of (Euclid n) by TOPREAL3:8;
A2: TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) = TopSpaceMetr (Euclid n) by EUCLID:def_8;
then reconsider AA = A ` as Subset of (TopSpaceMetr (Euclid n)) ;
AA is open by A2, PRE_TOPC:30;
then consider r being real number such that
A3: r > 0 and
A4: Ball (e,r) c= A ` by A1, TOPMETR:15;
reconsider r = r as positive real number by A3;
take r ; ::_thesis: Ball (p,r) misses A
Ball (p,r) = Ball (e,r) by TOPREAL9:13;
hence Ball (p,r) misses A by A4, SUBSET_1:23; ::_thesis: verum
end;
theorem Th26: :: JORDAN:26
for n being Element of NAT
for A being bounded Subset of (TOP-REAL n)
for a being Point of (TOP-REAL n) ex r being positive real number st A c= Ball (a,r)
proof
let n be Element of NAT ; ::_thesis: for A being bounded Subset of (TOP-REAL n)
for a being Point of (TOP-REAL n) ex r being positive real number st A c= Ball (a,r)
let A be bounded Subset of (TOP-REAL n); ::_thesis: for a being Point of (TOP-REAL n) ex r being positive real number st A c= Ball (a,r)
let a be Point of (TOP-REAL n); ::_thesis: ex r being positive real number st A c= Ball (a,r)
reconsider C = A as bounded Subset of (Euclid n) by JORDAN2C:11;
consider r being Real, x being Element of (Euclid n) such that
A1: 0 < r and
A2: C c= Ball (x,r) by METRIC_6:def_3;
reconsider r = r as positive real number by A1;
reconsider x1 = x as Point of (TOP-REAL n) by TOPREAL3:8;
take s = r + |.(x1 - a).|; ::_thesis: A c= Ball (a,s)
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in A or p in Ball (a,s) )
assume A3: p in A ; ::_thesis: p in Ball (a,s)
then reconsider p1 = p as Point of (TOP-REAL n) ;
p = p1 ;
then reconsider p = p as Point of (Euclid n) by TOPREAL3:8;
A4: dist (p,x) < r by A2, A3, METRIC_1:11;
A5: |.(p1 - x1).| = dist (p,x) by SPPOL_1:39;
A6: |.(p1 - a).| <= |.(p1 - x1).| + |.(x1 - a).| by TOPRNS_1:34;
|.(p1 - x1).| + |.(x1 - a).| < s by A4, A5, XREAL_1:6;
then |.(p1 - a).| < s by A6, XXREAL_0:2;
hence p in Ball (a,s) by TOPREAL9:7; ::_thesis: verum
end;
theorem :: JORDAN:27
for S, T being TopStruct
for f being Function of S,T st f is being_homeomorphism holds
f is onto ;
registration
let T be non empty T_2 TopSpace;
cluster non empty -> non empty T_2 for SubSpace of T;
coherence
for b1 being non empty SubSpace of T holds b1 is T_2 ;
end;
registration
let p be Point of (TOP-REAL 2);
let r be real number ;
cluster Tdisk (p,r) -> closed ;
coherence
Tdisk (p,r) is closed
proof
let A be Subset of (TOP-REAL 2); :: according to BORSUK_1:def_11 ::_thesis: ( not A = the carrier of (Tdisk (p,r)) or A is closed )
assume A = the carrier of (Tdisk (p,r)) ; ::_thesis: A is closed
then A = cl_Ball (p,r) by BROUWER:3;
hence A is closed ; ::_thesis: verum
end;
end;
registration
let p be Point of (TOP-REAL 2);
let r be real number ;
cluster Tdisk (p,r) -> compact ;
coherence
Tdisk (p,r) is compact
proof
set D = Tdisk (p,r);
reconsider Q = [#] (Tdisk (p,r)) as Subset of (TOP-REAL 2) by TSEP_1:1;
[#] (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
then Q is compact by TOPREAL6:79;
then [#] (Tdisk (p,r)) is compact by COMPTS_1:2;
hence Tdisk (p,r) is compact by COMPTS_1:1; ::_thesis: verum
end;
end;
begin
theorem :: JORDAN:28
for T being non empty TopSpace
for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f is connected
proof
let T be non empty TopSpace; ::_thesis: for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f is connected
let a, b be Point of T; ::_thesis: for f being Path of a,b st a,b are_connected holds
rng f is connected
let f be Path of a,b; ::_thesis: ( a,b are_connected implies rng f is connected )
assume A1: a,b are_connected ; ::_thesis: rng f is connected
A2: dom f = the carrier of I[01] by FUNCT_2:def_1;
reconsider A = [.0,1.] as interval Subset of R^1 by TOPMETR:17;
reconsider B = A as Subset of I[01] by A2, BORSUK_1:40;
A3: B is connected by CONNSP_1:23;
A4: f is continuous by A1, BORSUK_2:def_2;
f .: B = rng f by A2, BORSUK_1:40, RELAT_1:113;
hence rng f is connected by A3, A4, TOPS_2:61; ::_thesis: verum
end;
theorem Th29: :: JORDAN:29
for X being non empty TopSpace
for Y being non empty SubSpace of X
for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
proof
let X be non empty TopSpace; ::_thesis: for Y being non empty SubSpace of X
for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let Y be non empty SubSpace of X; ::_thesis: for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let x1, x2 be Point of X; ::_thesis: for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let y1, y2 be Point of Y; ::_thesis: for f being Path of x1,x2 st x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let f be Path of x1,x2; ::_thesis: ( x1 = y1 & x2 = y2 & x1,x2 are_connected & rng f c= the carrier of Y implies ( y1,y2 are_connected & f is Path of y1,y2 ) )
assume that
A1: x1 = y1 and
A2: x2 = y2 and
A3: x1,x2 are_connected ; ::_thesis: ( not rng f c= the carrier of Y or ( y1,y2 are_connected & f is Path of y1,y2 ) )
assume rng f c= the carrier of Y ; ::_thesis: ( y1,y2 are_connected & f is Path of y1,y2 )
then reconsider g = f as Function of I[01],Y by FUNCT_2:6;
A4: f is continuous by A3, BORSUK_2:def_2;
A5: ( f . 0 = y1 & f . 1 = y2 ) by A1, A2, A3, BORSUK_2:def_2;
A6: g is continuous by A4, PRE_TOPC:27;
thus ex f being Function of I[01],Y st
( f is continuous & f . 0 = y1 & f . 1 = y2 ) :: according to BORSUK_2:def_1 ::_thesis: f is Path of y1,y2
proof
take g ; ::_thesis: ( g is continuous & g . 0 = y1 & g . 1 = y2 )
thus g is continuous by A4, PRE_TOPC:27; ::_thesis: ( g . 0 = y1 & g . 1 = y2 )
thus ( g . 0 = y1 & g . 1 = y2 ) by A1, A2, A3, BORSUK_2:def_2; ::_thesis: verum
end;
y1,y2 are_connected by A5, A6, BORSUK_2:def_1;
hence f is Path of y1,y2 by A5, A6, BORSUK_2:def_2; ::_thesis: verum
end;
theorem Th30: :: JORDAN:30
for X being non empty pathwise_connected TopSpace
for Y being non empty SubSpace of X
for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
proof
let X be non empty pathwise_connected TopSpace; ::_thesis: for Y being non empty SubSpace of X
for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let Y be non empty SubSpace of X; ::_thesis: for x1, x2 being Point of X
for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let x1, x2 be Point of X; ::_thesis: for y1, y2 being Point of Y
for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
let y1, y2 be Point of Y; ::_thesis: for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 )
x1,x2 are_connected by BORSUK_2:def_3;
hence for f being Path of x1,x2 st x1 = y1 & x2 = y2 & rng f c= the carrier of Y holds
( y1,y2 are_connected & f is Path of y1,y2 ) by Th29; ::_thesis: verum
end;
Lm7: for T being non empty TopSpace
for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f c= rng (- f)
proof
let T be non empty TopSpace; ::_thesis: for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f c= rng (- f)
let a, b be Point of T; ::_thesis: for f being Path of a,b st a,b are_connected holds
rng f c= rng (- f)
let f be Path of a,b; ::_thesis: ( a,b are_connected implies rng f c= rng (- f) )
assume A1: a,b are_connected ; ::_thesis: rng f c= rng (- f)
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in rng (- f) )
assume y in rng f ; ::_thesis: y in rng (- f)
then consider x being set such that
A2: x in dom f and
A3: f . x = y by FUNCT_1:def_3;
reconsider x = x as Point of I[01] by A2;
A4: dom (- f) = the carrier of I[01] by FUNCT_2:def_1;
A5: 1 - x is Point of I[01] by JORDAN5B:4;
then (- f) . (1 - x) = f . (1 - (1 - x)) by A1, BORSUK_2:def_6;
hence y in rng (- f) by A3, A4, A5, FUNCT_1:def_3; ::_thesis: verum
end;
theorem Th31: :: JORDAN:31
for T being non empty TopSpace
for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f = rng (- f)
proof
let T be non empty TopSpace; ::_thesis: for a, b being Point of T
for f being Path of a,b st a,b are_connected holds
rng f = rng (- f)
let a, b be Point of T; ::_thesis: for f being Path of a,b st a,b are_connected holds
rng f = rng (- f)
let f be Path of a,b; ::_thesis: ( a,b are_connected implies rng f = rng (- f) )
assume A1: a,b are_connected ; ::_thesis: rng f = rng (- f)
hence rng f c= rng (- f) by Lm7; :: according to XBOOLE_0:def_10 ::_thesis: rng (- f) c= rng f
f = - (- f) by A1, BORSUK_6:43;
hence rng (- f) c= rng f by A1, Lm7; ::_thesis: verum
end;
theorem Th32: :: JORDAN:32
for T being non empty pathwise_connected TopSpace
for a, b being Point of T
for f being Path of a,b holds rng f = rng (- f)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b being Point of T
for f being Path of a,b holds rng f = rng (- f)
let a, b be Point of T; ::_thesis: for f being Path of a,b holds rng f = rng (- f)
let f be Path of a,b; ::_thesis: rng f = rng (- f)
a,b are_connected by BORSUK_2:def_3;
hence rng f = rng (- f) by Th31; ::_thesis: verum
end;
theorem Th33: :: JORDAN:33
for T being non empty TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng f c= rng (f + g)
proof
let T be non empty TopSpace; ::_thesis: for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng f c= rng (f + g)
let a, b, c be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng f c= rng (f + g)
let f be Path of a,b; ::_thesis: for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng f c= rng (f + g)
let g be Path of b,c; ::_thesis: ( a,b are_connected & b,c are_connected implies rng f c= rng (f + g) )
assume that
A1: a,b are_connected and
A2: b,c are_connected ; ::_thesis: rng f c= rng (f + g)
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in rng (f + g) )
assume y in rng f ; ::_thesis: y in rng (f + g)
then consider x being set such that
A3: x in dom f and
A4: f . x = y by FUNCT_1:def_3;
A5: dom (f + g) = the carrier of I[01] by FUNCT_2:def_1;
reconsider x = x as Point of I[01] by A3;
(1 / 2) * x = x / 2 ;
then A6: x / 2 is Point of I[01] by BORSUK_6:6;
x <= 1 by BORSUK_1:43;
then x / 2 <= 1 / 2 by XREAL_1:72;
then (f + g) . (x / 2) = f . (2 * (x / 2)) by A1, A2, A6, BORSUK_2:def_5;
hence y in rng (f + g) by A4, A5, A6, FUNCT_1:def_3; ::_thesis: verum
end;
theorem :: JORDAN:34
for T being non empty pathwise_connected TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c holds rng f c= rng (f + g)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c holds rng f c= rng (f + g)
let a, b, c be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c holds rng f c= rng (f + g)
let f be Path of a,b; ::_thesis: for g being Path of b,c holds rng f c= rng (f + g)
let g be Path of b,c; ::_thesis: rng f c= rng (f + g)
A1: a,b are_connected by BORSUK_2:def_3;
b,c are_connected by BORSUK_2:def_3;
hence rng f c= rng (f + g) by A1, Th33; ::_thesis: verum
end;
theorem Th35: :: JORDAN:35
for T being non empty TopSpace
for a, b, c being Point of T
for f being Path of b,c
for g being Path of a,b st a,b are_connected & b,c are_connected holds
rng f c= rng (g + f)
proof
let T be non empty TopSpace; ::_thesis: for a, b, c being Point of T
for f being Path of b,c
for g being Path of a,b st a,b are_connected & b,c are_connected holds
rng f c= rng (g + f)
let a, b, c be Point of T; ::_thesis: for f being Path of b,c
for g being Path of a,b st a,b are_connected & b,c are_connected holds
rng f c= rng (g + f)
let f be Path of b,c; ::_thesis: for g being Path of a,b st a,b are_connected & b,c are_connected holds
rng f c= rng (g + f)
let g be Path of a,b; ::_thesis: ( a,b are_connected & b,c are_connected implies rng f c= rng (g + f) )
assume that
A1: a,b are_connected and
A2: b,c are_connected ; ::_thesis: rng f c= rng (g + f)
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in rng (g + f) )
assume y in rng f ; ::_thesis: y in rng (g + f)
then consider x being set such that
A3: x in dom f and
A4: f . x = y by FUNCT_1:def_3;
A5: dom (g + f) = the carrier of I[01] by FUNCT_2:def_1;
reconsider x = x as Point of I[01] by A3;
A6: 0 <= x by BORSUK_1:43;
then A7: 0 + (1 / 2) <= (x / 2) + (1 / 2) by XREAL_1:6;
x <= 1 by BORSUK_1:43;
then x + 1 <= 1 + 1 by XREAL_1:6;
then (x + 1) / 2 <= 2 / 2 by XREAL_1:72;
then A8: (x / 2) + (1 / 2) is Point of I[01] by A6, BORSUK_1:43;
then (g + f) . ((x / 2) + (1 / 2)) = f . ((2 * ((x / 2) + (1 / 2))) - 1) by A1, A2, A7, BORSUK_2:def_5;
hence y in rng (g + f) by A4, A5, A8, FUNCT_1:def_3; ::_thesis: verum
end;
theorem :: JORDAN:36
for T being non empty pathwise_connected TopSpace
for a, b, c being Point of T
for f being Path of b,c
for g being Path of a,b holds rng f c= rng (g + f)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b, c being Point of T
for f being Path of b,c
for g being Path of a,b holds rng f c= rng (g + f)
let a, b, c be Point of T; ::_thesis: for f being Path of b,c
for g being Path of a,b holds rng f c= rng (g + f)
let f be Path of b,c; ::_thesis: for g being Path of a,b holds rng f c= rng (g + f)
let g be Path of a,b; ::_thesis: rng f c= rng (g + f)
A1: a,b are_connected by BORSUK_2:def_3;
b,c are_connected by BORSUK_2:def_3;
hence rng f c= rng (g + f) by A1, Th35; ::_thesis: verum
end;
theorem Th37: :: JORDAN:37
for T being non empty TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng (f + g) = (rng f) \/ (rng g)
proof
let T be non empty TopSpace; ::_thesis: for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng (f + g) = (rng f) \/ (rng g)
let a, b, c be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng (f + g) = (rng f) \/ (rng g)
let f be Path of a,b; ::_thesis: for g being Path of b,c st a,b are_connected & b,c are_connected holds
rng (f + g) = (rng f) \/ (rng g)
let g be Path of b,c; ::_thesis: ( a,b are_connected & b,c are_connected implies rng (f + g) = (rng f) \/ (rng g) )
assume that
A1: a,b are_connected and
A2: b,c are_connected ; ::_thesis: rng (f + g) = (rng f) \/ (rng g)
thus rng (f + g) c= (rng f) \/ (rng g) :: according to XBOOLE_0:def_10 ::_thesis: (rng f) \/ (rng g) c= rng (f + g)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (f + g) or y in (rng f) \/ (rng g) )
assume y in rng (f + g) ; ::_thesis: y in (rng f) \/ (rng g)
then consider x being set such that
A3: x in dom (f + g) and
A4: y = (f + g) . x by FUNCT_1:def_3;
reconsider x = x as Point of I[01] by A3;
percases ( x <= 1 / 2 or 1 / 2 <= x ) ;
supposeA5: x <= 1 / 2 ; ::_thesis: y in (rng f) \/ (rng g)
then A6: (f + g) . x = f . (2 * x) by A1, A2, BORSUK_2:def_5;
A7: rng f c= (rng f) \/ (rng g) by XBOOLE_1:7;
A8: dom f = the carrier of I[01] by FUNCT_2:def_1;
2 * x is Point of I[01] by A5, BORSUK_6:3;
then y in rng f by A4, A6, A8, FUNCT_1:def_3;
hence y in (rng f) \/ (rng g) by A7; ::_thesis: verum
end;
supposeA9: 1 / 2 <= x ; ::_thesis: y in (rng f) \/ (rng g)
then A10: (f + g) . x = g . ((2 * x) - 1) by A1, A2, BORSUK_2:def_5;
A11: rng g c= (rng f) \/ (rng g) by XBOOLE_1:7;
A12: dom g = the carrier of I[01] by FUNCT_2:def_1;
(2 * x) - 1 is Point of I[01] by A9, BORSUK_6:4;
then y in rng g by A4, A10, A12, FUNCT_1:def_3;
hence y in (rng f) \/ (rng g) by A11; ::_thesis: verum
end;
end;
end;
A13: rng f c= rng (f + g) by A1, A2, Th33;
rng g c= rng (f + g) by A1, A2, Th35;
hence (rng f) \/ (rng g) c= rng (f + g) by A13, XBOOLE_1:8; ::_thesis: verum
end;
theorem :: JORDAN:38
for T being non empty pathwise_connected TopSpace
for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c holds rng (f + g) = (rng f) \/ (rng g)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b, c being Point of T
for f being Path of a,b
for g being Path of b,c holds rng (f + g) = (rng f) \/ (rng g)
let a, b, c be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c holds rng (f + g) = (rng f) \/ (rng g)
let f be Path of a,b; ::_thesis: for g being Path of b,c holds rng (f + g) = (rng f) \/ (rng g)
let g be Path of b,c; ::_thesis: rng (f + g) = (rng f) \/ (rng g)
A1: a,b are_connected by BORSUK_2:def_3;
b,c are_connected by BORSUK_2:def_3;
hence rng (f + g) = (rng f) \/ (rng g) by A1, Th37; ::_thesis: verum
end;
theorem Th39: :: JORDAN:39
for T being non empty TopSpace
for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
proof
let T be non empty TopSpace; ::_thesis: for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let a, b, c, d be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let f be Path of a,b; ::_thesis: for g being Path of b,c
for h being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let g be Path of b,c; ::_thesis: for h being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let h be Path of c,d; ::_thesis: ( a,b are_connected & b,c are_connected & c,d are_connected implies rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h) )
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected ; ::_thesis: rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
a,c are_connected by A1, A2, BORSUK_6:42;
hence rng ((f + g) + h) = (rng (f + g)) \/ (rng h) by A3, Th37
.= ((rng f) \/ (rng g)) \/ (rng h) by A1, A2, Th37 ;
::_thesis: verum
end;
theorem Th40: :: JORDAN:40
for T being non empty pathwise_connected TopSpace
for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let a, b, c, d be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let f be Path of a,b; ::_thesis: for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let g be Path of b,c; ::_thesis: for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let h be Path of c,d; ::_thesis: rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
A1: a,b are_connected by BORSUK_2:def_3;
A2: b,c are_connected by BORSUK_2:def_3;
c,d are_connected by BORSUK_2:def_3;
hence rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h) by A1, A2, Th39; ::_thesis: verum
end;
Lm8: for T being non empty TopSpace
for a, b, c, d, e being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
proof
let T be non empty TopSpace; ::_thesis: for a, b, c, d, e being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let a, b, c, d, e be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let f be Path of a,b; ::_thesis: for g being Path of b,c
for h being Path of c,d
for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let g be Path of b,c; ::_thesis: for h being Path of c,d
for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let h be Path of c,d; ::_thesis: for i being Path of d,e st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected holds
rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let i be Path of d,e; ::_thesis: ( a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected implies rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i) )
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected and
A4: d,e are_connected ; ::_thesis: rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
a,c are_connected by A1, A2, BORSUK_6:42;
then a,d are_connected by A3, BORSUK_6:42;
hence rng (((f + g) + h) + i) = (rng ((f + g) + h)) \/ (rng i) by A4, Th37
.= (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i) by A1, A2, A3, Th39 ;
::_thesis: verum
end;
Lm9: for T being non empty pathwise_connected TopSpace
for a, b, c, d, e being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b, c, d, e being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let a, b, c, d, e be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let f be Path of a,b; ::_thesis: for g being Path of b,c
for h being Path of c,d
for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let g be Path of b,c; ::_thesis: for h being Path of c,d
for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let h be Path of c,d; ::_thesis: for i being Path of d,e holds rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
let i be Path of d,e; ::_thesis: rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)
A1: a,b are_connected by BORSUK_2:def_3;
A2: b,c are_connected by BORSUK_2:def_3;
A3: c,d are_connected by BORSUK_2:def_3;
d,e are_connected by BORSUK_2:def_3;
hence rng (((f + g) + h) + i) = (((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i) by A1, A2, A3, Lm8; ::_thesis: verum
end;
Lm10: for T being non empty TopSpace
for a, b, c, d, e, z being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
proof
let T be non empty TopSpace; ::_thesis: for a, b, c, d, e, z being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let a, b, c, d, e, z be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let f be Path of a,b; ::_thesis: for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let g be Path of b,c; ::_thesis: for h being Path of c,d
for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let h be Path of c,d; ::_thesis: for i being Path of d,e
for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let i be Path of d,e; ::_thesis: for j being Path of e,z st a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected holds
rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let j be Path of e,z; ::_thesis: ( a,b are_connected & b,c are_connected & c,d are_connected & d,e are_connected & e,z are_connected implies rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j) )
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected and
A4: d,e are_connected and
A5: e,z are_connected ; ::_thesis: rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
a,c are_connected by A1, A2, BORSUK_6:42;
then a,d are_connected by A3, BORSUK_6:42;
then a,e are_connected by A4, BORSUK_6:42;
hence rng ((((f + g) + h) + i) + j) = (rng (((f + g) + h) + i)) \/ (rng j) by A5, Th37
.= ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j) by A1, A2, A3, A4, Lm8 ;
::_thesis: verum
end;
Lm11: for T being non empty pathwise_connected TopSpace
for a, b, c, d, e, z being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b, c, d, e, z being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let a, b, c, d, e, z be Point of T; ::_thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let f be Path of a,b; ::_thesis: for g being Path of b,c
for h being Path of c,d
for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let g be Path of b,c; ::_thesis: for h being Path of c,d
for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let h be Path of c,d; ::_thesis: for i being Path of d,e
for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let i be Path of d,e; ::_thesis: for j being Path of e,z holds rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
let j be Path of e,z; ::_thesis: rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j)
A1: a,b are_connected by BORSUK_2:def_3;
A2: b,c are_connected by BORSUK_2:def_3;
A3: c,d are_connected by BORSUK_2:def_3;
A4: d,e are_connected by BORSUK_2:def_3;
e,z are_connected by BORSUK_2:def_3;
hence rng ((((f + g) + h) + i) + j) = ((((rng f) \/ (rng g)) \/ (rng h)) \/ (rng i)) \/ (rng j) by A1, A2, A3, A4, Lm10; ::_thesis: verum
end;
theorem Th41: :: JORDAN:41
for T being non empty TopSpace
for a being Point of T holds I[01] --> a is Path of a,a
proof
let T be non empty TopSpace; ::_thesis: for a being Point of T holds I[01] --> a is Path of a,a
let a be Point of T; ::_thesis: I[01] --> a is Path of a,a
thus a,a are_connected ; :: according to BORSUK_2:def_2 ::_thesis: ( I[01] --> a is continuous & (I[01] --> a) . 0 = a & (I[01] --> a) . 1 = a )
thus ( I[01] --> a is continuous & (I[01] --> a) . 0 = a & (I[01] --> a) . 1 = a ) by BORSUK_1:def_14, BORSUK_1:def_15, TOPALG_3:4; ::_thesis: verum
end;
theorem Th42: :: JORDAN:42
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | P) st
( rng f = P & F = f )
proof
let n be Element of NAT ; ::_thesis: for p1, p2 being Point of (TOP-REAL n)
for P being Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | P) st
( rng f = P & F = f )
let p1, p2 be Point of (TOP-REAL n); ::_thesis: for P being Subset of (TOP-REAL n) st P is_an_arc_of p1,p2 holds
ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | P) st
( rng f = P & F = f )
let P be Subset of (TOP-REAL n); ::_thesis: ( P is_an_arc_of p1,p2 implies ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | P) st
( rng f = P & F = f ) )
assume A1: P is_an_arc_of p1,p2 ; ::_thesis: ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | P) st
( rng f = P & F = f )
then reconsider P1 = P as non empty Subset of (TOP-REAL n) by TOPREAL1:1;
consider h being Function of I[01],((TOP-REAL n) | P) such that
A2: h is being_homeomorphism and
A3: h . 0 = p1 and
A4: h . 1 = p2 by A1, TOPREAL1:def_1;
h is Function of I[01],((TOP-REAL n) | P1) ;
then reconsider h1 = h as Function of I[01],(TOP-REAL n) by TOPREALA:7;
h1 is continuous by A2, PRE_TOPC:26;
then reconsider f = h as Path of p1,p2 by A3, A4, BORSUK_2:def_4;
take f ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | P) st
( rng f = P & f = f )
take h ; ::_thesis: ( rng h = P & f = h )
thus rng h = [#] ((TOP-REAL n) | P) by A2, TOPS_2:def_5
.= P by PRE_TOPC:8 ; ::_thesis: f = h
thus f = h ; ::_thesis: verum
end;
theorem Th43: :: JORDAN:43
for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n) ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & F = f )
proof
let n be Element of NAT ; ::_thesis: for p1, p2 being Point of (TOP-REAL n) ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & F = f )
let p1, p2 be Point of (TOP-REAL n); ::_thesis: ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & F = f )
percases ( p1 = p2 or p1 <> p2 ) ;
supposeA1: p1 = p2 ; ::_thesis: ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & F = f )
then reconsider g = I[01] --> p1 as Path of p1,p2 by Th41;
take g ; ::_thesis: ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & g = f )
A2: LSeg (p1,p2) = {p1} by A1, RLTOPSP1:70;
A3: rng g = {p1} by FUNCOP_1:8;
the carrier of ((TOP-REAL n) | (LSeg (p1,p2))) = LSeg (p1,p2) by PRE_TOPC:8;
then reconsider f = g as Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) by A2, A3, FUNCT_2:6;
take f ; ::_thesis: ( rng f = LSeg (p1,p2) & g = f )
thus ( rng f = LSeg (p1,p2) & g = f ) by A1, A3, RLTOPSP1:70; ::_thesis: verum
end;
suppose p1 <> p2 ; ::_thesis: ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & F = f )
hence ex F being Path of p1,p2 ex f being Function of I[01],((TOP-REAL n) | (LSeg (p1,p2))) st
( rng f = LSeg (p1,p2) & F = f ) by Th42, TOPREAL1:9; ::_thesis: verum
end;
end;
end;
theorem Th44: :: JORDAN:44
for P being Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2 holds
ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} )
proof
let P be Subset of (TOP-REAL 2); ::_thesis: for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & q1 in P & q2 in P & q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2 holds
ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} )
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( P is_an_arc_of p1,p2 & q1 in P & q2 in P & q1 <> p1 & q1 <> p2 & q2 <> p1 & q2 <> p2 implies ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} ) )
assume that
A1: P is_an_arc_of p1,p2 and
A2: q1 in P and
A3: q2 in P and
A4: q1 <> p1 and
A5: q1 <> p2 and
A6: q2 <> p1 and
A7: q2 <> p2 ; ::_thesis: ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} )
percases ( q1 = q2 or q1 <> q2 ) ;
suppose q1 = q2 ; ::_thesis: ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} )
then reconsider f = I[01] --> q1 as Path of q1,q2 by Th41;
take f ; ::_thesis: ( rng f c= P & rng f misses {p1,p2} )
A8: rng f = {q1} by FUNCOP_1:8;
thus rng f c= P ::_thesis: rng f misses {p1,p2}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng f or x in P )
assume x in rng f ; ::_thesis: x in P
hence x in P by A2, A8, TARSKI:def_1; ::_thesis: verum
end;
A9: not p1 in {q1} by A4, TARSKI:def_1;
not p2 in {q1} by A5, TARSKI:def_1;
hence rng f misses {p1,p2} by A8, A9, ZFMISC_1:51; ::_thesis: verum
end;
suppose q1 <> q2 ; ::_thesis: ex f being Path of q1,q2 st
( rng f c= P & rng f misses {p1,p2} )
then consider Q being non empty Subset of (TOP-REAL 2) such that
A10: Q is_an_arc_of q1,q2 and
A11: Q c= P and
A12: Q misses {p1,p2} by A1, A2, A3, A4, A5, A6, A7, JORDAN16:23;
consider g being Path of q1,q2, f being Function of I[01],((TOP-REAL 2) | Q) such that
A13: rng f = Q and
A14: g = f by A10, Th42;
reconsider h = f as Function of I[01],(TOP-REAL 2) by TOPREALA:7;
the carrier of ((TOP-REAL 2) | Q) = Q by PRE_TOPC:8;
then reconsider z1 = q1, z2 = q2 as Point of ((TOP-REAL 2) | Q) by A10, TOPREAL1:1;
A15: z1,z2 are_connected
proof
take f ; :: according to BORSUK_2:def_1 ::_thesis: ( f is continuous & f . 0 = z1 & f . 1 = z2 )
thus f is continuous by A14, PRE_TOPC:27; ::_thesis: ( f . 0 = z1 & f . 1 = z2 )
thus ( f . 0 = z1 & f . 1 = z2 ) by A14, BORSUK_2:def_4; ::_thesis: verum
end;
A16: f is continuous by A14, PRE_TOPC:27;
( f . 0 = z1 & f . 1 = z2 ) by A14, BORSUK_2:def_4;
then f is Path of z1,z2 by A15, A16, BORSUK_2:def_2;
then reconsider h = h as Path of q1,q2 by A15, TOPALG_2:1;
take h ; ::_thesis: ( rng h c= P & rng h misses {p1,p2} )
thus ( rng h c= P & rng h misses {p1,p2} ) by A11, A12, A13; ::_thesis: verum
end;
end;
end;
begin
theorem Th45: :: JORDAN:45
for a, b, c, d being real number st a <= b & c <= d holds
rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d) )
assume that
A1: a <= b and
A2: c <= d ; ::_thesis: rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rectangle (a,b,c,d) or x in closed_inside_of_rectangle (a,b,c,d) )
assume x in rectangle (a,b,c,d) ; ::_thesis: x in closed_inside_of_rectangle (a,b,c,d)
then x in { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) } by A1, A2, SPPOL_2:54;
then ex p being Point of (TOP-REAL 2) st
( x = p & ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) ) ;
hence x in closed_inside_of_rectangle (a,b,c,d) by A1, A2; ::_thesis: verum
end;
theorem Th46: :: JORDAN:46
for a, b, c, d being real number holds inside_of_rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: inside_of_rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in inside_of_rectangle (a,b,c,d) or x in closed_inside_of_rectangle (a,b,c,d) )
assume x in inside_of_rectangle (a,b,c,d) ; ::_thesis: x in closed_inside_of_rectangle (a,b,c,d)
then ex p being Point of (TOP-REAL 2) st
( x = p & a < p `1 & p `1 < b & c < p `2 & p `2 < d ) ;
hence x in closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: verum
end;
theorem Th47: :: JORDAN:47
for a, b, c, d being real number holds closed_inside_of_rectangle (a,b,c,d) = (outside_of_rectangle (a,b,c,d)) `
proof
let a, b, c, d be real number ; ::_thesis: closed_inside_of_rectangle (a,b,c,d) = (outside_of_rectangle (a,b,c,d)) `
set R = closed_inside_of_rectangle (a,b,c,d);
set O = outside_of_rectangle (a,b,c,d);
thus closed_inside_of_rectangle (a,b,c,d) c= (outside_of_rectangle (a,b,c,d)) ` :: according to XBOOLE_0:def_10 ::_thesis: (outside_of_rectangle (a,b,c,d)) ` c= closed_inside_of_rectangle (a,b,c,d)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in closed_inside_of_rectangle (a,b,c,d) or x in (outside_of_rectangle (a,b,c,d)) ` )
assume x in closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: x in (outside_of_rectangle (a,b,c,d)) `
then consider p being Point of (TOP-REAL 2) such that
A1: x = p and
A2: a <= p `1 and
A3: p `1 <= b and
A4: c <= p `2 and
A5: p `2 <= d ;
now__::_thesis:_not_p_in_outside_of_rectangle_(a,b,c,d)
assume p in outside_of_rectangle (a,b,c,d) ; ::_thesis: contradiction
then ex p1 being Point of (TOP-REAL 2) st
( p1 = p & ( not a <= p1 `1 or not p1 `1 <= b or not c <= p1 `2 or not p1 `2 <= d ) ) ;
hence contradiction by A2, A3, A4, A5; ::_thesis: verum
end;
hence x in (outside_of_rectangle (a,b,c,d)) ` by A1, SUBSET_1:29; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (outside_of_rectangle (a,b,c,d)) ` or x in closed_inside_of_rectangle (a,b,c,d) )
assume A6: x in (outside_of_rectangle (a,b,c,d)) ` ; ::_thesis: x in closed_inside_of_rectangle (a,b,c,d)
then A7: not x in outside_of_rectangle (a,b,c,d) by XBOOLE_0:def_5;
reconsider x = x as Point of (TOP-REAL 2) by A6;
A8: a <= x `1 by A7;
A9: x `1 <= b by A7;
A10: c <= x `2 by A7;
x `2 <= d by A7;
hence x in closed_inside_of_rectangle (a,b,c,d) by A8, A9, A10; ::_thesis: verum
end;
registration
let a, b, c, d be real number ;
cluster closed_inside_of_rectangle (a,b,c,d) -> closed ;
coherence
closed_inside_of_rectangle (a,b,c,d) is closed
proof
set P2 = outside_of_rectangle (a,b,c,d);
A1: a is Real by XREAL_0:def_1;
A2: b is Real by XREAL_0:def_1;
A3: c is Real by XREAL_0:def_1;
d is Real by XREAL_0:def_1;
then reconsider P2 = outside_of_rectangle (a,b,c,d) as open Subset of (TOP-REAL 2) by A1, A2, A3, JORDAN1:34;
P2 ` is closed ;
hence closed_inside_of_rectangle (a,b,c,d) is closed by Th47; ::_thesis: verum
end;
end;
theorem Th48: :: JORDAN:48
for a, b, c, d being real number holds closed_inside_of_rectangle (a,b,c,d) misses outside_of_rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: closed_inside_of_rectangle (a,b,c,d) misses outside_of_rectangle (a,b,c,d)
set R = closed_inside_of_rectangle (a,b,c,d);
set P2 = outside_of_rectangle (a,b,c,d);
assume closed_inside_of_rectangle (a,b,c,d) meets outside_of_rectangle (a,b,c,d) ; ::_thesis: contradiction
then consider x being set such that
A1: x in closed_inside_of_rectangle (a,b,c,d) and
A2: x in outside_of_rectangle (a,b,c,d) by XBOOLE_0:3;
A3: ex p being Point of (TOP-REAL 2) st
( x = p & a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) by A1;
ex p being Point of (TOP-REAL 2) st
( x = p & ( not a <= p `1 or not p `1 <= b or not c <= p `2 or not p `2 <= d ) ) by A2;
hence contradiction by A3; ::_thesis: verum
end;
theorem Th49: :: JORDAN:49
for a, b, c, d being real number holds (closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d)) = inside_of_rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: (closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d)) = inside_of_rectangle (a,b,c,d)
set R = closed_inside_of_rectangle (a,b,c,d);
set P1 = inside_of_rectangle (a,b,c,d);
thus (closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d)) c= inside_of_rectangle (a,b,c,d) by XBOOLE_1:17; :: according to XBOOLE_0:def_10 ::_thesis: inside_of_rectangle (a,b,c,d) c= (closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d))
(inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d)) c= (inside_of_rectangle (a,b,c,d)) /\ (closed_inside_of_rectangle (a,b,c,d)) by Th46, XBOOLE_1:26;
hence inside_of_rectangle (a,b,c,d) c= (closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d)) ; ::_thesis: verum
end;
theorem Th50: :: JORDAN:50
for a, b, c, d being real number st a < b & c < d holds
Int (closed_inside_of_rectangle (a,b,c,d)) = inside_of_rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies Int (closed_inside_of_rectangle (a,b,c,d)) = inside_of_rectangle (a,b,c,d) )
assume that
A1: a < b and
A2: c < d ; ::_thesis: Int (closed_inside_of_rectangle (a,b,c,d)) = inside_of_rectangle (a,b,c,d)
set P = rectangle (a,b,c,d);
set R = closed_inside_of_rectangle (a,b,c,d);
set P1 = inside_of_rectangle (a,b,c,d);
set P2 = outside_of_rectangle (a,b,c,d);
A3: a is Real by XREAL_0:def_1;
A4: b is Real by XREAL_0:def_1;
A5: c is Real by XREAL_0:def_1;
A6: d is Real by XREAL_0:def_1;
A7: rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) } by A1, A2, SPPOL_2:54;
A8: closed_inside_of_rectangle (a,b,c,d) misses outside_of_rectangle (a,b,c,d) by Th48;
thus Int (closed_inside_of_rectangle (a,b,c,d)) = (Cl (((outside_of_rectangle (a,b,c,d)) `) `)) ` by Th47
.= ((outside_of_rectangle (a,b,c,d)) \/ (rectangle (a,b,c,d))) ` by A1, A2, A3, A4, A5, A6, A7, JORDAN1:44
.= ((outside_of_rectangle (a,b,c,d)) `) /\ ((rectangle (a,b,c,d)) `) by XBOOLE_1:53
.= (closed_inside_of_rectangle (a,b,c,d)) /\ ((rectangle (a,b,c,d)) `) by Th47
.= (closed_inside_of_rectangle (a,b,c,d)) /\ ((inside_of_rectangle (a,b,c,d)) \/ (outside_of_rectangle (a,b,c,d))) by A1, A2, A3, A4, A5, A6, A7, JORDAN1:36
.= ((closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d))) \/ ((closed_inside_of_rectangle (a,b,c,d)) /\ (outside_of_rectangle (a,b,c,d))) by XBOOLE_1:23
.= ((closed_inside_of_rectangle (a,b,c,d)) /\ (inside_of_rectangle (a,b,c,d))) \/ {} by A8, XBOOLE_0:def_7
.= inside_of_rectangle (a,b,c,d) by Th49 ; ::_thesis: verum
end;
theorem Th51: :: JORDAN:51
for a, b, c, d being real number st a <= b & c <= d holds
(closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d) )
assume that
A1: a <= b and
A2: c <= d ; ::_thesis: (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d)
set R = rectangle (a,b,c,d);
set P = closed_inside_of_rectangle (a,b,c,d);
set P1 = inside_of_rectangle (a,b,c,d);
A3: rectangle (a,b,c,d) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) } by A1, A2, SPPOL_2:54;
thus (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) c= rectangle (a,b,c,d) :: according to XBOOLE_0:def_10 ::_thesis: rectangle (a,b,c,d) c= (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) or x in rectangle (a,b,c,d) )
assume A4: x in (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) ; ::_thesis: x in rectangle (a,b,c,d)
then A5: not x in inside_of_rectangle (a,b,c,d) by XBOOLE_0:def_5;
x in closed_inside_of_rectangle (a,b,c,d) by A4, XBOOLE_0:def_5;
then consider p being Point of (TOP-REAL 2) such that
A6: x = p and
A7: a <= p `1 and
A8: p `1 <= b and
A9: c <= p `2 and
A10: p `2 <= d ;
( not a < p `1 or not p `1 < b or not c < p `2 or not p `2 < d ) by A5, A6;
then ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) by A7, A8, A9, A10, XXREAL_0:1;
hence x in rectangle (a,b,c,d) by A3, A6; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rectangle (a,b,c,d) or x in (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) )
assume A11: x in rectangle (a,b,c,d) ; ::_thesis: x in (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d))
then A12: ex p being Point of (TOP-REAL 2) st
( p = x & ( ( p `1 = a & p `2 <= d & p `2 >= c ) or ( p `1 <= b & p `1 >= a & p `2 = d ) or ( p `1 <= b & p `1 >= a & p `2 = c ) or ( p `1 = b & p `2 <= d & p `2 >= c ) ) ) by A3;
A13: rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d) by A1, A2, Th45;
now__::_thesis:_not_x_in_inside_of_rectangle_(a,b,c,d)
assume x in inside_of_rectangle (a,b,c,d) ; ::_thesis: contradiction
then ex p being Point of (TOP-REAL 2) st
( x = p & a < p `1 & p `1 < b & c < p `2 & p `2 < d ) ;
hence contradiction by A12; ::_thesis: verum
end;
hence x in (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) by A11, A13, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th52: :: JORDAN:52
for a, b, c, d being real number st a < b & c < d holds
Fr (closed_inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies Fr (closed_inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d) )
assume that
A1: a < b and
A2: c < d ; ::_thesis: Fr (closed_inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d)
set P = closed_inside_of_rectangle (a,b,c,d);
thus Fr (closed_inside_of_rectangle (a,b,c,d)) = (closed_inside_of_rectangle (a,b,c,d)) \ (Int (closed_inside_of_rectangle (a,b,c,d))) by TOPS_1:43
.= (closed_inside_of_rectangle (a,b,c,d)) \ (inside_of_rectangle (a,b,c,d)) by A1, A2, Th50
.= rectangle (a,b,c,d) by A1, A2, Th51 ; ::_thesis: verum
end;
theorem :: JORDAN:53
for a, b, c, d being real number st a <= b & c <= d holds
W-bound (closed_inside_of_rectangle (a,b,c,d)) = a
proof
let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies W-bound (closed_inside_of_rectangle (a,b,c,d)) = a )
assume that
A1: a <= b and
A2: c <= d ; ::_thesis: W-bound (closed_inside_of_rectangle (a,b,c,d)) = a
set X = closed_inside_of_rectangle (a,b,c,d);
reconsider Z = (proj1 | (closed_inside_of_rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) as Subset of REAL ;
A3: closed_inside_of_rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) by PRE_TOPC:8;
A4: |[a,c]| in closed_inside_of_rectangle (a,b,c,d) by A1, A2, TOPREALA:31;
A5: for p being real number st p in Z holds
p >= a
proof
let p be real number ; ::_thesis: ( p in Z implies p >= a )
assume p in Z ; ::_thesis: p >= a
then consider p0 being set such that
A6: p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
A7: p = (proj1 | (closed_inside_of_rectangle (a,b,c,d))) . p0 by FUNCT_2:64;
ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & a <= p1 `1 & p1 `1 <= b & c <= p1 `2 & p1 `2 <= d ) by A3, A6;
hence p >= a by A3, A6, A7, PSCOMP_1:22; ::_thesis: verum
end;
for q being real number st ( for p being real number st p in Z holds
p >= q ) holds
a >= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p >= q ) implies a >= q )
assume A8: for p being real number st p in Z holds
p >= q ; ::_thesis: a >= q
A9: |[a,c]| `1 = a by EUCLID:52;
(proj1 | (closed_inside_of_rectangle (a,b,c,d))) . |[a,c]| = |[a,c]| `1 by A1, A2, PSCOMP_1:22, TOPREALA:31;
hence a >= q by A3, A4, A8, A9, FUNCT_2:35; ::_thesis: verum
end;
hence W-bound (closed_inside_of_rectangle (a,b,c,d)) = a by A4, A5, SEQ_4:44; ::_thesis: verum
end;
theorem :: JORDAN:54
for a, b, c, d being real number st a <= b & c <= d holds
S-bound (closed_inside_of_rectangle (a,b,c,d)) = c
proof
let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies S-bound (closed_inside_of_rectangle (a,b,c,d)) = c )
assume that
A1: a <= b and
A2: c <= d ; ::_thesis: S-bound (closed_inside_of_rectangle (a,b,c,d)) = c
set X = closed_inside_of_rectangle (a,b,c,d);
reconsider Z = (proj2 | (closed_inside_of_rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) as Subset of REAL ;
A3: closed_inside_of_rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) by PRE_TOPC:8;
A4: |[a,c]| in closed_inside_of_rectangle (a,b,c,d) by A1, A2, TOPREALA:31;
A5: for p being real number st p in Z holds
p >= c
proof
let p be real number ; ::_thesis: ( p in Z implies p >= c )
assume p in Z ; ::_thesis: p >= c
then consider p0 being set such that
A6: p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
A7: p = (proj2 | (closed_inside_of_rectangle (a,b,c,d))) . p0 by FUNCT_2:64;
ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & a <= p1 `1 & p1 `1 <= b & c <= p1 `2 & p1 `2 <= d ) by A3, A6;
hence p >= c by A3, A6, A7, PSCOMP_1:23; ::_thesis: verum
end;
for q being real number st ( for p being real number st p in Z holds
p >= q ) holds
c >= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p >= q ) implies c >= q )
assume A8: for p being real number st p in Z holds
p >= q ; ::_thesis: c >= q
A9: |[a,c]| `2 = c by EUCLID:52;
(proj2 | (closed_inside_of_rectangle (a,b,c,d))) . |[a,c]| = |[a,c]| `2 by A1, A2, PSCOMP_1:23, TOPREALA:31;
hence c >= q by A3, A4, A8, A9, FUNCT_2:35; ::_thesis: verum
end;
hence S-bound (closed_inside_of_rectangle (a,b,c,d)) = c by A4, A5, SEQ_4:44; ::_thesis: verum
end;
theorem :: JORDAN:55
for a, b, c, d being real number st a <= b & c <= d holds
E-bound (closed_inside_of_rectangle (a,b,c,d)) = b
proof
let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies E-bound (closed_inside_of_rectangle (a,b,c,d)) = b )
assume that
A1: a <= b and
A2: c <= d ; ::_thesis: E-bound (closed_inside_of_rectangle (a,b,c,d)) = b
set X = closed_inside_of_rectangle (a,b,c,d);
reconsider Z = (proj1 | (closed_inside_of_rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) as Subset of REAL ;
A3: closed_inside_of_rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) by PRE_TOPC:8;
A4: for p being real number st p in Z holds
p <= b
proof
let p be real number ; ::_thesis: ( p in Z implies p <= b )
assume p in Z ; ::_thesis: p <= b
then consider p0 being set such that
A5: p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
A6: p = (proj1 | (closed_inside_of_rectangle (a,b,c,d))) . p0 by FUNCT_2:64;
ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & a <= p1 `1 & p1 `1 <= b & c <= p1 `2 & p1 `2 <= d ) by A3, A5;
hence p <= b by A3, A5, A6, PSCOMP_1:22; ::_thesis: verum
end;
A7: for q being real number st ( for p being real number st p in Z holds
p <= q ) holds
b <= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p <= q ) implies b <= q )
assume A8: for p being real number st p in Z holds
p <= q ; ::_thesis: b <= q
A9: |[b,d]| `1 = b by EUCLID:52;
|[b,d]| `2 = d by EUCLID:52;
then A10: |[b,d]| in closed_inside_of_rectangle (a,b,c,d) by A1, A2, A9;
then (proj1 | (closed_inside_of_rectangle (a,b,c,d))) . |[b,d]| = |[b,d]| `1 by PSCOMP_1:22;
hence b <= q by A3, A8, A9, A10, FUNCT_2:35; ::_thesis: verum
end;
|[a,c]| in closed_inside_of_rectangle (a,b,c,d) by A1, A2, TOPREALA:31;
hence E-bound (closed_inside_of_rectangle (a,b,c,d)) = b by A4, A7, SEQ_4:46; ::_thesis: verum
end;
theorem :: JORDAN:56
for a, b, c, d being real number st a <= b & c <= d holds
N-bound (closed_inside_of_rectangle (a,b,c,d)) = d
proof
let a, b, c, d be real number ; ::_thesis: ( a <= b & c <= d implies N-bound (closed_inside_of_rectangle (a,b,c,d)) = d )
assume that
A1: a <= b and
A2: c <= d ; ::_thesis: N-bound (closed_inside_of_rectangle (a,b,c,d)) = d
set X = closed_inside_of_rectangle (a,b,c,d);
reconsider Z = (proj2 | (closed_inside_of_rectangle (a,b,c,d))) .: the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) as Subset of REAL ;
A3: closed_inside_of_rectangle (a,b,c,d) = the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) by PRE_TOPC:8;
A4: for p being real number st p in Z holds
p <= d
proof
let p be real number ; ::_thesis: ( p in Z implies p <= d )
assume p in Z ; ::_thesis: p <= d
then consider p0 being set such that
A5: p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
p0 in the carrier of ((TOP-REAL 2) | (closed_inside_of_rectangle (a,b,c,d))) and
A6: p = (proj2 | (closed_inside_of_rectangle (a,b,c,d))) . p0 by FUNCT_2:64;
ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & a <= p1 `1 & p1 `1 <= b & c <= p1 `2 & p1 `2 <= d ) by A3, A5;
hence p <= d by A3, A5, A6, PSCOMP_1:23; ::_thesis: verum
end;
A7: for q being real number st ( for p being real number st p in Z holds
p <= q ) holds
d <= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p <= q ) implies d <= q )
assume A8: for p being real number st p in Z holds
p <= q ; ::_thesis: d <= q
A9: |[b,d]| `1 = b by EUCLID:52;
A10: |[b,d]| `2 = d by EUCLID:52;
then A11: |[b,d]| in closed_inside_of_rectangle (a,b,c,d) by A1, A2, A9;
then (proj2 | (closed_inside_of_rectangle (a,b,c,d))) . |[b,d]| = |[b,d]| `2 by PSCOMP_1:23;
hence d <= q by A3, A8, A10, A11, FUNCT_2:35; ::_thesis: verum
end;
|[a,c]| in closed_inside_of_rectangle (a,b,c,d) by A1, A2, TOPREALA:31;
hence N-bound (closed_inside_of_rectangle (a,b,c,d)) = d by A4, A7, SEQ_4:46; ::_thesis: verum
end;
theorem Th57: :: JORDAN:57
for a, b, c, d being real number
for p1, p2 being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st a < b & c < d & p1 in closed_inside_of_rectangle (a,b,c,d) & not p2 in closed_inside_of_rectangle (a,b,c,d) & P is_an_arc_of p1,p2 holds
Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) c= closed_inside_of_rectangle (a,b,c,d)
proof
let a, b, c, d be real number ; ::_thesis: for p1, p2 being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st a < b & c < d & p1 in closed_inside_of_rectangle (a,b,c,d) & not p2 in closed_inside_of_rectangle (a,b,c,d) & P is_an_arc_of p1,p2 holds
Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) c= closed_inside_of_rectangle (a,b,c,d)
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: for P being Subset of (TOP-REAL 2) st a < b & c < d & p1 in closed_inside_of_rectangle (a,b,c,d) & not p2 in closed_inside_of_rectangle (a,b,c,d) & P is_an_arc_of p1,p2 holds
Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) c= closed_inside_of_rectangle (a,b,c,d)
let P be Subset of (TOP-REAL 2); ::_thesis: ( a < b & c < d & p1 in closed_inside_of_rectangle (a,b,c,d) & not p2 in closed_inside_of_rectangle (a,b,c,d) & P is_an_arc_of p1,p2 implies Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) c= closed_inside_of_rectangle (a,b,c,d) )
set R = closed_inside_of_rectangle (a,b,c,d);
set dR = rectangle (a,b,c,d);
set n = First_Point (P,p1,p2,(rectangle (a,b,c,d)));
assume that
A1: a < b and
A2: c < d and
A3: p1 in closed_inside_of_rectangle (a,b,c,d) and
A4: not p2 in closed_inside_of_rectangle (a,b,c,d) and
A5: P is_an_arc_of p1,p2 ; ::_thesis: Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) c= closed_inside_of_rectangle (a,b,c,d)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) or x in closed_inside_of_rectangle (a,b,c,d) )
assume that
A6: x in Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) and
A7: not x in closed_inside_of_rectangle (a,b,c,d) ; ::_thesis: contradiction
reconsider x = x as Point of (TOP-REAL 2) by A6;
A8: Fr (closed_inside_of_rectangle (a,b,c,d)) = rectangle (a,b,c,d) by A1, A2, Th52;
p1 in P by A5, TOPREAL1:1;
then A9: P meets closed_inside_of_rectangle (a,b,c,d) by A3, XBOOLE_0:3;
p2 in P by A5, TOPREAL1:1;
then P \ (closed_inside_of_rectangle (a,b,c,d)) <> {} (TOP-REAL 2) by A4, XBOOLE_0:def_5;
then A10: P meets rectangle (a,b,c,d) by A5, A8, A9, CONNSP_1:22, JORDAN6:10;
A11: P is closed by A5, JORDAN6:11;
then A12: P /\ (rectangle (a,b,c,d)) is closed ;
A13: First_Point (P,p1,p2,(rectangle (a,b,c,d))) in P /\ (rectangle (a,b,c,d)) by A5, A10, A11, JORDAN5C:def_1;
percases ( x = First_Point (P,p1,p2,(rectangle (a,b,c,d))) or x <> First_Point (P,p1,p2,(rectangle (a,b,c,d))) ) ;
suppose x = First_Point (P,p1,p2,(rectangle (a,b,c,d))) ; ::_thesis: contradiction
then A14: x in rectangle (a,b,c,d) by A13, XBOOLE_0:def_4;
rectangle (a,b,c,d) c= closed_inside_of_rectangle (a,b,c,d) by A1, A2, Th45;
hence contradiction by A7, A14; ::_thesis: verum
end;
supposeA15: x <> First_Point (P,p1,p2,(rectangle (a,b,c,d))) ; ::_thesis: contradiction
reconsider P = P as non empty Subset of (TOP-REAL 2) by A5, TOPREAL1:1;
consider f being Function of I[01],((TOP-REAL 2) | P) such that
A16: f is being_homeomorphism and
A17: f . 0 = p1 and
A18: f . 1 = p2 by A5, TOPREAL1:def_1;
A19: rng f = [#] ((TOP-REAL 2) | P) by A16, TOPS_2:def_5
.= P by PRE_TOPC:def_5 ;
First_Point (P,p1,p2,(rectangle (a,b,c,d))) in P by A13, XBOOLE_0:def_4;
then consider na being set such that
A20: na in dom f and
A21: f . na = First_Point (P,p1,p2,(rectangle (a,b,c,d))) by A19, FUNCT_1:def_3;
reconsider na = na as Real by A20, XREAL_0:def_1;
A22: 0 <= na by A20, BORSUK_1:43;
A23: na <= 1 by A20, BORSUK_1:43;
A24: Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) c= P by JORDAN16:2;
then consider xa being set such that
A25: xa in dom f and
A26: f . xa = x by A6, A19, FUNCT_1:def_3;
reconsider xa = xa as Real by A25, XREAL_0:def_1;
A27: 0 <= xa by A25, BORSUK_1:43;
A28: xa <= 1 by A25, BORSUK_1:43;
A29: Segment (P,p1,p2,p1,x) is_an_arc_of p1,x by A3, A5, A6, A7, A24, JORDAN16:24;
then p1 in Segment (P,p1,p2,p1,x) by TOPREAL1:1;
then A30: Segment (P,p1,p2,p1,x) meets closed_inside_of_rectangle (a,b,c,d) by A3, XBOOLE_0:3;
x in Segment (P,p1,p2,p1,x) by A29, TOPREAL1:1;
then (Segment (P,p1,p2,p1,x)) \ (closed_inside_of_rectangle (a,b,c,d)) <> {} (TOP-REAL 2) by A7, XBOOLE_0:def_5;
then Segment (P,p1,p2,p1,x) meets Fr (closed_inside_of_rectangle (a,b,c,d)) by A29, A30, CONNSP_1:22, JORDAN6:10;
then consider z being set such that
A31: z in Segment (P,p1,p2,p1,x) and
A32: z in rectangle (a,b,c,d) by A8, XBOOLE_0:3;
reconsider z = z as Point of (TOP-REAL 2) by A31;
Segment (P,p1,p2,p1,x) = { p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p,x,P,p1,p2 ) } by JORDAN6:26;
then A33: ex zz being Point of (TOP-REAL 2) st
( zz = z & LE p1,zz,P,p1,p2 & LE zz,x,P,p1,p2 ) by A31;
Segment (P,p1,p2,p1,x) c= P by JORDAN16:2;
then consider za being set such that
A34: za in dom f and
A35: f . za = z by A19, A31, FUNCT_1:def_3;
reconsider za = za as Real by A34, XREAL_0:def_1;
A36: 0 <= za by A34, BORSUK_1:43;
A37: za <= 1 by A34, BORSUK_1:43;
A38: na <= za by A5, A10, A12, A16, A17, A18, A21, A23, A32, A35, A36, JORDAN5C:def_1;
A39: za <= xa by A16, A17, A18, A26, A27, A28, A33, A35, A37, JORDAN5C:def_3;
Segment (P,p1,p2,p1,(First_Point (P,p1,p2,(rectangle (a,b,c,d))))) = { p where p is Point of (TOP-REAL 2) : ( LE p1,p,P,p1,p2 & LE p, First_Point (P,p1,p2,(rectangle (a,b,c,d))),P,p1,p2 ) } by JORDAN6:26;
then ex xx being Point of (TOP-REAL 2) st
( xx = x & LE p1,xx,P,p1,p2 & LE xx, First_Point (P,p1,p2,(rectangle (a,b,c,d))),P,p1,p2 ) by A6;
then xa <= na by A16, A17, A18, A21, A22, A23, A26, A28, JORDAN5C:def_3;
then xa < na by A15, A21, A26, XXREAL_0:1;
hence contradiction by A38, A39, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
begin
definition
let S, T be non empty TopSpace;
let x be Point of [:S,T:];
:: original: `1
redefine funcx `1 -> Element of S;
coherence
x `1 is Element of S
proof
the carrier of [:S,T:] = [: the carrier of S, the carrier of T:] by BORSUK_1:def_2;
hence x `1 is Element of S by MCART_1:10; ::_thesis: verum
end;
:: original: `2
redefine funcx `2 -> Element of T;
coherence
x `2 is Element of T
proof
the carrier of [:S,T:] = [: the carrier of S, the carrier of T:] by BORSUK_1:def_2;
hence x `2 is Element of T by MCART_1:10; ::_thesis: verum
end;
end;
definition
let o be Point of (TOP-REAL 2);
func diffX2_1 o -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def1: :: JORDAN:def 1
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `2) `1) - (o `1);
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2) `1) - (o `1)
proof
deffunc H1( Point of [:(TOP-REAL 2),(TOP-REAL 2):]) -> Element of REAL = (($1 `2) `1) - (o `1);
consider xo being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A1: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = H1(x) from FUNCT_2:sch_4();
take xo ; ::_thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `2) `1) - (o `1)
thus for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `2) `1) - (o `1) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2) `1) - (o `1) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2) `1) - (o `1) ) holds
b1 = b2
proof
let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `2) `1) - (o `1) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `2) `1) - (o `1) ) implies f = g )
assume that
A2: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `2) `1) - (o `1) and
A3: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `2) `1) - (o `1) ; ::_thesis: f = g
now__::_thesis:_for_x_being_Point_of_[:(TOP-REAL_2),(TOP-REAL_2):]_holds_f_._x_=_g_._x
let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: f . x = g . x
thus f . x = ((x `2) `1) - (o `1) by A2
.= g . x by A3 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
func diffX2_2 o -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def2: :: JORDAN:def 2
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `2) `2) - (o `2);
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2) `2) - (o `2)
proof
deffunc H1( Point of [:(TOP-REAL 2),(TOP-REAL 2):]) -> Element of REAL = (($1 `2) `2) - (o `2);
consider xo being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A4: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = H1(x) from FUNCT_2:sch_4();
take xo ; ::_thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `2) `2) - (o `2)
thus for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `2) `2) - (o `2) by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `2) `2) - (o `2) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2) `2) - (o `2) ) holds
b1 = b2
proof
let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `2) `2) - (o `2) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `2) `2) - (o `2) ) implies f = g )
assume that
A5: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `2) `2) - (o `2) and
A6: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `2) `2) - (o `2) ; ::_thesis: f = g
now__::_thesis:_for_x_being_Point_of_[:(TOP-REAL_2),(TOP-REAL_2):]_holds_f_._x_=_g_._x
let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: f . x = g . x
thus f . x = ((x `2) `2) - (o `2) by A5
.= g . x by A6 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines diffX2_1 JORDAN:def_1_:_
for o being Point of (TOP-REAL 2)
for b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b2 = diffX2_1 o iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2) `1) - (o `1) );
:: deftheorem Def2 defines diffX2_2 JORDAN:def_2_:_
for o being Point of (TOP-REAL 2)
for b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b2 = diffX2_2 o iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `2) `2) - (o `2) );
definition
func diffX1_X2_1 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def3: :: JORDAN:def 3
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `1) `1) - ((x `2) `1);
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1) `1) - ((x `2) `1)
proof
deffunc H1( Point of [:(TOP-REAL 2),(TOP-REAL 2):]) -> Element of REAL = (($1 `1) `1) - (($1 `2) `1);
consider xo being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A1: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = H1(x) from FUNCT_2:sch_4();
take xo ; ::_thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `1) `1) - ((x `2) `1)
thus for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `1) `1) - ((x `2) `1) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1) `1) - ((x `2) `1) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `1) `1) - ((x `2) `1) ) holds
b1 = b2
proof
let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1) `1) - ((x `2) `1) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1) `1) - ((x `2) `1) ) implies f = g )
assume that
A2: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1) `1) - ((x `2) `1) and
A3: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1) `1) - ((x `2) `1) ; ::_thesis: f = g
now__::_thesis:_for_x_being_Point_of_[:(TOP-REAL_2),(TOP-REAL_2):]_holds_f_._x_=_g_._x
let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: f . x = g . x
thus f . x = ((x `1) `1) - ((x `2) `1) by A2
.= g . x by A3 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
func diffX1_X2_2 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def4: :: JORDAN:def 4
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = ((x `1) `2) - ((x `2) `2);
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1) `2) - ((x `2) `2)
proof
deffunc H1( Point of [:(TOP-REAL 2),(TOP-REAL 2):]) -> Element of REAL = (($1 `1) `2) - (($1 `2) `2);
consider xo being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A4: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = H1(x) from FUNCT_2:sch_4();
take xo ; ::_thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `1) `2) - ((x `2) `2)
thus for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = ((x `1) `2) - ((x `2) `2) by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1) `2) - ((x `2) `2) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = ((x `1) `2) - ((x `2) `2) ) holds
b1 = b2
proof
let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1) `2) - ((x `2) `2) ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1) `2) - ((x `2) `2) ) implies f = g )
assume that
A5: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = ((x `1) `2) - ((x `2) `2) and
A6: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = ((x `1) `2) - ((x `2) `2) ; ::_thesis: f = g
now__::_thesis:_for_x_being_Point_of_[:(TOP-REAL_2),(TOP-REAL_2):]_holds_f_._x_=_g_._x
let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: f . x = g . x
thus f . x = ((x `1) `2) - ((x `2) `2) by A5
.= g . x by A6 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
func Proj2_1 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def5: :: JORDAN:def 5
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = (x `2) `1 ;
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2) `1
proof
deffunc H1( Point of [:(TOP-REAL 2),(TOP-REAL 2):]) -> Element of REAL = ($1 `2) `1 ;
consider xo being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A7: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = H1(x) from FUNCT_2:sch_4();
take xo ; ::_thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = (x `2) `1
thus for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = (x `2) `1 by A7; ::_thesis: verum
end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2) `1 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = (x `2) `1 ) holds
b1 = b2
proof
let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = (x `2) `1 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `1 ) implies f = g )
assume that
A8: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = (x `2) `1 and
A9: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `1 ; ::_thesis: f = g
now__::_thesis:_for_x_being_Point_of_[:(TOP-REAL_2),(TOP-REAL_2):]_holds_f_._x_=_g_._x
let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: f . x = g . x
thus f . x = (x `2) `1 by A8
.= g . x by A9 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
func Proj2_2 -> RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] means :Def6: :: JORDAN:def 6
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds it . x = (x `2) `2 ;
existence
ex b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st
for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2) `2
proof
deffunc H1( Point of [:(TOP-REAL 2),(TOP-REAL 2):]) -> Element of REAL = ($1 `2) `2 ;
consider xo being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A10: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = H1(x) from FUNCT_2:sch_4();
take xo ; ::_thesis: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = (x `2) `2
thus for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds xo . x = (x `2) `2 by A10; ::_thesis: verum
end;
uniqueness
for b1, b2 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] st ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2) `2 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b2 . x = (x `2) `2 ) holds
b1 = b2
proof
let f, g be RealMap of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: ( ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = (x `2) `2 ) & ( for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `2 ) implies f = g )
assume that
A11: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds f . x = (x `2) `2 and
A12: for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds g . x = (x `2) `2 ; ::_thesis: f = g
now__::_thesis:_for_x_being_Point_of_[:(TOP-REAL_2),(TOP-REAL_2):]_holds_f_._x_=_g_._x
let x be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: f . x = g . x
thus f . x = (x `2) `2 by A11
.= g . x by A12 ; ::_thesis: verum
end;
hence f = g by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines diffX1_X2_1 JORDAN:def_3_:_
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = diffX1_X2_1 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1) `1) - ((x `2) `1) );
:: deftheorem Def4 defines diffX1_X2_2 JORDAN:def_4_:_
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = diffX1_X2_2 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = ((x `1) `2) - ((x `2) `2) );
:: deftheorem Def5 defines Proj2_1 JORDAN:def_5_:_
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = Proj2_1 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2) `1 );
:: deftheorem Def6 defines Proj2_2 JORDAN:def_6_:_
for b1 being RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] holds
( b1 = Proj2_2 iff for x being Point of [:(TOP-REAL 2),(TOP-REAL 2):] holds b1 . x = (x `2) `2 );
theorem Th58: :: JORDAN:58
for o being Point of (TOP-REAL 2) holds diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof
let o be Point of (TOP-REAL 2); ::_thesis: diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
reconsider Xo = diffX2_1 o as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:17;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st Xo . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V )
proof
let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: for V being Subset of R^1 st Xo . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V )
let V be Subset of R^1; ::_thesis: ( Xo . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V ) )
assume that
A1: Xo . p in V and
A2: V is open ; ::_thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Xo .: W c= V )
A3: Xo . p = ((p `2) `1) - (o `1) by Def1;
set r = ((p `2) `1) - (o `1);
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:39, TOPMETR:17;
consider g being real number such that
A4: 0 < g and
A5: ].((((p `2) `1) - (o `1)) - g),((((p `2) `1) - (o `1)) + g).[ c= V1 by A1, A3, RCOMP_1:19;
reconsider g = g as Element of REAL by XREAL_0:def_1;
set W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2) `1) - g < x & x < ((p `2) `1) + g ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } as Subset of (TOP-REAL 2) ;
take [:([#] (TOP-REAL 2)),W2:] ; ::_thesis: ( p in [:([#] (TOP-REAL 2)),W2:] & [:([#] (TOP-REAL 2)),W2:] is open & Xo .: [:([#] (TOP-REAL 2)),W2:] c= V )
A6: p `2 = |[((p `2) `1),((p `2) `2)]| by EUCLID:53;
A7: p = [(p `1),(p `2)] by Lm5, MCART_1:21;
A8: ((p `2) `1) - g < ((p `2) `1) - 0 by A4, XREAL_1:15;
((p `2) `1) + 0 < ((p `2) `1) + g by A4, XREAL_1:6;
then p `2 in W2 by A6, A8;
hence p in [:([#] (TOP-REAL 2)),W2:] by A7, ZFMISC_1:def_2; ::_thesis: ( [:([#] (TOP-REAL 2)),W2:] is open & Xo .: [:([#] (TOP-REAL 2)),W2:] c= V )
W2 is open by PSCOMP_1:19;
hence [:([#] (TOP-REAL 2)),W2:] is open by BORSUK_1:6; ::_thesis: Xo .: [:([#] (TOP-REAL 2)),W2:] c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in Xo .: [:([#] (TOP-REAL 2)),W2:] or b in V )
assume b in Xo .: [:([#] (TOP-REAL 2)),W2:] ; ::_thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A9: a in [:([#] (TOP-REAL 2)),W2:] and
A10: Xo . a = b by FUNCT_2:65;
A11: a = [(a `1),(a `2)] by Lm5, MCART_1:21;
A12: (diffX2_1 o) . a = ((a `2) `1) - (o `1) by Def1;
a `2 in W2 by A9, A11, ZFMISC_1:87;
then consider x2, y2 being Element of REAL such that
A13: a `2 = |[x2,y2]| and
A14: ((p `2) `1) - g < x2 and
A15: x2 < ((p `2) `1) + g ;
A16: (a `2) `1 = x2 by A13, EUCLID:52;
then A17: (((p `2) `1) - g) - (o `1) < ((a `2) `1) - (o `1) by A14, XREAL_1:9;
((a `2) `1) - (o `1) < (((p `2) `1) + g) - (o `1) by A15, A16, XREAL_1:9;
then ((a `2) `1) - (o `1) in ].((((p `2) `1) - (o `1)) - g),((((p `2) `1) - (o `1)) + g).[ by A17, XXREAL_1:4;
hence b in V by A5, A10, A12; ::_thesis: verum
end;
hence diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:10; ::_thesis: verum
end;
theorem Th59: :: JORDAN:59
for o being Point of (TOP-REAL 2) holds diffX2_2 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof
let o be Point of (TOP-REAL 2); ::_thesis: diffX2_2 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
reconsider Yo = diffX2_2 o as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:17;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st Yo . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Yo .: W c= V )
proof
let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: for V being Subset of R^1 st Yo . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Yo .: W c= V )
let V be Subset of R^1; ::_thesis: ( Yo . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Yo .: W c= V ) )
assume that
A1: Yo . p in V and
A2: V is open ; ::_thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Yo .: W c= V )
A3: p = [(p `1),(p `2)] by Lm5, MCART_1:21;
A4: Yo . p = ((p `2) `2) - (o `2) by Def2;
set r = ((p `2) `2) - (o `2);
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:39, TOPMETR:17;
consider g being real number such that
A5: 0 < g and
A6: ].((((p `2) `2) - (o `2)) - g),((((p `2) `2) - (o `2)) + g).[ c= V1 by A1, A4, RCOMP_1:19;
reconsider g = g as Element of REAL by XREAL_0:def_1;
set W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2) `2) - g < y & y < ((p `2) `2) + g ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } as Subset of (TOP-REAL 2) ;
take [:([#] (TOP-REAL 2)),W2:] ; ::_thesis: ( p in [:([#] (TOP-REAL 2)),W2:] & [:([#] (TOP-REAL 2)),W2:] is open & Yo .: [:([#] (TOP-REAL 2)),W2:] c= V )
A7: p `2 = |[((p `2) `1),((p `2) `2)]| by EUCLID:53;
A8: ((p `2) `2) - g < ((p `2) `2) - 0 by A5, XREAL_1:15;
((p `2) `2) + 0 < ((p `2) `2) + g by A5, XREAL_1:6;
then p `2 in W2 by A7, A8;
hence p in [:([#] (TOP-REAL 2)),W2:] by A3, ZFMISC_1:def_2; ::_thesis: ( [:([#] (TOP-REAL 2)),W2:] is open & Yo .: [:([#] (TOP-REAL 2)),W2:] c= V )
W2 is open by PSCOMP_1:21;
hence [:([#] (TOP-REAL 2)),W2:] is open by BORSUK_1:6; ::_thesis: Yo .: [:([#] (TOP-REAL 2)),W2:] c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in Yo .: [:([#] (TOP-REAL 2)),W2:] or b in V )
assume b in Yo .: [:([#] (TOP-REAL 2)),W2:] ; ::_thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A9: a in [:([#] (TOP-REAL 2)),W2:] and
A10: Yo . a = b by FUNCT_2:65;
A11: a = [(a `1),(a `2)] by Lm5, MCART_1:21;
A12: (diffX2_2 o) . a = ((a `2) `2) - (o `2) by Def2;
a `2 in W2 by A9, A11, ZFMISC_1:87;
then consider x2, y2 being Element of REAL such that
A13: a `2 = |[x2,y2]| and
A14: ((p `2) `2) - g < y2 and
A15: y2 < ((p `2) `2) + g ;
A16: (a `2) `2 = y2 by A13, EUCLID:52;
then A17: (((p `2) `2) - g) - (o `2) < ((a `2) `2) - (o `2) by A14, XREAL_1:9;
((a `2) `2) - (o `2) < (((p `2) `2) + g) - (o `2) by A15, A16, XREAL_1:9;
then ((a `2) `2) - (o `2) in ].((((p `2) `2) - (o `2)) - g),((((p `2) `2) - (o `2)) + g).[ by A17, XXREAL_1:4;
hence b in V by A6, A10, A12; ::_thesis: verum
end;
hence diffX2_2 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:10; ::_thesis: verum
end;
theorem Th60: :: JORDAN:60
diffX1_X2_1 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof
reconsider Dx = diffX1_X2_1 as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:17;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st Dx . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dx .: W c= V )
proof
let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: for V being Subset of R^1 st Dx . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dx .: W c= V )
let V be Subset of R^1; ::_thesis: ( Dx . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dx .: W c= V ) )
assume that
A1: Dx . p in V and
A2: V is open ; ::_thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dx .: W c= V )
A3: p = [(p `1),(p `2)] by Lm5, MCART_1:21;
A4: diffX1_X2_1 . p = ((p `1) `1) - ((p `2) `1) by Def3;
set r = ((p `1) `1) - ((p `2) `1);
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:39, TOPMETR:17;
consider g being real number such that
A5: 0 < g and
A6: ].((((p `1) `1) - ((p `2) `1)) - g),((((p `1) `1) - ((p `2) `1)) + g).[ c= V1 by A1, A4, RCOMP_1:19;
reconsider g = g as Element of REAL by XREAL_0:def_1;
set W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `1) `1) - (g / 2) < x & x < ((p `1) `1) + (g / 2) ) } ;
set W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - (g / 2) < x & x < ((p `2) `1) + (g / 2) ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `1) `1) - (g / 2) < x & x < ((p `1) `1) + (g / 2) ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `1) `1) - (g / 2) < x & x < ((p `1) `1) + (g / 2) ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `1) `1) - (g / 2) < x & x < ((p `1) `1) + (g / 2) ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `1) `1) - (g / 2) < x & x < ((p `1) `1) + (g / 2) ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `1) `1) - (g / 2) < x & x < ((p `1) `1) + (g / 2) ) } as Subset of (TOP-REAL 2) ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - (g / 2) < x & x < ((p `2) `1) + (g / 2) ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - (g / 2) < x & x < ((p `2) `1) + (g / 2) ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - (g / 2) < x & x < ((p `2) `1) + (g / 2) ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2) `1) - (g / 2) < x & x < ((p `2) `1) + (g / 2) ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - (g / 2) < x & x < ((p `2) `1) + (g / 2) ) } as Subset of (TOP-REAL 2) ;
take [:W1,W2:] ; ::_thesis: ( p in [:W1,W2:] & [:W1,W2:] is open & Dx .: [:W1,W2:] c= V )
A7: p `1 = |[((p `1) `1),((p `1) `2)]| by EUCLID:53;
A8: 0 / 2 < g / 2 by A5, XREAL_1:74;
then A9: ((p `1) `1) - (g / 2) < ((p `1) `1) - 0 by XREAL_1:15;
((p `1) `1) + 0 < ((p `1) `1) + (g / 2) by A8, XREAL_1:6;
then A10: p `1 in W1 by A7, A9;
A11: p `2 = |[((p `2) `1),((p `2) `2)]| by EUCLID:53;
A12: ((p `2) `1) - (g / 2) < ((p `2) `1) - 0 by A8, XREAL_1:15;
((p `2) `1) + 0 < ((p `2) `1) + (g / 2) by A8, XREAL_1:6;
then p `2 in W2 by A11, A12;
hence p in [:W1,W2:] by A3, A10, ZFMISC_1:def_2; ::_thesis: ( [:W1,W2:] is open & Dx .: [:W1,W2:] c= V )
A13: W1 is open by PSCOMP_1:19;
W2 is open by PSCOMP_1:19;
hence [:W1,W2:] is open by A13, BORSUK_1:6; ::_thesis: Dx .: [:W1,W2:] c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in Dx .: [:W1,W2:] or b in V )
assume b in Dx .: [:W1,W2:] ; ::_thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A14: a in [:W1,W2:] and
A15: Dx . a = b by FUNCT_2:65;
A16: a = [(a `1),(a `2)] by Lm5, MCART_1:21;
A17: diffX1_X2_1 . a = ((a `1) `1) - ((a `2) `1) by Def3;
a `1 in W1 by A14, A16, ZFMISC_1:87;
then consider x1, y1 being Element of REAL such that
A18: a `1 = |[x1,y1]| and
A19: ((p `1) `1) - (g / 2) < x1 and
A20: x1 < ((p `1) `1) + (g / 2) ;
A21: (a `1) `1 = x1 by A18, EUCLID:52;
A22: (((p `1) `1) - (g / 2)) + (g / 2) < x1 + (g / 2) by A19, XREAL_1:6;
A23: ((p `1) `1) - x1 > ((p `1) `1) - (((p `1) `1) + (g / 2)) by A20, XREAL_1:15;
A24: ((p `1) `1) - x1 < (x1 + (g / 2)) - x1 by A22, XREAL_1:9;
((p `1) `1) - x1 > - (g / 2) by A23;
then A25: abs (((p `1) `1) - x1) < g / 2 by A24, SEQ_2:1;
a `2 in W2 by A14, A16, ZFMISC_1:87;
then consider x2, y2 being Element of REAL such that
A26: a `2 = |[x2,y2]| and
A27: ((p `2) `1) - (g / 2) < x2 and
A28: x2 < ((p `2) `1) + (g / 2) ;
A29: (a `2) `1 = x2 by A26, EUCLID:52;
A30: (((p `2) `1) - (g / 2)) + (g / 2) < x2 + (g / 2) by A27, XREAL_1:6;
A31: ((p `2) `1) - x2 > ((p `2) `1) - (((p `2) `1) + (g / 2)) by A28, XREAL_1:15;
A32: ((p `2) `1) - x2 < (x2 + (g / 2)) - x2 by A30, XREAL_1:9;
((p `2) `1) - x2 > - (g / 2) by A31;
then abs (((p `2) `1) - x2) < g / 2 by A32, SEQ_2:1;
then A33: (abs (((p `1) `1) - x1)) + (abs (((p `2) `1) - x2)) < (g / 2) + (g / 2) by A25, XREAL_1:8;
abs ((((p `1) `1) - x1) - (((p `2) `1) - x2)) <= (abs (((p `1) `1) - x1)) + (abs (((p `2) `1) - x2)) by COMPLEX1:57;
then abs (- ((((p `1) `1) - x1) - (((p `2) `1) - x2))) <= (abs (((p `1) `1) - x1)) + (abs (((p `2) `1) - x2)) by COMPLEX1:52;
then abs ((x1 - x2) - (((p `1) `1) - ((p `2) `1))) < g by A33, XXREAL_0:2;
then ((a `1) `1) - ((a `2) `1) in ].((((p `1) `1) - ((p `2) `1)) - g),((((p `1) `1) - ((p `2) `1)) + g).[ by A21, A29, RCOMP_1:1;
hence b in V by A6, A15, A17; ::_thesis: verum
end;
hence diffX1_X2_1 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:10; ::_thesis: verum
end;
theorem Th61: :: JORDAN:61
diffX1_X2_2 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof
reconsider Dy = diffX1_X2_2 as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:17;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st Dy . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dy .: W c= V )
proof
let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: for V being Subset of R^1 st Dy . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dy .: W c= V )
let V be Subset of R^1; ::_thesis: ( Dy . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dy .: W c= V ) )
assume that
A1: Dy . p in V and
A2: V is open ; ::_thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & Dy .: W c= V )
A3: p = [(p `1),(p `2)] by Lm5, MCART_1:21;
A4: diffX1_X2_2 . p = ((p `1) `2) - ((p `2) `2) by Def4;
set r = ((p `1) `2) - ((p `2) `2);
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:39, TOPMETR:17;
consider g being real number such that
A5: 0 < g and
A6: ].((((p `1) `2) - ((p `2) `2)) - g),((((p `1) `2) - ((p `2) `2)) + g).[ c= V1 by A1, A4, RCOMP_1:19;
reconsider g = g as Element of REAL by XREAL_0:def_1;
set W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `1) `2) - (g / 2) < y & y < ((p `1) `2) + (g / 2) ) } ;
set W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - (g / 2) < y & y < ((p `2) `2) + (g / 2) ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `1) `2) - (g / 2) < y & y < ((p `1) `2) + (g / 2) ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `1) `2) - (g / 2) < y & y < ((p `1) `2) + (g / 2) ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `1) `2) - (g / 2) < y & y < ((p `1) `2) + (g / 2) ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `1) `2) - (g / 2) < y & y < ((p `1) `2) + (g / 2) ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `1) `2) - (g / 2) < y & y < ((p `1) `2) + (g / 2) ) } as Subset of (TOP-REAL 2) ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - (g / 2) < y & y < ((p `2) `2) + (g / 2) ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - (g / 2) < y & y < ((p `2) `2) + (g / 2) ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - (g / 2) < y & y < ((p `2) `2) + (g / 2) ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2) `2) - (g / 2) < y & y < ((p `2) `2) + (g / 2) ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W2 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - (g / 2) < y & y < ((p `2) `2) + (g / 2) ) } as Subset of (TOP-REAL 2) ;
take [:W1,W2:] ; ::_thesis: ( p in [:W1,W2:] & [:W1,W2:] is open & Dy .: [:W1,W2:] c= V )
A7: p `1 = |[((p `1) `1),((p `1) `2)]| by EUCLID:53;
A8: 0 / 2 < g / 2 by A5, XREAL_1:74;
then A9: ((p `1) `2) - (g / 2) < ((p `1) `2) - 0 by XREAL_1:15;
((p `1) `2) + 0 < ((p `1) `2) + (g / 2) by A8, XREAL_1:6;
then A10: p `1 in W1 by A7, A9;
A11: p `2 = |[((p `2) `1),((p `2) `2)]| by EUCLID:53;
A12: ((p `2) `2) - (g / 2) < ((p `2) `2) - 0 by A8, XREAL_1:15;
((p `2) `2) + 0 < ((p `2) `2) + (g / 2) by A8, XREAL_1:6;
then p `2 in W2 by A11, A12;
hence p in [:W1,W2:] by A3, A10, ZFMISC_1:def_2; ::_thesis: ( [:W1,W2:] is open & Dy .: [:W1,W2:] c= V )
A13: W1 is open by PSCOMP_1:21;
W2 is open by PSCOMP_1:21;
hence [:W1,W2:] is open by A13, BORSUK_1:6; ::_thesis: Dy .: [:W1,W2:] c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in Dy .: [:W1,W2:] or b in V )
assume b in Dy .: [:W1,W2:] ; ::_thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A14: a in [:W1,W2:] and
A15: Dy . a = b by FUNCT_2:65;
A16: a = [(a `1),(a `2)] by Lm5, MCART_1:21;
A17: diffX1_X2_2 . a = ((a `1) `2) - ((a `2) `2) by Def4;
a `1 in W1 by A14, A16, ZFMISC_1:87;
then consider x1, y1 being Element of REAL such that
A18: a `1 = |[x1,y1]| and
A19: ((p `1) `2) - (g / 2) < y1 and
A20: y1 < ((p `1) `2) + (g / 2) ;
A21: (a `1) `2 = y1 by A18, EUCLID:52;
A22: (((p `1) `2) - (g / 2)) + (g / 2) < y1 + (g / 2) by A19, XREAL_1:6;
A23: ((p `1) `2) - y1 > ((p `1) `2) - (((p `1) `2) + (g / 2)) by A20, XREAL_1:15;
A24: ((p `1) `2) - y1 < (y1 + (g / 2)) - y1 by A22, XREAL_1:9;
((p `1) `2) - y1 > - (g / 2) by A23;
then A25: abs (((p `1) `2) - y1) < g / 2 by A24, SEQ_2:1;
a `2 in W2 by A14, A16, ZFMISC_1:87;
then consider x2, y2 being Element of REAL such that
A26: a `2 = |[x2,y2]| and
A27: ((p `2) `2) - (g / 2) < y2 and
A28: y2 < ((p `2) `2) + (g / 2) ;
A29: (a `2) `2 = y2 by A26, EUCLID:52;
A30: (((p `2) `2) - (g / 2)) + (g / 2) < y2 + (g / 2) by A27, XREAL_1:6;
A31: ((p `2) `2) - y2 > ((p `2) `2) - (((p `2) `2) + (g / 2)) by A28, XREAL_1:15;
A32: ((p `2) `2) - y2 < (y2 + (g / 2)) - y2 by A30, XREAL_1:9;
((p `2) `2) - y2 > - (g / 2) by A31;
then abs (((p `2) `2) - y2) < g / 2 by A32, SEQ_2:1;
then A33: (abs (((p `1) `2) - y1)) + (abs (((p `2) `2) - y2)) < (g / 2) + (g / 2) by A25, XREAL_1:8;
abs ((((p `1) `2) - y1) - (((p `2) `2) - y2)) <= (abs (((p `1) `2) - y1)) + (abs (((p `2) `2) - y2)) by COMPLEX1:57;
then abs (- ((((p `1) `2) - y1) - (((p `2) `2) - y2))) <= (abs (((p `1) `2) - y1)) + (abs (((p `2) `2) - y2)) by COMPLEX1:52;
then abs ((y1 - y2) - (((p `1) `2) - ((p `2) `2))) < g by A33, XXREAL_0:2;
then ((a `1) `2) - ((a `2) `2) in ].((((p `1) `2) - ((p `2) `2)) - g),((((p `1) `2) - ((p `2) `2)) + g).[ by A21, A29, RCOMP_1:1;
hence b in V by A6, A15, A17; ::_thesis: verum
end;
hence diffX1_X2_2 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:10; ::_thesis: verum
end;
theorem Th62: :: JORDAN:62
Proj2_1 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof
reconsider fX2 = Proj2_1 as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:17;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st fX2 . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fX2 .: W c= V )
proof
let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: for V being Subset of R^1 st fX2 . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fX2 .: W c= V )
let V be Subset of R^1; ::_thesis: ( fX2 . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fX2 .: W c= V ) )
assume that
A1: fX2 . p in V and
A2: V is open ; ::_thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fX2 .: W c= V )
A3: p = [(p `1),(p `2)] by Lm5, MCART_1:21;
A4: fX2 . p = (p `2) `1 by Def5;
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:39, TOPMETR:17;
consider g being real number such that
A5: 0 < g and
A6: ].(((p `2) `1) - g),(((p `2) `1) + g).[ c= V1 by A1, A4, RCOMP_1:19;
reconsider g = g as Element of REAL by XREAL_0:def_1;
set W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2) `1) - g < x & x < ((p `2) `1) + g ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `1) - g < x & x < ((p `2) `1) + g ) } as Subset of (TOP-REAL 2) ;
take [:([#] (TOP-REAL 2)),W1:] ; ::_thesis: ( p in [:([#] (TOP-REAL 2)),W1:] & [:([#] (TOP-REAL 2)),W1:] is open & fX2 .: [:([#] (TOP-REAL 2)),W1:] c= V )
A7: p `2 = |[((p `2) `1),((p `2) `2)]| by EUCLID:53;
A8: ((p `2) `1) - g < ((p `2) `1) - 0 by A5, XREAL_1:15;
((p `2) `1) + 0 < ((p `2) `1) + g by A5, XREAL_1:6;
then p `2 in W1 by A7, A8;
hence p in [:([#] (TOP-REAL 2)),W1:] by A3, ZFMISC_1:def_2; ::_thesis: ( [:([#] (TOP-REAL 2)),W1:] is open & fX2 .: [:([#] (TOP-REAL 2)),W1:] c= V )
W1 is open by PSCOMP_1:19;
hence [:([#] (TOP-REAL 2)),W1:] is open by BORSUK_1:6; ::_thesis: fX2 .: [:([#] (TOP-REAL 2)),W1:] c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in fX2 .: [:([#] (TOP-REAL 2)),W1:] or b in V )
assume b in fX2 .: [:([#] (TOP-REAL 2)),W1:] ; ::_thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A9: a in [:([#] (TOP-REAL 2)),W1:] and
A10: fX2 . a = b by FUNCT_2:65;
A11: a = [(a `1),(a `2)] by Lm5, MCART_1:21;
A12: fX2 . a = (a `2) `1 by Def5;
a `2 in W1 by A9, A11, ZFMISC_1:87;
then consider x1, y1 being Element of REAL such that
A13: a `2 = |[x1,y1]| and
A14: ((p `2) `1) - g < x1 and
A15: x1 < ((p `2) `1) + g ;
A16: (a `2) `1 = x1 by A13, EUCLID:52;
A17: (((p `2) `1) - g) + g < x1 + g by A14, XREAL_1:6;
A18: ((p `2) `1) - x1 > ((p `2) `1) - (((p `2) `1) + g) by A15, XREAL_1:15;
A19: ((p `2) `1) - x1 < (x1 + g) - x1 by A17, XREAL_1:9;
((p `2) `1) - x1 > - g by A18;
then abs (((p `2) `1) - x1) < g by A19, SEQ_2:1;
then abs (- (((p `2) `1) - x1)) < g by COMPLEX1:52;
then abs (x1 - ((p `2) `1)) < g ;
then (a `2) `1 in ].(((p `2) `1) - g),(((p `2) `1) + g).[ by A16, RCOMP_1:1;
hence b in V by A6, A10, A12; ::_thesis: verum
end;
hence Proj2_1 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:10; ::_thesis: verum
end;
theorem Th63: :: JORDAN:63
Proj2_2 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1
proof
reconsider fY2 = Proj2_2 as Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by TOPMETR:17;
for p being Point of [:(TOP-REAL 2),(TOP-REAL 2):]
for V being Subset of R^1 st fY2 . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fY2 .: W c= V )
proof
let p be Point of [:(TOP-REAL 2),(TOP-REAL 2):]; ::_thesis: for V being Subset of R^1 st fY2 . p in V & V is open holds
ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fY2 .: W c= V )
let V be Subset of R^1; ::_thesis: ( fY2 . p in V & V is open implies ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fY2 .: W c= V ) )
assume that
A1: fY2 . p in V and
A2: V is open ; ::_thesis: ex W being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] st
( p in W & W is open & fY2 .: W c= V )
A3: p = [(p `1),(p `2)] by Lm5, MCART_1:21;
A4: fY2 . p = (p `2) `2 by Def6;
reconsider V1 = V as open Subset of REAL by A2, BORSUK_5:39, TOPMETR:17;
consider g being real number such that
A5: 0 < g and
A6: ].(((p `2) `2) - g),(((p `2) `2) + g).[ c= V1 by A1, A4, RCOMP_1:19;
reconsider g = g as Element of REAL by XREAL_0:def_1;
set W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } ;
{ |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } or a in the carrier of (TOP-REAL 2) )
assume a in { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then ex x, y being Element of REAL st
( a = |[x,y]| & ((p `2) `2) - g < y & y < ((p `2) `2) + g ) ;
hence a in the carrier of (TOP-REAL 2) ; ::_thesis: verum
end;
then reconsider W1 = { |[x,y]| where x, y is Element of REAL : ( ((p `2) `2) - g < y & y < ((p `2) `2) + g ) } as Subset of (TOP-REAL 2) ;
take [:([#] (TOP-REAL 2)),W1:] ; ::_thesis: ( p in [:([#] (TOP-REAL 2)),W1:] & [:([#] (TOP-REAL 2)),W1:] is open & fY2 .: [:([#] (TOP-REAL 2)),W1:] c= V )
A7: p `2 = |[((p `2) `1),((p `2) `2)]| by EUCLID:53;
A8: ((p `2) `2) - g < ((p `2) `2) - 0 by A5, XREAL_1:15;
((p `2) `2) + 0 < ((p `2) `2) + g by A5, XREAL_1:6;
then p `2 in W1 by A7, A8;
hence p in [:([#] (TOP-REAL 2)),W1:] by A3, ZFMISC_1:def_2; ::_thesis: ( [:([#] (TOP-REAL 2)),W1:] is open & fY2 .: [:([#] (TOP-REAL 2)),W1:] c= V )
W1 is open by PSCOMP_1:21;
hence [:([#] (TOP-REAL 2)),W1:] is open by BORSUK_1:6; ::_thesis: fY2 .: [:([#] (TOP-REAL 2)),W1:] c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in fY2 .: [:([#] (TOP-REAL 2)),W1:] or b in V )
assume b in fY2 .: [:([#] (TOP-REAL 2)),W1:] ; ::_thesis: b in V
then consider a being Point of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A9: a in [:([#] (TOP-REAL 2)),W1:] and
A10: fY2 . a = b by FUNCT_2:65;
A11: a = [(a `1),(a `2)] by Lm5, MCART_1:21;
A12: fY2 . a = (a `2) `2 by Def6;
a `2 in W1 by A9, A11, ZFMISC_1:87;
then consider x1, y1 being Element of REAL such that
A13: a `2 = |[x1,y1]| and
A14: ((p `2) `2) - g < y1 and
A15: y1 < ((p `2) `2) + g ;
A16: (a `2) `2 = y1 by A13, EUCLID:52;
A17: (((p `2) `2) - g) + g < y1 + g by A14, XREAL_1:6;
A18: ((p `2) `2) - y1 > ((p `2) `2) - (((p `2) `2) + g) by A15, XREAL_1:15;
A19: ((p `2) `2) - y1 < (y1 + g) - y1 by A17, XREAL_1:9;
((p `2) `2) - y1 > - g by A18;
then abs (((p `2) `2) - y1) < g by A19, SEQ_2:1;
then abs (- (((p `2) `2) - y1)) < g by COMPLEX1:52;
then abs (y1 - ((p `2) `2)) < g ;
then (a `2) `2 in ].(((p `2) `2) - g),(((p `2) `2) + g).[ by A16, RCOMP_1:1;
hence b in V by A6, A10, A12; ::_thesis: verum
end;
hence Proj2_2 is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by JGRAPH_2:10; ::_thesis: verum
end;
registration
let o be Point of (TOP-REAL 2);
cluster diffX2_1 o -> continuous ;
coherence
diffX2_1 o is continuous
proof
diffX2_1 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by Th58;
hence diffX2_1 o is continuous by JORDAN5A:27; ::_thesis: verum
end;
cluster diffX2_2 o -> continuous ;
coherence
diffX2_2 o is continuous
proof
diffX2_2 o is continuous Function of [:(TOP-REAL 2),(TOP-REAL 2):],R^1 by Th59;
hence diffX2_2 o is continuous by JORDAN5A:27; ::_thesis: verum
end;
end;
registration
cluster diffX1_X2_1 -> continuous ;
coherence
diffX1_X2_1 is continuous by Th60, JORDAN5A:27;
cluster diffX1_X2_2 -> continuous ;
coherence
diffX1_X2_2 is continuous by Th61, JORDAN5A:27;
cluster Proj2_1 -> continuous ;
coherence
Proj2_1 is continuous by Th62, JORDAN5A:27;
cluster Proj2_2 -> continuous ;
coherence
Proj2_2 is continuous by Th63, JORDAN5A:27;
end;
definition
let n be non empty Element of NAT ;
let o, p be Point of (TOP-REAL n);
let r be positive real number ;
assume B1: p is Point of (Tdisk (o,r)) ;
set X = (TOP-REAL n) | ((cl_Ball (o,r)) \ {p});
func DiskProj (o,r,p) -> Function of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})),(Tcircle (o,r)) means :Def7: :: JORDAN:def 7
for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & it . x = HC (p,y,o,r) );
existence
ex b1 being Function of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})),(Tcircle (o,r)) st
for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC (p,y,o,r) )
proof
A1: the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) = (cl_Ball (o,r)) \ {p} by PRE_TOPC:8;
defpred S1[ set , set ] means ex z being Point of (TOP-REAL n) st
( $1 = z & $2 = HC (p,z,o,r) );
A2: for x being set st x in the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) holds
ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) implies ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] ) )
assume A3: x in the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ; ::_thesis: ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] )
reconsider z = x as Point of (TOP-REAL n) by A3, PRE_TOPC:25;
z in cl_Ball (o,r) by A1, A3, XBOOLE_0:def_5;
then A4: z is Point of (Tdisk (o,r)) by BROUWER:3;
p <> z by A1, A3, ZFMISC_1:56;
then HC (p,z,o,r) is Point of (Tcircle (o,r)) by B1, A4, BROUWER:6;
hence ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] ) ; ::_thesis: verum
end;
consider f being Function of the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})), the carrier of (Tcircle (o,r)) such that
A5: for x being set st x in the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) holds
S1[x,f . x] from FUNCT_2:sch_1(A2);
reconsider f = f as Function of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})),(Tcircle (o,r)) ;
take f ; ::_thesis: for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (p,y,o,r) )
let x be Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})); ::_thesis: ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (p,y,o,r) )
thus ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (p,y,o,r) ) by A5; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})),(Tcircle (o,r)) st ( for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC (p,y,o,r) ) ) & ( for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b2 . x = HC (p,y,o,r) ) ) holds
b1 = b2
proof
let f, g be Function of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})),(Tcircle (o,r)); ::_thesis: ( ( for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (p,y,o,r) ) ) & ( for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC (p,y,o,r) ) ) implies f = g )
assume that
A6: for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (p,y,o,r) ) and
A7: for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC (p,y,o,r) ) ; ::_thesis: f = g
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_((TOP-REAL_n)_|_((cl_Ball_(o,r))_\_{p}))_holds_
f_._x_=_g_._x
let x be set ; ::_thesis: ( x in the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) implies f . x = g . x )
assume A8: x in the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ; ::_thesis: f . x = g . x
A9: ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (p,y,o,r) ) by A6, A8;
ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC (p,y,o,r) ) by A7, A8;
hence f . x = g . x by A9; ::_thesis: verum
end;
hence f = g by FUNCT_2:12; ::_thesis: verum
end;
end;
:: deftheorem Def7 defines DiskProj JORDAN:def_7_:_
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p is Point of (Tdisk (o,r)) holds
for b5 being Function of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})),(Tcircle (o,r)) holds
( b5 = DiskProj (o,r,p) iff for x being Point of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) ex y being Point of (TOP-REAL n) st
( x = y & b5 . x = HC (p,y,o,r) ) );
theorem Th64: :: JORDAN:64
for o, p being Point of (TOP-REAL 2)
for r being positive real number st p is Point of (Tdisk (o,r)) holds
DiskProj (o,r,p) is continuous
proof
let o, p be Point of (TOP-REAL 2); ::_thesis: for r being positive real number st p is Point of (Tdisk (o,r)) holds
DiskProj (o,r,p) is continuous
let r be positive real number ; ::_thesis: ( p is Point of (Tdisk (o,r)) implies DiskProj (o,r,p) is continuous )
assume A1: p is Point of (Tdisk (o,r)) ; ::_thesis: DiskProj (o,r,p) is continuous
set D = Tdisk (o,r);
set cB = cl_Ball (o,r);
set Bp = (cl_Ball (o,r)) \ {p};
set OK = [:((cl_Ball (o,r)) \ {p}),{p}:];
set D1 = (TOP-REAL 2) | ((cl_Ball (o,r)) \ {p});
set D2 = (TOP-REAL 2) | {p};
set S1 = Tcircle (o,r);
A2: p in {p} by TARSKI:def_1;
A3: the carrier of (Tdisk (o,r)) = cl_Ball (o,r) by BROUWER:3;
A4: the carrier of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) = (cl_Ball (o,r)) \ {p} by PRE_TOPC:8;
A5: the carrier of ((TOP-REAL 2) | {p}) = {p} by PRE_TOPC:8;
set TD = [:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:];
set gg = DiskProj (o,r,p);
set xo = diffX2_1 o;
set yo = diffX2_2 o;
set dx = diffX1_X2_1 ;
set dy = diffX1_X2_2 ;
set fx2 = Proj2_1 ;
set fy2 = Proj2_2 ;
reconsider rr = r ^2 as Real by XREAL_0:def_1;
set f1 = the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] --> rr;
reconsider f1 = the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] --> rr as continuous RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] by Lm6;
set Zf1 = f1 | [:((cl_Ball (o,r)) \ {p}),{p}:];
set Zfx2 = Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:];
set Zfy2 = Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:];
set Zdx = diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:];
set Zdy = diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:];
set Zxo = (diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:];
set Zyo = (diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:];
set xx = ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]);
set yy = ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]);
set m = ((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]));
A6: the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) = [:((cl_Ball (o,r)) \ {p}),{p}:] by PRE_TOPC:8;
A7: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A8: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A9: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: (Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence (Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_1 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A10: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: (Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence (Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = Proj2_2 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A11: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = f1 . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = f1 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A12: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A13: for y being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
proof
let y be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for z being Point of ((TOP-REAL 2) | {p}) holds ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
let z be Point of ((TOP-REAL 2) | {p}); ::_thesis: ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z]
[y,z] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A4, A5, ZFMISC_1:def_2;
hence ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
now__::_thesis:_for_b_being_real_number_st_b_in_rng_(((diffX1_X2_1_|_[:((cl_Ball_(o,r))_\_{p}),{p}:])_(#)_(diffX1_X2_1_|_[:((cl_Ball_(o,r))_\_{p}),{p}:]))_+_((diffX1_X2_2_|_[:((cl_Ball_(o,r))_\_{p}),{p}:])_(#)_(diffX1_X2_2_|_[:((cl_Ball_(o,r))_\_{p}),{p}:])))_holds_
0_<_b
let b be real number ; ::_thesis: ( b in rng (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) implies 0 < b )
assume b in rng (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) ; ::_thesis: 0 < b
then consider a being set such that
A14: a in dom (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) and
A15: (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . a = b by FUNCT_1:def_3;
consider y, z being set such that
A16: y in (cl_Ball (o,r)) \ {p} and
A17: z in {p} and
A18: a = [y,z] by A14, ZFMISC_1:def_2;
A19: z = p by A17, TARSKI:def_1;
reconsider y = y, z = z as Point of (TOP-REAL 2) by A16, A17;
A20: y <> z by A16, A19, ZFMISC_1:56;
A21: diffX1_X2_1 . [y,z] = (([y,z] `1) `1) - (([y,z] `2) `1) by Def3;
A22: diffX1_X2_2 . [y,z] = (([y,z] `1) `2) - (([y,z] `2) `2) by Def4;
set r1 = (y `1) - (z `1);
set r2 = (y `2) - (z `2);
A23: (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_1 . [y,z] by A4, A5, A7, A16, A17;
A24: (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = diffX1_X2_2 . [y,z] by A4, A5, A8, A16, A17;
dom (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) c= the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by RELAT_1:def_18;
then a in the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by A14;
then A25: (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . [y,z] = (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z]) + (((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z]) by A18, VALUED_1:1
.= (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z]) * ((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z])) + (((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z]) by VALUED_1:5
.= (((y `1) - (z `1)) ^2) + (((y `2) - (z `2)) ^2) by A21, A22, A23, A24, VALUED_1:5 ;
now__::_thesis:_not_(((y_`1)_-_(z_`1))_^2)_+_(((y_`2)_-_(z_`2))_^2)_=_0
assume A26: (((y `1) - (z `1)) ^2) + (((y `2) - (z `2)) ^2) = 0 ; ::_thesis: contradiction
then A27: (y `1) - (z `1) = 0 by COMPLEX1:1;
(y `2) - (z `2) = 0 by A26, COMPLEX1:1;
hence contradiction by A20, A27, TOPREAL3:6; ::_thesis: verum
end;
hence 0 < b by A15, A18, A25; ::_thesis: verum
end;
then reconsider m = ((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) as positive-yielding continuous RealMap of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by PARTFUN3:def_1;
set p1 = ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])));
set p2 = ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]);
A28: dom (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
now__::_thesis:_for_b_being_real_number_st_b_in_rng_(((((diffX2_1_o)_|_[:((cl_Ball_(o,r))_\_{p}),{p}:])_(#)_((diffX2_1_o)_|_[:((cl_Ball_(o,r))_\_{p}),{p}:]))_+_(((diffX2_2_o)_|_[:((cl_Ball_(o,r))_\_{p}),{p}:])_(#)_((diffX2_2_o)_|_[:((cl_Ball_(o,r))_\_{p}),{p}:])))_-_(f1_|_[:((cl_Ball_(o,r))_\_{p}),{p}:]))_holds_
0_>=_b
let b be real number ; ::_thesis: ( b in rng (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) implies 0 >= b )
assume b in rng (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) ; ::_thesis: 0 >= b
then consider a being set such that
A29: a in dom (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) and
A30: (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . a = b by FUNCT_1:def_3;
consider y, z being set such that
A31: y in (cl_Ball (o,r)) \ {p} and
A32: z in {p} and
A33: a = [y,z] by A29, ZFMISC_1:def_2;
reconsider y = y, z = z as Point of (TOP-REAL 2) by A31, A32;
set r3 = (z `1) - (o `1);
set r4 = (z `2) - (o `2);
A34: (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = f1 . [y,z] by A4, A5, A11, A31, A32;
A35: ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_1 o) . [y,z] by A4, A5, A12, A31, A32;
A36: ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z] = (diffX2_2 o) . [y,z] by A4, A5, A13, A31, A32;
A37: (diffX2_1 o) . [y,z] = (([y,z] `2) `1) - (o `1) by Def1;
A38: (diffX2_2 o) . [y,z] = (([y,z] `2) `2) - (o `2) by Def2;
dom (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) c= the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by RELAT_1:def_18;
then A39: a in the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by A29;
A40: (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z] = (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . [y,z]) - ((f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z]) by A29, A33, VALUED_1:13
.= (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . [y,z]) - (r ^2) by A34, FUNCOP_1:7
.= (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z]) + ((((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z])) - (r ^2) by A33, A39, VALUED_1:1
.= (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z]) * (((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,z])) + ((((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,z])) - (r ^2) by VALUED_1:5
.= ((((z `1) - (o `1)) ^2) + (((z `2) - (o `2)) ^2)) - (r ^2) by A35, A36, A37, A38, VALUED_1:5 ;
z = p by A32, TARSKI:def_1;
then |.(z - o).| <= r by A1, A3, TOPREAL9:8;
then A41: |.(z - o).| ^2 <= r ^2 by SQUARE_1:15;
|.(z - o).| ^2 = (((z - o) `1) ^2) + (((z - o) `2) ^2) by JGRAPH_1:29
.= (((z `1) - (o `1)) ^2) + (((z - o) `2) ^2) by TOPREAL3:3
.= (((z `1) - (o `1)) ^2) + (((z `2) - (o `2)) ^2) by TOPREAL3:3 ;
then ((((z `1) - (o `1)) ^2) + (((z `2) - (o `2)) ^2)) - (r ^2) <= (r ^2) - (r ^2) by A41, XREAL_1:9;
hence 0 >= b by A30, A33, A40; ::_thesis: verum
end;
then reconsider p2 = ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) - (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) as nonpositive-yielding continuous RealMap of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by PARTFUN3:def_3;
set pp = (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2);
set k = ((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m;
set x3 = (Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]));
set y3 = (Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]));
reconsider X3 = (Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])), Y3 = (Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) + ((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) as Function of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]),R^1 by TOPMETR:17;
set F = <:X3,Y3:>;
set R = R2Homeomorphism ;
A42: for x being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) holds (DiskProj (o,r,p)) . x = (R2Homeomorphism * <:X3,Y3:>) . [x,p]
proof
let x be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: (DiskProj (o,r,p)) . x = (R2Homeomorphism * <:X3,Y3:>) . [x,p]
consider y being Point of (TOP-REAL 2) such that
A43: x = y and
A44: (DiskProj (o,r,p)) . x = HC (p,y,o,r) by A1, Def7;
A45: x <> p by A4, ZFMISC_1:56;
A46: [y,p] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A2, A4, A43, ZFMISC_1:def_2;
set r1 = (y `1) - (p `1);
set r2 = (y `2) - (p `2);
set r3 = (p `1) - (o `1);
set r4 = (p `2) - (o `2);
set l = ((- ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2))))) + (sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))))) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2));
A47: Proj2_1 . [y,p] = ([y,p] `2) `1 by Def5;
A48: Proj2_2 . [y,p] = ([y,p] `2) `2 by Def6;
A49: diffX1_X2_1 . [y,p] = (([y,p] `1) `1) - (([y,p] `2) `1) by Def3;
A50: diffX1_X2_2 . [y,p] = (([y,p] `1) `2) - (([y,p] `2) `2) by Def4;
A51: (diffX2_1 o) . [y,p] = (([y,p] `2) `1) - (o `1) by Def1;
A52: (diffX2_2 o) . [y,p] = (([y,p] `2) `2) - (o `2) by Def2;
A53: dom X3 = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
A54: dom Y3 = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
A55: dom ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
A56: p is Point of ((TOP-REAL 2) | {p}) by A5, TARSKI:def_1;
then A57: (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p] = diffX1_X2_1 . [y,p] by A7, A43;
A58: (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p] = diffX1_X2_2 . [y,p] by A8, A43, A56;
A59: (f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p] = f1 . [y,p] by A11, A43, A56;
A60: ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p] = (diffX2_1 o) . [y,p] by A12, A43, A56;
A61: ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p] = (diffX2_2 o) . [y,p] by A13, A43, A56;
A62: m . [y,p] = (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) + (((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by A6, A46, VALUED_1:1
.= (((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) * ((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p])) + (((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by VALUED_1:5
.= (((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2) by A49, A50, A57, A58, VALUED_1:5 ;
A63: (((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p] = (((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) * ((diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) by VALUED_1:5;
A64: (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p] = (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) * ((diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) by VALUED_1:5;
A65: ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . [y,p] = ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) + ((((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by A6, A46, VALUED_1:1;
then A66: (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) . [y,p] = ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2 by A49, A50, A51, A52, A57, A58, A60, A61, A63, A64, VALUED_1:5;
A67: p2 . [y,p] = (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . [y,p]) - ((f1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) by A6, A28, A46, VALUED_1:13
.= (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]))) . [y,p]) - (r ^2) by A59, FUNCOP_1:7
.= (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) + ((((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p])) - (r ^2) by A6, A46, VALUED_1:1
.= (((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) * (((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p])) + ((((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) ((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p])) - (r ^2) by VALUED_1:5
.= ((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2) by A51, A52, A60, A61, VALUED_1:5 ;
dom (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2))) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
then A68: (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2))) . [y,p] = sqrt (((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)) . [y,p]) by A6, A46, PARTFUN3:def_5
.= sqrt (((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) . [y,p]) - ((m (#) p2) . [y,p])) by A6, A46, A55, VALUED_1:13
.= sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))) by A62, A66, A67, VALUED_1:5 ;
dom (((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
then A69: (((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) . [y,p] = (((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) . [y,p]) * ((m . [y,p]) ") by A6, A46, RFUNCT_1:def_1
.= (((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) . [y,p]) / (m . [y,p]) by XCMPLX_0:def_9
.= (((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) . [y,p]) + ((sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2))) . [y,p])) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) by A6, A46, A62, VALUED_1:1
.= ((- ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2))))) + (sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))))) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) by A49, A50, A51, A52, A57, A58, A60, A61, A63, A64, A65, A68, VALUED_1:8 ;
A70: X3 . [y,p] = ((Proj2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) + (((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by A6, A46, VALUED_1:1
.= (p `1) + (((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by A9, A43, A47, A56
.= (p `1) + ((((- ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2))))) + (sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))))) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2))) * ((y `1) - (p `1))) by A49, A57, A69, VALUED_1:5 ;
A71: Y3 . [y,p] = ((Proj2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]) . [y,p]) + (((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by A6, A46, VALUED_1:1
.= (p `2) + (((((- ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) + (sqrt ((((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:]))) (#) ((((diffX2_1 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_1 | [:((cl_Ball (o,r)) \ {p}),{p}:])) + (((diffX2_2 o) | [:((cl_Ball (o,r)) \ {p}),{p}:]) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:((cl_Ball (o,r)) \ {p}),{p}:])) . [y,p]) by A10, A43, A48, A56
.= (p `2) + ((((- ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2))))) + (sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))))) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2))) * ((y `2) - (p `2))) by A50, A58, A69, VALUED_1:5 ;
A72: y in (cl_Ball (o,r)) \ {p} by A4, A43;
(cl_Ball (o,r)) \ {p} c= cl_Ball (o,r) by XBOOLE_1:36;
hence (DiskProj (o,r,p)) . x = |[((p `1) + ((((- ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2))))) + (sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))))) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2))) * ((y `1) - (p `1)))),((p `2) + ((((- ((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2))))) + (sqrt ((((((p `1) - (o `1)) * ((y `1) - (p `1))) + (((p `2) - (o `2)) * ((y `2) - (p `2)))) ^2) - (((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2)) * (((((p `1) - (o `1)) ^2) + (((p `2) - (o `2)) ^2)) - (r ^2)))))) / ((((y `1) - (p `1)) ^2) + (((y `2) - (p `2)) ^2))) * ((y `2) - (p `2))))]| by A1, A3, A43, A44, A45, A72, BROUWER:8
.= R2Homeomorphism . [(X3 . [y,p]),(Y3 . [y,p])] by A70, A71, TOPREALA:def_2
.= R2Homeomorphism . (<:X3,Y3:> . [y,p]) by A6, A46, A53, A54, FUNCT_3:49
.= (R2Homeomorphism * <:X3,Y3:>) . [x,p] by A6, A43, A46, FUNCT_2:15 ;
::_thesis: verum
end;
A73: X3 is continuous by JORDAN5A:27;
Y3 is continuous by JORDAN5A:27;
then reconsider F = <:X3,Y3:> as continuous Function of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]),[:R^1,R^1:] by A73, YELLOW12:41;
for pp being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))
for V being Subset of (Tcircle (o,r)) st (DiskProj (o,r,p)) . pp in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) st
( pp in W & W is open & (DiskProj (o,r,p)) .: W c= V )
proof
let pp be Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})); ::_thesis: for V being Subset of (Tcircle (o,r)) st (DiskProj (o,r,p)) . pp in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) st
( pp in W & W is open & (DiskProj (o,r,p)) .: W c= V )
let V be Subset of (Tcircle (o,r)); ::_thesis: ( (DiskProj (o,r,p)) . pp in V & V is open implies ex W being Subset of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) st
( pp in W & W is open & (DiskProj (o,r,p)) .: W c= V ) )
assume that
A74: (DiskProj (o,r,p)) . pp in V and
A75: V is open ; ::_thesis: ex W being Subset of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) st
( pp in W & W is open & (DiskProj (o,r,p)) .: W c= V )
reconsider p1 = pp, fp = p as Point of (TOP-REAL 2) by PRE_TOPC:25;
A76: [pp,p] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A2, A4, ZFMISC_1:def_2;
consider V1 being Subset of (TOP-REAL 2) such that
A77: V1 is open and
A78: V1 /\ ([#] (Tcircle (o,r))) = V by A75, TOPS_2:24;
A79: (DiskProj (o,r,p)) . pp = (R2Homeomorphism * F) . [pp,p] by A42;
R2Homeomorphism " is being_homeomorphism by TOPREALA:34, TOPS_2:56;
then A80: (R2Homeomorphism ") .: V1 is open by A77, TOPGRP_1:25;
A81: dom F = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) by FUNCT_2:def_1;
A82: dom R2Homeomorphism = the carrier of [:R^1,R^1:] by FUNCT_2:def_1;
then A83: rng F c= dom R2Homeomorphism ;
then A84: dom (R2Homeomorphism * F) = dom F by RELAT_1:27;
A85: rng R2Homeomorphism = [#] (TOP-REAL 2) by TOPREALA:34, TOPS_2:def_5;
A86: (R2Homeomorphism ") * (R2Homeomorphism * F) = ((R2Homeomorphism ") * R2Homeomorphism) * F by RELAT_1:36
.= (id (dom R2Homeomorphism)) * F by A85, TOPREALA:34, TOPS_2:52 ;
dom (id (dom R2Homeomorphism)) = dom R2Homeomorphism ;
then A87: dom ((id (dom R2Homeomorphism)) * F) = dom F by A83, RELAT_1:27;
for x being set st x in dom F holds
((id (dom R2Homeomorphism)) * F) . x = F . x
proof
let x be set ; ::_thesis: ( x in dom F implies ((id (dom R2Homeomorphism)) * F) . x = F . x )
assume A88: x in dom F ; ::_thesis: ((id (dom R2Homeomorphism)) * F) . x = F . x
A89: F . x in rng F by A88, FUNCT_1:def_3;
thus ((id (dom R2Homeomorphism)) * F) . x = (id (dom R2Homeomorphism)) . (F . x) by A88, FUNCT_1:13
.= F . x by A82, A89, FUNCT_1:18 ; ::_thesis: verum
end;
then A90: (id (dom R2Homeomorphism)) * F = F by A87, FUNCT_1:2;
(R2Homeomorphism * F) . [p1,fp] in V1 by A74, A78, A79, XBOOLE_0:def_4;
then (R2Homeomorphism ") . ((R2Homeomorphism * F) . [p1,fp]) in (R2Homeomorphism ") .: V1 by FUNCT_2:35;
then ((R2Homeomorphism ") * (R2Homeomorphism * F)) . [p1,fp] in (R2Homeomorphism ") .: V1 by A6, A76, A81, A84, FUNCT_1:13;
then consider W being Subset of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]) such that
A91: [p1,fp] in W and
A92: W is open and
A93: F .: W c= (R2Homeomorphism ") .: V1 by A6, A76, A80, A86, A90, JGRAPH_2:10;
consider WW being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A94: WW is open and
A95: WW /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:])) = W by A92, TOPS_2:24;
consider SF being Subset-Family of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A96: WW = union SF and
A97: for e being set st e in SF holds
ex X1, Y1 being Subset of (TOP-REAL 2) st
( e = [:X1,Y1:] & X1 is open & Y1 is open ) by A94, BORSUK_1:5;
[p1,fp] in WW by A91, A95, XBOOLE_0:def_4;
then consider Z being set such that
A98: [p1,fp] in Z and
A99: Z in SF by A96, TARSKI:def_4;
consider X1, Y1 being Subset of (TOP-REAL 2) such that
A100: Z = [:X1,Y1:] and
A101: X1 is open and
Y1 is open by A97, A99;
set ZZ = Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]));
reconsider XX = X1 /\ ([#] ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p}))) as open Subset of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) by A101, TOPS_2:24;
take XX ; ::_thesis: ( pp in XX & XX is open & (DiskProj (o,r,p)) .: XX c= V )
pp in X1 by A98, A100, ZFMISC_1:87;
hence pp in XX by XBOOLE_0:def_4; ::_thesis: ( XX is open & (DiskProj (o,r,p)) .: XX c= V )
thus XX is open ; ::_thesis: (DiskProj (o,r,p)) .: XX c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in (DiskProj (o,r,p)) .: XX or b in V )
assume b in (DiskProj (o,r,p)) .: XX ; ::_thesis: b in V
then consider a being Point of ((TOP-REAL 2) | ((cl_Ball (o,r)) \ {p})) such that
A102: a in XX and
A103: b = (DiskProj (o,r,p)) . a by FUNCT_2:65;
reconsider a1 = a, fa = fp as Point of (TOP-REAL 2) by PRE_TOPC:25;
A104: a in X1 by A102, XBOOLE_0:def_4;
A105: [a,p] in [:((cl_Ball (o,r)) \ {p}),{p}:] by A2, A4, ZFMISC_1:def_2;
fa in Y1 by A98, A100, ZFMISC_1:87;
then [a,fa] in Z by A100, A104, ZFMISC_1:def_2;
then [a,fa] in Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:])) by A6, A105, XBOOLE_0:def_4;
then A106: F . [a1,fa] in F .: (Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]))) by FUNCT_2:35;
A107: R2Homeomorphism " = R2Homeomorphism " by TOPREALA:34, TOPS_2:def_4;
A108: dom (R2Homeomorphism ") = [#] (TOP-REAL 2) by A85, TOPREALA:34, TOPS_2:49;
Z c= WW by A96, A99, ZFMISC_1:74;
then Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:])) c= WW /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:])) by XBOOLE_1:27;
then F .: (Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:((cl_Ball (o,r)) \ {p}),{p}:]))) c= F .: W by A95, RELAT_1:123;
then F . [a1,fa] in F .: W by A106;
then R2Homeomorphism . (F . [a1,fa]) in R2Homeomorphism .: ((R2Homeomorphism ") .: V1) by A93, FUNCT_2:35;
then (R2Homeomorphism * F) . [a1,fa] in R2Homeomorphism .: ((R2Homeomorphism ") .: V1) by A6, A105, FUNCT_2:15;
then (R2Homeomorphism * F) . [a1,fa] in V1 by A107, A108, PARTFUN3:1, TOPREALA:34;
then (DiskProj (o,r,p)) . a in V1 by A42;
hence b in V by A78, A103, XBOOLE_0:def_4; ::_thesis: verum
end;
hence DiskProj (o,r,p) is continuous by JGRAPH_2:10; ::_thesis: verum
end;
theorem Th65: :: JORDAN:65
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball (o,r) holds
(DiskProj (o,r,p)) | (Sphere (o,r)) = id (Sphere (o,r))
proof
let n be non empty Element of NAT ; ::_thesis: for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball (o,r) holds
(DiskProj (o,r,p)) | (Sphere (o,r)) = id (Sphere (o,r))
let o, p be Point of (TOP-REAL n); ::_thesis: for r being positive real number st p in Ball (o,r) holds
(DiskProj (o,r,p)) | (Sphere (o,r)) = id (Sphere (o,r))
let r be positive real number ; ::_thesis: ( p in Ball (o,r) implies (DiskProj (o,r,p)) | (Sphere (o,r)) = id (Sphere (o,r)) )
assume A1: p in Ball (o,r) ; ::_thesis: (DiskProj (o,r,p)) | (Sphere (o,r)) = id (Sphere (o,r))
A2: the carrier of (Tdisk (o,r)) = cl_Ball (o,r) by BROUWER:3;
A3: the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) = (cl_Ball (o,r)) \ {p} by PRE_TOPC:8;
A4: dom (DiskProj (o,r,p)) = the carrier of ((TOP-REAL n) | ((cl_Ball (o,r)) \ {p})) by FUNCT_2:def_1;
A5: Sphere (o,r) misses Ball (o,r) by TOPREAL9:19;
A6: Sphere (o,r) c= cl_Ball (o,r) by TOPREAL9:17;
A7: Ball (o,r) c= cl_Ball (o,r) by TOPREAL9:16;
A8: Sphere (o,r) c= (cl_Ball (o,r)) \ {p}
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Sphere (o,r) or a in (cl_Ball (o,r)) \ {p} )
assume A9: a in Sphere (o,r) ; ::_thesis: a in (cl_Ball (o,r)) \ {p}
then a <> p by A1, A5, XBOOLE_0:3;
hence a in (cl_Ball (o,r)) \ {p} by A6, A9, ZFMISC_1:56; ::_thesis: verum
end;
then A10: dom ((DiskProj (o,r,p)) | (Sphere (o,r))) = Sphere (o,r) by A3, A4, RELAT_1:62;
A11: dom (id (Sphere (o,r))) = Sphere (o,r) ;
now__::_thesis:_for_x_being_set_st_x_in_dom_((DiskProj_(o,r,p))_|_(Sphere_(o,r)))_holds_
((DiskProj_(o,r,p))_|_(Sphere_(o,r)))_._x_=_(id_(Sphere_(o,r)))_._x
let x be set ; ::_thesis: ( x in dom ((DiskProj (o,r,p)) | (Sphere (o,r))) implies ((DiskProj (o,r,p)) | (Sphere (o,r))) . x = (id (Sphere (o,r))) . x )
assume A12: x in dom ((DiskProj (o,r,p)) | (Sphere (o,r))) ; ::_thesis: ((DiskProj (o,r,p)) | (Sphere (o,r))) . x = (id (Sphere (o,r))) . x
then x in dom (DiskProj (o,r,p)) by RELAT_1:57;
then consider y being Point of (TOP-REAL n) such that
A13: x = y and
A14: (DiskProj (o,r,p)) . x = HC (p,y,o,r) by A1, A2, A7, Def7;
y in halfline (p,y) by TOPREAL9:28;
then A15: x in (halfline (p,y)) /\ (Sphere (o,r)) by A12, A13, XBOOLE_0:def_4;
A16: x <> p by A1, A5, A12, XBOOLE_0:3;
thus ((DiskProj (o,r,p)) | (Sphere (o,r))) . x = (DiskProj (o,r,p)) . x by A12, FUNCT_1:47
.= x by A1, A2, A6, A7, A10, A12, A13, A14, A15, A16, BROUWER:def_3
.= (id (Sphere (o,r))) . x by A12, FUNCT_1:18 ; ::_thesis: verum
end;
hence (DiskProj (o,r,p)) | (Sphere (o,r)) = id (Sphere (o,r)) by A3, A4, A8, A11, FUNCT_1:2, RELAT_1:62; ::_thesis: verum
end;
definition
let n be non empty Element of NAT ;
let o, p be Point of (TOP-REAL n);
let r be positive real number ;
assume B1: p in Ball (o,r) ;
set X = Tcircle (o,r);
func RotateCircle (o,r,p) -> Function of (Tcircle (o,r)),(Tcircle (o,r)) means :Def8: :: JORDAN:def 8
for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & it . x = HC (y,p,o,r) );
existence
ex b1 being Function of (Tcircle (o,r)),(Tcircle (o,r)) st
for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC (y,p,o,r) )
proof
A1: the carrier of (Tcircle (o,r)) = Sphere (o,r) by TOPREALB:9;
defpred S1[ set , set ] means ex z being Point of (TOP-REAL n) st
( $1 = z & $2 = HC (z,p,o,r) );
A2: for x being set st x in the carrier of (Tcircle (o,r)) holds
ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of (Tcircle (o,r)) implies ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] ) )
assume A3: x in the carrier of (Tcircle (o,r)) ; ::_thesis: ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] )
reconsider z = x as Point of (TOP-REAL n) by A3, PRE_TOPC:25;
Sphere (o,r) c= cl_Ball (o,r) by TOPREAL9:17;
then A4: z is Point of (Tdisk (o,r)) by A1, A3, BROUWER:3;
Ball (o,r) c= cl_Ball (o,r) by TOPREAL9:16;
then A5: p is Point of (Tdisk (o,r)) by B1, BROUWER:3;
Ball (o,r) misses Sphere (o,r) by TOPREAL9:19;
then p <> z by B1, A1, A3, XBOOLE_0:3;
then HC (z,p,o,r) is Point of (Tcircle (o,r)) by A4, A5, BROUWER:6;
hence ex y being set st
( y in the carrier of (Tcircle (o,r)) & S1[x,y] ) ; ::_thesis: verum
end;
consider f being Function of the carrier of (Tcircle (o,r)), the carrier of (Tcircle (o,r)) such that
A6: for x being set st x in the carrier of (Tcircle (o,r)) holds
S1[x,f . x] from FUNCT_2:sch_1(A2);
reconsider f = f as Function of (Tcircle (o,r)),(Tcircle (o,r)) ;
take f ; ::_thesis: for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (y,p,o,r) )
let x be Point of (Tcircle (o,r)); ::_thesis: ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (y,p,o,r) )
thus ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (y,p,o,r) ) by A6; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (Tcircle (o,r)),(Tcircle (o,r)) st ( for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & b1 . x = HC (y,p,o,r) ) ) & ( for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & b2 . x = HC (y,p,o,r) ) ) holds
b1 = b2
proof
let f, g be Function of (Tcircle (o,r)),(Tcircle (o,r)); ::_thesis: ( ( for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (y,p,o,r) ) ) & ( for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC (y,p,o,r) ) ) implies f = g )
assume that
A7: for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (y,p,o,r) ) and
A8: for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC (y,p,o,r) ) ; ::_thesis: f = g
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(Tcircle_(o,r))_holds_
f_._x_=_g_._x
let x be set ; ::_thesis: ( x in the carrier of (Tcircle (o,r)) implies f . x = g . x )
assume A9: x in the carrier of (Tcircle (o,r)) ; ::_thesis: f . x = g . x
A10: ex y being Point of (TOP-REAL n) st
( x = y & f . x = HC (y,p,o,r) ) by A7, A9;
ex y being Point of (TOP-REAL n) st
( x = y & g . x = HC (y,p,o,r) ) by A8, A9;
hence f . x = g . x by A10; ::_thesis: verum
end;
hence f = g by FUNCT_2:12; ::_thesis: verum
end;
end;
:: deftheorem Def8 defines RotateCircle JORDAN:def_8_:_
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball (o,r) holds
for b5 being Function of (Tcircle (o,r)),(Tcircle (o,r)) holds
( b5 = RotateCircle (o,r,p) iff for x being Point of (Tcircle (o,r)) ex y being Point of (TOP-REAL n) st
( x = y & b5 . x = HC (y,p,o,r) ) );
theorem Th66: :: JORDAN:66
for o, p being Point of (TOP-REAL 2)
for r being positive real number st p in Ball (o,r) holds
RotateCircle (o,r,p) is continuous
proof
let o, p be Point of (TOP-REAL 2); ::_thesis: for r being positive real number st p in Ball (o,r) holds
RotateCircle (o,r,p) is continuous
let r be positive real number ; ::_thesis: ( p in Ball (o,r) implies RotateCircle (o,r,p) is continuous )
assume A1: p in Ball (o,r) ; ::_thesis: RotateCircle (o,r,p) is continuous
set D = Tdisk (o,r);
set cB = cl_Ball (o,r);
set Bp = Sphere (o,r);
set OK = [:{p},(Sphere (o,r)):];
set D1 = (TOP-REAL 2) | {p};
set D2 = (TOP-REAL 2) | (Sphere (o,r));
set S1 = Tcircle (o,r);
A2: (TOP-REAL 2) | (Sphere (o,r)) = Tcircle (o,r) by TOPREALB:def_6;
A3: Ball (o,r) misses Sphere (o,r) by TOPREAL9:19;
A4: p in {p} by TARSKI:def_1;
A5: Sphere (o,r) c= cl_Ball (o,r) by TOPREAL9:17;
A6: Ball (o,r) c= cl_Ball (o,r) by TOPREAL9:16;
A7: the carrier of (Tdisk (o,r)) = cl_Ball (o,r) by BROUWER:3;
A8: the carrier of ((TOP-REAL 2) | {p}) = {p} by PRE_TOPC:8;
A9: the carrier of ((TOP-REAL 2) | (Sphere (o,r))) = Sphere (o,r) by PRE_TOPC:8;
set TD = [:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):];
set gg = RotateCircle (o,r,p);
set xo = diffX2_1 o;
set yo = diffX2_2 o;
set dx = diffX1_X2_1 ;
set dy = diffX1_X2_2 ;
set fx2 = Proj2_1 ;
set fy2 = Proj2_2 ;
reconsider rr = r ^2 as Real by XREAL_0:def_1;
set f1 = the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] --> rr;
reconsider f1 = the carrier of [:(TOP-REAL 2),(TOP-REAL 2):] --> rr as continuous RealMap of [:(TOP-REAL 2),(TOP-REAL 2):] by Lm6;
set Zf1 = f1 | [:{p},(Sphere (o,r)):];
set Zfx2 = Proj2_1 | [:{p},(Sphere (o,r)):];
set Zfy2 = Proj2_2 | [:{p},(Sphere (o,r)):];
set Zdx = diffX1_X2_1 | [:{p},(Sphere (o,r)):];
set Zdy = diffX1_X2_2 | [:{p},(Sphere (o,r)):];
set Zxo = (diffX2_1 o) | [:{p},(Sphere (o,r)):];
set Zyo = (diffX2_2 o) | [:{p},(Sphere (o,r)):];
set xx = ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):]);
set yy = ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]);
set m = ((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]));
A10: the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) = [:{p},(Sphere (o,r)):] by PRE_TOPC:8;
A11: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_1 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_1 . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: (diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_1 . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence (diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_1 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A12: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_2 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_2 . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: (diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_2 . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence (diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_2 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A13: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (Proj2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_1 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (Proj2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_1 . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: (Proj2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_1 . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence (Proj2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_1 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A14: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (Proj2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_2 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (Proj2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_2 . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: (Proj2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_2 . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence (Proj2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = Proj2_2 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A15: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (f1 | [:{p},(Sphere (o,r)):]) . [y,z] = f1 . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (f1 | [:{p},(Sphere (o,r)):]) . [y,z] = f1 . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: (f1 | [:{p},(Sphere (o,r)):]) . [y,z] = f1 . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence (f1 | [:{p},(Sphere (o,r)):]) . [y,z] = f1 . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A16: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_1 o) . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_1 o) . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_1 o) . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_1 o) . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
A17: for y being Point of ((TOP-REAL 2) | {p})
for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_2 o) . [y,z]
proof
let y be Point of ((TOP-REAL 2) | {p}); ::_thesis: for z being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_2 o) . [y,z]
let z be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_2 o) . [y,z]
[y,z] in [:{p},(Sphere (o,r)):] by A8, A9, ZFMISC_1:def_2;
hence ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_2 o) . [y,z] by FUNCT_1:49; ::_thesis: verum
end;
now__::_thesis:_for_b_being_real_number_st_b_in_rng_(((diffX1_X2_1_|_[:{p},(Sphere_(o,r)):])_(#)_(diffX1_X2_1_|_[:{p},(Sphere_(o,r)):]))_+_((diffX1_X2_2_|_[:{p},(Sphere_(o,r)):])_(#)_(diffX1_X2_2_|_[:{p},(Sphere_(o,r)):])))_holds_
0_<_b
let b be real number ; ::_thesis: ( b in rng (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) implies 0 < b )
assume b in rng (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) ; ::_thesis: 0 < b
then consider a being set such that
A18: a in dom (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) and
A19: (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) . a = b by FUNCT_1:def_3;
consider y, z being set such that
A20: y in {p} and
A21: z in Sphere (o,r) and
A22: a = [y,z] by A18, ZFMISC_1:def_2;
A23: y = p by A20, TARSKI:def_1;
reconsider y = y, z = z as Point of (TOP-REAL 2) by A20, A21;
A24: y <> z by A1, A3, A21, A23, XBOOLE_0:3;
A25: diffX1_X2_1 . [y,z] = (([y,z] `1) `1) - (([y,z] `2) `1) by Def3;
A26: diffX1_X2_2 . [y,z] = (([y,z] `1) `2) - (([y,z] `2) `2) by Def4;
set r1 = (y `1) - (z `1);
set r2 = (y `2) - (z `2);
A27: (diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_1 . [y,z] by A8, A9, A11, A20, A21;
A28: (diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [y,z] = diffX1_X2_2 . [y,z] by A8, A9, A12, A20, A21;
dom (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) c= the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by RELAT_1:def_18;
then a in the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by A18;
then A29: (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) . [y,z] = (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) . [y,z]) + (((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [y,z]) by A22, VALUED_1:1
.= (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z]) * ((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [y,z])) + (((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [y,z]) by VALUED_1:5
.= (((y `1) - (z `1)) ^2) + (((y `2) - (z `2)) ^2) by A25, A26, A27, A28, VALUED_1:5 ;
now__::_thesis:_not_(((y_`1)_-_(z_`1))_^2)_+_(((y_`2)_-_(z_`2))_^2)_=_0
assume A30: (((y `1) - (z `1)) ^2) + (((y `2) - (z `2)) ^2) = 0 ; ::_thesis: contradiction
then A31: (y `1) - (z `1) = 0 by COMPLEX1:1;
(y `2) - (z `2) = 0 by A30, COMPLEX1:1;
hence contradiction by A24, A31, TOPREAL3:6; ::_thesis: verum
end;
hence 0 < b by A19, A22, A29; ::_thesis: verum
end;
then reconsider m = ((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) as positive-yielding continuous RealMap of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by PARTFUN3:def_1;
set p1 = ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])));
set p2 = ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):]);
A32: dom (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
now__::_thesis:_for_b_being_real_number_st_b_in_rng_(((((diffX2_1_o)_|_[:{p},(Sphere_(o,r)):])_(#)_((diffX2_1_o)_|_[:{p},(Sphere_(o,r)):]))_+_(((diffX2_2_o)_|_[:{p},(Sphere_(o,r)):])_(#)_((diffX2_2_o)_|_[:{p},(Sphere_(o,r)):])))_-_(f1_|_[:{p},(Sphere_(o,r)):]))_holds_
0_>=_b
let b be real number ; ::_thesis: ( b in rng (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) implies 0 >= b )
assume b in rng (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) ; ::_thesis: 0 >= b
then consider a being set such that
A33: a in dom (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) and
A34: (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) . a = b by FUNCT_1:def_3;
consider y, z being set such that
A35: y in {p} and
A36: z in Sphere (o,r) and
A37: a = [y,z] by A33, ZFMISC_1:def_2;
reconsider y = y, z = z as Point of (TOP-REAL 2) by A35, A36;
set r3 = (z `1) - (o `1);
set r4 = (z `2) - (o `2);
A38: (f1 | [:{p},(Sphere (o,r)):]) . [y,z] = f1 . [y,z] by A8, A9, A15, A35, A36;
A39: ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_1 o) . [y,z] by A8, A9, A16, A35, A36;
A40: ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [y,z] = (diffX2_2 o) . [y,z] by A8, A9, A17, A35, A36;
A41: (diffX2_1 o) . [y,z] = (([y,z] `2) `1) - (o `1) by Def1;
A42: (diffX2_2 o) . [y,z] = (([y,z] `2) `2) - (o `2) by Def2;
dom (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) c= the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by RELAT_1:def_18;
then A43: a in the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by A33;
A44: (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):])) . [y,z] = (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) . [y,z]) - ((f1 | [:{p},(Sphere (o,r)):]) . [y,z]) by A33, A37, VALUED_1:13
.= (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) . [y,z]) - (r ^2) by A38, FUNCOP_1:7
.= (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) . [y,z]) + ((((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):])) . [y,z])) - (r ^2) by A37, A43, VALUED_1:1
.= (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z]) * (((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [y,z])) + ((((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):])) . [y,z])) - (r ^2) by VALUED_1:5
.= ((((z `1) - (o `1)) ^2) + (((z `2) - (o `2)) ^2)) - (r ^2) by A39, A40, A41, A42, VALUED_1:5 ;
|.(z - o).| <= r by A5, A36, TOPREAL9:8;
then A45: |.(z - o).| ^2 <= r ^2 by SQUARE_1:15;
|.(z - o).| ^2 = (((z - o) `1) ^2) + (((z - o) `2) ^2) by JGRAPH_1:29
.= (((z `1) - (o `1)) ^2) + (((z - o) `2) ^2) by TOPREAL3:3
.= (((z `1) - (o `1)) ^2) + (((z `2) - (o `2)) ^2) by TOPREAL3:3 ;
then ((((z `1) - (o `1)) ^2) + (((z `2) - (o `2)) ^2)) - (r ^2) <= (r ^2) - (r ^2) by A45, XREAL_1:9;
hence 0 >= b by A34, A37, A44; ::_thesis: verum
end;
then reconsider p2 = ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) - (f1 | [:{p},(Sphere (o,r)):]) as nonpositive-yielding continuous RealMap of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by PARTFUN3:def_3;
set pp = (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2);
set k = ((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m;
set x3 = (Proj2_1 | [:{p},(Sphere (o,r)):]) + ((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):]));
set y3 = (Proj2_2 | [:{p},(Sphere (o,r)):]) + ((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]));
reconsider X3 = (Proj2_1 | [:{p},(Sphere (o,r)):]) + ((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])), Y3 = (Proj2_2 | [:{p},(Sphere (o,r)):]) + ((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) as Function of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]),R^1 by TOPMETR:17;
set F = <:X3,Y3:>;
set R = R2Homeomorphism ;
A46: for x being Point of ((TOP-REAL 2) | (Sphere (o,r))) holds (RotateCircle (o,r,p)) . x = (R2Homeomorphism * <:X3,Y3:>) . [p,x]
proof
let x be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: (RotateCircle (o,r,p)) . x = (R2Homeomorphism * <:X3,Y3:>) . [p,x]
consider y being Point of (TOP-REAL 2) such that
A47: x = y and
A48: (RotateCircle (o,r,p)) . x = HC (y,p,o,r) by A1, A2, Def8;
A49: x <> p by A1, A3, A9, XBOOLE_0:3;
A50: [p,y] in [:{p},(Sphere (o,r)):] by A4, A9, A47, ZFMISC_1:def_2;
set r1 = (p `1) - (y `1);
set r2 = (p `2) - (y `2);
set r3 = (y `1) - (o `1);
set r4 = (y `2) - (o `2);
set l = ((- ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2))))) + (sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))))) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2));
A51: Proj2_1 . [p,y] = ([p,y] `2) `1 by Def5;
A52: Proj2_2 . [p,y] = ([p,y] `2) `2 by Def6;
A53: diffX1_X2_1 . [p,y] = (([p,y] `1) `1) - (([p,y] `2) `1) by Def3;
A54: diffX1_X2_2 . [p,y] = (([p,y] `1) `2) - (([p,y] `2) `2) by Def4;
A55: (diffX2_1 o) . [p,y] = (([p,y] `2) `1) - (o `1) by Def1;
A56: (diffX2_2 o) . [p,y] = (([p,y] `2) `2) - (o `2) by Def2;
A57: dom X3 = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
A58: dom Y3 = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
A59: dom ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
A60: p is Point of ((TOP-REAL 2) | {p}) by A8, TARSKI:def_1;
then A61: (diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [p,y] = diffX1_X2_1 . [p,y] by A11, A47;
A62: (diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [p,y] = diffX1_X2_2 . [p,y] by A12, A47, A60;
A63: (f1 | [:{p},(Sphere (o,r)):]) . [p,y] = f1 . [p,y] by A15, A47, A60;
A64: ((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [p,y] = (diffX2_1 o) . [p,y] by A16, A47, A60;
A65: ((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [p,y] = (diffX2_2 o) . [p,y] by A17, A47, A60;
A66: m . [p,y] = (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) . [p,y]) + (((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [p,y]) by A10, A50, VALUED_1:1
.= (((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [p,y]) * ((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [p,y])) + (((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [p,y]) by VALUED_1:5
.= (((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2) by A53, A54, A61, A62, VALUED_1:5 ;
A67: (((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) . [p,y] = (((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [p,y]) * ((diffX1_X2_1 | [:{p},(Sphere (o,r)):]) . [p,y]) by VALUED_1:5;
A68: (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [p,y] = (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) . [p,y]) * ((diffX1_X2_2 | [:{p},(Sphere (o,r)):]) . [p,y]) by VALUED_1:5;
A69: ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) . [p,y] = ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) . [p,y]) + ((((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [p,y]) by A10, A50, VALUED_1:1;
then A70: (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) . [p,y] = ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2 by A53, A54, A55, A56, A61, A62, A64, A65, A67, A68, VALUED_1:5;
A71: p2 . [p,y] = (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) . [p,y]) - ((f1 | [:{p},(Sphere (o,r)):]) . [p,y]) by A10, A32, A50, VALUED_1:13
.= (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):]))) . [p,y]) - (r ^2) by A63, FUNCOP_1:7
.= (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_1 o) | [:{p},(Sphere (o,r)):])) . [p,y]) + ((((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):])) . [p,y])) - (r ^2) by A10, A50, VALUED_1:1
.= (((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [p,y]) * (((diffX2_1 o) | [:{p},(Sphere (o,r)):]) . [p,y])) + ((((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) ((diffX2_2 o) | [:{p},(Sphere (o,r)):])) . [p,y])) - (r ^2) by VALUED_1:5
.= ((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2) by A55, A56, A64, A65, VALUED_1:5 ;
dom (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2))) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
then A72: (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2))) . [p,y] = sqrt (((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)) . [p,y]) by A10, A50, PARTFUN3:def_5
.= sqrt (((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) . [p,y]) - ((m (#) p2) . [p,y])) by A10, A50, A59, VALUED_1:13
.= sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))) by A66, A70, A71, VALUED_1:5 ;
dom (((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
then A73: (((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) . [p,y] = (((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) . [p,y]) * ((m . [p,y]) ") by A10, A50, RFUNCT_1:def_1
.= (((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) . [p,y]) / (m . [p,y]) by XCMPLX_0:def_9
.= (((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) . [p,y]) + ((sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2))) . [p,y])) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) by A10, A50, A66, VALUED_1:1
.= ((- ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2))))) + (sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))))) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) by A53, A54, A55, A56, A61, A62, A64, A65, A67, A68, A69, A72, VALUED_1:8 ;
A74: X3 . [p,y] = ((Proj2_1 | [:{p},(Sphere (o,r)):]) . [p,y]) + (((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) . [p,y]) by A10, A50, VALUED_1:1
.= (y `1) + (((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) . [p,y]) by A13, A47, A51, A60
.= (y `1) + ((((- ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2))))) + (sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))))) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2))) * ((p `1) - (y `1))) by A53, A61, A73, VALUED_1:5 ;
A75: Y3 . [p,y] = ((Proj2_2 | [:{p},(Sphere (o,r)):]) . [p,y]) + (((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [p,y]) by A10, A50, VALUED_1:1
.= (y `2) + (((((- ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) + (sqrt ((((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):]))) (#) ((((diffX2_1 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_1 | [:{p},(Sphere (o,r)):])) + (((diffX2_2 o) | [:{p},(Sphere (o,r)):]) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])))) - (m (#) p2)))) / m) (#) (diffX1_X2_2 | [:{p},(Sphere (o,r)):])) . [p,y]) by A14, A47, A52, A60
.= (y `2) + ((((- ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2))))) + (sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))))) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2))) * ((p `2) - (y `2))) by A54, A62, A73, VALUED_1:5 ;
y in the carrier of ((TOP-REAL 2) | (Sphere (o,r))) by A47;
hence (RotateCircle (o,r,p)) . x = |[((y `1) + ((((- ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2))))) + (sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))))) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2))) * ((p `1) - (y `1)))),((y `2) + ((((- ((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2))))) + (sqrt ((((((y `1) - (o `1)) * ((p `1) - (y `1))) + (((y `2) - (o `2)) * ((p `2) - (y `2)))) ^2) - (((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2)) * (((((y `1) - (o `1)) ^2) + (((y `2) - (o `2)) ^2)) - (r ^2)))))) / ((((p `1) - (y `1)) ^2) + (((p `2) - (y `2)) ^2))) * ((p `2) - (y `2))))]| by A1, A5, A6, A7, A9, A47, A48, A49, BROUWER:8
.= R2Homeomorphism . [(X3 . [p,y]),(Y3 . [p,y])] by A74, A75, TOPREALA:def_2
.= R2Homeomorphism . (<:X3,Y3:> . [p,y]) by A10, A50, A57, A58, FUNCT_3:49
.= (R2Homeomorphism * <:X3,Y3:>) . [p,x] by A10, A47, A50, FUNCT_2:15 ;
::_thesis: verum
end;
A76: X3 is continuous by JORDAN5A:27;
Y3 is continuous by JORDAN5A:27;
then reconsider F = <:X3,Y3:> as continuous Function of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]),[:R^1,R^1:] by A76, YELLOW12:41;
for pp being Point of ((TOP-REAL 2) | (Sphere (o,r)))
for V being Subset of (Tcircle (o,r)) st (RotateCircle (o,r,p)) . pp in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | (Sphere (o,r))) st
( pp in W & W is open & (RotateCircle (o,r,p)) .: W c= V )
proof
let pp be Point of ((TOP-REAL 2) | (Sphere (o,r))); ::_thesis: for V being Subset of (Tcircle (o,r)) st (RotateCircle (o,r,p)) . pp in V & V is open holds
ex W being Subset of ((TOP-REAL 2) | (Sphere (o,r))) st
( pp in W & W is open & (RotateCircle (o,r,p)) .: W c= V )
let V be Subset of (Tcircle (o,r)); ::_thesis: ( (RotateCircle (o,r,p)) . pp in V & V is open implies ex W being Subset of ((TOP-REAL 2) | (Sphere (o,r))) st
( pp in W & W is open & (RotateCircle (o,r,p)) .: W c= V ) )
assume that
A77: (RotateCircle (o,r,p)) . pp in V and
A78: V is open ; ::_thesis: ex W being Subset of ((TOP-REAL 2) | (Sphere (o,r))) st
( pp in W & W is open & (RotateCircle (o,r,p)) .: W c= V )
reconsider p1 = pp, fp = p as Point of (TOP-REAL 2) by PRE_TOPC:25;
A79: [p,pp] in [:{p},(Sphere (o,r)):] by A4, A9, ZFMISC_1:def_2;
consider V1 being Subset of (TOP-REAL 2) such that
A80: V1 is open and
A81: V1 /\ ([#] (Tcircle (o,r))) = V by A78, TOPS_2:24;
A82: (RotateCircle (o,r,p)) . pp = (R2Homeomorphism * F) . [p,pp] by A46;
R2Homeomorphism " is being_homeomorphism by TOPREALA:34, TOPS_2:56;
then A83: (R2Homeomorphism ") .: V1 is open by A80, TOPGRP_1:25;
A84: dom F = the carrier of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) by FUNCT_2:def_1;
A85: dom R2Homeomorphism = the carrier of [:R^1,R^1:] by FUNCT_2:def_1;
then A86: rng F c= dom R2Homeomorphism ;
then A87: dom (R2Homeomorphism * F) = dom F by RELAT_1:27;
A88: rng R2Homeomorphism = [#] (TOP-REAL 2) by TOPREALA:34, TOPS_2:def_5;
A89: (R2Homeomorphism ") * (R2Homeomorphism * F) = ((R2Homeomorphism ") * R2Homeomorphism) * F by RELAT_1:36
.= (id (dom R2Homeomorphism)) * F by A88, TOPREALA:34, TOPS_2:52 ;
dom (id (dom R2Homeomorphism)) = dom R2Homeomorphism ;
then A90: dom ((id (dom R2Homeomorphism)) * F) = dom F by A86, RELAT_1:27;
for x being set st x in dom F holds
((id (dom R2Homeomorphism)) * F) . x = F . x
proof
let x be set ; ::_thesis: ( x in dom F implies ((id (dom R2Homeomorphism)) * F) . x = F . x )
assume A91: x in dom F ; ::_thesis: ((id (dom R2Homeomorphism)) * F) . x = F . x
A92: F . x in rng F by A91, FUNCT_1:def_3;
thus ((id (dom R2Homeomorphism)) * F) . x = (id (dom R2Homeomorphism)) . (F . x) by A91, FUNCT_1:13
.= F . x by A85, A92, FUNCT_1:18 ; ::_thesis: verum
end;
then A93: (id (dom R2Homeomorphism)) * F = F by A90, FUNCT_1:2;
(R2Homeomorphism * F) . [fp,p1] in V1 by A77, A81, A82, XBOOLE_0:def_4;
then (R2Homeomorphism ") . ((R2Homeomorphism * F) . [fp,p1]) in (R2Homeomorphism ") .: V1 by FUNCT_2:35;
then ((R2Homeomorphism ") * (R2Homeomorphism * F)) . [fp,p1] in (R2Homeomorphism ") .: V1 by A10, A79, A84, A87, FUNCT_1:13;
then consider W being Subset of ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]) such that
A94: [fp,p1] in W and
A95: W is open and
A96: F .: W c= (R2Homeomorphism ") .: V1 by A10, A79, A83, A89, A93, JGRAPH_2:10;
consider WW being Subset of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A97: WW is open and
A98: WW /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):])) = W by A95, TOPS_2:24;
consider SF being Subset-Family of [:(TOP-REAL 2),(TOP-REAL 2):] such that
A99: WW = union SF and
A100: for e being set st e in SF holds
ex X1, Y1 being Subset of (TOP-REAL 2) st
( e = [:X1,Y1:] & X1 is open & Y1 is open ) by A97, BORSUK_1:5;
[fp,p1] in WW by A94, A98, XBOOLE_0:def_4;
then consider Z being set such that
A101: [fp,p1] in Z and
A102: Z in SF by A99, TARSKI:def_4;
consider X1, Y1 being Subset of (TOP-REAL 2) such that
A103: Z = [:X1,Y1:] and
X1 is open and
A104: Y1 is open by A100, A102;
set ZZ = Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]));
reconsider XX = Y1 /\ ([#] ((TOP-REAL 2) | (Sphere (o,r)))) as open Subset of ((TOP-REAL 2) | (Sphere (o,r))) by A104, TOPS_2:24;
take XX ; ::_thesis: ( pp in XX & XX is open & (RotateCircle (o,r,p)) .: XX c= V )
pp in Y1 by A101, A103, ZFMISC_1:87;
hence pp in XX by XBOOLE_0:def_4; ::_thesis: ( XX is open & (RotateCircle (o,r,p)) .: XX c= V )
thus XX is open ; ::_thesis: (RotateCircle (o,r,p)) .: XX c= V
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in (RotateCircle (o,r,p)) .: XX or b in V )
assume b in (RotateCircle (o,r,p)) .: XX ; ::_thesis: b in V
then consider a being Point of ((TOP-REAL 2) | (Sphere (o,r))) such that
A105: a in XX and
A106: b = (RotateCircle (o,r,p)) . a by A2, FUNCT_2:65;
reconsider a1 = a, fa = fp as Point of (TOP-REAL 2) by PRE_TOPC:25;
A107: a in Y1 by A105, XBOOLE_0:def_4;
A108: [p,a] in [:{p},(Sphere (o,r)):] by A4, A9, ZFMISC_1:def_2;
fa in X1 by A101, A103, ZFMISC_1:87;
then [fa,a] in Z by A103, A107, ZFMISC_1:def_2;
then [fa,a] in Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):])) by A10, A108, XBOOLE_0:def_4;
then A109: F . [fa,a1] in F .: (Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]))) by FUNCT_2:35;
A110: R2Homeomorphism " = R2Homeomorphism " by TOPREALA:34, TOPS_2:def_4;
A111: dom (R2Homeomorphism ") = [#] (TOP-REAL 2) by A88, TOPREALA:34, TOPS_2:49;
A112: (RotateCircle (o,r,p)) . a1 in the carrier of (Tcircle (o,r)) by A2, FUNCT_2:5;
Z c= WW by A99, A102, ZFMISC_1:74;
then Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):])) c= WW /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):])) by XBOOLE_1:27;
then F .: (Z /\ ([#] ([:(TOP-REAL 2),(TOP-REAL 2):] | [:{p},(Sphere (o,r)):]))) c= F .: W by A98, RELAT_1:123;
then F . [fa,a1] in F .: W by A109;
then R2Homeomorphism . (F . [fa,a1]) in R2Homeomorphism .: ((R2Homeomorphism ") .: V1) by A96, FUNCT_2:35;
then (R2Homeomorphism * F) . [fa,a1] in R2Homeomorphism .: ((R2Homeomorphism ") .: V1) by A10, A108, FUNCT_2:15;
then (R2Homeomorphism * F) . [fa,a1] in V1 by A110, A111, PARTFUN3:1, TOPREALA:34;
then (RotateCircle (o,r,p)) . a in V1 by A46;
hence b in V by A81, A106, A112, XBOOLE_0:def_4; ::_thesis: verum
end;
hence RotateCircle (o,r,p) is continuous by A2, JGRAPH_2:10; ::_thesis: verum
end;
theorem Th67: :: JORDAN:67
for n being non empty Element of NAT
for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball (o,r) holds
RotateCircle (o,r,p) is without_fixpoints
proof
let n be non empty Element of NAT ; ::_thesis: for o, p being Point of (TOP-REAL n)
for r being positive real number st p in Ball (o,r) holds
RotateCircle (o,r,p) is without_fixpoints
let o, p be Point of (TOP-REAL n); ::_thesis: for r being positive real number st p in Ball (o,r) holds
RotateCircle (o,r,p) is without_fixpoints
let r be positive real number ; ::_thesis: ( p in Ball (o,r) implies RotateCircle (o,r,p) is without_fixpoints )
assume A1: p in Ball (o,r) ; ::_thesis: RotateCircle (o,r,p) is without_fixpoints
set f = RotateCircle (o,r,p);
let x be set ; :: according to ABIAN:def_5 ::_thesis: not x is_a_fixpoint_of RotateCircle (o,r,p)
assume A2: x in dom (RotateCircle (o,r,p)) ; :: according to ABIAN:def_3 ::_thesis: not x = (RotateCircle (o,r,p)) . x
set S = Tcircle (o,r);
A3: dom (RotateCircle (o,r,p)) = the carrier of (Tcircle (o,r)) by FUNCT_2:def_1;
consider y being Point of (TOP-REAL n) such that
A4: x = y and
A5: (RotateCircle (o,r,p)) . x = HC (y,p,o,r) by A1, A2, Def8;
A6: the carrier of (Tcircle (o,r)) = Sphere (o,r) by TOPREALB:9;
Sphere (o,r) c= cl_Ball (o,r) by TOPREAL9:17;
then A7: y is Point of (Tdisk (o,r)) by A2, A3, A4, A6, BROUWER:3;
Ball (o,r) c= cl_Ball (o,r) by TOPREAL9:16;
then A8: p is Point of (Tdisk (o,r)) by A1, BROUWER:3;
Ball (o,r) misses Sphere (o,r) by TOPREAL9:19;
then y <> p by A1, A2, A4, A6, XBOOLE_0:3;
hence not x = (RotateCircle (o,r,p)) . x by A4, A5, A7, A8, BROUWER:def_3; ::_thesis: verum
end;
begin
theorem Th68: :: JORDAN:68
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C `)) st U = P & U is a_component & V is a_component & U <> V holds
Cl P misses V
proof
let C be Simple_closed_curve; ::_thesis: for P being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C `)) st U = P & U is a_component & V is a_component & U <> V holds
Cl P misses V
let P be Subset of (TOP-REAL 2); ::_thesis: for U, V being Subset of ((TOP-REAL 2) | (C `)) st U = P & U is a_component & V is a_component & U <> V holds
Cl P misses V
let U, V be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: ( U = P & U is a_component & V is a_component & U <> V implies Cl P misses V )
assume that
A1: U = P and
A2: U is a_component and
A3: V is a_component and
A4: U <> V ; ::_thesis: Cl P misses V
assume Cl P meets V ; ::_thesis: contradiction
then A5: ex x being set st
( x in Cl P & x in V ) by XBOOLE_0:3;
the carrier of ((TOP-REAL 2) | (C `)) = C ` by PRE_TOPC:8;
then reconsider V1 = V as Subset of (TOP-REAL 2) by XBOOLE_1:1;
reconsider T2C = (TOP-REAL 2) | (C `) as non empty SubSpace of TOP-REAL 2 ;
T2C is locally_connected by JORDAN2C:81;
then V is open by A3, CONNSP_2:15;
then V1 is open by TSEP_1:17;
then P meets V1 by A5, PRE_TOPC:def_7;
hence contradiction by A1, A2, A3, A4, CONNSP_1:35; ::_thesis: verum
end;
theorem Th69: :: JORDAN:69
for C being Simple_closed_curve
for U being Subset of ((TOP-REAL 2) | (C `)) st U is a_component holds
((TOP-REAL 2) | (C `)) | U is pathwise_connected
proof
let C be Simple_closed_curve; ::_thesis: for U being Subset of ((TOP-REAL 2) | (C `)) st U is a_component holds
((TOP-REAL 2) | (C `)) | U is pathwise_connected
let U be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: ( U is a_component implies ((TOP-REAL 2) | (C `)) | U is pathwise_connected )
set T = (TOP-REAL 2) | (C `);
assume A1: U is a_component ; ::_thesis: ((TOP-REAL 2) | (C `)) | U is pathwise_connected
let a, b be Point of (((TOP-REAL 2) | (C `)) | U); :: according to BORSUK_2:def_3 ::_thesis: a,b are_connected
A2: the carrier of (((TOP-REAL 2) | (C `)) | U) = U by PRE_TOPC:8;
A3: U <> {} ((TOP-REAL 2) | (C `)) by A1, CONNSP_1:32;
percases ( a = b or a <> b ) ;
supposeA4: a = b ; ::_thesis: a,b are_connected
reconsider TU = ((TOP-REAL 2) | (C `)) | U as non empty TopSpace by A3;
reconsider a = a as Point of TU ;
reconsider f = I[01] --> a as Function of I[01],(((TOP-REAL 2) | (C `)) | U) ;
take f ; :: according to BORSUK_2:def_1 ::_thesis: ( f is continuous & f . 0 = a & f . 1 = b )
thus ( f is continuous & f . 0 = a & f . 1 = b ) by A4, BORSUK_1:def_14, BORSUK_1:def_15, TOPALG_3:4; ::_thesis: verum
end;
supposeA5: a <> b ; ::_thesis: a,b are_connected
A6: ((TOP-REAL 2) | (C `)) | U is SubSpace of TOP-REAL 2 by TSEP_1:7;
then reconsider a1 = a, b1 = b as Point of (TOP-REAL 2) by A3, PRE_TOPC:25;
reconsider V = U as Subset of (TOP-REAL 2) by PRE_TOPC:11;
V is_a_component_of C ` by A1, CONNSP_1:def_6;
then A7: V is open by SPRECT_3:8;
U is connected by A1, CONNSP_1:def_5;
then V is connected by CONNSP_1:23;
then consider P being Subset of (TOP-REAL 2) such that
A8: P is_S-P_arc_joining a1,b1 and
A9: P c= V by A2, A3, A5, A7, TOPREAL4:29;
A10: a1 in P by A8, TOPREAL4:3;
P is_an_arc_of a1,b1 by A8, TOPREAL4:2;
then consider g being Function of I[01],((TOP-REAL 2) | P) such that
A11: g is being_homeomorphism and
A12: g . 0 = a and
A13: g . 1 = b by TOPREAL1:def_1;
A14: the carrier of ((TOP-REAL 2) | P) = P by PRE_TOPC:8;
then reconsider f = g as Function of I[01],(((TOP-REAL 2) | (C `)) | U) by A2, A9, A10, FUNCT_2:7;
take f ; :: according to BORSUK_2:def_1 ::_thesis: ( f is continuous & f . 0 = a & f . 1 = b )
(TOP-REAL 2) | P is SubSpace of ((TOP-REAL 2) | (C `)) | U by A2, A6, A9, A14, TSEP_1:4;
hence f is continuous by A11, PRE_TOPC:26; ::_thesis: ( f . 0 = a & f . 1 = b )
thus ( f . 0 = a & f . 1 = b ) by A12, A13; ::_thesis: verum
end;
end;
end;
Lm12: for p1, p2, p being Point of (TOP-REAL 2)
for A being Subset of (TOP-REAL 2)
for r being non negative real number st A is_an_arc_of p1,p2 & A is Subset of (Tdisk (p,r)) holds
ex f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st
( f is continuous & f | A = id A )
proof
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for A being Subset of (TOP-REAL 2)
for r being non negative real number st A is_an_arc_of p1,p2 & A is Subset of (Tdisk (p,r)) holds
ex f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st
( f is continuous & f | A = id A )
let A be Subset of (TOP-REAL 2); ::_thesis: for r being non negative real number st A is_an_arc_of p1,p2 & A is Subset of (Tdisk (p,r)) holds
ex f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st
( f is continuous & f | A = id A )
let r be non negative real number ; ::_thesis: ( A is_an_arc_of p1,p2 & A is Subset of (Tdisk (p,r)) implies ex f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st
( f is continuous & f | A = id A ) )
set D = Tdisk (p,r);
assume that
A1: A is_an_arc_of p1,p2 and
A2: A is Subset of (Tdisk (p,r)) ; ::_thesis: ex f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st
( f is continuous & f | A = id A )
reconsider A1 = A as non empty Subset of (Tdisk (p,r)) by A1, A2, TOPREAL1:1;
reconsider A2 = A as non empty Subset of (TOP-REAL 2) by A1, TOPREAL1:1;
set TA = (TOP-REAL 2) | A2;
consider h being Function of I[01],((TOP-REAL 2) | A2) such that
A3: h is being_homeomorphism and
h . 0 = p1 and
h . 1 = p2 by A1, TOPREAL1:def_1;
A4: L[01] (((#) ((- 1),1)),(((- 1),1) (#))) is being_homeomorphism by TREAL_1:17;
reconsider hh = h as Function of (Closed-Interval-TSpace (0,1)),((TOP-REAL 2) | A2) by TOPMETR:20;
A5: (TOP-REAL 2) | A2 = (Tdisk (p,r)) | A1 by TOPALG_5:4;
then reconsider f = (L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) * (hh ") as Function of ((Tdisk (p,r)) | A1),(Closed-Interval-TSpace ((- 1),1)) ;
A is closed by A1, JORDAN6:11;
then A6: A1 is closed by TSEP_1:12;
hh " is continuous by A3, TOPMETR:20, TOPS_2:def_5;
then consider g being continuous Function of (Tdisk (p,r)),(Closed-Interval-TSpace ((- 1),1)) such that
A7: g | A1 = f by A4, A5, A6, TIETZE:23;
reconsider R = (hh * ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) ")) * g as Function of (Tdisk (p,r)),((TOP-REAL 2) | A) ;
take R ; ::_thesis: ( R is continuous & R | A = id A )
(L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) " is continuous by A4, TOPS_2:def_5;
hence R is continuous by A3, TOPMETR:20; ::_thesis: R | A = id A
A8: the carrier of ((TOP-REAL 2) | A2) = A1 by PRE_TOPC:8;
A9: dom R = the carrier of (Tdisk (p,r)) by FUNCT_2:def_1;
A10: dom (id A) = A ;
now__::_thesis:_for_a_being_set_st_a_in_dom_(R_|_A)_holds_
(R_|_A)_._a_=_(id_A)_._a
let a be set ; ::_thesis: ( a in dom (R | A) implies (R | A) . a = (id A) . a )
assume A11: a in dom (R | A) ; ::_thesis: (R | A) . a = (id A) . a
then A12: a in dom R by RELAT_1:57;
A13: dom g = the carrier of (Tdisk (p,r)) by FUNCT_2:def_1;
A14: dom ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) * (hh ")) = the carrier of ((TOP-REAL 2) | A2) by FUNCT_2:def_1;
A15: (hh * ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) ")) * ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) * (hh ")) = ((hh * ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) ")) * (L[01] (((#) ((- 1),1)),(((- 1),1) (#))))) * (hh ") by RELAT_1:36
.= (hh * (((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) ") * (L[01] (((#) ((- 1),1)),(((- 1),1) (#)))))) * (hh ") by RELAT_1:36
.= (hh * (id (Closed-Interval-TSpace (0,1)))) * (hh ") by A4, GRCAT_1:41
.= hh * (hh ") by FUNCT_2:17
.= id ((TOP-REAL 2) | A2) by A3, GRCAT_1:41 ;
thus (R | A) . a = R . a by A11, FUNCT_1:49
.= (hh * ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) ")) . (g . a) by A13, A12, FUNCT_1:13
.= (hh * ((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) ")) . (((L[01] (((#) ((- 1),1)),(((- 1),1) (#)))) * (hh ")) . a) by A7, A11, FUNCT_1:49
.= (id A) . a by A8, A11, A14, A15, FUNCT_1:13 ; ::_thesis: verum
end;
hence R | A = id A by A2, A9, A10, FUNCT_1:2, RELAT_1:62; ::_thesis: verum
end;
Lm13: for p1, p2, p being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for A, P, B being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `))
for r being positive real number st A is_an_arc_of p1,p2 & A c= C & C c= Ball (p,r) & p in U & (Cl P) /\ (P `) c= A & P c= Ball (p,r) holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
proof
let p1, p2, p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve
for A, P, B being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `))
for r being positive real number st A is_an_arc_of p1,p2 & A c= C & C c= Ball (p,r) & p in U & (Cl P) /\ (P `) c= A & P c= Ball (p,r) holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
let C be Simple_closed_curve; ::_thesis: for A, P, B being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `))
for r being positive real number st A is_an_arc_of p1,p2 & A c= C & C c= Ball (p,r) & p in U & (Cl P) /\ (P `) c= A & P c= Ball (p,r) holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
let A, P, B be Subset of (TOP-REAL 2); ::_thesis: for U being Subset of ((TOP-REAL 2) | (C `))
for r being positive real number st A is_an_arc_of p1,p2 & A c= C & C c= Ball (p,r) & p in U & (Cl P) /\ (P `) c= A & P c= Ball (p,r) holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
let U be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: for r being positive real number st A is_an_arc_of p1,p2 & A c= C & C c= Ball (p,r) & p in U & (Cl P) /\ (P `) c= A & P c= Ball (p,r) holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
let r be positive real number ; ::_thesis: ( A is_an_arc_of p1,p2 & A c= C & C c= Ball (p,r) & p in U & (Cl P) /\ (P `) c= A & P c= Ball (p,r) implies for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) ) )
set D = Tdisk (p,r);
assume that
A1: A is_an_arc_of p1,p2 and
A2: A c= C and
A3: C c= Ball (p,r) and
A4: p in U and
A5: (Cl P) /\ (P `) c= A and
A6: P c= Ball (p,r) ; ::_thesis: for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
let f be Function of (Tdisk (p,r)),((TOP-REAL 2) | A); ::_thesis: ( f is continuous & f | A = id A & U = P & U is a_component & B = (cl_Ball (p,r)) \ {p} implies ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) ) )
assume that
A7: f is continuous and
A8: f | A = id A and
A9: U = P and
A10: U is a_component and
A11: B = (cl_Ball (p,r)) \ {p} ; ::_thesis: ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
reconsider B1 = B as non empty Subset of (TOP-REAL 2) by A11;
reconsider T2B1 = (TOP-REAL 2) | B1 as non empty SubSpace of TOP-REAL 2 ;
A12: the carrier of ((TOP-REAL 2) | (C `)) = C ` by PRE_TOPC:8;
A13: the carrier of ((TOP-REAL 2) | A) = A by PRE_TOPC:8;
A14: the carrier of (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
A15: Ball (p,r) c= cl_Ball (p,r) by TOPREAL9:16;
A16: A <> {} by A1, TOPREAL1:1;
reconsider A1 = A as non empty Subset of (TOP-REAL 2) by A1, TOPREAL1:1;
A17: not p in C by A4, A12, XBOOLE_0:def_5;
|.(p - p).| = 0 by TOPRNS_1:28;
then A18: p in [#] (Tdisk (p,r)) by A14, TOPREAL9:8;
A19: P c= Cl P by PRE_TOPC:18;
then reconsider F1 = (Cl P) /\ ([#] (Tdisk (p,r))) as non empty Subset of (Tdisk (p,r)) by A4, A9, A18, XBOOLE_0:def_4;
A20: Sphere (p,r) c= cl_Ball (p,r) by TOPREAL9:17;
A21: Ball (p,r) misses Sphere (p,r) by TOPREAL9:19;
consider e being Point of (TOP-REAL 2) such that
A22: e in Sphere (p,r) by SUBSET_1:4;
not e in P by A6, A21, A22, XBOOLE_0:3;
then e in P ` by SUBSET_1:29;
then reconsider F3 = (P `) /\ ([#] (Tdisk (p,r))) as non empty Subset of (Tdisk (p,r)) by A14, A20, A22, XBOOLE_0:def_4;
reconsider T1 = (Tdisk (p,r)) | F1 as non empty SubSpace of Tdisk (p,r) ;
reconsider T3 = (Tdisk (p,r)) | F3 as non empty SubSpace of Tdisk (p,r) ;
A23: the carrier of T1 = F1 by PRE_TOPC:8;
A24: the carrier of T3 = F3 by PRE_TOPC:8;
A25: the carrier of T2B1 = B1 by PRE_TOPC:8;
A26: A c= B
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A or a in B )
assume a in A ; ::_thesis: a in B
then A27: a in C by A2;
then a in Ball (p,r) by A3;
hence a in B by A11, A15, A17, A27, ZFMISC_1:56; ::_thesis: verum
end;
A28: F3 c= B
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in F3 or a in B )
assume A29: a in F3 ; ::_thesis: a in B
then a in P ` by XBOOLE_0:def_4;
then not a in P by XBOOLE_0:def_5;
hence a in B by A4, A9, A11, A14, A29, ZFMISC_1:56; ::_thesis: verum
end;
f | F1 is Function of F1,A by A13, A16, FUNCT_2:32;
then reconsider f1 = f | F1 as Function of T1,T2B1 by A16, A23, A25, A26, FUNCT_2:7;
reconsider g1 = id F3 as Function of T3,T2B1 by A24, A25, A28, FUNCT_2:7;
A30: F1 = [#] T1 by PRE_TOPC:8;
A31: F3 = [#] T3 by PRE_TOPC:8;
A32: ([#] T1) \/ ([#] T3) = [#] (Tdisk (p,r))
proof
thus ([#] T1) \/ ([#] T3) c= [#] (Tdisk (p,r)) by A30, A31, XBOOLE_1:8; :: according to XBOOLE_0:def_10 ::_thesis: [#] (Tdisk (p,r)) c= ([#] T1) \/ ([#] T3)
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in [#] (Tdisk (p,r)) or p in ([#] T1) \/ ([#] T3) )
assume A33: p in [#] (Tdisk (p,r)) ; ::_thesis: p in ([#] T1) \/ ([#] T3)
percases ( p in P or not p in P ) ;
suppose p in P ; ::_thesis: p in ([#] T1) \/ ([#] T3)
then p in F1 by A19, A33, XBOOLE_0:def_4;
hence p in ([#] T1) \/ ([#] T3) by A30, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not p in P ; ::_thesis: p in ([#] T1) \/ ([#] T3)
then p in P ` by A14, A33, SUBSET_1:29;
then p in F3 by A33, XBOOLE_0:def_4;
hence p in ([#] T1) \/ ([#] T3) by A31, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
reconsider DT = [#] (Tdisk (p,r)) as closed Subset of (TOP-REAL 2) by BORSUK_1:def_11, TSEP_1:1;
DT /\ (Cl P) is closed ;
then A34: F1 is closed by TSEP_1:8;
P is_a_component_of C `
proof
take U ; :: according to CONNSP_1:def_6 ::_thesis: ( U = P & U is a_component )
thus ( U = P & U is a_component ) by A9, A10; ::_thesis: verum
end;
then P is open by SPRECT_3:8;
then DT /\ (P `) is closed ;
then A35: F3 is closed by TSEP_1:8;
reconsider f2 = f | F1 as Function of T1,((TOP-REAL 2) | A1) by A23, FUNCT_2:32;
A36: (TOP-REAL 2) | A1 is SubSpace of T2B1 by A13, A25, A26, TSEP_1:4;
T3 is SubSpace of TOP-REAL 2 by TSEP_1:7;
then A37: T3 is SubSpace of T2B1 by A24, A25, A28, TSEP_1:4;
f2 is continuous by A7, TOPMETR:7;
then A38: f1 is continuous by A36, PRE_TOPC:26;
reconsider g2 = id F3 as Function of T3,T3 by A24;
g2 = id T3 by PRE_TOPC:8;
then A39: g1 is continuous by A37, PRE_TOPC:26;
A40: for x being set st x in Cl P & x in P ` holds
f . x = x
proof
let x be set ; ::_thesis: ( x in Cl P & x in P ` implies f . x = x )
assume that
A41: x in Cl P and
A42: x in P ` ; ::_thesis: f . x = x
A43: x in (Cl P) /\ (P `) by A41, A42, XBOOLE_0:def_4;
then (id A) . x = x by A5, FUNCT_1:18;
hence f . x = x by A5, A8, A43, FUNCT_1:49; ::_thesis: verum
end;
for x being set st x in ([#] T1) /\ ([#] T3) holds
f1 . x = g1 . x
proof
let x be set ; ::_thesis: ( x in ([#] T1) /\ ([#] T3) implies f1 . x = g1 . x )
assume A44: x in ([#] T1) /\ ([#] T3) ; ::_thesis: f1 . x = g1 . x
then A45: x in [#] T1 by XBOOLE_0:def_4;
then A46: x in Cl P by A30, XBOOLE_0:def_4;
x in P ` by A31, A44, XBOOLE_0:def_4;
then A47: f . x = x by A40, A46;
thus f1 . x = f . x by A30, A45, FUNCT_1:49
.= g1 . x by A31, A44, A47, FUNCT_1:18 ; ::_thesis: verum
end;
then consider g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) such that
A48: g = f1 +* g1 and
A49: g is continuous by A30, A31, A32, A34, A35, A38, A39, JGRAPH_2:1;
take g ; ::_thesis: ( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) ) ) )
thus g is continuous by A49; ::_thesis: for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) )
let x be Point of (Tdisk (p,r)); ::_thesis: ( ( x in Cl P implies g . x = f . x ) & ( x in P ` implies g . x = x ) )
A50: dom g1 = the carrier of T3 by FUNCT_2:def_1;
hereby ::_thesis: ( x in P ` implies g . x = x )
assume A51: x in Cl P ; ::_thesis: g . x = f . x
then A52: x in F1 by XBOOLE_0:def_4;
percases ( not x in dom g1 or x in dom g1 ) ;
suppose not x in dom g1 ; ::_thesis: g . x = f . x
hence g . x = f1 . x by A48, FUNCT_4:11
.= f . x by A52, FUNCT_1:49 ;
::_thesis: verum
end;
supposeA53: x in dom g1 ; ::_thesis: g . x = f . x
then A54: x in P ` by A24, XBOOLE_0:def_4;
thus g . x = g1 . x by A48, A53, FUNCT_4:13
.= x by A24, A53, FUNCT_1:18
.= f . x by A40, A51, A54 ; ::_thesis: verum
end;
end;
end;
assume x in P ` ; ::_thesis: g . x = x
then A55: x in F3 by XBOOLE_0:def_4;
hence g . x = g1 . x by A24, A48, A50, FUNCT_4:13
.= x by A55, FUNCT_1:18 ;
::_thesis: verum
end;
Lm14: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve
for P, B being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C `))
for A being non empty Subset of (TOP-REAL 2) st U <> V holds
for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
proof
let p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve
for P, B being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C `))
for A being non empty Subset of (TOP-REAL 2) st U <> V holds
for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
let C be Simple_closed_curve; ::_thesis: for P, B being Subset of (TOP-REAL 2)
for U, V being Subset of ((TOP-REAL 2) | (C `))
for A being non empty Subset of (TOP-REAL 2) st U <> V holds
for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
let P, B be Subset of (TOP-REAL 2); ::_thesis: for U, V being Subset of ((TOP-REAL 2) | (C `))
for A being non empty Subset of (TOP-REAL 2) st U <> V holds
for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
let U, V be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: for A being non empty Subset of (TOP-REAL 2) st U <> V holds
for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
let A be non empty Subset of (TOP-REAL 2); ::_thesis: ( U <> V implies for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) ) )
assume A1: U <> V ; ::_thesis: for r being positive real number st A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P holds
for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
let r be positive real number ; ::_thesis: ( A c= C & C c= Ball (p,r) & p in V & (Cl P) /\ (P `) c= A & Ball (p,r) meets P implies for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) ) )
set D = Tdisk (p,r);
assume that
A2: A c= C and
A3: C c= Ball (p,r) and
A4: p in V and
A5: (Cl P) /\ (P `) c= A and
A6: Ball (p,r) meets P ; ::_thesis: for f being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) st f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} holds
ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
let f be Function of (Tdisk (p,r)),((TOP-REAL 2) | A); ::_thesis: ( f is continuous & f | A = id A & U = P & U is a_component & V is a_component & B = (cl_Ball (p,r)) \ {p} implies ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) ) )
assume that
A7: f is continuous and
A8: f | A = id A and
A9: U = P and
A10: U is a_component and
A11: V is a_component and
A12: B = (cl_Ball (p,r)) \ {p} ; ::_thesis: ex g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) st
( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
reconsider B1 = B as non empty Subset of (TOP-REAL 2) by A12;
reconsider T2B1 = (TOP-REAL 2) | B1 as non empty SubSpace of TOP-REAL 2 ;
A13: the carrier of ((TOP-REAL 2) | (C `)) = C ` by PRE_TOPC:8;
A14: the carrier of ((TOP-REAL 2) | A) = A by PRE_TOPC:8;
A15: the carrier of (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
A16: Ball (p,r) c= cl_Ball (p,r) by TOPREAL9:16;
A17: not p in C by A4, A13, XBOOLE_0:def_5;
|.(p - p).| = 0 by TOPRNS_1:28;
then A18: p in [#] (Tdisk (p,r)) by A15, TOPREAL9:8;
A19: P c= Cl P by PRE_TOPC:18;
ex j being set st
( j in Ball (p,r) & j in P ) by A6, XBOOLE_0:3;
then reconsider F1 = (Cl P) /\ ([#] (Tdisk (p,r))) as non empty Subset of (Tdisk (p,r)) by A15, A16, A19, XBOOLE_0:def_4;
now__::_thesis:_not_p_in_P
assume A20: p in P ; ::_thesis: contradiction
U misses V by A1, A10, A11, CONNSP_1:35;
hence contradiction by A4, A9, A20, XBOOLE_0:3; ::_thesis: verum
end;
then p in P ` by SUBSET_1:29;
then reconsider F3 = (P `) /\ ([#] (Tdisk (p,r))) as non empty Subset of (Tdisk (p,r)) by A18, XBOOLE_0:def_4;
set T1 = (Tdisk (p,r)) | F1;
set T3 = (Tdisk (p,r)) | F3;
A21: the carrier of ((Tdisk (p,r)) | F1) = F1 by PRE_TOPC:8;
A22: the carrier of ((Tdisk (p,r)) | F3) = F3 by PRE_TOPC:8;
A23: the carrier of ((TOP-REAL 2) | B1) = B1 by PRE_TOPC:8;
A24: A c= B
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in A or a in B )
assume a in A ; ::_thesis: a in B
then A25: a in C by A2;
then a in Ball (p,r) by A3;
hence a in B by A12, A16, A17, A25, ZFMISC_1:56; ::_thesis: verum
end;
A26: F1 c= B
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in F1 or a in B )
assume A27: a in F1 ; ::_thesis: a in B
then A28: a in Cl P by XBOOLE_0:def_4;
now__::_thesis:_not_p_in_Cl_P
assume A29: p in Cl P ; ::_thesis: contradiction
Cl P misses V by A1, A9, A10, A11, Th68;
hence contradiction by A4, A29, XBOOLE_0:3; ::_thesis: verum
end;
hence a in B by A12, A15, A27, A28, ZFMISC_1:56; ::_thesis: verum
end;
then reconsider f1 = id F1 as Function of ((Tdisk (p,r)) | F1),T2B1 by A21, A23, FUNCT_2:7;
f | F3 is Function of F3,A by A14, FUNCT_2:32;
then reconsider g1 = f | F3 as Function of ((Tdisk (p,r)) | F3),T2B1 by A22, A23, A24, FUNCT_2:7;
A30: F1 = [#] ((Tdisk (p,r)) | F1) by PRE_TOPC:8;
A31: F3 = [#] ((Tdisk (p,r)) | F3) by PRE_TOPC:8;
A32: ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3)) = [#] (Tdisk (p,r))
proof
thus ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3)) c= [#] (Tdisk (p,r)) by A30, A31, XBOOLE_1:8; :: according to XBOOLE_0:def_10 ::_thesis: [#] (Tdisk (p,r)) c= ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3))
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in [#] (Tdisk (p,r)) or p in ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3)) )
assume A33: p in [#] (Tdisk (p,r)) ; ::_thesis: p in ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3))
percases ( p in P or not p in P ) ;
suppose p in P ; ::_thesis: p in ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3))
then p in F1 by A19, A33, XBOOLE_0:def_4;
hence p in ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3)) by A30, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not p in P ; ::_thesis: p in ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3))
then p in P ` by A15, A33, SUBSET_1:29;
then p in F3 by A33, XBOOLE_0:def_4;
hence p in ([#] ((Tdisk (p,r)) | F1)) \/ ([#] ((Tdisk (p,r)) | F3)) by A31, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
reconsider DT = [#] (Tdisk (p,r)) as closed Subset of (TOP-REAL 2) by BORSUK_1:def_11, TSEP_1:1;
DT /\ (Cl P) is closed ;
then A34: F1 is closed by TSEP_1:8;
P is_a_component_of C `
proof
take U ; :: according to CONNSP_1:def_6 ::_thesis: ( U = P & U is a_component )
thus ( U = P & U is a_component ) by A9, A10; ::_thesis: verum
end;
then P is open by SPRECT_3:8;
then DT /\ (P `) is closed ;
then A35: F3 is closed by TSEP_1:8;
A36: id ((Tdisk (p,r)) | F1) = id F1 by PRE_TOPC:8;
(Tdisk (p,r)) | F1 is SubSpace of TOP-REAL 2 by TSEP_1:7;
then (Tdisk (p,r)) | F1 is SubSpace of T2B1 by A21, A23, A26, TSEP_1:4;
then A37: f1 is continuous by A36, PRE_TOPC:26;
A38: (TOP-REAL 2) | A is SubSpace of T2B1 by A14, A23, A24, TSEP_1:4;
reconsider g2 = g1 as Function of ((Tdisk (p,r)) | F3),((TOP-REAL 2) | A) by A22, FUNCT_2:32;
g2 is continuous by A7, TOPMETR:7;
then A39: g1 is continuous by A38, PRE_TOPC:26;
A40: for x being set st x in Cl P & x in P ` holds
f . x = x
proof
let x be set ; ::_thesis: ( x in Cl P & x in P ` implies f . x = x )
assume that
A41: x in Cl P and
A42: x in P ` ; ::_thesis: f . x = x
A43: x in (Cl P) /\ (P `) by A41, A42, XBOOLE_0:def_4;
then (id A) . x = x by A5, FUNCT_1:18;
hence f . x = x by A5, A8, A43, FUNCT_1:49; ::_thesis: verum
end;
for x being set st x in ([#] ((Tdisk (p,r)) | F1)) /\ ([#] ((Tdisk (p,r)) | F3)) holds
f1 . x = g1 . x
proof
let x be set ; ::_thesis: ( x in ([#] ((Tdisk (p,r)) | F1)) /\ ([#] ((Tdisk (p,r)) | F3)) implies f1 . x = g1 . x )
assume A44: x in ([#] ((Tdisk (p,r)) | F1)) /\ ([#] ((Tdisk (p,r)) | F3)) ; ::_thesis: f1 . x = g1 . x
then A45: x in [#] ((Tdisk (p,r)) | F1) by XBOOLE_0:def_4;
then A46: x in Cl P by A30, XBOOLE_0:def_4;
x in P ` by A31, A44, XBOOLE_0:def_4;
then A47: f . x = x by A40, A46;
thus f1 . x = x by A30, A45, FUNCT_1:18
.= g1 . x by A31, A44, A47, FUNCT_1:49 ; ::_thesis: verum
end;
then consider g being Function of (Tdisk (p,r)),((TOP-REAL 2) | B) such that
A48: g = f1 +* g1 and
A49: g is continuous by A30, A31, A32, A34, A35, A37, A39, JGRAPH_2:1;
take g ; ::_thesis: ( g is continuous & ( for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) ) ) )
thus g is continuous by A49; ::_thesis: for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) )
let x be Point of (Tdisk (p,r)); ::_thesis: ( ( x in Cl P implies g . x = x ) & ( x in P ` implies g . x = f . x ) )
A50: dom g1 = the carrier of ((Tdisk (p,r)) | F3) by FUNCT_2:def_1;
hereby ::_thesis: ( x in P ` implies g . x = f . x )
assume A51: x in Cl P ; ::_thesis: g . x = x
then A52: x in F1 by XBOOLE_0:def_4;
percases ( not x in dom g1 or x in dom g1 ) ;
suppose not x in dom g1 ; ::_thesis: g . x = x
hence g . x = f1 . x by A48, FUNCT_4:11
.= x by A52, FUNCT_1:18 ;
::_thesis: verum
end;
supposeA53: x in dom g1 ; ::_thesis: g . x = x
then A54: x in P ` by A22, XBOOLE_0:def_4;
thus g . x = g1 . x by A48, A53, FUNCT_4:13
.= f . x by A22, A53, FUNCT_1:49
.= x by A40, A51, A54 ; ::_thesis: verum
end;
end;
end;
assume x in P ` ; ::_thesis: g . x = f . x
then A55: x in F3 by XBOOLE_0:def_4;
hence g . x = g1 . x by A22, A48, A50, FUNCT_4:13
.= f . x by A55, FUNCT_1:49 ;
::_thesis: verum
end;
Lm15: for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `)) st not BDD C is empty & U = P & U is a_component holds
C = Fr P
proof
let C be Simple_closed_curve; ::_thesis: for P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `)) st not BDD C is empty & U = P & U is a_component holds
C = Fr P
let P be Subset of (TOP-REAL 2); ::_thesis: for U being Subset of ((TOP-REAL 2) | (C `)) st not BDD C is empty & U = P & U is a_component holds
C = Fr P
let U be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: ( not BDD C is empty & U = P & U is a_component implies C = Fr P )
assume that
A1: not BDD C is empty and
A2: U = P and
A3: U is a_component and
A4: C <> Fr P ; ::_thesis: contradiction
A5: the carrier of ((TOP-REAL 2) | (C `)) = C ` by PRE_TOPC:8;
reconsider T2C = (TOP-REAL 2) | (C `) as non empty SubSpace of TOP-REAL 2 ;
A6: T2C is locally_connected by JORDAN2C:81;
then U is open by A3, CONNSP_2:15;
then reconsider P = P as open Subset of (TOP-REAL 2) by A2, TSEP_1:17;
A7: Fr P = (Cl P) /\ (P `) by PRE_TOPC:22;
set Z = { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ;
set V = union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ;
A8: ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C = the carrier of (TOP-REAL 2)
proof
A9: union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } c= the carrier of (TOP-REAL 2)
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } or a in the carrier of (TOP-REAL 2) )
assume a in union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ; ::_thesis: a in the carrier of (TOP-REAL 2)
then consider A being set such that
A10: a in A and
A11: A in { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } by TARSKI:def_4;
ex X being Subset of ((TOP-REAL 2) | (C `)) st
( X = A & X is a_component & X <> U ) by A11;
hence a in the carrier of (TOP-REAL 2) by A5, A10, TARSKI:def_3; ::_thesis: verum
end;
U c= the carrier of (TOP-REAL 2) by A5, XBOOLE_1:1;
then (union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U c= the carrier of (TOP-REAL 2) by A9, XBOOLE_1:8;
hence ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C c= the carrier of (TOP-REAL 2) by XBOOLE_1:8; :: according to XBOOLE_0:def_10 ::_thesis: the carrier of (TOP-REAL 2) c= ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in the carrier of (TOP-REAL 2) or a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C )
assume A12: a in the carrier of (TOP-REAL 2) ; ::_thesis: a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C
percases ( a in C or not a in C ) ;
suppose a in C ; ::_thesis: a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C
hence a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C by XBOOLE_0:def_3; ::_thesis: verum
end;
suppose not a in C ; ::_thesis: a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C
then reconsider a = a as Point of ((TOP-REAL 2) | (C `)) by A5, A12, SUBSET_1:29;
A13: a in Component_of a by CONNSP_1:38;
percases ( Component_of a = U or Component_of a <> U ) ;
suppose Component_of a = U ; ::_thesis: a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C
then a in (union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U by A13, XBOOLE_0:def_3;
hence a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C by XBOOLE_0:def_3; ::_thesis: verum
end;
supposeA14: Component_of a <> U ; ::_thesis: a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C
Component_of a is a_component by CONNSP_1:40;
then Component_of a in { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } by A14;
then a in union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } by A13, TARSKI:def_4;
then a in (union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U by XBOOLE_0:def_3;
hence a in ((union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U) \/ C by XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
end;
end;
A15: P misses P ` by XBOOLE_1:79;
Fr P c= C
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Fr P or a in C )
assume A16: a in Fr P ; ::_thesis: a in C
then A17: a in Cl P by XBOOLE_0:def_4;
A18: a in P ` by A7, A16, XBOOLE_0:def_4;
assume not a in C ; ::_thesis: contradiction
then a in (union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } ) \/ U by A8, A16, XBOOLE_0:def_3;
then ( a in union { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } or a in U ) by XBOOLE_0:def_3;
then consider O being set such that
A19: a in O and
A20: O in { X where X is Subset of ((TOP-REAL 2) | (C `)) : ( X is a_component & X <> U ) } by A2, A15, A18, TARSKI:def_4, XBOOLE_0:3;
consider X being Subset of ((TOP-REAL 2) | (C `)) such that
A21: X = O and
A22: X is a_component and
A23: X <> U by A20;
Cl P misses X by A2, A3, A22, A23, Th68;
hence contradiction by A17, A19, A21, XBOOLE_0:3; ::_thesis: verum
end;
then Fr P c< C by A4, XBOOLE_0:def_8;
then consider p1, p2 being Point of (TOP-REAL 2), A being Subset of (TOP-REAL 2) such that
A24: A is_an_arc_of p1,p2 and
A25: Fr P c= A and
A26: A c= C by BORSUK_4:59;
A27: U <> {} ((TOP-REAL 2) | (C `)) by A3, CONNSP_1:32;
percases ( P is bounded or not P is bounded ) ;
suppose P is bounded ; ::_thesis: contradiction
then reconsider P = P as bounded Subset of (TOP-REAL 2) ;
consider p being set such that
A28: p in U by A27, XBOOLE_0:def_1;
reconsider p = p as Point of (TOP-REAL 2) by A2, A28;
A29: P \/ C is bounded by TOPREAL6:67;
then reconsider PC = P \/ C as bounded Subset of (Euclid 2) by JORDAN2C:11;
consider r being positive real number such that
A30: PC c= Ball (p,r) by A29, Th26;
C c= PC by XBOOLE_1:7;
then A31: C c= Ball (p,r) by A30, XBOOLE_1:1;
set D = Tdisk (p,r);
set S = Tcircle (p,r);
set B = (cl_Ball (p,r)) \ {p};
A32: the carrier of (Tcircle (p,r)) = Sphere (p,r) by TOPREALB:9;
A33: the carrier of (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
A34: Sphere (p,r) c= cl_Ball (p,r) by TOPREAL9:17;
A35: Ball (p,r) misses Sphere (p,r) by TOPREAL9:19;
A36: Ball (p,r) c= cl_Ball (p,r) by TOPREAL9:16;
A c= Ball (p,r) by A26, A31, XBOOLE_1:1;
then A is Subset of (Tdisk (p,r)) by A33, A36, XBOOLE_1:1;
then consider R being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) such that
A37: R is continuous and
A38: R | A = id A by A24, Lm12;
P c= PC by XBOOLE_1:7;
then A39: P c= Ball (p,r) by A30, XBOOLE_1:1;
then consider f being Function of (Tdisk (p,r)),((TOP-REAL 2) | ((cl_Ball (p,r)) \ {p})) such that
A40: f is continuous and
A41: for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies f . x = R . x ) & ( x in P ` implies f . x = x ) ) by A2, A3, A7, A24, A25, A26, A28, A31, A37, A38, Lm13;
set g = DiskProj (p,r,p);
set h = RotateCircle (p,r,p);
A42: Tcircle (p,r) is SubSpace of Tdisk (p,r) by A32, A33, A34, TSEP_1:4;
reconsider F = (RotateCircle (p,r,p)) * ((DiskProj (p,r,p)) * f) as Function of (Tdisk (p,r)),(Tdisk (p,r)) by A32, A33, A34, FUNCT_2:7;
p is Point of (Tdisk (p,r)) by Th17;
then A43: DiskProj (p,r,p) is continuous by Th64;
|.(p - p).| = 0 by TOPRNS_1:28;
then A44: p in Ball (p,r) by TOPREAL9:7;
then RotateCircle (p,r,p) is continuous by Th66;
then A45: F is continuous by A40, A42, A43, PRE_TOPC:26;
now__::_thesis:_for_x_being_set_holds_not_x_is_a_fixpoint_of_F
let x be set ; ::_thesis: not b1 is_a_fixpoint_of F
percases ( x in dom F or not x in dom F ) ;
supposeA46: x in dom F ; ::_thesis: not b1 is_a_fixpoint_of F
A47: (Ball (p,r)) \/ (Sphere (p,r)) = cl_Ball (p,r) by TOPREAL9:18;
now__::_thesis:_F_._x_<>_x
percases ( x in Ball (p,r) or x in Sphere (p,r) ) by A33, A46, A47, XBOOLE_0:def_3;
supposeA48: x in Ball (p,r) ; ::_thesis: F . x <> x
F . x in the carrier of (Tcircle (p,r)) by A46, FUNCT_2:5;
hence F . x <> x by A32, A35, A48, XBOOLE_0:3; ::_thesis: verum
end;
supposeA49: x in Sphere (p,r) ; ::_thesis: F . x <> x
A50: dom f = the carrier of (Tdisk (p,r)) by FUNCT_2:def_1;
not x in P by A35, A39, A49, XBOOLE_0:3;
then A51: x in P ` by A49, SUBSET_1:29;
A52: (DiskProj (p,r,p)) | (Sphere (p,r)) = id (Sphere (p,r)) by A44, Th65;
RotateCircle (p,r,p) is without_fixpoints by A44, Th67;
then A53: not x is_a_fixpoint_of RotateCircle (p,r,p) by ABIAN:def_5;
A54: dom (RotateCircle (p,r,p)) = the carrier of (Tcircle (p,r)) by FUNCT_2:def_1;
F . x = (RotateCircle (p,r,p)) . (((DiskProj (p,r,p)) * f) . x) by A46, FUNCT_1:12
.= (RotateCircle (p,r,p)) . ((DiskProj (p,r,p)) . (f . x)) by A33, A34, A49, A50, FUNCT_1:13
.= (RotateCircle (p,r,p)) . ((DiskProj (p,r,p)) . x) by A33, A34, A41, A49, A51
.= (RotateCircle (p,r,p)) . ((id (Sphere (p,r))) . x) by A49, A52, FUNCT_1:49
.= (RotateCircle (p,r,p)) . x by A49, FUNCT_1:18 ;
hence F . x <> x by A32, A49, A53, A54, ABIAN:def_3; ::_thesis: verum
end;
end;
end;
hence not x is_a_fixpoint_of F by ABIAN:def_3; ::_thesis: verum
end;
suppose not x in dom F ; ::_thesis: not b1 is_a_fixpoint_of F
hence not x is_a_fixpoint_of F by ABIAN:def_3; ::_thesis: verum
end;
end;
end;
then not F is with_fixpoint by ABIAN:def_5;
hence contradiction by A45, BROUWER:14; ::_thesis: verum
end;
supposeA55: not P is bounded ; ::_thesis: contradiction
consider p being set such that
A56: p in BDD C by A1, XBOOLE_0:def_1;
consider Z being set such that
A57: p in Z and
A58: Z in { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } by A56, TARSKI:def_4;
consider P1 being Subset of (TOP-REAL 2) such that
A59: Z = P1 and
A60: P1 is_inside_component_of C by A58;
consider U1 being Subset of ((TOP-REAL 2) | (C `)) such that
A61: U1 = P1 and
A62: U1 is a_component and
U1 is bounded Subset of (Euclid 2) by A60, JORDAN2C:13;
U1 is open by A6, A62, CONNSP_2:15;
then reconsider P1 = P1 as non empty open bounded Subset of (TOP-REAL 2) by A57, A59, A60, A61, JORDAN2C:def_2, TSEP_1:17;
reconsider p = p as Point of (TOP-REAL 2) by A57, A59;
A63: p in P1 by A57, A59;
A64: P1 \/ C is bounded by TOPREAL6:67;
then reconsider PC = P1 \/ C as bounded Subset of (Euclid 2) by JORDAN2C:11;
consider rv being positive real number such that
A65: PC c= Ball (p,rv) by A64, Th26;
not P c= Ball (p,rv) by A55, RLTOPSP1:42;
then consider u being set such that
A66: u in P and
A67: not u in Ball (p,rv) by TARSKI:def_3;
reconsider u = u as Point of (TOP-REAL 2) by A66;
set r = |.(u - p).|;
P misses P1 by A2, A3, A55, A61, A62, CONNSP_1:35;
then p <> u by A57, A59, A66, XBOOLE_0:3;
then reconsider r = |.(u - p).| as non empty non negative real number by TOPRNS_1:28;
A68: r >= rv by A67, TOPREAL9:7;
then Ball (p,rv) c= Ball (p,r) by Th18;
then A69: PC c= Ball (p,r) by A65, XBOOLE_1:1;
A70: Fr (Ball (p,r)) = Sphere (p,r) by Th24;
u in Sphere (p,r) by TOPREAL9:9;
then A71: P meets Ball (p,r) by A66, A70, TOPS_1:28;
A72: C c= PC by XBOOLE_1:7;
then A73: C c= Ball (p,r) by A69, XBOOLE_1:1;
set D = Tdisk (p,r);
set S = Tcircle (p,r);
set B = (cl_Ball (p,r)) \ {p};
A74: the carrier of (Tcircle (p,r)) = Sphere (p,r) by TOPREALB:9;
A75: the carrier of (Tdisk (p,r)) = cl_Ball (p,r) by BROUWER:3;
A76: Sphere (p,r) c= cl_Ball (p,r) by TOPREAL9:17;
A77: Ball (p,r) misses Sphere (p,r) by TOPREAL9:19;
A78: Ball (p,r) c= cl_Ball (p,r) by TOPREAL9:16;
A c= Ball (p,r) by A26, A73, XBOOLE_1:1;
then A is Subset of (Tdisk (p,r)) by A75, A78, XBOOLE_1:1;
then consider R being Function of (Tdisk (p,r)),((TOP-REAL 2) | A) such that
A79: R is continuous and
A80: R | A = id A by A24, Lm12;
p1 in A by A24, TOPREAL1:1;
then consider f being Function of (Tdisk (p,r)),((TOP-REAL 2) | ((cl_Ball (p,r)) \ {p})) such that
A81: f is continuous and
A82: for x being Point of (Tdisk (p,r)) holds
( ( x in Cl P implies f . x = x ) & ( x in P ` implies f . x = R . x ) ) by A2, A3, A7, A25, A26, A55, A61, A62, A63, A71, A73, A79, A80, Lm14;
set g = DiskProj (p,r,p);
set h = RotateCircle (p,r,p);
A83: Tcircle (p,r) is SubSpace of Tdisk (p,r) by A74, A75, A76, TSEP_1:4;
reconsider F = (RotateCircle (p,r,p)) * ((DiskProj (p,r,p)) * f) as Function of (Tdisk (p,r)),(Tdisk (p,r)) by A74, A75, A76, FUNCT_2:7;
p is Point of (Tdisk (p,r)) by Th17;
then A84: DiskProj (p,r,p) is continuous by Th64;
|.(p - p).| = 0 by TOPRNS_1:28;
then A85: p in Ball (p,r) by TOPREAL9:7;
then RotateCircle (p,r,p) is continuous by Th66;
then A86: F is continuous by A81, A83, A84, PRE_TOPC:26;
now__::_thesis:_for_x_being_set_holds_not_x_is_a_fixpoint_of_F
let x be set ; ::_thesis: not b1 is_a_fixpoint_of F
percases ( x in dom F or not x in dom F ) ;
supposeA87: x in dom F ; ::_thesis: not b1 is_a_fixpoint_of F
A88: (Ball (p,r)) \/ (Sphere (p,r)) = cl_Ball (p,r) by TOPREAL9:18;
now__::_thesis:_F_._x_<>_x
percases ( x in Ball (p,r) or x in Sphere (p,r) ) by A75, A87, A88, XBOOLE_0:def_3;
supposeA89: x in Ball (p,r) ; ::_thesis: F . x <> x
F . x in the carrier of (Tcircle (p,r)) by A87, FUNCT_2:5;
hence F . x <> x by A74, A77, A89, XBOOLE_0:3; ::_thesis: verum
end;
supposeA90: x in Sphere (p,r) ; ::_thesis: F . x <> x
A91: dom f = the carrier of (Tdisk (p,r)) by FUNCT_2:def_1;
A92: P c= Cl P by PRE_TOPC:18;
set SS = Sphere (p,r);
Sphere (p,r) c= C `
proof
let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in Sphere (p,r) or a in C ` )
assume A93: a in Sphere (p,r) ; ::_thesis: a in C `
assume not a in C ` ; ::_thesis: contradiction
then A94: a in C by A93, SUBSET_1:29;
reconsider a = a as Point of (TOP-REAL 2) by A93;
a in PC by A72, A94;
then |.(a - p).| < rv by A65, TOPREAL9:7;
hence contradiction by A68, A93, TOPREAL9:9; ::_thesis: verum
end;
then reconsider SS = Sphere (p,r) as Subset of ((TOP-REAL 2) | (C `)) by PRE_TOPC:8;
A95: u in SS by TOPREAL9:9;
SS is connected by CONNSP_1:23;
then ( SS misses U or SS c= U ) by A3, CONNSP_1:36;
then A96: x in P by A2, A66, A90, A95, XBOOLE_0:3;
A97: (DiskProj (p,r,p)) | (Sphere (p,r)) = id (Sphere (p,r)) by A85, Th65;
RotateCircle (p,r,p) is without_fixpoints by A85, Th67;
then A98: not x is_a_fixpoint_of RotateCircle (p,r,p) by ABIAN:def_5;
A99: dom (RotateCircle (p,r,p)) = the carrier of (Tcircle (p,r)) by FUNCT_2:def_1;
F . x = (RotateCircle (p,r,p)) . (((DiskProj (p,r,p)) * f) . x) by A87, FUNCT_1:12
.= (RotateCircle (p,r,p)) . ((DiskProj (p,r,p)) . (f . x)) by A75, A76, A90, A91, FUNCT_1:13
.= (RotateCircle (p,r,p)) . ((DiskProj (p,r,p)) . x) by A75, A76, A82, A90, A92, A96
.= (RotateCircle (p,r,p)) . ((id (Sphere (p,r))) . x) by A90, A97, FUNCT_1:49
.= (RotateCircle (p,r,p)) . x by A90, FUNCT_1:18 ;
hence F . x <> x by A74, A90, A98, A99, ABIAN:def_3; ::_thesis: verum
end;
end;
end;
hence not x is_a_fixpoint_of F by ABIAN:def_3; ::_thesis: verum
end;
suppose not x in dom F ; ::_thesis: not b1 is_a_fixpoint_of F
hence not x is_a_fixpoint_of F by ABIAN:def_3; ::_thesis: verum
end;
end;
end;
then F is without_fixpoints by ABIAN:def_5;
hence contradiction by A86, BROUWER:14; ::_thesis: verum
end;
end;
end;
set rp = 1;
set rl = - 1;
set rg = 3;
set rd = - 3;
set a = |[(- 1),0]|;
set b = |[1,0]|;
set c = |[0,3]|;
set d = |[0,(- 3)]|;
set lg = |[(- 1),3]|;
set pg = |[1,3]|;
set ld = |[(- 1),(- 3)]|;
set pd = |[1,(- 3)]|;
set R = closed_inside_of_rectangle ((- 1),1,(- 3),3);
set dR = rectangle ((- 1),1,(- 3),3);
set TR = Trectangle ((- 1),1,(- 3),3);
Lm16: |[(- 1),0]| `1 = - 1
by EUCLID:52;
Lm17: |[1,0]| `1 = 1
by EUCLID:52;
Lm18: |[(- 1),0]| `2 = 0
by EUCLID:52;
Lm19: |[1,0]| `2 = 0
by EUCLID:52;
Lm20: |[0,3]| `1 = 0
by EUCLID:52;
Lm21: |[0,3]| `2 = 3
by EUCLID:52;
Lm22: |[0,(- 3)]| `1 = 0
by EUCLID:52;
Lm23: |[0,(- 3)]| `2 = - 3
by EUCLID:52;
Lm24: |[(- 1),3]| `1 = - 1
by EUCLID:52;
Lm25: |[(- 1),3]| `2 = 3
by EUCLID:52;
Lm26: |[(- 1),(- 3)]| `1 = - 1
by EUCLID:52;
Lm27: |[(- 1),(- 3)]| `2 = - 3
by EUCLID:52;
Lm28: |[1,3]| `1 = 1
by EUCLID:52;
Lm29: |[1,3]| `2 = 3
by EUCLID:52;
Lm30: |[1,(- 3)]| `1 = 1
by EUCLID:52;
Lm31: |[1,(- 3)]| `2 = - 3
by EUCLID:52;
Lm32: |[(- 1),(- 3)]| = |[(|[(- 1),(- 3)]| `1),(|[(- 1),(- 3)]| `2)]|
by EUCLID:53;
Lm33: |[(- 1),3]| = |[(|[(- 1),3]| `1),(|[(- 1),3]| `2)]|
by EUCLID:53;
Lm34: |[1,(- 3)]| = |[(|[1,(- 3)]| `1),(|[1,(- 3)]| `2)]|
by EUCLID:53;
Lm35: |[1,3]| = |[(|[1,3]| `1),(|[1,3]| `2)]|
by EUCLID:53;
Lm36: rectangle ((- 1),1,(- 3),3) = ((LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)) \/ (LSeg (|[(- 1),3]|,|[1,3]|))) \/ ((LSeg (|[1,3]|,|[1,(- 3)]|)) \/ (LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|)))
by SPPOL_2:def_3;
Lm37: LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) c= (LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)) \/ (LSeg (|[(- 1),3]|,|[1,3]|))
by XBOOLE_1:7;
(LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)) \/ (LSeg (|[(- 1),3]|,|[1,3]|)) c= rectangle ((- 1),1,(- 3),3)
by Lm36, XBOOLE_1:7;
then Lm38: LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) c= rectangle ((- 1),1,(- 3),3)
by Lm37, XBOOLE_1:1;
Lm39: LSeg (|[(- 1),3]|,|[1,3]|) c= (LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)) \/ (LSeg (|[(- 1),3]|,|[1,3]|))
by XBOOLE_1:7;
(LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)) \/ (LSeg (|[(- 1),3]|,|[1,3]|)) c= rectangle ((- 1),1,(- 3),3)
by Lm36, XBOOLE_1:7;
then Lm40: LSeg (|[(- 1),3]|,|[1,3]|) c= rectangle ((- 1),1,(- 3),3)
by Lm39, XBOOLE_1:1;
Lm41: LSeg (|[1,3]|,|[1,(- 3)]|) c= (LSeg (|[1,3]|,|[1,(- 3)]|)) \/ (LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|))
by XBOOLE_1:7;
(LSeg (|[1,3]|,|[1,(- 3)]|)) \/ (LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|)) c= rectangle ((- 1),1,(- 3),3)
by Lm36, XBOOLE_1:7;
then Lm42: LSeg (|[1,3]|,|[1,(- 3)]|) c= rectangle ((- 1),1,(- 3),3)
by Lm41, XBOOLE_1:1;
Lm43: LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|) c= (LSeg (|[1,3]|,|[1,(- 3)]|)) \/ (LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|))
by XBOOLE_1:7;
(LSeg (|[1,3]|,|[1,(- 3)]|)) \/ (LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|)) c= rectangle ((- 1),1,(- 3),3)
by Lm36, XBOOLE_1:7;
then Lm44: LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|) c= rectangle ((- 1),1,(- 3),3)
by Lm43, XBOOLE_1:1;
Lm45: LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) is vertical
by Lm24, Lm26, SPPOL_1:16;
Lm46: LSeg (|[1,(- 3)]|,|[1,3]|) is vertical
by Lm28, Lm30, SPPOL_1:16;
Lm47: LSeg (|[(- 1),0]|,|[(- 1),3]|) is vertical
by Lm16, Lm24, SPPOL_1:16;
Lm48: LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) is vertical
by Lm16, Lm26, SPPOL_1:16;
Lm49: LSeg (|[1,0]|,|[1,3]|) is vertical
by Lm17, Lm28, SPPOL_1:16;
Lm50: LSeg (|[1,0]|,|[1,(- 3)]|) is vertical
by Lm17, Lm30, SPPOL_1:16;
Lm51: LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) is horizontal
by Lm23, Lm27, SPPOL_1:15;
Lm52: LSeg (|[1,(- 3)]|,|[0,(- 3)]|) is horizontal
by Lm23, Lm31, SPPOL_1:15;
Lm53: LSeg (|[(- 1),3]|,|[0,3]|) is horizontal
by Lm21, Lm25, SPPOL_1:15;
Lm54: LSeg (|[1,3]|,|[0,3]|) is horizontal
by Lm21, Lm29, SPPOL_1:15;
Lm55: LSeg (|[(- 1),3]|,|[1,3]|) is horizontal
by Lm25, Lm29, SPPOL_1:15;
Lm56: LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) is horizontal
by Lm27, Lm31, SPPOL_1:15;
Lm57: LSeg (|[(- 1),0]|,|[(- 1),3]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)
by Lm16, Lm18, Lm25, Lm26, Lm27, Lm45, Lm47, GOBOARD7:63;
Lm58: LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)
by Lm18, Lm25, Lm26, Lm27, Lm45, Lm48, GOBOARD7:63;
Lm59: LSeg (|[1,0]|,|[1,3]|) c= LSeg (|[1,(- 3)]|,|[1,3]|)
by Lm17, Lm19, Lm29, Lm30, Lm31, Lm46, Lm49, GOBOARD7:63;
Lm60: LSeg (|[1,0]|,|[1,(- 3)]|) c= LSeg (|[1,(- 3)]|,|[1,3]|)
by Lm19, Lm29, Lm30, Lm31, Lm46, Lm50, GOBOARD7:63;
Lm61: rectangle ((- 1),1,(- 3),3) = { p where p is Point of (TOP-REAL 2) : ( ( p `1 = - 1 & p `2 <= 3 & p `2 >= - 3 ) or ( p `1 <= 1 & p `1 >= - 1 & p `2 = 3 ) or ( p `1 <= 1 & p `1 >= - 1 & p `2 = - 3 ) or ( p `1 = 1 & p `2 <= 3 & p `2 >= - 3 ) ) }
by SPPOL_2:54;
then Lm62: |[0,3]| in rectangle ((- 1),1,(- 3),3)
by Lm20, Lm21;
Lm63: |[0,(- 3)]| in rectangle ((- 1),1,(- 3),3)
by Lm22, Lm23, Lm61;
Lm64: (2 + 1) ^2 = (4 + 4) + 1
;
then Lm65: sqrt 9 = 3
by SQUARE_1:def_2;
Lm66: dist (|[(- 1),0]|,|[1,0]|) = sqrt ((((|[(- 1),0]| `1) - (|[1,0]| `1)) ^2) + (((|[(- 1),0]| `2) - (|[1,0]| `2)) ^2)) by TOPREAL6:92
.= - (- 2) by Lm16, Lm17, Lm18, Lm19, SQUARE_1:23 ;
theorem Th70: :: JORDAN:70
for C being Simple_closed_curve
for h being Homeomorphism of TOP-REAL 2 holds h .: C is being_simple_closed_curve
proof
let C be Simple_closed_curve; ::_thesis: for h being Homeomorphism of TOP-REAL 2 holds h .: C is being_simple_closed_curve
let h be Homeomorphism of TOP-REAL 2; ::_thesis: h .: C is being_simple_closed_curve
consider f being Function of ((TOP-REAL 2) | R^2-unit_square),((TOP-REAL 2) | C) such that
A1: f is being_homeomorphism by TOPREAL2:def_1;
reconsider g = h | C as Function of ((TOP-REAL 2) | C),((TOP-REAL 2) | (h .: C)) by JORDAN24:12;
take g * f ; :: according to TOPREAL2:def_1 ::_thesis: g * f is being_homeomorphism
g is being_homeomorphism by JORDAN24:14;
hence g * f is being_homeomorphism by A1, TOPS_2:57; ::_thesis: verum
end;
theorem Th71: :: JORDAN:71
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
P c= closed_inside_of_rectangle ((- 1),1,(- 3),3)
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) )
assume that
A1: |[(- 1),0]| in P and
A2: |[1,0]| in P and
A3: for x, y being Point of (TOP-REAL 2) st x in P & y in P holds
dist (|[(- 1),0]|,|[1,0]|) >= dist (x,y) ; :: according to JORDAN24:def_1 ::_thesis: P c= closed_inside_of_rectangle ((- 1),1,(- 3),3)
let p be set ; :: according to TARSKI:def_3 ::_thesis: ( not p in P or p in closed_inside_of_rectangle ((- 1),1,(- 3),3) )
assume A4: p in P ; ::_thesis: p in closed_inside_of_rectangle ((- 1),1,(- 3),3)
then reconsider p = p as Point of (TOP-REAL 2) ;
A5: dist (|[(- 1),0]|,p) = sqrt ((((- 1) - (p `1)) ^2) + ((0 - (p `2)) ^2)) by Lm16, Lm18, TOPREAL6:92
.= sqrt ((((- 1) - (p `1)) ^2) + ((p `2) ^2)) ;
A6: now__::_thesis:_not_9_<_(p_`2)_^2
assume 9 < (p `2) ^2 ; ::_thesis: contradiction
then 0 + 9 < (((- 1) - (p `1)) ^2) + ((p `2) ^2) by XREAL_1:8;
then 3 < sqrt ((((- 1) - (p `1)) ^2) + ((p `2) ^2)) by Lm65, SQUARE_1:27;
then 2 < sqrt ((((- 1) - (p `1)) ^2) + ((p `2) ^2)) by XXREAL_0:2;
hence contradiction by A1, A3, A4, A5, Lm66; ::_thesis: verum
end;
A7: now__::_thesis:_not_-_1_>_p_`1
assume A8: - 1 > p `1 ; ::_thesis: contradiction
then LSeg (p,|[1,0]|) meets Vertical_Line (- 1) by Lm17, Th8;
then consider x being set such that
A9: x in LSeg (p,|[1,0]|) and
A10: x in Vertical_Line (- 1) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A9;
A11: x `1 = - 1 by A10, JORDAN6:31;
A12: dist (p,|[1,0]|) = (dist (p,x)) + (dist (x,|[1,0]|)) by A9, JORDAN1K:29;
A13: dist (x,|[1,0]|) = sqrt ((((x `1) - (|[1,0]| `1)) ^2) + (((x `2) - (|[1,0]| `2)) ^2)) by TOPREAL6:92
.= sqrt (((- 2) ^2) + (((x `2) - 0) ^2)) by A11, Lm17, EUCLID:52
.= sqrt (4 + ((x `2) ^2)) ;
now__::_thesis:_not_dist_(x,|[1,0]|)_<_dist_(|[(-_1),0]|,|[1,0]|)
assume dist (x,|[1,0]|) < dist (|[(- 1),0]|,|[1,0]|) ; ::_thesis: contradiction
then 4 + ((x `2) ^2) < 4 + 0 by A13, Lm66, SQUARE_1:20, SQUARE_1:26;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
then (dist (p,|[1,0]|)) + 0 > (dist (|[(- 1),0]|,|[1,0]|)) + 0 by A8, A11, A12, JORDAN1K:22, XREAL_1:8;
hence contradiction by A2, A3, A4; ::_thesis: verum
end;
A14: now__::_thesis:_not_p_`1_>_1
assume A15: p `1 > 1 ; ::_thesis: contradiction
then LSeg (p,|[(- 1),0]|) meets Vertical_Line 1 by Lm16, Th8;
then consider x being set such that
A16: x in LSeg (p,|[(- 1),0]|) and
A17: x in Vertical_Line 1 by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A16;
A18: x `1 = 1 by A17, JORDAN6:31;
A19: dist (p,|[(- 1),0]|) = (dist (p,x)) + (dist (x,|[(- 1),0]|)) by A16, JORDAN1K:29;
A20: dist (x,|[(- 1),0]|) = sqrt ((((x `1) - (|[(- 1),0]| `1)) ^2) + (((x `2) - (|[(- 1),0]| `2)) ^2)) by TOPREAL6:92
.= sqrt (4 + ((x `2) ^2)) by A18, Lm16, Lm18 ;
now__::_thesis:_not_dist_(x,|[(-_1),0]|)_<_dist_(|[(-_1),0]|,|[1,0]|)
assume dist (x,|[(- 1),0]|) < dist (|[(- 1),0]|,|[1,0]|) ; ::_thesis: contradiction
then 4 + ((x `2) ^2) < 4 + 0 by A20, Lm66, SQUARE_1:20, SQUARE_1:26;
hence contradiction by XREAL_1:6; ::_thesis: verum
end;
then (dist (p,|[(- 1),0]|)) + 0 > (dist (|[(- 1),0]|,|[1,0]|)) + 0 by A15, A18, A19, JORDAN1K:22, XREAL_1:8;
hence contradiction by A1, A3, A4; ::_thesis: verum
end;
A21: now__::_thesis:_not_-_3_>_p_`2
assume - 3 > p `2 ; ::_thesis: contradiction
then (p `2) ^2 > (- 3) ^2 by SQUARE_1:44;
hence contradiction by A6; ::_thesis: verum
end;
3 >= p `2 by A6, Lm64, SQUARE_1:16;
hence p in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A7, A14, A21; ::_thesis: verum
end;
Lm67: rectangle ((- 1),1,(- 3),3) c= closed_inside_of_rectangle ((- 1),1,(- 3),3)
by Th45;
Lm68: |[(- 1),3]| `2 = |[(- 1),3]| `2
;
Lm69: |[(- 1),3]| `1 <= |[0,3]| `1
by Lm24, EUCLID:52;
|[0,3]| `1 <= |[1,3]| `1
by Lm28, EUCLID:52;
then LSeg (|[(- 1),3]|,|[0,3]|) c= LSeg (|[(- 1),3]|,|[1,3]|)
by Lm53, Lm55, Lm68, Lm69, GOBOARD7:64;
then Lm70: LSeg (|[(- 1),3]|,|[0,3]|) c= rectangle ((- 1),1,(- 3),3)
by Lm40, XBOOLE_1:1;
LSeg (|[1,3]|,|[0,3]|) c= LSeg (|[(- 1),3]|,|[1,3]|)
by Lm20, Lm21, Lm24, Lm25, Lm28, Lm54, Lm55, GOBOARD7:64;
then Lm71: LSeg (|[1,3]|,|[0,3]|) c= rectangle ((- 1),1,(- 3),3)
by Lm40, XBOOLE_1:1;
Lm72: |[(- 1),(- 3)]| `2 = |[(- 1),(- 3)]| `2
;
Lm73: |[(- 1),(- 3)]| `1 <= |[0,(- 3)]| `1
by Lm26, EUCLID:52;
|[0,(- 3)]| `1 <= |[1,(- 3)]| `1
by Lm30, EUCLID:52;
then LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|)
by Lm51, Lm56, Lm72, Lm73, GOBOARD7:64;
then Lm74: LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) c= rectangle ((- 1),1,(- 3),3)
by Lm44, XBOOLE_1:1;
LSeg (|[1,(- 3)]|,|[0,(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|)
by Lm22, Lm23, Lm26, Lm27, Lm30, Lm52, Lm56, GOBOARD7:64;
then Lm75: LSeg (|[1,(- 3)]|,|[0,(- 3)]|) c= rectangle ((- 1),1,(- 3),3)
by Lm44, XBOOLE_1:1;
Lm76: for p being Point of (TOP-REAL 2) st 0 <= p `2 & p in rectangle ((- 1),1,(- 3),3) & not p in LSeg (|[(- 1),0]|,|[(- 1),3]|) & not p in LSeg (|[(- 1),3]|,|[0,3]|) & not p in LSeg (|[0,3]|,|[1,3]|) holds
p in LSeg (|[1,3]|,|[1,0]|)
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( 0 <= p `2 & p in rectangle ((- 1),1,(- 3),3) & not p in LSeg (|[(- 1),0]|,|[(- 1),3]|) & not p in LSeg (|[(- 1),3]|,|[0,3]|) & not p in LSeg (|[0,3]|,|[1,3]|) implies p in LSeg (|[1,3]|,|[1,0]|) )
assume A1: 0 <= p `2 ; ::_thesis: ( not p in rectangle ((- 1),1,(- 3),3) or p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
assume p in rectangle ((- 1),1,(- 3),3) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
then consider p1 being Point of (TOP-REAL 2) such that
A2: p1 = p and
A3: ( ( p1 `1 = - 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = - 3 ) or ( p1 `1 = 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ) by Lm61;
percases ( ( p1 `1 = - 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = - 3 ) or ( p1 `1 = 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ) by A3;
suppose ( p1 `1 = - 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) ) by A1, A2, Lm16, Lm18, Lm24, Lm25, GOBOARD7:7; ::_thesis: verum
end;
supposeA4: ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
percases ( p1 `1 <= |[0,3]| `1 or |[0,3]| `1 <= p1 `1 ) ;
suppose p1 `1 <= |[0,3]| `1 ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) ) by A2, A4, Lm21, Lm24, Lm25, GOBOARD7:8; ::_thesis: verum
end;
suppose |[0,3]| `1 <= p1 `1 ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) ) by A2, A4, Lm21, Lm28, Lm29, GOBOARD7:8; ::_thesis: verum
end;
end;
end;
suppose ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = - 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) ) by A1, A2; ::_thesis: verum
end;
suppose ( p1 `1 = 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),3]|) or p in LSeg (|[(- 1),3]|,|[0,3]|) or p in LSeg (|[0,3]|,|[1,3]|) or p in LSeg (|[1,3]|,|[1,0]|) ) by A1, A2, Lm17, Lm19, Lm28, Lm29, GOBOARD7:7; ::_thesis: verum
end;
end;
end;
Lm77: for p being Point of (TOP-REAL 2) st p `2 <= 0 & p in rectangle ((- 1),1,(- 3),3) & not p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) & not p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) & not p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) holds
p in LSeg (|[1,(- 3)]|,|[1,0]|)
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p `2 <= 0 & p in rectangle ((- 1),1,(- 3),3) & not p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) & not p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) & not p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) implies p in LSeg (|[1,(- 3)]|,|[1,0]|) )
assume A1: p `2 <= 0 ; ::_thesis: ( not p in rectangle ((- 1),1,(- 3),3) or p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
assume p in rectangle ((- 1),1,(- 3),3) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
then consider p1 being Point of (TOP-REAL 2) such that
A2: p1 = p and
A3: ( ( p1 `1 = - 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = - 3 ) or ( p1 `1 = 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ) by Lm61;
percases ( ( p1 `1 = - 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = 3 ) or ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = - 3 ) or ( p1 `1 = 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ) by A3;
suppose ( p1 `1 = - 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) ) by A1, A2, Lm16, Lm18, Lm26, Lm27, GOBOARD7:7; ::_thesis: verum
end;
suppose ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) ) by A1, A2; ::_thesis: verum
end;
supposeA4: ( p1 `1 <= 1 & p1 `1 >= - 1 & p1 `2 = - 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
percases ( p1 `1 <= |[0,(- 3)]| `1 or |[0,(- 3)]| `1 <= p1 `1 ) ;
suppose p1 `1 <= |[0,(- 3)]| `1 ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) ) by A2, A4, Lm23, Lm26, Lm27, GOBOARD7:8; ::_thesis: verum
end;
suppose |[0,(- 3)]| `1 <= p1 `1 ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) ) by A2, A4, Lm23, Lm30, Lm31, GOBOARD7:8; ::_thesis: verum
end;
end;
end;
suppose ( p1 `1 = 1 & p1 `2 <= 3 & p1 `2 >= - 3 ) ; ::_thesis: ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) )
hence ( p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or p in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or p in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or p in LSeg (|[1,(- 3)]|,|[1,0]|) ) by A1, A2, Lm17, Lm19, Lm30, Lm31, GOBOARD7:7; ::_thesis: verum
end;
end;
end;
theorem Th72: :: JORDAN:72
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
P misses LSeg (|[(- 1),3]|,|[1,3]|)
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies P misses LSeg (|[(- 1),3]|,|[1,3]|) )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: P misses LSeg (|[(- 1),3]|,|[1,3]|)
assume P meets LSeg (|[(- 1),3]|,|[1,3]|) ; ::_thesis: contradiction
then consider x being set such that
A2: x in P and
A3: x in LSeg (|[(- 1),3]|,|[1,3]|) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A2;
|[(- 1),3]| in LSeg (|[(- 1),3]|,|[1,3]|) by RLTOPSP1:68;
then A4: x `2 = 3 by A3, Lm25, Lm55, SPPOL_1:def_2;
A5: |[(- 1),0]| in P by A1, JORDAN24:def_1;
A6: dist (|[(- 1),0]|,x) = sqrt ((((|[(- 1),0]| `1) - (x `1)) ^2) + (((|[(- 1),0]| `2) - (x `2)) ^2)) by TOPREAL6:92
.= sqrt ((((- 1) - (x `1)) ^2) + (3 ^2)) by A4, Lm18, EUCLID:52 ;
0 + 4 < (((- 1) - (x `1)) ^2) + 9 by XREAL_1:8;
then 2 < dist (|[(- 1),0]|,x) by A6, SQUARE_1:20, SQUARE_1:27;
hence contradiction by A1, A2, A5, Lm66, JORDAN24:def_1; ::_thesis: verum
end;
theorem Th73: :: JORDAN:73
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
P misses LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|)
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies P misses LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: P misses LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|)
assume P meets LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) ; ::_thesis: contradiction
then consider x being set such that
A2: x in P and
A3: x in LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A2;
|[(- 1),(- 3)]| in LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) by RLTOPSP1:68;
then A4: x `2 = - 3 by A3, Lm27, Lm56, SPPOL_1:def_2;
A5: |[(- 1),0]| in P by A1, JORDAN24:def_1;
A6: dist (|[(- 1),0]|,x) = sqrt ((((|[(- 1),0]| `1) - (x `1)) ^2) + (((|[(- 1),0]| `2) - (x `2)) ^2)) by TOPREAL6:92
.= sqrt ((((- 1) - (x `1)) ^2) + ((- (- 3)) ^2)) by A4, Lm18, EUCLID:52 ;
0 + 4 < (((- 1) - (x `1)) ^2) + 9 by XREAL_1:8;
then 2 < dist (|[(- 1),0]|,x) by A6, SQUARE_1:20, SQUARE_1:27;
hence contradiction by A1, A2, A5, Lm66, JORDAN24:def_1; ::_thesis: verum
end;
theorem Th74: :: JORDAN:74
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
P /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|}
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies P /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: P /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|}
then A2: |[(- 1),0]| in P by JORDAN24:def_1;
A3: |[1,0]| in P by A1, JORDAN24:def_1;
thus P /\ (rectangle ((- 1),1,(- 3),3)) c= {|[(- 1),0]|,|[1,0]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[(- 1),0]|,|[1,0]|} c= P /\ (rectangle ((- 1),1,(- 3),3))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in P /\ (rectangle ((- 1),1,(- 3),3)) or x in {|[(- 1),0]|,|[1,0]|} )
assume A4: x in P /\ (rectangle ((- 1),1,(- 3),3)) ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then A5: x in P by XBOOLE_0:def_4;
x in rectangle ((- 1),1,(- 3),3) by A4, XBOOLE_0:def_4;
then A6: ( x in (LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|)) \/ (LSeg (|[(- 1),3]|,|[1,3]|)) or x in (LSeg (|[1,3]|,|[1,(- 3)]|)) \/ (LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|)) ) by Lm36, XBOOLE_0:def_3;
reconsider x = x as Point of (TOP-REAL 2) by A4;
percases ( x in LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) or x in LSeg (|[(- 1),3]|,|[1,3]|) or x in LSeg (|[1,3]|,|[1,(- 3)]|) or x in LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|) ) by A6, XBOOLE_0:def_3;
supposeA7: x in LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
|[(- 1),(- 3)]| in LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) by RLTOPSP1:68;
then A8: x `1 = - 1 by A7, Lm26, Lm45, SPPOL_1:def_3;
percases ( x `2 = 0 or x `2 <> 0 ) ;
suppose x `2 = 0 ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then x = |[(- 1),0]| by A8, Lm16, Lm18, TOPREAL3:6;
hence x in {|[(- 1),0]|,|[1,0]|} by TARSKI:def_2; ::_thesis: verum
end;
suppose x `2 <> 0 ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then A9: (x `2) ^2 > 0 by SQUARE_1:12;
A10: dist (|[1,0]|,x) = sqrt (((1 - (- 1)) ^2) + ((0 - (x `2)) ^2)) by A8, Lm17, Lm19, TOPREAL6:92
.= sqrt (4 + ((x `2) ^2)) ;
0 + 4 < ((x `2) ^2) + 4 by A9, XREAL_1:6;
then 2 < sqrt (((x `2) ^2) + 4) by SQUARE_1:20, SQUARE_1:27;
hence x in {|[(- 1),0]|,|[1,0]|} by A1, A3, A5, A10, Lm66, JORDAN24:def_1; ::_thesis: verum
end;
end;
end;
suppose x in LSeg (|[(- 1),3]|,|[1,3]|) ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then LSeg (|[(- 1),3]|,|[1,3]|) meets P by A5, XBOOLE_0:3;
hence x in {|[(- 1),0]|,|[1,0]|} by A1, Th72; ::_thesis: verum
end;
supposeA11: x in LSeg (|[1,3]|,|[1,(- 3)]|) ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
|[1,(- 3)]| in LSeg (|[1,(- 3)]|,|[1,3]|) by RLTOPSP1:68;
then A12: x `1 = 1 by A11, Lm30, Lm46, SPPOL_1:def_3;
percases ( x `2 = 0 or x `2 <> 0 ) ;
suppose x `2 = 0 ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then x = |[1,0]| by A12, Lm17, Lm19, TOPREAL3:6;
hence x in {|[(- 1),0]|,|[1,0]|} by TARSKI:def_2; ::_thesis: verum
end;
suppose x `2 <> 0 ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then A13: (x `2) ^2 > 0 by SQUARE_1:12;
A14: dist (x,|[(- 1),0]|) = sqrt ((((x `1) - (|[(- 1),0]| `1)) ^2) + (((x `2) - (|[(- 1),0]| `2)) ^2)) by TOPREAL6:92
.= sqrt (4 + ((x `2) ^2)) by A12, Lm16, Lm18 ;
0 + 4 < ((x `2) ^2) + 4 by A13, XREAL_1:6;
then 2 < sqrt (((x `2) ^2) + 4) by SQUARE_1:20, SQUARE_1:27;
hence x in {|[(- 1),0]|,|[1,0]|} by A1, A2, A5, A14, Lm66, JORDAN24:def_1; ::_thesis: verum
end;
end;
end;
suppose x in LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|) ; ::_thesis: x in {|[(- 1),0]|,|[1,0]|}
then LSeg (|[1,(- 3)]|,|[(- 1),(- 3)]|) meets P by A5, XBOOLE_0:3;
hence x in {|[(- 1),0]|,|[1,0]|} by A1, Th73; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[(- 1),0]|,|[1,0]|} or x in P /\ (rectangle ((- 1),1,(- 3),3)) )
assume x in {|[(- 1),0]|,|[1,0]|} ; ::_thesis: x in P /\ (rectangle ((- 1),1,(- 3),3))
then A15: ( x = |[(- 1),0]| or x = |[1,0]| ) by TARSKI:def_2;
A16: |[(- 1),0]| in rectangle ((- 1),1,(- 3),3) by Lm16, Lm18, Lm61;
|[1,0]| in rectangle ((- 1),1,(- 3),3) by Lm17, Lm19, Lm61;
hence x in P /\ (rectangle ((- 1),1,(- 3),3)) by A2, A3, A15, A16, XBOOLE_0:def_4; ::_thesis: verum
end;
Lm78: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
LSeg (|[(- 1),3]|,|[0,3]|) misses C
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies LSeg (|[(- 1),3]|,|[0,3]|) misses C )
assume |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: LSeg (|[(- 1),3]|,|[0,3]|) misses C
then A1: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by Th74;
assume LSeg (|[(- 1),3]|,|[0,3]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A2: q in LSeg (|[(- 1),3]|,|[0,3]|) and
A3: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A3;
q in (rectangle ((- 1),1,(- 3),3)) /\ C by A2, A3, Lm70, XBOOLE_0:def_4;
then ( q = |[(- 1),0]| or q = |[1,0]| ) by A1, TARSKI:def_2;
hence contradiction by A2, Lm18, Lm19, TOPREAL3:12; ::_thesis: verum
end;
Lm79: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
LSeg (|[1,3]|,|[0,3]|) misses C
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies LSeg (|[1,3]|,|[0,3]|) misses C )
assume |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: LSeg (|[1,3]|,|[0,3]|) misses C
then A1: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by Th74;
assume LSeg (|[1,3]|,|[0,3]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A2: q in LSeg (|[1,3]|,|[0,3]|) and
A3: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A3;
q in (rectangle ((- 1),1,(- 3),3)) /\ C by A2, A3, Lm71, XBOOLE_0:def_4;
then ( q = |[(- 1),0]| or q = |[1,0]| ) by A1, TARSKI:def_2;
hence contradiction by A2, Lm18, Lm19, TOPREAL3:12; ::_thesis: verum
end;
Lm80: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) misses C
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) misses C )
assume |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) misses C
then A1: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by Th74;
assume LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A2: q in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) and
A3: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A3;
q in (rectangle ((- 1),1,(- 3),3)) /\ C by A2, A3, Lm74, XBOOLE_0:def_4;
then ( q = |[(- 1),0]| or q = |[1,0]| ) by A1, TARSKI:def_2;
hence contradiction by A2, Lm18, Lm19, TOPREAL3:12; ::_thesis: verum
end;
Lm81: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
LSeg (|[1,(- 3)]|,|[0,(- 3)]|) misses C
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies LSeg (|[1,(- 3)]|,|[0,(- 3)]|) misses C )
assume |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: LSeg (|[1,(- 3)]|,|[0,(- 3)]|) misses C
then A1: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by Th74;
assume LSeg (|[1,(- 3)]|,|[0,(- 3)]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A2: q in LSeg (|[1,(- 3)]|,|[0,(- 3)]|) and
A3: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A3;
q in (rectangle ((- 1),1,(- 3),3)) /\ C by A2, A3, Lm75, XBOOLE_0:def_4;
then ( q = |[(- 1),0]| or q = |[1,0]| ) by A1, TARSKI:def_2;
hence contradiction by A2, Lm18, Lm19, TOPREAL3:12; ::_thesis: verum
end;
Lm82: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[(- 1),0]|,|[(- 1),3]|) holds
LSeg (p,|[(- 1),3]|) misses C
proof
let p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[(- 1),0]|,|[(- 1),3]|) holds
LSeg (p,|[(- 1),3]|) misses C
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[(- 1),0]|,|[(- 1),3]|) implies LSeg (p,|[(- 1),3]|) misses C )
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C and
A2: p in C ` and
A3: p in LSeg (|[(- 1),0]|,|[(- 1),3]|) ; ::_thesis: LSeg (p,|[(- 1),3]|) misses C
A4: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by A1, Th74;
assume LSeg (p,|[(- 1),3]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A5: q in LSeg (p,|[(- 1),3]|) and
A6: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A6;
|[(- 1),3]| in LSeg (|[(- 1),0]|,|[(- 1),3]|) by RLTOPSP1:68;
then A7: p `1 = |[(- 1),3]| `1 by A3, Lm47, SPPOL_1:def_3;
A8: p `2 <= |[(- 1),3]| `2 by A3, Lm25, JGRAPH_6:1;
A9: LSeg (p,|[(- 1),3]|) is vertical by A7, SPPOL_1:16;
|[(- 1),0]| `2 <= p `2 by A3, Lm18, JGRAPH_6:1;
then LSeg (p,|[(- 1),3]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) by A7, A8, A9, Lm18, Lm24, Lm26, Lm27, Lm45, GOBOARD7:63;
then LSeg (p,|[(- 1),3]|) c= rectangle ((- 1),1,(- 3),3) by Lm38, XBOOLE_1:1;
then q in (rectangle ((- 1),1,(- 3),3)) /\ C by A5, A6, XBOOLE_0:def_4;
then A10: ( q = |[(- 1),0]| or q = |[1,0]| ) by A4, TARSKI:def_2;
|[(- 1),0]| in LSeg (|[(- 1),0]|,|[(- 1),3]|) by RLTOPSP1:68;
then A11: |[(- 1),0]| `1 = p `1 by A3, Lm47, SPPOL_1:def_3;
A12: |[(- 1),0]| in C by A1, JORDAN24:def_1;
not p in C by A2, XBOOLE_0:def_5;
then |[(- 1),0]| `2 <> p `2 by A11, A12, TOPREAL3:6;
then A13: |[(- 1),0]| `2 < p `2 by A3, Lm18, JGRAPH_6:1;
p = |[(p `1),(p `2)]| by EUCLID:53;
hence contradiction by A5, A7, A8, A10, A13, Lm17, Lm24, Lm33, JGRAPH_6:1; ::_thesis: verum
end;
Lm83: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[1,0]|,|[1,3]|) holds
LSeg (p,|[1,3]|) misses C
proof
let p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[1,0]|,|[1,3]|) holds
LSeg (p,|[1,3]|) misses C
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[1,0]|,|[1,3]|) implies LSeg (p,|[1,3]|) misses C )
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C and
A2: p in C ` and
A3: p in LSeg (|[1,0]|,|[1,3]|) ; ::_thesis: LSeg (p,|[1,3]|) misses C
A4: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by A1, Th74;
assume LSeg (p,|[1,3]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A5: q in LSeg (p,|[1,3]|) and
A6: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A6;
|[1,3]| in LSeg (|[1,0]|,|[1,3]|) by RLTOPSP1:68;
then A7: p `1 = |[1,3]| `1 by A3, Lm49, SPPOL_1:def_3;
A8: p `2 <= |[1,3]| `2 by A3, Lm29, JGRAPH_6:1;
A9: LSeg (p,|[1,3]|) is vertical by A7, SPPOL_1:16;
|[1,0]| `2 <= p `2 by A3, Lm19, JGRAPH_6:1;
then LSeg (p,|[1,3]|) c= LSeg (|[1,(- 3)]|,|[1,3]|) by A7, A8, A9, Lm19, Lm28, Lm30, Lm31, Lm46, GOBOARD7:63;
then LSeg (p,|[1,3]|) c= rectangle ((- 1),1,(- 3),3) by Lm42, XBOOLE_1:1;
then q in (rectangle ((- 1),1,(- 3),3)) /\ C by A5, A6, XBOOLE_0:def_4;
then A10: ( q = |[(- 1),0]| or q = |[1,0]| ) by A4, TARSKI:def_2;
|[1,0]| in LSeg (|[1,0]|,|[1,3]|) by RLTOPSP1:68;
then A11: |[1,0]| `1 = p `1 by A3, Lm49, SPPOL_1:def_3;
A12: |[1,0]| in C by A1, JORDAN24:def_1;
not p in C by A2, XBOOLE_0:def_5;
then |[1,0]| `2 <> p `2 by A11, A12, TOPREAL3:6;
then A13: |[1,0]| `2 < p `2 by A3, Lm19, JGRAPH_6:1;
p = |[(p `1),(p `2)]| by EUCLID:53;
hence contradiction by A5, A7, A8, A10, A13, Lm16, Lm28, Lm35, JGRAPH_6:1; ::_thesis: verum
end;
Lm84: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) holds
LSeg (p,|[(- 1),(- 3)]|) misses C
proof
let p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) holds
LSeg (p,|[(- 1),(- 3)]|) misses C
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) implies LSeg (p,|[(- 1),(- 3)]|) misses C )
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C and
A2: p in C ` and
A3: p in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) ; ::_thesis: LSeg (p,|[(- 1),(- 3)]|) misses C
A4: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by A1, Th74;
assume LSeg (p,|[(- 1),(- 3)]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A5: q in LSeg (p,|[(- 1),(- 3)]|) and
A6: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A6;
|[(- 1),(- 3)]| in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) by RLTOPSP1:68;
then A7: p `1 = |[(- 1),(- 3)]| `1 by A3, Lm48, SPPOL_1:def_3;
A8: |[(- 1),(- 3)]| `2 <= p `2 by A3, Lm27, JGRAPH_6:1;
A9: LSeg (p,|[(- 1),(- 3)]|) is vertical by A7, SPPOL_1:16;
p `2 <= |[(- 1),0]| `2 by A3, Lm18, JGRAPH_6:1;
then LSeg (p,|[(- 1),(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) by A7, A8, A9, Lm18, Lm25, Lm45, GOBOARD7:63;
then LSeg (p,|[(- 1),(- 3)]|) c= rectangle ((- 1),1,(- 3),3) by Lm38, XBOOLE_1:1;
then q in (rectangle ((- 1),1,(- 3),3)) /\ C by A5, A6, XBOOLE_0:def_4;
then A10: ( q = |[(- 1),0]| or q = |[1,0]| ) by A4, TARSKI:def_2;
|[(- 1),0]| in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) by RLTOPSP1:68;
then A11: |[(- 1),0]| `1 = p `1 by A3, Lm48, SPPOL_1:def_3;
A12: |[(- 1),0]| in C by A1, JORDAN24:def_1;
not p in C by A2, XBOOLE_0:def_5;
then |[(- 1),0]| `2 <> p `2 by A11, A12, TOPREAL3:6;
then A13: p `2 < |[(- 1),0]| `2 by A3, Lm18, JGRAPH_6:1;
p = |[(p `1),(p `2)]| by EUCLID:53;
hence contradiction by A5, A7, A8, A10, A13, Lm17, Lm26, Lm32, JGRAPH_6:1; ::_thesis: verum
end;
Lm85: for p being Point of (TOP-REAL 2)
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[1,0]|,|[1,(- 3)]|) holds
LSeg (p,|[1,(- 3)]|) misses C
proof
let p be Point of (TOP-REAL 2); ::_thesis: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[1,0]|,|[1,(- 3)]|) holds
LSeg (p,|[1,(- 3)]|) misses C
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C & p in C ` & p in LSeg (|[1,0]|,|[1,(- 3)]|) implies LSeg (p,|[1,(- 3)]|) misses C )
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C and
A2: p in C ` and
A3: p in LSeg (|[1,0]|,|[1,(- 3)]|) ; ::_thesis: LSeg (p,|[1,(- 3)]|) misses C
A4: C /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by A1, Th74;
assume LSeg (p,|[1,(- 3)]|) meets C ; ::_thesis: contradiction
then consider q being set such that
A5: q in LSeg (p,|[1,(- 3)]|) and
A6: q in C by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A6;
|[1,(- 3)]| in LSeg (|[1,0]|,|[1,(- 3)]|) by RLTOPSP1:68;
then A7: p `1 = |[1,(- 3)]| `1 by A3, Lm50, SPPOL_1:def_3;
A8: |[1,(- 3)]| `2 <= p `2 by A3, Lm31, JGRAPH_6:1;
A9: LSeg (p,|[1,(- 3)]|) is vertical by A7, SPPOL_1:16;
p `2 <= |[1,0]| `2 by A3, Lm19, JGRAPH_6:1;
then LSeg (p,|[1,(- 3)]|) c= LSeg (|[1,(- 3)]|,|[1,3]|) by A7, A8, A9, Lm19, Lm29, Lm46, GOBOARD7:63;
then LSeg (p,|[1,(- 3)]|) c= rectangle ((- 1),1,(- 3),3) by Lm42, XBOOLE_1:1;
then q in (rectangle ((- 1),1,(- 3),3)) /\ C by A5, A6, XBOOLE_0:def_4;
then A10: ( q = |[(- 1),0]| or q = |[1,0]| ) by A4, TARSKI:def_2;
|[1,0]| in LSeg (|[1,0]|,|[1,(- 3)]|) by RLTOPSP1:68;
then A11: |[1,0]| `1 = p `1 by A3, Lm50, SPPOL_1:def_3;
A12: |[1,0]| in C by A1, JORDAN24:def_1;
not p in C by A2, XBOOLE_0:def_5;
then |[1,0]| `2 <> p `2 by A11, A12, TOPREAL3:6;
then A13: p `2 < |[1,0]| `2 by A3, Lm19, JGRAPH_6:1;
p = |[(p `1),(p `2)]| by EUCLID:53;
hence contradiction by A5, A7, A8, A10, A13, Lm16, Lm30, Lm34, JGRAPH_6:1; ::_thesis: verum
end;
Lm86: for r being real number holds
( not |[0,r]| in rectangle ((- 1),1,(- 3),3) or r = - 3 or r = 3 )
proof
let r be real number ; ::_thesis: ( not |[0,r]| in rectangle ((- 1),1,(- 3),3) or r = - 3 or r = 3 )
assume |[0,r]| in rectangle ((- 1),1,(- 3),3) ; ::_thesis: ( r = - 3 or r = 3 )
then ex p being Point of (TOP-REAL 2) st
( p = |[0,r]| & ( ( p `1 = - 1 & p `2 <= 3 & p `2 >= - 3 ) or ( p `1 <= 1 & p `1 >= - 1 & p `2 = 3 ) or ( p `1 <= 1 & p `1 >= - 1 & p `2 = - 3 ) or ( p `1 = 1 & p `2 <= 3 & p `2 >= - 3 ) ) ) by Lm61;
hence ( r = - 3 or r = 3 ) by EUCLID:52; ::_thesis: verum
end;
theorem Th75: :: JORDAN:75
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
W-bound P = - 1
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies W-bound P = - 1 )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: W-bound P = - 1
then A2: P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Th71;
A3: P = the carrier of ((TOP-REAL 2) | P) by PRE_TOPC:8;
A4: |[(- 1),0]| in P by A1, JORDAN24:def_1;
reconsider P = P as non empty Subset of (TOP-REAL 2) by A1, JORDAN24:def_1;
reconsider Z = (proj1 | P) .: the carrier of ((TOP-REAL 2) | P) as Subset of REAL ;
A5: for p being real number st p in Z holds
p >= - 1
proof
let p be real number ; ::_thesis: ( p in Z implies p >= - 1 )
assume p in Z ; ::_thesis: p >= - 1
then consider p0 being set such that
A6: p0 in the carrier of ((TOP-REAL 2) | P) and
p0 in the carrier of ((TOP-REAL 2) | P) and
A7: p = (proj1 | P) . p0 by FUNCT_2:64;
p0 in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2, A3, A6;
then ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & - 1 <= p1 `1 & p1 `1 <= 1 & - 3 <= p1 `2 & p1 `2 <= 3 ) ;
hence p >= - 1 by A3, A6, A7, PSCOMP_1:22; ::_thesis: verum
end;
for q being real number st ( for p being real number st p in Z holds
p >= q ) holds
- 1 >= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p >= q ) implies - 1 >= q )
assume A8: for p being real number st p in Z holds
p >= q ; ::_thesis: - 1 >= q
(proj1 | P) . |[(- 1),0]| = |[(- 1),0]| `1 by A4, PSCOMP_1:22;
hence - 1 >= q by A3, A4, A8, Lm16, FUNCT_2:35; ::_thesis: verum
end;
hence W-bound P = - 1 by A5, SEQ_4:44; ::_thesis: verum
end;
theorem Th76: :: JORDAN:76
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
E-bound P = 1
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies E-bound P = 1 )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: E-bound P = 1
then A2: P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Th71;
A3: |[1,0]| in P by A1, JORDAN24:def_1;
reconsider P = P as non empty Subset of (TOP-REAL 2) by A1, JORDAN24:def_1;
reconsider Z = (proj1 | P) .: the carrier of ((TOP-REAL 2) | P) as Subset of REAL ;
A4: P = the carrier of ((TOP-REAL 2) | P) by PRE_TOPC:8;
A5: for p being real number st p in Z holds
p <= 1
proof
let p be real number ; ::_thesis: ( p in Z implies p <= 1 )
assume p in Z ; ::_thesis: p <= 1
then consider p0 being set such that
A6: p0 in the carrier of ((TOP-REAL 2) | P) and
p0 in the carrier of ((TOP-REAL 2) | P) and
A7: p = (proj1 | P) . p0 by FUNCT_2:64;
p0 in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2, A4, A6;
then ex p1 being Point of (TOP-REAL 2) st
( p0 = p1 & - 1 <= p1 `1 & p1 `1 <= 1 & - 3 <= p1 `2 & p1 `2 <= 3 ) ;
hence p <= 1 by A4, A6, A7, PSCOMP_1:22; ::_thesis: verum
end;
for q being real number st ( for p being real number st p in Z holds
p <= q ) holds
1 <= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p <= q ) implies 1 <= q )
assume A8: for p being real number st p in Z holds
p <= q ; ::_thesis: 1 <= q
(proj1 | P) . |[1,0]| = |[1,0]| `1 by A3, PSCOMP_1:22;
hence 1 <= q by A3, A4, A8, Lm17, FUNCT_2:35; ::_thesis: verum
end;
hence E-bound P = 1 by A5, SEQ_4:46; ::_thesis: verum
end;
theorem Th77: :: JORDAN:77
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
W-most P = {|[(- 1),0]|}
proof
let P be compact Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies W-most P = {|[(- 1),0]|} )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: W-most P = {|[(- 1),0]|}
then A2: P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Th71;
set L = LSeg ((SW-corner P),(NW-corner P));
A3: |[(- 1),0]| in P by A1, JORDAN24:def_1;
A4: (SW-corner P) `1 = |[(- 1),(S-bound P)]| `1 by A1, Th75
.= - 1 by EUCLID:52 ;
A5: (NW-corner P) `1 = |[(- 1),(N-bound P)]| `1 by A1, Th75
.= - 1 by EUCLID:52 ;
thus W-most P c= {|[(- 1),0]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[(- 1),0]|} c= W-most P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in W-most P or x in {|[(- 1),0]|} )
assume A6: x in W-most P ; ::_thesis: x in {|[(- 1),0]|}
then A7: x in P by XBOOLE_0:def_4;
reconsider x = x as Point of (TOP-REAL 2) by A6;
A8: x in LSeg ((SW-corner P),(NW-corner P)) by A6, XBOOLE_0:def_4;
SW-corner P in LSeg ((SW-corner P),(NW-corner P)) by RLTOPSP1:68;
then A9: x `1 = - 1 by A4, A8, SPPOL_1:def_3;
x in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2, A7;
then ex p being Point of (TOP-REAL 2) st
( x = p & - 1 <= p `1 & p `1 <= 1 & - 3 <= p `2 & p `2 <= 3 ) ;
then x in rectangle ((- 1),1,(- 3),3) by A9, Lm61;
then x in P /\ (rectangle ((- 1),1,(- 3),3)) by A7, XBOOLE_0:def_4;
then x in {|[(- 1),0]|,|[1,0]|} by A1, Th74;
then ( x = |[(- 1),0]| or x = |[1,0]| ) by TARSKI:def_2;
hence x in {|[(- 1),0]|} by A9, EUCLID:52, TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[(- 1),0]|} or x in W-most P )
assume x in {|[(- 1),0]|} ; ::_thesis: x in W-most P
then A10: x = |[(- 1),0]| by TARSKI:def_1;
A11: (SW-corner P) `2 = S-bound P by EUCLID:52;
A12: (NW-corner P) `2 = N-bound P by EUCLID:52;
A13: (SW-corner P) `2 <= |[(- 1),0]| `2 by A3, A11, PSCOMP_1:24;
|[(- 1),0]| `2 <= (NW-corner P) `2 by A3, A12, PSCOMP_1:24;
then |[(- 1),0]| in LSeg ((SW-corner P),(NW-corner P)) by A4, A5, A13, Lm16, GOBOARD7:7;
hence x in W-most P by A3, A10, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th78: :: JORDAN:78
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
E-most P = {|[1,0]|}
proof
let P be compact Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies E-most P = {|[1,0]|} )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: E-most P = {|[1,0]|}
then A2: P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Th71;
set L = LSeg ((SE-corner P),(NE-corner P));
A3: |[1,0]| in P by A1, JORDAN24:def_1;
A4: (SE-corner P) `1 = |[1,(S-bound P)]| `1 by A1, Th76
.= 1 by EUCLID:52 ;
A5: (NE-corner P) `1 = |[1,(N-bound P)]| `1 by A1, Th76
.= 1 by EUCLID:52 ;
thus E-most P c= {|[1,0]|} :: according to XBOOLE_0:def_10 ::_thesis: {|[1,0]|} c= E-most P
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in E-most P or x in {|[1,0]|} )
assume A6: x in E-most P ; ::_thesis: x in {|[1,0]|}
then A7: x in P by XBOOLE_0:def_4;
reconsider x = x as Point of (TOP-REAL 2) by A6;
A8: x in LSeg ((SE-corner P),(NE-corner P)) by A6, XBOOLE_0:def_4;
SE-corner P in LSeg ((SE-corner P),(NE-corner P)) by RLTOPSP1:68;
then A9: x `1 = 1 by A4, A8, SPPOL_1:def_3;
x in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2, A7;
then ex p being Point of (TOP-REAL 2) st
( x = p & - 1 <= p `1 & p `1 <= 1 & - 3 <= p `2 & p `2 <= 3 ) ;
then x in rectangle ((- 1),1,(- 3),3) by A9, Lm61;
then x in P /\ (rectangle ((- 1),1,(- 3),3)) by A7, XBOOLE_0:def_4;
then x in {|[(- 1),0]|,|[1,0]|} by A1, Th74;
then ( x = |[(- 1),0]| or x = |[1,0]| ) by TARSKI:def_2;
hence x in {|[1,0]|} by A9, EUCLID:52, TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {|[1,0]|} or x in E-most P )
assume x in {|[1,0]|} ; ::_thesis: x in E-most P
then A10: x = |[1,0]| by TARSKI:def_1;
A11: (SE-corner P) `2 = S-bound P by EUCLID:52;
A12: (NE-corner P) `2 = N-bound P by EUCLID:52;
A13: (SE-corner P) `2 <= |[1,0]| `2 by A3, A11, PSCOMP_1:24;
|[1,0]| `2 <= (NE-corner P) `2 by A3, A12, PSCOMP_1:24;
then |[1,0]| in LSeg ((SE-corner P),(NE-corner P)) by A4, A5, A13, Lm17, GOBOARD7:7;
hence x in E-most P by A3, A10, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th79: :: JORDAN:79
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
( W-min P = |[(- 1),0]| & W-max P = |[(- 1),0]| )
proof
let P be compact Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies ( W-min P = |[(- 1),0]| & W-max P = |[(- 1),0]| ) )
set M = W-most P;
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: ( W-min P = |[(- 1),0]| & W-max P = |[(- 1),0]| )
then A2: W-most P = {|[(- 1),0]|} by Th77;
set f = proj2 | (W-most P);
A3: dom (proj2 | (W-most P)) = the carrier of ((TOP-REAL 2) | (W-most P)) by FUNCT_2:def_1;
A4: the carrier of ((TOP-REAL 2) | (W-most P)) = W-most P by PRE_TOPC:8;
A5: |[(- 1),0]| in {|[(- 1),0]|} by TARSKI:def_1;
A6: (proj2 | (W-most P)) .: the carrier of ((TOP-REAL 2) | (W-most P)) = Im ((proj2 | (W-most P)),|[(- 1),0]|) by A1, A4, Th77
.= {((proj2 | (W-most P)) . |[(- 1),0]|)} by A2, A3, A4, A5, FUNCT_1:59
.= {(proj2 . |[(- 1),0]|)} by A2, A5, FUNCT_1:49
.= {(|[(- 1),0]| `2)} by PSCOMP_1:def_6 ;
then A7: lower_bound (proj2 | (W-most P)) = |[(- 1),0]| `2 by SEQ_4:9;
A8: upper_bound (proj2 | (W-most P)) = |[(- 1),0]| `2 by A6, SEQ_4:9;
|[(- 1),0]| = |[(|[(- 1),0]| `1),(|[(- 1),0]| `2)]| by EUCLID:53;
hence ( W-min P = |[(- 1),0]| & W-max P = |[(- 1),0]| ) by A1, A7, A8, Lm16, Th75; ::_thesis: verum
end;
theorem Th80: :: JORDAN:80
for P being compact Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
( E-min P = |[1,0]| & E-max P = |[1,0]| )
proof
let P be compact Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies ( E-min P = |[1,0]| & E-max P = |[1,0]| ) )
set M = E-most P;
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: ( E-min P = |[1,0]| & E-max P = |[1,0]| )
then A2: E-most P = {|[1,0]|} by Th78;
set f = proj2 | (E-most P);
A3: dom (proj2 | (E-most P)) = the carrier of ((TOP-REAL 2) | (E-most P)) by FUNCT_2:def_1;
A4: the carrier of ((TOP-REAL 2) | (E-most P)) = E-most P by PRE_TOPC:8;
A5: |[1,0]| in {|[1,0]|} by TARSKI:def_1;
A6: (proj2 | (E-most P)) .: the carrier of ((TOP-REAL 2) | (E-most P)) = Im ((proj2 | (E-most P)),|[1,0]|) by A1, A4, Th78
.= {((proj2 | (E-most P)) . |[1,0]|)} by A2, A3, A4, A5, FUNCT_1:59
.= {(proj2 . |[1,0]|)} by A2, A5, FUNCT_1:49
.= {(|[1,0]| `2)} by PSCOMP_1:def_6 ;
then A7: lower_bound (proj2 | (E-most P)) = |[1,0]| `2 by SEQ_4:9;
A8: upper_bound (proj2 | (E-most P)) = |[1,0]| `2 by A6, SEQ_4:9;
|[1,0]| = |[(|[1,0]| `1),(|[1,0]| `2)]| by EUCLID:53;
hence ( E-min P = |[1,0]| & E-max P = |[1,0]| ) by A1, A7, A8, Lm17, Th76; ::_thesis: verum
end;
Lm87: for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
|[0,3]| `1 = ((W-bound P) + (E-bound P)) / 2
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies |[0,3]| `1 = ((W-bound P) + (E-bound P)) / 2 )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: |[0,3]| `1 = ((W-bound P) + (E-bound P)) / 2
then A2: W-bound P = - 1 by Th75;
E-bound P = 1 by A1, Th76;
hence |[0,3]| `1 = ((W-bound P) + (E-bound P)) / 2 by A2, EUCLID:52; ::_thesis: verum
end;
Lm88: for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
|[0,(- 3)]| `1 = ((W-bound P) + (E-bound P)) / 2
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies |[0,(- 3)]| `1 = ((W-bound P) + (E-bound P)) / 2 )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: |[0,(- 3)]| `1 = ((W-bound P) + (E-bound P)) / 2
then A2: W-bound P = - 1 by Th75;
E-bound P = 1 by A1, Th76;
hence |[0,(- 3)]| `1 = ((W-bound P) + (E-bound P)) / 2 by A2, EUCLID:52; ::_thesis: verum
end;
theorem Th81: :: JORDAN:81
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
LSeg (|[0,3]|,(UMP P)) is vertical
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies LSeg (|[0,3]|,(UMP P)) is vertical )
assume |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: LSeg (|[0,3]|,(UMP P)) is vertical
then |[0,3]| `1 = ((W-bound P) + (E-bound P)) / 2 by Lm87
.= (UMP P) `1 by EUCLID:52 ;
hence LSeg (|[0,3]|,(UMP P)) is vertical by SPPOL_1:16; ::_thesis: verum
end;
theorem Th82: :: JORDAN:82
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P holds
LSeg ((LMP P),|[0,(- 3)]|) is vertical
proof
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P implies LSeg ((LMP P),|[0,(- 3)]|) is vertical )
assume |[(- 1),0]|,|[1,0]| realize-max-dist-in P ; ::_thesis: LSeg ((LMP P),|[0,(- 3)]|) is vertical
then |[0,(- 3)]| `1 = ((W-bound P) + (E-bound P)) / 2 by Lm88
.= (LMP P) `1 by EUCLID:52 ;
hence LSeg ((LMP P),|[0,(- 3)]|) is vertical by SPPOL_1:16; ::_thesis: verum
end;
theorem Th83: :: JORDAN:83
for p being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P & p in P holds
p `2 < 3
proof
let p be Point of (TOP-REAL 2); ::_thesis: for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P & p in P holds
p `2 < 3
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P & p in P implies p `2 < 3 )
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P and
A2: p in P ; ::_thesis: p `2 < 3
A3: P /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by A1, Th74;
P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A1, Th71;
then p in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2;
then A4: ex p1 being Point of (TOP-REAL 2) st
( p1 = p & - 1 <= p1 `1 & p1 `1 <= 1 & - 3 <= p1 `2 & p1 `2 <= 3 ) ;
now__::_thesis:_not_p_`2_=_|[0,3]|_`2
assume A5: p `2 = |[0,3]| `2 ; ::_thesis: contradiction
then p in LSeg (|[(- 1),3]|,|[1,3]|) by A4, Lm21, Lm24, Lm25, Lm28, Lm29, GOBOARD7:8;
then p in P /\ (rectangle ((- 1),1,(- 3),3)) by A2, Lm40, XBOOLE_0:def_4;
hence contradiction by A3, A5, Lm18, Lm19, Lm21, TARSKI:def_2; ::_thesis: verum
end;
hence p `2 < 3 by A4, Lm21, XXREAL_0:1; ::_thesis: verum
end;
theorem Th84: :: JORDAN:84
for p being Point of (TOP-REAL 2)
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P & p in P holds
- 3 < p `2
proof
let p be Point of (TOP-REAL 2); ::_thesis: for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in P & p in P holds
- 3 < p `2
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in P & p in P implies - 3 < p `2 )
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in P and
A2: p in P ; ::_thesis: - 3 < p `2
A3: P /\ (rectangle ((- 1),1,(- 3),3)) = {|[(- 1),0]|,|[1,0]|} by A1, Th74;
P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A1, Th71;
then p in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2;
then A4: ex p1 being Point of (TOP-REAL 2) st
( p1 = p & - 1 <= p1 `1 & p1 `1 <= 1 & - 3 <= p1 `2 & p1 `2 <= 3 ) ;
now__::_thesis:_not_p_`2_=_|[0,(-_3)]|_`2
assume A5: p `2 = |[0,(- 3)]| `2 ; ::_thesis: contradiction
then p in LSeg (|[(- 1),(- 3)]|,|[1,(- 3)]|) by A4, Lm23, Lm26, Lm27, Lm30, Lm31, GOBOARD7:8;
then p in P /\ (rectangle ((- 1),1,(- 3),3)) by A2, Lm44, XBOOLE_0:def_4;
then ( p = |[(- 1),0]| or p = |[1,0]| ) by A3, TARSKI:def_2;
hence contradiction by A5, Lm23, EUCLID:52; ::_thesis: verum
end;
hence - 3 < p `2 by A4, Lm23, XXREAL_0:1; ::_thesis: verum
end;
theorem Th85: :: JORDAN:85
for p being Point of (TOP-REAL 2)
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg (|[0,3]|,(UMP D)) holds
(UMP D) `2 <= p `2
proof
let p be Point of (TOP-REAL 2); ::_thesis: for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg (|[0,3]|,(UMP D)) holds
(UMP D) `2 <= p `2
let D be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg (|[0,3]|,(UMP D)) implies (UMP D) `2 <= p `2 )
set x = UMP D;
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D and
A2: p in LSeg (|[0,3]|,(UMP D)) ; ::_thesis: (UMP D) `2 <= p `2
A3: UMP D in LSeg (|[0,3]|,(UMP D)) by RLTOPSP1:68;
A4: LSeg (|[0,3]|,(UMP D)) is vertical by A1, Th81;
A5: |[0,3]| = |[(|[0,3]| `1),(|[0,3]| `2)]| by EUCLID:53;
A6: UMP D = |[((UMP D) `1),((UMP D) `2)]| by EUCLID:53;
|[0,3]| in LSeg (|[0,3]|,(UMP D)) by RLTOPSP1:68;
then A7: |[0,3]| `1 = (UMP D) `1 by A3, A4, SPPOL_1:def_3;
(UMP D) `2 <= |[0,3]| `2 by A1, Lm21, Th83, JORDAN21:30;
hence (UMP D) `2 <= p `2 by A2, A5, A6, A7, JGRAPH_6:1; ::_thesis: verum
end;
theorem Th86: :: JORDAN:86
for p being Point of (TOP-REAL 2)
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg ((LMP D),|[0,(- 3)]|) holds
p `2 <= (LMP D) `2
proof
let p be Point of (TOP-REAL 2); ::_thesis: for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg ((LMP D),|[0,(- 3)]|) holds
p `2 <= (LMP D) `2
let D be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D & p in LSeg ((LMP D),|[0,(- 3)]|) implies p `2 <= (LMP D) `2 )
set x = LMP D;
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D and
A2: p in LSeg ((LMP D),|[0,(- 3)]|) ; ::_thesis: p `2 <= (LMP D) `2
A3: LMP D in LSeg ((LMP D),|[0,(- 3)]|) by RLTOPSP1:68;
A4: LSeg ((LMP D),|[0,(- 3)]|) is vertical by A1, Th82;
A5: |[0,(- 3)]| = |[(|[0,(- 3)]| `1),(|[0,(- 3)]| `2)]| by EUCLID:53;
A6: LMP D = |[((LMP D) `1),((LMP D) `2)]| by EUCLID:53;
|[0,(- 3)]| in LSeg ((LMP D),|[0,(- 3)]|) by RLTOPSP1:68;
then A7: |[0,(- 3)]| `1 = (LMP D) `1 by A3, A4, SPPOL_1:def_3;
|[0,(- 3)]| `2 <= (LMP D) `2 by A1, Lm23, Th84, JORDAN21:31;
hence p `2 <= (LMP D) `2 by A2, A5, A6, A7, JGRAPH_6:1; ::_thesis: verum
end;
theorem Th87: :: JORDAN:87
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D holds
LSeg (|[0,3]|,(UMP D)) c= north_halfline (UMP D)
proof
let D be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D implies LSeg (|[0,3]|,(UMP D)) c= north_halfline (UMP D) )
set p = UMP D;
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D ; ::_thesis: LSeg (|[0,3]|,(UMP D)) c= north_halfline (UMP D)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg (|[0,3]|,(UMP D)) or x in north_halfline (UMP D) )
assume A2: x in LSeg (|[0,3]|,(UMP D)) ; ::_thesis: x in north_halfline (UMP D)
then reconsider x = x as Point of (TOP-REAL 2) ;
A3: UMP D in LSeg (|[0,3]|,(UMP D)) by RLTOPSP1:68;
LSeg (|[0,3]|,(UMP D)) is vertical by A1, Th81;
then A4: x `1 = (UMP D) `1 by A2, A3, SPPOL_1:def_3;
(UMP D) `2 <= x `2 by A1, A2, Th85;
hence x in north_halfline (UMP D) by A4, TOPREAL1:def_10; ::_thesis: verum
end;
theorem Th88: :: JORDAN:88
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D holds
LSeg ((LMP D),|[0,(- 3)]|) c= south_halfline (LMP D)
proof
let D be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D implies LSeg ((LMP D),|[0,(- 3)]|) c= south_halfline (LMP D) )
set p = LMP D;
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D ; ::_thesis: LSeg ((LMP D),|[0,(- 3)]|) c= south_halfline (LMP D)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in LSeg ((LMP D),|[0,(- 3)]|) or x in south_halfline (LMP D) )
assume A2: x in LSeg ((LMP D),|[0,(- 3)]|) ; ::_thesis: x in south_halfline (LMP D)
then reconsider x = x as Point of (TOP-REAL 2) ;
A3: LMP D in LSeg ((LMP D),|[0,(- 3)]|) by RLTOPSP1:68;
A4: LSeg ((LMP D),|[0,(- 3)]|) is vertical by A1, Th82;
then A5: x `1 = (LMP D) `1 by A2, A3, SPPOL_1:def_3;
A6: |[0,(- 3)]| = |[(|[0,(- 3)]| `1),(|[0,(- 3)]| `2)]| by EUCLID:53;
A7: LMP D = |[((LMP D) `1),((LMP D) `2)]| by EUCLID:53;
|[0,(- 3)]| in LSeg ((LMP D),|[0,(- 3)]|) by RLTOPSP1:68;
then A8: |[0,(- 3)]| `1 = (LMP D) `1 by A3, A4, SPPOL_1:def_3;
|[0,(- 3)]| `2 <= (LMP D) `2 by A1, Lm23, Th84, JORDAN21:31;
then x `2 <= (LMP D) `2 by A2, A6, A7, A8, JGRAPH_6:1;
hence x in south_halfline (LMP D) by A5, TOPREAL1:def_12; ::_thesis: verum
end;
theorem Th89: :: JORDAN:89
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C holds
LSeg (|[0,3]|,(UMP C)) misses P
proof
let C be Simple_closed_curve; ::_thesis: for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C holds
LSeg (|[0,3]|,(UMP C)) misses P
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C implies LSeg (|[0,3]|,(UMP C)) misses P )
set m = UMP C;
set L = LSeg (|[0,3]|,(UMP C));
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C and
A2: P is_inside_component_of C ; ::_thesis: LSeg (|[0,3]|,(UMP C)) misses P
A3: ex VP being Subset of ((TOP-REAL 2) | (C `)) st
( VP = P & VP is a_component & VP is bounded Subset of (Euclid 2) ) by A2, JORDAN2C:13;
UMP C in LSeg (|[0,3]|,(UMP C)) by RLTOPSP1:68;
then {(UMP C)} c= LSeg (|[0,3]|,(UMP C)) by ZFMISC_1:31;
then A4: LSeg (|[0,3]|,(UMP C)) = ((LSeg (|[0,3]|,(UMP C))) \ {(UMP C)}) \/ {(UMP C)} by XBOOLE_1:45;
A5: (LSeg (|[0,3]|,(UMP C))) \ {(UMP C)} c= (north_halfline (UMP C)) \ {(UMP C)} by A1, Th87, XBOOLE_1:33;
(north_halfline (UMP C)) \ {(UMP C)} c= UBD C by Th12;
then (LSeg (|[0,3]|,(UMP C))) \ {(UMP C)} c= UBD C by A5, XBOOLE_1:1;
then A6: (LSeg (|[0,3]|,(UMP C))) \ {(UMP C)} misses P by A2, Th14, XBOOLE_1:63;
{(UMP C)} misses P by A3, Lm4, JORDAN21:30;
hence LSeg (|[0,3]|,(UMP C)) misses P by A4, A6, XBOOLE_1:70; ::_thesis: verum
end;
theorem Th90: :: JORDAN:90
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C holds
LSeg ((LMP C),|[0,(- 3)]|) misses P
proof
let C be Simple_closed_curve; ::_thesis: for P being Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C holds
LSeg ((LMP C),|[0,(- 3)]|) misses P
let P be Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C & P is_inside_component_of C implies LSeg ((LMP C),|[0,(- 3)]|) misses P )
set m = LMP C;
set L = LSeg ((LMP C),|[0,(- 3)]|);
assume that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C and
A2: P is_inside_component_of C ; ::_thesis: LSeg ((LMP C),|[0,(- 3)]|) misses P
A3: ex VP being Subset of ((TOP-REAL 2) | (C `)) st
( VP = P & VP is a_component & VP is bounded Subset of (Euclid 2) ) by A2, JORDAN2C:13;
LMP C in LSeg ((LMP C),|[0,(- 3)]|) by RLTOPSP1:68;
then {(LMP C)} c= LSeg ((LMP C),|[0,(- 3)]|) by ZFMISC_1:31;
then A4: LSeg ((LMP C),|[0,(- 3)]|) = ((LSeg ((LMP C),|[0,(- 3)]|)) \ {(LMP C)}) \/ {(LMP C)} by XBOOLE_1:45;
A5: (LSeg ((LMP C),|[0,(- 3)]|)) \ {(LMP C)} c= (south_halfline (LMP C)) \ {(LMP C)} by A1, Th88, XBOOLE_1:33;
(south_halfline (LMP C)) \ {(LMP C)} c= UBD C by Th13;
then (LSeg ((LMP C),|[0,(- 3)]|)) \ {(LMP C)} c= UBD C by A5, XBOOLE_1:1;
then A6: (LSeg ((LMP C),|[0,(- 3)]|)) \ {(LMP C)} misses P by A2, Th14, XBOOLE_1:63;
{(LMP C)} misses P by A3, Lm4, JORDAN21:31;
hence LSeg ((LMP C),|[0,(- 3)]|) misses P by A4, A6, XBOOLE_1:70; ::_thesis: verum
end;
theorem Th91: :: JORDAN:91
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D holds
(LSeg (|[0,3]|,(UMP D))) /\ D = {(UMP D)}
proof
let D be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D implies (LSeg (|[0,3]|,(UMP D))) /\ D = {(UMP D)} )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D ; ::_thesis: (LSeg (|[0,3]|,(UMP D))) /\ D = {(UMP D)}
set m = UMP D;
set w = ((W-bound D) + (E-bound D)) / 2;
A2: |[0,3]| `1 = ((W-bound D) + (E-bound D)) / 2 by A1, Lm87;
A3: (UMP D) `1 = ((W-bound D) + (E-bound D)) / 2 by EUCLID:52;
A4: UMP D in LSeg (|[0,3]|,(UMP D)) by RLTOPSP1:68;
A5: UMP D in D by JORDAN21:30;
thus (LSeg (|[0,3]|,(UMP D))) /\ D c= {(UMP D)} :: according to XBOOLE_0:def_10 ::_thesis: {(UMP D)} c= (LSeg (|[0,3]|,(UMP D))) /\ D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg (|[0,3]|,(UMP D))) /\ D or x in {(UMP D)} )
assume A6: x in (LSeg (|[0,3]|,(UMP D))) /\ D ; ::_thesis: x in {(UMP D)}
then A7: x in LSeg (|[0,3]|,(UMP D)) by XBOOLE_0:def_4;
A8: x in D by A6, XBOOLE_0:def_4;
reconsider x = x as Point of (TOP-REAL 2) by A6;
LSeg (|[0,3]|,(UMP D)) is vertical by A2, A3, SPPOL_1:16;
then A9: x `1 = (UMP D) `1 by A4, A7, SPPOL_1:def_3;
then x in Vertical_Line (((W-bound D) + (E-bound D)) / 2) by A3, JORDAN6:31;
then x in D /\ (Vertical_Line (((W-bound D) + (E-bound D)) / 2)) by A8, XBOOLE_0:def_4;
then A10: x `2 <= (UMP D) `2 by JORDAN21:28;
(UMP D) `2 <= x `2 by A1, A7, Th85;
then x `2 = (UMP D) `2 by A10, XXREAL_0:1;
then x = UMP D by A9, TOPREAL3:6;
hence x in {(UMP D)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(UMP D)} or x in (LSeg (|[0,3]|,(UMP D))) /\ D )
assume x in {(UMP D)} ; ::_thesis: x in (LSeg (|[0,3]|,(UMP D))) /\ D
then x = UMP D by TARSKI:def_1;
hence x in (LSeg (|[0,3]|,(UMP D))) /\ D by A4, A5, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem :: JORDAN:92
for D being compact with_the_max_arc Subset of (TOP-REAL 2) st |[(- 1),0]|,|[1,0]| realize-max-dist-in D holds
(LSeg (|[0,(- 3)]|,(LMP D))) /\ D = {(LMP D)}
proof
let D be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in D implies (LSeg (|[0,(- 3)]|,(LMP D))) /\ D = {(LMP D)} )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in D ; ::_thesis: (LSeg (|[0,(- 3)]|,(LMP D))) /\ D = {(LMP D)}
set m = LMP D;
set w = ((W-bound D) + (E-bound D)) / 2;
A2: |[0,(- 3)]| `1 = ((W-bound D) + (E-bound D)) / 2 by A1, Lm88;
A3: (LMP D) `1 = ((W-bound D) + (E-bound D)) / 2 by EUCLID:52;
A4: LMP D in LSeg (|[0,(- 3)]|,(LMP D)) by RLTOPSP1:68;
A5: LMP D in D by JORDAN21:31;
thus (LSeg (|[0,(- 3)]|,(LMP D))) /\ D c= {(LMP D)} :: according to XBOOLE_0:def_10 ::_thesis: {(LMP D)} c= (LSeg (|[0,(- 3)]|,(LMP D))) /\ D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (LSeg (|[0,(- 3)]|,(LMP D))) /\ D or x in {(LMP D)} )
assume A6: x in (LSeg (|[0,(- 3)]|,(LMP D))) /\ D ; ::_thesis: x in {(LMP D)}
then A7: x in LSeg (|[0,(- 3)]|,(LMP D)) by XBOOLE_0:def_4;
A8: x in D by A6, XBOOLE_0:def_4;
reconsider x = x as Point of (TOP-REAL 2) by A6;
LSeg (|[0,(- 3)]|,(LMP D)) is vertical by A2, A3, SPPOL_1:16;
then A9: x `1 = (LMP D) `1 by A4, A7, SPPOL_1:def_3;
then x in Vertical_Line (((W-bound D) + (E-bound D)) / 2) by A3, JORDAN6:31;
then x in D /\ (Vertical_Line (((W-bound D) + (E-bound D)) / 2)) by A8, XBOOLE_0:def_4;
then A10: (LMP D) `2 <= x `2 by JORDAN21:29;
x `2 <= (LMP D) `2 by A1, A7, Th86;
then x `2 = (LMP D) `2 by A10, XXREAL_0:1;
then x = LMP D by A9, TOPREAL3:6;
hence x in {(LMP D)} by TARSKI:def_1; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(LMP D)} or x in (LSeg (|[0,(- 3)]|,(LMP D))) /\ D )
assume x in {(LMP D)} ; ::_thesis: x in (LSeg (|[0,(- 3)]|,(LMP D))) /\ D
then x = LMP D by TARSKI:def_1;
hence x in (LSeg (|[0,(- 3)]|,(LMP D))) /\ D by A4, A5, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th93: :: JORDAN:93
for P, A being Subset of (TOP-REAL 2) st P is compact & |[(- 1),0]|,|[1,0]| realize-max-dist-in P & A is_inside_component_of P holds
A c= closed_inside_of_rectangle ((- 1),1,(- 3),3)
proof
let P, A be Subset of (TOP-REAL 2); ::_thesis: ( P is compact & |[(- 1),0]|,|[1,0]| realize-max-dist-in P & A is_inside_component_of P implies A c= closed_inside_of_rectangle ((- 1),1,(- 3),3) )
assume that
A1: P is compact and
A2: |[(- 1),0]|,|[1,0]| realize-max-dist-in P and
A3: A is_inside_component_of P ; ::_thesis: A c= closed_inside_of_rectangle ((- 1),1,(- 3),3)
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in A or x in closed_inside_of_rectangle ((- 1),1,(- 3),3) )
assume that
A4: x in A and
A5: not x in closed_inside_of_rectangle ((- 1),1,(- 3),3) ; ::_thesis: contradiction
P c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A2, Th71;
then A6: (closed_inside_of_rectangle ((- 1),1,(- 3),3)) ` c= P ` by SUBSET_1:12;
reconsider x = x as Point of (TOP-REAL 2) by A4;
A7: ( not - 1 <= x `1 or not x `1 <= 1 or not - 3 <= x `2 or not x `2 <= 3 ) by A5;
percases ( 0 <= x `1 or x `1 < 0 ) ;
supposeA8: 0 <= x `1 ; ::_thesis: contradiction
set E = east_halfline x;
east_halfline x c= (closed_inside_of_rectangle ((- 1),1,(- 3),3)) `
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in east_halfline x or e in (closed_inside_of_rectangle ((- 1),1,(- 3),3)) ` )
assume A9: e in east_halfline x ; ::_thesis: e in (closed_inside_of_rectangle ((- 1),1,(- 3),3)) `
then reconsider e = e as Point of (TOP-REAL 2) ;
A10: e `1 >= x `1 by A9, TOPREAL1:def_11;
now__::_thesis:_not_e_in_closed_inside_of_rectangle_((-_1),1,(-_3),3)
assume e in closed_inside_of_rectangle ((- 1),1,(- 3),3) ; ::_thesis: contradiction
then ex p being Point of (TOP-REAL 2) st
( e = p & - 1 <= p `1 & p `1 <= 1 & - 3 <= p `2 & p `2 <= 3 ) ;
hence contradiction by A7, A8, A9, A10, TOPREAL1:def_11, XXREAL_0:2; ::_thesis: verum
end;
hence e in (closed_inside_of_rectangle ((- 1),1,(- 3),3)) ` by SUBSET_1:29; ::_thesis: verum
end;
then east_halfline x c= P ` by A6, XBOOLE_1:1;
then east_halfline x misses P by SUBSET_1:23;
then A11: east_halfline x c= UBD P by A1, JORDAN2C:127;
x in east_halfline x by TOPREAL1:38;
then A meets UBD P by A4, A11, XBOOLE_0:3;
hence contradiction by A3, Th14; ::_thesis: verum
end;
supposeA12: x `1 < 0 ; ::_thesis: contradiction
set E = west_halfline x;
west_halfline x c= (closed_inside_of_rectangle ((- 1),1,(- 3),3)) `
proof
let e be set ; :: according to TARSKI:def_3 ::_thesis: ( not e in west_halfline x or e in (closed_inside_of_rectangle ((- 1),1,(- 3),3)) ` )
assume A13: e in west_halfline x ; ::_thesis: e in (closed_inside_of_rectangle ((- 1),1,(- 3),3)) `
then reconsider e = e as Point of (TOP-REAL 2) ;
A14: e `1 <= x `1 by A13, TOPREAL1:def_13;
now__::_thesis:_not_e_in_closed_inside_of_rectangle_((-_1),1,(-_3),3)
assume e in closed_inside_of_rectangle ((- 1),1,(- 3),3) ; ::_thesis: contradiction
then ex p being Point of (TOP-REAL 2) st
( e = p & - 1 <= p `1 & p `1 <= 1 & - 3 <= p `2 & p `2 <= 3 ) ;
hence contradiction by A7, A12, A13, A14, TOPREAL1:def_13, XXREAL_0:2; ::_thesis: verum
end;
hence e in (closed_inside_of_rectangle ((- 1),1,(- 3),3)) ` by SUBSET_1:29; ::_thesis: verum
end;
then west_halfline x c= P ` by A6, XBOOLE_1:1;
then west_halfline x misses P by SUBSET_1:23;
then A15: west_halfline x c= UBD P by A1, JORDAN2C:126;
x in west_halfline x by TOPREAL1:38;
then A meets UBD P by A4, A15, XBOOLE_0:3;
hence contradiction by A3, Th14; ::_thesis: verum
end;
end;
end;
Lm89: for p being Point of (TOP-REAL 2) st p in closed_inside_of_rectangle ((- 1),1,(- 3),3) holds
closed_inside_of_rectangle ((- 1),1,(- 3),3) c= Ball (p,10)
proof
let p be Point of (TOP-REAL 2); ::_thesis: ( p in closed_inside_of_rectangle ((- 1),1,(- 3),3) implies closed_inside_of_rectangle ((- 1),1,(- 3),3) c= Ball (p,10) )
assume p in closed_inside_of_rectangle ((- 1),1,(- 3),3) ; ::_thesis: closed_inside_of_rectangle ((- 1),1,(- 3),3) c= Ball (p,10)
then consider p1 being Point of (TOP-REAL 2) such that
A1: p1 = p and
A2: - 1 <= p1 `1 and
A3: p1 `1 <= 1 and
A4: - 3 <= p1 `2 and
A5: p1 `2 <= 3 ;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in closed_inside_of_rectangle ((- 1),1,(- 3),3) or x in Ball (p,10) )
assume A6: x in closed_inside_of_rectangle ((- 1),1,(- 3),3) ; ::_thesis: x in Ball (p,10)
then reconsider x = x as Point of (TOP-REAL 2) ;
consider p2 being Point of (TOP-REAL 2) such that
A7: p2 = x and
A8: - 1 <= p2 `1 and
A9: p2 `1 <= 1 and
A10: - 3 <= p2 `2 and
A11: p2 `2 <= 3 by A6;
A12: ex s, t being Point of (Euclid 2) st
( s = p1 & t = p2 & dist (p1,p2) = dist (s,t) ) by TOPREAL6:def_1;
dist (p1,p2) <= (1 - (- 1)) + (3 - (- 3)) by A2, A3, A4, A5, A8, A9, A10, A11, TOPREAL6:95;
then dist (p1,p2) < 10 by XXREAL_0:2;
then |.(x - p).| < 10 by A1, A7, A12, SPPOL_1:39;
hence x in Ball (p,10) by TOPREAL9:7; ::_thesis: verum
end;
theorem :: JORDAN:94
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
LSeg (|[0,3]|,|[0,(- 3)]|) meets C
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies LSeg (|[0,3]|,|[0,(- 3)]|) meets C )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: LSeg (|[0,3]|,|[0,(- 3)]|) meets C
set Jc = Upper_Arc C;
consider Pf being Path of |[0,3]|,|[0,(- 3)]|, f being Function of I[01],((TOP-REAL 2) | (LSeg (|[0,3]|,|[0,(- 3)]|))) such that
A2: rng f = LSeg (|[0,3]|,|[0,(- 3)]|) and
A3: Pf = f by Th43;
A4: |[(- 1),0]| = W-min C by A1, Th79;
|[1,0]| = E-max C by A1, Th80;
then Upper_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| by A4, JORDAN6:def_8;
then consider Pg being Path of |[(- 1),0]|,|[1,0]|, g being Function of I[01],((TOP-REAL 2) | (Upper_Arc C)) such that
A5: rng g = Upper_Arc C and
A6: Pg = g by Th42;
A7: Upper_Arc C c= C by JORDAN6:61;
A8: C c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A1, Th71;
A9: |[(- 1),0]| in C by A1, JORDAN24:def_1;
A10: |[1,0]| in C by A1, JORDAN24:def_1;
A11: the carrier of (Trectangle ((- 1),1,(- 3),3)) = closed_inside_of_rectangle ((- 1),1,(- 3),3) by PRE_TOPC:8;
reconsider AR = |[(- 1),0]|, BR = |[1,0]|, CR = |[0,3]|, DR = |[0,(- 3)]| as Point of (Trectangle ((- 1),1,(- 3),3)) by A8, A9, A10, Lm62, Lm63, Lm67, PRE_TOPC:8;
rng Pg c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A5, A6, A7, A8, A11, XBOOLE_1:1;
then reconsider h = Pg as Path of AR,BR by Th30;
LSeg (|[0,3]|,|[0,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Lm62, Lm63, Lm67, JORDAN1:def_1;
then reconsider v = Pf as Path of CR,DR by A2, A3, A11, Th30;
consider s, t being Point of I[01] such that
A12: h . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A13: dom h = the carrier of I[01] by FUNCT_2:def_1;
dom v = the carrier of I[01] by FUNCT_2:def_1;
then A14: v . t in rng Pf by FUNCT_1:def_3;
h . s in rng Pg by A13, FUNCT_1:def_3;
hence LSeg (|[0,3]|,|[0,(- 3)]|) meets C by A2, A3, A5, A6, A7, A12, A14, XBOOLE_0:3; ::_thesis: verum
end;
Lm90: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
ex Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc )
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies ex Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc ) )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: ex Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc )
set U = Upper_Arc C;
set L = Lower_Arc C;
A2: (Upper_Arc C) \/ (Lower_Arc C) = C by JORDAN6:def_9;
A3: UMP C in C by JORDAN21:30;
LMP C in C by JORDAN21:31;
then A4: ( LMP C in Upper_Arc C or LMP C in Lower_Arc C ) by A2, XBOOLE_0:def_3;
A5: W-min C = |[(- 1),0]| by A1, Th79;
A6: E-max C = |[1,0]| by A1, Th80;
percases ( UMP C in Upper_Arc C or UMP C in Lower_Arc C ) by A2, A3, XBOOLE_0:def_3;
supposeA7: UMP C in Upper_Arc C ; ::_thesis: ex Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc )
take Upper_Arc C ; ::_thesis: ex Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Upper_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = (Upper_Arc C) \/ Jd & (Upper_Arc C) /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Upper_Arc C & LMP C in Jd & W-bound C = W-bound (Upper_Arc C) & E-bound C = E-bound (Upper_Arc C) )
take Lower_Arc C ; ::_thesis: ( Upper_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & Lower_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & C = (Upper_Arc C) \/ (Lower_Arc C) & (Upper_Arc C) /\ (Lower_Arc C) = {|[(- 1),0]|,|[1,0]|} & UMP C in Upper_Arc C & LMP C in Lower_Arc C & W-bound C = W-bound (Upper_Arc C) & E-bound C = E-bound (Upper_Arc C) )
thus ( Upper_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & Lower_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & C = (Upper_Arc C) \/ (Lower_Arc C) & (Upper_Arc C) /\ (Lower_Arc C) = {|[(- 1),0]|,|[1,0]|} & UMP C in Upper_Arc C & LMP C in Lower_Arc C & W-bound C = W-bound (Upper_Arc C) & E-bound C = E-bound (Upper_Arc C) ) by A4, A5, A6, A7, JORDAN21:17, JORDAN21:18, JORDAN21:50, JORDAN6:50; ::_thesis: verum
end;
supposeA8: UMP C in Lower_Arc C ; ::_thesis: ex Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc )
take Lower_Arc C ; ::_thesis: ex Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st
( Lower_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = (Lower_Arc C) \/ Jd & (Lower_Arc C) /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Lower_Arc C & LMP C in Jd & W-bound C = W-bound (Lower_Arc C) & E-bound C = E-bound (Lower_Arc C) )
take Upper_Arc C ; ::_thesis: ( Lower_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & Upper_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & C = (Lower_Arc C) \/ (Upper_Arc C) & (Lower_Arc C) /\ (Upper_Arc C) = {|[(- 1),0]|,|[1,0]|} & UMP C in Lower_Arc C & LMP C in Upper_Arc C & W-bound C = W-bound (Lower_Arc C) & E-bound C = E-bound (Lower_Arc C) )
thus ( Lower_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & Upper_Arc C is_an_arc_of |[(- 1),0]|,|[1,0]| & C = (Lower_Arc C) \/ (Upper_Arc C) & (Lower_Arc C) /\ (Upper_Arc C) = {|[(- 1),0]|,|[1,0]|} & UMP C in Lower_Arc C & LMP C in Upper_Arc C & W-bound C = W-bound (Lower_Arc C) & E-bound C = E-bound (Lower_Arc C) ) by A4, A5, A6, A8, JORDAN21:19, JORDAN21:20, JORDAN21:49, JORDAN6:50; ::_thesis: verum
end;
end;
end;
theorem Th95: :: JORDAN:95
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
for Ux being Subset of (TOP-REAL 2) st Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) holds
( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) )
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
for Ux being Subset of (TOP-REAL 2) st Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) holds
( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) ) )
set m = UMP C;
set j = LMP C;
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
for Ux being Subset of (TOP-REAL 2) st Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) holds
( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) )
let Jc, Jd be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc implies for Ux being Subset of (TOP-REAL 2) st Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) holds
( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) ) )
assume that
A2: Jc is_an_arc_of |[(- 1),0]|,|[1,0]| and
A3: Jd is_an_arc_of |[(- 1),0]|,|[1,0]| and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc ; ::_thesis: for Ux being Subset of (TOP-REAL 2) st Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) holds
( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) )
set l = LMP Jc;
set LJ = (LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd;
set k = UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd);
set x = (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc));
set w = ((W-bound C) + (E-bound C)) / 2;
let Ux be Subset of (TOP-REAL 2); ::_thesis: ( Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) implies ( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) ) )
assume A10: Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) ; ::_thesis: ( Ux is_inside_component_of C & ( for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux ) )
A11: C c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A1, Th71;
A12: W-bound C = - 1 by A1, Th75;
A13: E-bound C = 1 by A1, Th76;
A14: |[(- 1),0]| in C by A1, JORDAN24:def_1;
A15: |[1,0]| in C by A1, JORDAN24:def_1;
A16: UMP C in C by JORDAN21:30;
A17: LMP Jc in Jc by JORDAN21:31;
A18: Jd c= C by A4, XBOOLE_1:7;
A19: Jc c= C by A4, XBOOLE_1:7;
then A20: LMP Jc in C by A17;
A21: (UMP C) `2 < |[0,3]| `2 by A1, Lm21, Th83, JORDAN21:30;
A22: (LMP Jc) `1 = 0 by A8, A9, A12, A13, EUCLID:52;
A23: |[0,3]| `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Lm87;
A24: (UMP C) `1 = ((W-bound C) + (E-bound C)) / 2 by EUCLID:52;
A25: UMP C <> |[(- 1),0]| by A12, A13, Lm16, EUCLID:52;
A26: UMP C <> |[1,0]| by A12, A13, Lm17, EUCLID:52;
A27: LMP Jc <> |[(- 1),0]| by A8, A9, A12, A13, Lm16, EUCLID:52;
A28: LMP Jc <> |[1,0]| by A8, A9, A12, A13, Lm17, EUCLID:52;
then consider Pml being Path of UMP C, LMP Jc such that
A29: rng Pml c= Jc and
A30: rng Pml misses {|[(- 1),0]|,|[1,0]|} by A2, A6, A17, A25, A26, A27, Th44;
set ml = rng Pml;
A31: rng Pml c= C by A19, A29, XBOOLE_1:1;
A32: LMP C in C by A7, A18;
A33: LSeg ((LMP Jc),|[0,(- 3)]|) is vertical by A22, Lm22, SPPOL_1:16;
A34: |[0,(- 3)]| `2 <= (LMP C) `2 by A1, A7, A18, Lm23, Th84;
A35: (LMP C) `1 = 0 by A12, A13, EUCLID:52;
LMP Jc in Vertical_Line (((W-bound C) + (E-bound C)) / 2) by A12, A13, A22, JORDAN6:31;
then A36: LMP Jc in C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) by A17, A19, XBOOLE_0:def_4;
then (LMP C) `2 <= (LMP Jc) `2 by JORDAN21:29;
then LMP C in LSeg ((LMP Jc),|[0,(- 3)]|) by A22, A34, A35, Lm22, GOBOARD7:7;
then A37: not (LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd is empty by A7, XBOOLE_0:def_4;
A38: (LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd is vertical by A33, Th4;
then A39: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in (LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd by A37, JORDAN21:30;
then A40: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in LSeg ((LMP Jc),|[0,(- 3)]|) by XBOOLE_0:def_4;
A41: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in Jd by A39, XBOOLE_0:def_4;
then A42: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in C by A18;
A43: |[0,(- 3)]| in LSeg ((LMP Jc),|[0,(- 3)]|) by RLTOPSP1:68;
then A44: (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `1 = 0 by A33, A40, Lm22, SPPOL_1:def_3;
then A45: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) <> |[(- 1),0]| by EUCLID:52;
A46: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) <> |[1,0]| by A44, EUCLID:52;
A47: LMP C <> |[(- 1),0]| by A35, EUCLID:52;
LMP C <> |[1,0]| by A35, EUCLID:52;
then consider Pkj being Path of UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd), LMP C such that
A48: rng Pkj c= Jd and
A49: rng Pkj misses {|[(- 1),0]|,|[1,0]|} by A3, A7, A41, A45, A46, A47, Th44;
set kj = rng Pkj;
A50: rng Pkj c= C by A18, A48, XBOOLE_1:1;
A51: (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in LSeg ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)),(LMP Jc)) by RLTOPSP1:69;
A52: Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) is a_component by CONNSP_1:40;
A53: the carrier of ((TOP-REAL 2) | (C `)) = C ` by PRE_TOPC:8;
A54: LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) is vertical by A22, A44, SPPOL_1:16;
A55: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) by RLTOPSP1:68;
A56: LMP Jc = |[((LMP Jc) `1),((LMP Jc) `2)]| by EUCLID:53;
A57: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) = |[((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `1),((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2)]| by EUCLID:53;
A58: |[0,(- 3)]| = |[(|[0,(- 3)]| `1),(|[0,(- 3)]| `2)]| by EUCLID:53;
|[0,(- 3)]| `2 <= (LMP Jc) `2 by A1, A17, A19, Lm23, Th84;
then A59: (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 <= (LMP Jc) `2 by A22, A40, A56, A58, Lm22, JGRAPH_6:1;
A60: |[(- 1),0]| <> UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) by A44, EUCLID:52;
|[1,0]| <> UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) by A44, EUCLID:52;
then not UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in {|[(- 1),0]|,|[1,0]|} by A60, TARSKI:def_2;
then A61: UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) <> LMP Jc by A5, A17, A41, XBOOLE_0:def_4;
then (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 <> (LMP Jc) `2 by A22, A44, TOPREAL3:6;
then A62: (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 < (LMP Jc) `2 by A59, XXREAL_0:1;
UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in Vertical_Line (((W-bound C) + (E-bound C)) / 2) by A12, A13, A44, JORDAN6:31;
then UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in C /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) by A18, A41, XBOOLE_0:def_4;
then (LMP C) `2 <= (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 by JORDAN21:29;
then |[0,(- 3)]| `2 <= (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 by A1, A7, A18, Lm23, Th84, XXREAL_0:2;
then A63: LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) c= LSeg ((LMP Jc),|[0,(- 3)]|) by A33, A44, A54, A59, Lm22, GOBOARD7:63;
A64: (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} c= C `
proof
let q be set ; :: according to TARSKI:def_3 ::_thesis: ( not q in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} or q in C ` )
assume that
A65: q in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} and
A66: not q in C ` ; ::_thesis: contradiction
A67: q in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) by A65, XBOOLE_0:def_5;
reconsider q = q as Point of (TOP-REAL 2) by A65;
A68: q in C by A66, SUBSET_1:29;
A69: q `1 = ((W-bound C) + (E-bound C)) / 2 by A12, A13, A44, A54, A55, A67, SPPOL_1:def_3;
then A70: q in Vertical_Line (((W-bound C) + (E-bound C)) / 2) by JORDAN6:31;
percases ( q in Jc or q in Jd ) by A4, A68, XBOOLE_0:def_3;
suppose q in Jc ; ::_thesis: contradiction
then q in Jc /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) by A70, XBOOLE_0:def_4;
then A71: (LMP Jc) `2 <= q `2 by A8, A9, JORDAN21:29;
q `2 <= (LMP Jc) `2 by A22, A44, A56, A57, A59, A67, JGRAPH_6:1;
then (LMP Jc) `2 = q `2 by A71, XXREAL_0:1;
then LMP Jc = q by A12, A13, A22, A69, TOPREAL3:6;
then q in {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} by TARSKI:def_2;
hence contradiction by A65, XBOOLE_0:def_5; ::_thesis: verum
end;
suppose q in Jd ; ::_thesis: contradiction
then A72: q in (LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd by A63, A67, XBOOLE_0:def_4;
A73: q `1 = |[0,(- 3)]| `1 by A33, A43, A63, A67, SPPOL_1:def_3;
A74: W-bound (LSeg ((LMP Jc),|[0,(- 3)]|)) <= W-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) by A72, PSCOMP_1:69, XBOOLE_1:17;
A75: E-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) <= E-bound (LSeg ((LMP Jc),|[0,(- 3)]|)) by A72, PSCOMP_1:67, XBOOLE_1:17;
A76: W-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) = E-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) by A37, A38, SPRECT_1:15;
A77: W-bound (LSeg ((LMP Jc),|[0,(- 3)]|)) = |[0,(- 3)]| `1 by A22, Lm22, SPRECT_1:54;
then W-bound (LSeg ((LMP Jc),|[0,(- 3)]|)) = W-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) by A22, A74, A75, A76, Lm22, SPRECT_1:57;
then q in Vertical_Line (((W-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (E-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) / 2) by A73, A76, A77, JORDAN6:31;
then q in ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) /\ (Vertical_Line (((W-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (E-bound ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) / 2)) by A72, XBOOLE_0:def_4;
then A78: q `2 <= (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 by JORDAN21:28;
(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 <= q `2 by A22, A44, A56, A57, A59, A67, JGRAPH_6:1;
then (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 = q `2 by A78, XXREAL_0:1;
then UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) = q by A12, A13, A44, A69, TOPREAL3:6;
then q in {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} by TARSKI:def_2;
hence contradiction by A65, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
then reconsider X = (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} as Subset of ((TOP-REAL 2) | (C `)) by PRE_TOPC:8;
now__::_thesis:_not_(1_/_2)_*_((UMP_((LSeg_((LMP_Jc),|[0,(-_3)]|))_/\_Jd))_+_(LMP_Jc))_in_{(LMP_Jc),(UMP_((LSeg_((LMP_Jc),|[0,(-_3)]|))_/\_Jd))}
assume (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} ; ::_thesis: contradiction
then ( (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) = LMP Jc or (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) = UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) ) by TARSKI:def_2;
hence contradiction by A61, Th1; ::_thesis: verum
end;
then A79: (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} by A51, XBOOLE_0:def_5;
then Component_of (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `)) = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) by A64, CONNSP_3:27;
then A80: (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) by A64, A79, CONNSP_3:26;
then A81: X meets Ux by A10, A79, XBOOLE_0:3;
(LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} is convex by JORDAN1:46;
then X is connected by CONNSP_1:23;
then A82: X c= Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) by A10, A52, A81, CONNSP_1:36;
A83: LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A20, A42, JORDAN1:def_1;
A84: the carrier of (Trectangle ((- 1),1,(- 3),3)) = closed_inside_of_rectangle ((- 1),1,(- 3),3) by PRE_TOPC:8;
reconsider AR = |[(- 1),0]|, BR = |[1,0]|, CR = |[0,3]|, DR = |[0,(- 3)]| as Point of (Trectangle ((- 1),1,(- 3),3)) by A11, A14, A15, Lm62, Lm63, Lm67, PRE_TOPC:8;
consider Pcm being Path of |[0,3]|, UMP C, fcm being Function of I[01],((TOP-REAL 2) | (LSeg (|[0,3]|,(UMP C)))) such that
A85: rng fcm = LSeg (|[0,3]|,(UMP C)) and
A86: Pcm = fcm by Th43;
A87: LSeg (|[0,3]|,(UMP C)) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A16, Lm62, Lm67, JORDAN1:def_1;
A88: rng Pml c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A31, XBOOLE_1:1;
thus Ux is_inside_component_of C ::_thesis: for V being Subset of (TOP-REAL 2) st V is_inside_component_of C holds
V = Ux
proof
thus A89: Ux is_a_component_of C ` by A10, A52, CONNSP_1:def_6; :: according to JORDAN2C:def_2 ::_thesis: Ux is bounded
assume not Ux is bounded ; ::_thesis: contradiction
then not Ux c= Ball (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),10) by RLTOPSP1:42;
then consider u being set such that
A90: u in Ux and
A91: not u in Ball (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),10) by TARSKI:def_3;
A92: closed_inside_of_rectangle ((- 1),1,(- 3),3) c= Ball (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),10) by A51, A83, Lm89;
reconsider u = u as Point of (TOP-REAL 2) by A90;
A93: Ux is open by A89, SPRECT_3:8;
Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) is connected by A52, CONNSP_1:def_5;
then A94: Ux is connected by A10, CONNSP_1:23;
(1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in Ball (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),10) by Th16;
then consider P1 being Subset of (TOP-REAL 2) such that
A95: P1 is_S-P_arc_joining (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)),u and
A96: P1 c= Ux by A10, A80, A90, A91, A93, A94, TOPREAL4:29;
A97: P1 is_an_arc_of (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)),u by A95, TOPREAL4:2;
reconsider P2 = P1 as Subset of ((TOP-REAL 2) | (C `)) by A10, A96, XBOOLE_1:1;
A98: P2 c= Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) by A10, A96;
A99: P2 misses C by A53, SUBSET_1:23;
then A100: P2 misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
A101: P2 misses Jd by A4, A99, XBOOLE_1:7, XBOOLE_1:63;
A102: ((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `1 = (1 / 2) * (((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) `1) by TOPREAL3:4
.= (1 / 2) * (((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `1) + ((LMP Jc) `1)) by TOPREAL3:2
.= 0 by A22, A44 ;
then A103: LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) is vertical by Lm22, SPPOL_1:16;
A104: (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) = |[(((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `1),(((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `2)]| by EUCLID:53;
A105: ((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `2 < (LMP Jc) `2 by A62, Th3;
A106: (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 < ((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `2 by A62, Th2;
then A107: |[0,(- 3)]| `2 <= ((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `2 by A1, A18, A41, Lm23, Th84, XXREAL_0:2;
|[0,(- 3)]| `1 = |[0,(- 3)]| `1 ;
then A108: LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) c= LSeg (|[0,(- 3)]|,(LMP Jc)) by A33, A103, A105, A107, GOBOARD7:63;
A109: LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) misses Jc
proof
assume not LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) misses Jc ; ::_thesis: contradiction
then consider q being set such that
A110: q in LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) and
A111: q in Jc by XBOOLE_0:3;
reconsider q = q as Point of (TOP-REAL 2) by A110;
q `2 <= ((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `2 by A58, A102, A104, A107, A110, Lm22, JGRAPH_6:1;
then A112: q `2 < (LMP Jc) `2 by A105, XXREAL_0:2;
q `1 = 0 by A33, A43, A108, A110, Lm22, SPPOL_1:def_3;
then q in Vertical_Line (((W-bound C) + (E-bound C)) / 2) by A12, A13, JORDAN6:31;
then q in Jc /\ (Vertical_Line (((W-bound C) + (E-bound C)) / 2)) by A111, XBOOLE_0:def_4;
hence contradiction by A8, A9, A112, JORDAN21:29; ::_thesis: verum
end;
set n = First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)));
A113: not u in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A91, A92;
A114: Fr (closed_inside_of_rectangle ((- 1),1,(- 3),3)) = rectangle ((- 1),1,(- 3),3) by Th52;
u in P1 by A97, TOPREAL1:1;
then A115: P1 \ (closed_inside_of_rectangle ((- 1),1,(- 3),3)) <> {} (TOP-REAL 2) by A113, XBOOLE_0:def_5;
(1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in P1 by A97, TOPREAL1:1;
then P1 meets closed_inside_of_rectangle ((- 1),1,(- 3),3) by A51, A83, XBOOLE_0:3;
then A116: P1 meets rectangle ((- 1),1,(- 3),3) by A97, A114, A115, CONNSP_1:22, JORDAN6:10;
P1 is closed by A95, JORDAN6:11, TOPREAL4:2;
then A117: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in P1 /\ (rectangle ((- 1),1,(- 3),3)) by A97, A116, JORDAN5C:def_1;
then A118: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in rectangle ((- 1),1,(- 3),3) by XBOOLE_0:def_4;
A119: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in P1 by A117, XBOOLE_0:def_4;
set alpha = Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))));
A120: - 3 < (UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) `2 by A1, A18, A41, Th84;
(LMP Jc) `2 <= (UMP C) `2 by A36, JORDAN21:28;
then ((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))) `2 < (UMP C) `2 by A105, XXREAL_0:2;
then not (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)) in rectangle ((- 1),1,(- 3),3) by A21, A102, A104, A106, A120, Lm86;
then A121: Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))) is_an_arc_of (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)), First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) by A95, A118, A119, JORDAN16:24, TOPREAL4:2;
A122: Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))) misses Jc by A100, JORDAN16:2, XBOOLE_1:63;
A123: Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))) misses Jd by A101, JORDAN16:2, XBOOLE_1:63;
consider Pdx being Path of |[0,(- 3)]|,(1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)), fdx being Function of I[01],((TOP-REAL 2) | (LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))))) such that
A124: rng fdx = LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) and
A125: Pdx = fdx by Th43;
consider PJc being Path of |[(- 1),0]|,|[1,0]|, fJc being Function of I[01],((TOP-REAL 2) | Jc) such that
A126: rng fJc = Jc and
A127: PJc = fJc by A2, Th42;
consider PJd being Path of |[(- 1),0]|,|[1,0]|, fJd being Function of I[01],((TOP-REAL 2) | Jd) such that
A128: rng fJd = Jd and
A129: PJd = fJd by A3, Th42;
consider Palpha being Path of (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)), First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))), falpha being Function of I[01],((TOP-REAL 2) | (Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))))) such that
A130: rng falpha = Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))) and
A131: Palpha = falpha by A121, Th42;
First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in closed_inside_of_rectangle ((- 1),1,(- 3),3) by A118, Lm67;
then A132: ex p being Point of (TOP-REAL 2) st
( p = First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) & - 1 <= p `1 & p `1 <= 1 & - 3 <= p `2 & p `2 <= 3 ) ;
rng PJc c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A11, A19, A84, A126, A127, XBOOLE_1:1;
then reconsider h = PJc as Path of AR,BR by Th30;
rng PJd c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A11, A18, A84, A128, A129, XBOOLE_1:1;
then reconsider H = PJd as Path of AR,BR by Th30;
A133: LSeg (|[0,(- 3)]|,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A51, A83, Lm63, Lm67, JORDAN1:def_1;
A134: Segment (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A51, A83, A95, A113, Th57, TOPREAL4:2;
A135: |[(- 1),(- 3)]| in LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) by RLTOPSP1:68;
A136: |[1,(- 3)]| in LSeg (|[1,(- 3)]|,|[1,3]|) by RLTOPSP1:68;
LSeg (|[(- 1),3]|,|[0,3]|) misses C by A1, Lm78;
then A137: LSeg (|[(- 1),3]|,|[0,3]|) misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
A138: LSeg (|[(- 1),3]|,|[0,3]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Lm67, Lm70, XBOOLE_1:1;
A139: LSeg (|[1,3]|,|[0,3]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Lm67, Lm71, XBOOLE_1:1;
LSeg (|[1,3]|,|[0,3]|) misses C by A1, Lm79;
then A140: LSeg (|[1,3]|,|[0,3]|) misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
consider Plx being Path of LMP Jc,(1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)), flx being Function of I[01],((TOP-REAL 2) | (LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))))) such that
A141: rng flx = LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) and
A142: Plx = flx by Th43;
set PCX = (Pcm + Pml) + Plx;
A143: rng ((Pcm + Pml) + Plx) = ((rng Pcm) \/ (rng Pml)) \/ (rng Plx) by Th40;
A144: rng Pml misses Jd
proof
assume rng Pml meets Jd ; ::_thesis: contradiction
then consider q being set such that
A145: q in rng Pml and
A146: q in Jd by XBOOLE_0:3;
q in {|[(- 1),0]|,|[1,0]|} by A5, A29, A145, A146, XBOOLE_0:def_4;
hence contradiction by A30, A145, XBOOLE_0:3; ::_thesis: verum
end;
A147: (LSeg (|[0,3]|,(UMP C))) /\ C = {(UMP C)} by A1, Th91;
A148: LSeg (|[0,3]|,(UMP C)) misses Jd
proof
assume LSeg (|[0,3]|,(UMP C)) meets Jd ; ::_thesis: contradiction
then consider q being set such that
A149: q in LSeg (|[0,3]|,(UMP C)) and
A150: q in Jd by XBOOLE_0:3;
q in {(UMP C)} by A18, A147, A149, A150, XBOOLE_0:def_4;
then q = UMP C by TARSKI:def_1;
then UMP C in {|[(- 1),0]|,|[1,0]|} by A5, A6, A150, XBOOLE_0:def_4;
hence contradiction by A25, A26, TARSKI:def_2; ::_thesis: verum
end;
LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) is vertical by A22, A102, SPPOL_1:16;
then A151: LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) c= LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) by A44, A54, A102, A105, A106, GOBOARD7:63;
LMP Jc in LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) by RLTOPSP1:68;
then {(LMP Jc)} c= LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) by ZFMISC_1:31;
then A152: LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) = ((LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)}) \/ {(LMP Jc)} by XBOOLE_1:45;
(LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)} c= (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))}
proof
let q be set ; :: according to TARSKI:def_3 ::_thesis: ( not q in (LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)} or q in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} )
assume A153: q in (LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)} ; ::_thesis: q in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))}
then A154: q in LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) by ZFMISC_1:56;
A155: q <> LMP Jc by A153, ZFMISC_1:56;
q <> UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) by A22, A56, A102, A104, A105, A106, A154, JGRAPH_6:1;
then not q in {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} by A155, TARSKI:def_2;
hence q in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} by A151, A154, XBOOLE_0:def_5; ::_thesis: verum
end;
then (LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)} c= C ` by A64, XBOOLE_1:1;
then (LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)} misses C by SUBSET_1:23;
then A156: (LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))))) \ {(LMP Jc)} misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
{(LMP Jc)} misses Jd
proof
assume {(LMP Jc)} meets Jd ; ::_thesis: contradiction
then LMP Jc in Jd by ZFMISC_1:50;
then LMP Jc in {|[(- 1),0]|,|[1,0]|} by A5, A17, XBOOLE_0:def_4;
hence contradiction by A27, A28, TARSKI:def_2; ::_thesis: verum
end;
then LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) misses Jd by A152, A156, XBOOLE_1:70;
then A157: rng ((Pcm + Pml) + Plx) misses Jd by A85, A86, A141, A142, A143, A144, A148, XBOOLE_1:114;
LSeg ((LMP Jc),((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc)))) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A83, A151, XBOOLE_1:1;
then A158: rng ((Pcm + Pml) + Plx) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A85, A86, A87, A88, A141, A142, A143, Lm1;
LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) misses C by A1, Lm80;
then A159: LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
LSeg (|[1,(- 3)]|,|[0,(- 3)]|) misses C by A1, Lm81;
then A160: LSeg (|[1,(- 3)]|,|[0,(- 3)]|) misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
percases ( (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 < 0 or (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 >= 0 ) ;
supposeA161: (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 < 0 ; ::_thesis: contradiction
percases ( First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) or First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) or First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) or First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[1,(- 3)]|,|[1,0]|) ) by A118, A161, Lm77;
supposeA162: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),0]|,|[(- 1),(- 3)]|) ; ::_thesis: contradiction
consider Pnld being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[(- 1),(- 3)]|, fnld being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|))) such that
A163: rng fnld = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) and
A164: Pnld = fnld by Th43;
consider Pldd being Path of |[(- 1),(- 3)]|,|[0,(- 3)]|, fldd being Function of I[01],((TOP-REAL 2) | (LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|))) such that
A165: rng fldd = LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) and
A166: Pldd = fldd by Th43;
A167: |[(- 1),(- 3)]| `1 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A135, A162, Lm45, Lm58, SPPOL_1:def_3;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) is vertical by SPPOL_1:16;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) by A132, A167, Lm25, Lm27, Lm45, GOBOARD7:63;
then A168: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) c= rectangle ((- 1),1,(- 3),3) by Lm38, XBOOLE_1:1;
set K1 = ((((Pcm + Pml) + Plx) + Palpha) + Pnld) + Pldd;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) misses C by A1, A53, A98, A119, A162, Lm84;
then A169: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
A170: rng (((((Pcm + Pml) + Plx) + Palpha) + Pnld) + Pldd) = (((rng ((Pcm + Pml) + Plx)) \/ (rng Palpha)) \/ (rng Pnld)) \/ (rng Pldd) by Lm9;
then A171: rng PJd misses rng (((((Pcm + Pml) + Plx) + Palpha) + Pnld) + Pldd) by A123, A128, A129, A130, A131, A157, A159, A163, A164, A165, A166, A169, Lm3;
A172: LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Lm67, Lm74, XBOOLE_1:1;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A168, Lm67, XBOOLE_1:1;
then rng (((((Pcm + Pml) + Plx) + Palpha) + Pnld) + Pldd) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A130, A131, A134, A158, A163, A164, A165, A166, A170, A172, Lm2;
then reconsider v = ((((Pcm + Pml) + Plx) + Palpha) + Pnld) + Pldd as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A173: H . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A174: dom H = the carrier of I[01] by FUNCT_2:def_1;
A175: dom v = the carrier of I[01] by FUNCT_2:def_1;
A176: H . s in rng PJd by A174, FUNCT_1:def_3;
v . t in rng (((((Pcm + Pml) + Plx) + Palpha) + Pnld) + Pldd) by A175, FUNCT_1:def_3;
hence contradiction by A171, A173, A176, XBOOLE_0:3; ::_thesis: verum
end;
supposeA177: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) ; ::_thesis: contradiction
consider Pnd being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[0,(- 3)]|, fnd being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|))) such that
A178: rng fnd = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) and
A179: Pnd = fnd by Th43;
set K1 = (((Pcm + Pml) + Plx) + Palpha) + Pnd;
|[(- 1),(- 3)]| in LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) by RLTOPSP1:68;
then A180: |[(- 1),(- 3)]| `2 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 by A177, Lm51, SPPOL_1:def_2;
then A181: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) is horizontal by Lm23, Lm27, SPPOL_1:15;
A182: |[(- 1),(- 3)]| `1 <= (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A177, Lm26, JGRAPH_6:3;
(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 <= |[0,(- 3)]| `1 by A177, Lm22, JGRAPH_6:3;
then A183: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) c= LSeg (|[(- 1),(- 3)]|,|[0,(- 3)]|) by A180, A181, A182, Lm51, GOBOARD7:64;
then A184: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) c= rectangle ((- 1),1,(- 3),3) by Lm74, XBOOLE_1:1;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) misses C by A1, A183, Lm80, XBOOLE_1:63;
then A185: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
A186: rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) = ((rng ((Pcm + Pml) + Plx)) \/ (rng Palpha)) \/ (rng Pnd) by Th40;
then A187: rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) misses Jd by A123, A130, A131, A157, A178, A179, A185, XBOOLE_1:114;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A184, Lm67, XBOOLE_1:1;
then rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A130, A131, A134, A158, A178, A179, A186, Lm1;
then reconsider v = (((Pcm + Pml) + Plx) + Palpha) + Pnd as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A188: H . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A189: dom H = the carrier of I[01] by FUNCT_2:def_1;
A190: dom v = the carrier of I[01] by FUNCT_2:def_1;
A191: H . s in rng PJd by A189, FUNCT_1:def_3;
v . t in rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) by A190, FUNCT_1:def_3;
hence contradiction by A128, A129, A187, A188, A191, XBOOLE_0:3; ::_thesis: verum
end;
supposeA192: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[0,(- 3)]|,|[1,(- 3)]|) ; ::_thesis: contradiction
consider Pnd being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[0,(- 3)]|, fnd being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|))) such that
A193: rng fnd = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) and
A194: Pnd = fnd by Th43;
set K1 = (((Pcm + Pml) + Plx) + Palpha) + Pnd;
|[1,(- 3)]| in LSeg (|[1,(- 3)]|,|[0,(- 3)]|) by RLTOPSP1:68;
then |[1,(- 3)]| `2 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 by A192, Lm52, SPPOL_1:def_2;
then A195: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) is horizontal by Lm23, Lm31, SPPOL_1:15;
A196: |[0,(- 3)]| `2 = |[0,(- 3)]| `2 ;
A197: |[0,(- 3)]| `1 <= (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A192, Lm22, JGRAPH_6:3;
(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 <= |[1,(- 3)]| `1 by A192, Lm30, JGRAPH_6:3;
then A198: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) c= LSeg (|[1,(- 3)]|,|[0,(- 3)]|) by A195, A196, A197, Lm52, GOBOARD7:64;
then A199: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) c= rectangle ((- 1),1,(- 3),3) by Lm75, XBOOLE_1:1;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) misses C by A1, A198, Lm81, XBOOLE_1:63;
then A200: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
A201: rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) = ((rng ((Pcm + Pml) + Plx)) \/ (rng Palpha)) \/ (rng Pnd) by Th40;
then A202: rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) misses Jd by A123, A130, A131, A157, A193, A194, A200, XBOOLE_1:114;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A199, Lm67, XBOOLE_1:1;
then rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A130, A131, A134, A158, A193, A194, A201, Lm1;
then reconsider v = (((Pcm + Pml) + Plx) + Palpha) + Pnd as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A203: H . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A204: dom H = the carrier of I[01] by FUNCT_2:def_1;
A205: dom v = the carrier of I[01] by FUNCT_2:def_1;
A206: H . s in rng PJd by A204, FUNCT_1:def_3;
v . t in rng ((((Pcm + Pml) + Plx) + Palpha) + Pnd) by A205, FUNCT_1:def_3;
hence contradiction by A128, A129, A202, A203, A206, XBOOLE_0:3; ::_thesis: verum
end;
supposeA207: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[1,(- 3)]|,|[1,0]|) ; ::_thesis: contradiction
consider Pnpd being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[1,(- 3)]|, fnpd being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|))) such that
A208: rng fnpd = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) and
A209: Pnpd = fnpd by Th43;
consider Ppdd being Path of |[1,(- 3)]|,|[0,(- 3)]|, fpdd being Function of I[01],((TOP-REAL 2) | (LSeg (|[1,(- 3)]|,|[0,(- 3)]|))) such that
A210: rng fpdd = LSeg (|[1,(- 3)]|,|[0,(- 3)]|) and
A211: Ppdd = fpdd by Th43;
A212: |[1,(- 3)]| `1 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A136, A207, Lm46, Lm60, SPPOL_1:def_3;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) is vertical by SPPOL_1:16;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) c= LSeg (|[1,(- 3)]|,|[1,3]|) by A132, A212, Lm29, Lm31, Lm46, GOBOARD7:63;
then A213: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) c= rectangle ((- 1),1,(- 3),3) by Lm42, XBOOLE_1:1;
set K1 = ((((Pcm + Pml) + Plx) + Palpha) + Pnpd) + Ppdd;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) misses C by A1, A53, A98, A119, A207, Lm85;
then A214: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) misses Jd by A4, XBOOLE_1:7, XBOOLE_1:63;
A215: rng (((((Pcm + Pml) + Plx) + Palpha) + Pnpd) + Ppdd) = (((rng ((Pcm + Pml) + Plx)) \/ (rng Palpha)) \/ (rng Pnpd)) \/ (rng Ppdd) by Lm9;
then A216: rng PJd misses rng (((((Pcm + Pml) + Plx) + Palpha) + Pnpd) + Ppdd) by A123, A128, A129, A130, A131, A157, A160, A208, A209, A210, A211, A214, Lm3;
A217: LSeg (|[1,(- 3)]|,|[0,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by Lm67, Lm75, XBOOLE_1:1;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A213, Lm67, XBOOLE_1:1;
then rng (((((Pcm + Pml) + Plx) + Palpha) + Pnpd) + Ppdd) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A130, A131, A134, A158, A208, A209, A210, A211, A215, A217, Lm2;
then reconsider v = ((((Pcm + Pml) + Plx) + Palpha) + Pnpd) + Ppdd as Path of CR,DR by Th30;
consider s, t being Point of I[01] such that
A218: H . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A219: dom H = the carrier of I[01] by FUNCT_2:def_1;
A220: dom v = the carrier of I[01] by FUNCT_2:def_1;
A221: H . s in rng PJd by A219, FUNCT_1:def_3;
v . t in rng (((((Pcm + Pml) + Plx) + Palpha) + Pnpd) + Ppdd) by A220, FUNCT_1:def_3;
hence contradiction by A216, A218, A221, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
supposeA222: (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 >= 0 ; ::_thesis: contradiction
percases ( First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),0]|,|[(- 1),3]|) or First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),3]|,|[0,3]|) or First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[0,3]|,|[1,3]|) or First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[1,3]|,|[1,0]|) ) by A118, A222, Lm76;
supposeA223: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),0]|,|[(- 1),3]|) ; ::_thesis: contradiction
consider Pnlg being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[(- 1),3]|, fnlg being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|))) such that
A224: rng fnlg = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) and
A225: Pnlg = fnlg by Th43;
consider Plgc being Path of |[(- 1),3]|,|[0,3]|, flgc being Function of I[01],((TOP-REAL 2) | (LSeg (|[(- 1),3]|,|[0,3]|))) such that
A226: rng flgc = LSeg (|[(- 1),3]|,|[0,3]|) and
A227: Plgc = flgc by Th43;
A228: |[(- 1),(- 3)]| `1 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A135, A223, Lm45, Lm57, SPPOL_1:def_3;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) is vertical by Lm24, Lm26, SPPOL_1:16;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) c= LSeg (|[(- 1),(- 3)]|,|[(- 1),3]|) by A132, A228, Lm25, Lm27, Lm45, GOBOARD7:63;
then A229: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) c= rectangle ((- 1),1,(- 3),3) by Lm38, XBOOLE_1:1;
set K1 = ((Pdx + Palpha) + Pnlg) + Plgc;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) misses C by A1, A53, A98, A119, A223, Lm82;
then A230: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
A231: rng (((Pdx + Palpha) + Pnlg) + Plgc) = (((rng Pdx) \/ (rng Palpha)) \/ (rng Pnlg)) \/ (rng Plgc) by Lm9;
then A232: rng (((Pdx + Palpha) + Pnlg) + Plgc) misses Jc by A109, A122, A124, A125, A130, A131, A137, A224, A225, A226, A227, A230, Lm3;
A233: rng (((Pdx + Palpha) + Pnlg) + Plgc) = rng (- (((Pdx + Palpha) + Pnlg) + Plgc)) by Th32;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[(- 1),3]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A229, Lm67, XBOOLE_1:1;
then rng (((Pdx + Palpha) + Pnlg) + Plgc) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A124, A125, A130, A131, A133, A134, A138, A224, A225, A226, A227, A231, Lm2;
then reconsider v = - (((Pdx + Palpha) + Pnlg) + Plgc) as Path of CR,DR by A233, Th30;
consider s, t being Point of I[01] such that
A234: h . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A235: dom h = the carrier of I[01] by FUNCT_2:def_1;
A236: dom v = the carrier of I[01] by FUNCT_2:def_1;
A237: h . s in rng PJc by A235, FUNCT_1:def_3;
v . t in rng (- (((Pdx + Palpha) + Pnlg) + Plgc)) by A236, FUNCT_1:def_3;
hence contradiction by A126, A127, A232, A233, A234, A237, XBOOLE_0:3; ::_thesis: verum
end;
supposeA238: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[(- 1),3]|,|[0,3]|) ; ::_thesis: contradiction
consider Pnc being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[0,3]|, fnc being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|))) such that
A239: rng fnc = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) and
A240: Pnc = fnc by Th43;
set K1 = (Pdx + Palpha) + Pnc;
|[(- 1),3]| in LSeg (|[(- 1),3]|,|[0,3]|) by RLTOPSP1:68;
then A241: |[(- 1),3]| `2 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 by A238, Lm53, SPPOL_1:def_2;
then A242: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) is horizontal by Lm21, Lm25, SPPOL_1:15;
A243: |[(- 1),3]| `1 <= (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A238, Lm24, JGRAPH_6:3;
(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 <= |[0,3]| `1 by A238, Lm20, JGRAPH_6:3;
then A244: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) c= LSeg (|[(- 1),3]|,|[0,3]|) by A241, A242, A243, Lm53, GOBOARD7:64;
then A245: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) c= rectangle ((- 1),1,(- 3),3) by Lm70, XBOOLE_1:1;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) misses C by A1, A244, Lm78, XBOOLE_1:63;
then A246: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
A247: rng ((Pdx + Palpha) + Pnc) = ((rng Pdx) \/ (rng Palpha)) \/ (rng Pnc) by Th40;
then A248: rng ((Pdx + Palpha) + Pnc) misses Jc by A109, A122, A124, A125, A130, A131, A239, A240, A246, XBOOLE_1:114;
A249: rng ((Pdx + Palpha) + Pnc) = rng (- ((Pdx + Palpha) + Pnc)) by Th32;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A245, Lm67, XBOOLE_1:1;
then rng ((Pdx + Palpha) + Pnc) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A124, A125, A130, A131, A133, A134, A239, A240, A247, Lm1;
then reconsider v = - ((Pdx + Palpha) + Pnc) as Path of CR,DR by A249, Th30;
consider s, t being Point of I[01] such that
A250: h . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A251: dom h = the carrier of I[01] by FUNCT_2:def_1;
A252: dom v = the carrier of I[01] by FUNCT_2:def_1;
A253: h . s in rng PJc by A251, FUNCT_1:def_3;
v . t in rng (- ((Pdx + Palpha) + Pnc)) by A252, FUNCT_1:def_3;
hence contradiction by A126, A127, A248, A249, A250, A253, XBOOLE_0:3; ::_thesis: verum
end;
supposeA254: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[0,3]|,|[1,3]|) ; ::_thesis: contradiction
consider Pnc being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[0,3]|, fnc being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|))) such that
A255: rng fnc = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) and
A256: Pnc = fnc by Th43;
set K1 = (Pdx + Palpha) + Pnc;
|[1,3]| in LSeg (|[1,3]|,|[0,3]|) by RLTOPSP1:68;
then |[1,3]| `2 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `2 by A254, Lm54, SPPOL_1:def_2;
then A257: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) is horizontal by Lm21, Lm29, SPPOL_1:15;
A258: |[0,3]| `2 = |[0,3]| `2 ;
A259: |[0,3]| `1 <= (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A254, Lm20, JGRAPH_6:3;
(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 <= |[1,3]| `1 by A254, Lm28, JGRAPH_6:3;
then A260: LSeg (|[0,3]|,(First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))))) c= LSeg (|[0,3]|,|[1,3]|) by A257, A258, A259, Lm54, GOBOARD7:64;
then A261: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) c= rectangle ((- 1),1,(- 3),3) by Lm71, XBOOLE_1:1;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) misses C by A1, A260, Lm79, XBOOLE_1:63;
then A262: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
A263: rng ((Pdx + Palpha) + Pnc) = ((rng Pdx) \/ (rng Palpha)) \/ (rng Pnc) by Th40;
then A264: rng ((Pdx + Palpha) + Pnc) misses Jc by A109, A122, A124, A125, A130, A131, A255, A256, A262, XBOOLE_1:114;
A265: rng ((Pdx + Palpha) + Pnc) = rng (- ((Pdx + Palpha) + Pnc)) by Th32;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[0,3]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A261, Lm67, XBOOLE_1:1;
then rng ((Pdx + Palpha) + Pnc) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A124, A125, A130, A131, A133, A134, A255, A256, A263, Lm1;
then reconsider v = - ((Pdx + Palpha) + Pnc) as Path of CR,DR by A265, Th30;
consider s, t being Point of I[01] such that
A266: h . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A267: dom h = the carrier of I[01] by FUNCT_2:def_1;
A268: dom v = the carrier of I[01] by FUNCT_2:def_1;
A269: h . s in rng PJc by A267, FUNCT_1:def_3;
v . t in rng (- ((Pdx + Palpha) + Pnc)) by A268, FUNCT_1:def_3;
hence contradiction by A126, A127, A264, A265, A266, A269, XBOOLE_0:3; ::_thesis: verum
end;
supposeA270: First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))) in LSeg (|[1,3]|,|[1,0]|) ; ::_thesis: contradiction
consider Pnpg being Path of First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3))),|[1,3]|, fnpg being Function of I[01],((TOP-REAL 2) | (LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|))) such that
A271: rng fnpg = LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) and
A272: Pnpg = fnpg by Th43;
consider Ppgc being Path of |[1,3]|,|[0,3]|, fpgc being Function of I[01],((TOP-REAL 2) | (LSeg (|[1,3]|,|[0,3]|))) such that
A273: rng fpgc = LSeg (|[1,3]|,|[0,3]|) and
A274: Ppgc = fpgc by Th43;
A275: |[1,(- 3)]| `1 = (First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))) `1 by A136, A270, Lm46, Lm59, SPPOL_1:def_3;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) is vertical by Lm28, Lm30, SPPOL_1:16;
then LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) c= LSeg (|[1,(- 3)]|,|[1,3]|) by A132, A275, Lm29, Lm31, Lm46, GOBOARD7:63;
then A276: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) c= rectangle ((- 1),1,(- 3),3) by Lm42, XBOOLE_1:1;
set K1 = ((Pdx + Palpha) + Pnpg) + Ppgc;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) misses C by A1, A53, A98, A119, A270, Lm83;
then A277: LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) misses Jc by A4, XBOOLE_1:7, XBOOLE_1:63;
A278: rng (((Pdx + Palpha) + Pnpg) + Ppgc) = (((rng Pdx) \/ (rng Palpha)) \/ (rng Pnpg)) \/ (rng Ppgc) by Lm9;
then A279: rng (((Pdx + Palpha) + Pnpg) + Ppgc) misses Jc by A109, A122, A124, A125, A130, A131, A140, A271, A272, A273, A274, A277, Lm3;
A280: rng (((Pdx + Palpha) + Pnpg) + Ppgc) = rng (- (((Pdx + Palpha) + Pnpg) + Ppgc)) by Th32;
LSeg ((First_Point (P1,((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),u,(rectangle ((- 1),1,(- 3),3)))),|[1,3]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A276, Lm67, XBOOLE_1:1;
then rng (((Pdx + Palpha) + Pnpg) + Ppgc) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A124, A125, A130, A131, A133, A134, A139, A271, A272, A273, A274, A278, Lm2;
then reconsider v = - (((Pdx + Palpha) + Pnpg) + Ppgc) as Path of CR,DR by A280, Th30;
consider s, t being Point of I[01] such that
A281: h . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A282: dom h = the carrier of I[01] by FUNCT_2:def_1;
A283: dom v = the carrier of I[01] by FUNCT_2:def_1;
A284: h . s in rng PJc by A282, FUNCT_1:def_3;
v . t in rng (- (((Pdx + Palpha) + Pnpg) + Ppgc)) by A283, FUNCT_1:def_3;
hence contradiction by A126, A127, A279, A280, A281, A284, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
end;
end;
let V be Subset of (TOP-REAL 2); ::_thesis: ( V is_inside_component_of C implies V = Ux )
assume A285: V is_inside_component_of C ; ::_thesis: V = Ux
assume A286: V <> Ux ; ::_thesis: contradiction
consider VP being Subset of ((TOP-REAL 2) | (C `)) such that
A287: VP = V and
A288: VP is a_component and
VP is bounded Subset of (Euclid 2) by A285, JORDAN2C:13;
reconsider T2C = (TOP-REAL 2) | (C `) as non empty SubSpace of TOP-REAL 2 ;
VP <> {} ((TOP-REAL 2) | (C `)) by A288, CONNSP_1:32;
then reconsider VP = VP as non empty Subset of T2C ;
A289: V misses C by A53, A287, SUBSET_1:23;
consider Pjd being Path of LMP C,|[0,(- 3)]|, fjd being Function of I[01],((TOP-REAL 2) | (LSeg ((LMP C),|[0,(- 3)]|))) such that
A290: rng fjd = LSeg ((LMP C),|[0,(- 3)]|) and
A291: Pjd = fjd by Th43;
consider Plk being Path of LMP Jc, UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd), flk being Function of I[01],((TOP-REAL 2) | (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))))) such that
A292: rng flk = LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) and
A293: Plk = flk by Th43;
set beta = (((Pcm + Pml) + Plk) + Pkj) + Pjd;
A294: rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) = ((((rng Pcm) \/ (rng Pml)) \/ (rng Plk)) \/ (rng Pkj)) \/ (rng Pjd) by Lm11;
dom ((((Pcm + Pml) + Plk) + Pkj) + Pjd) = [#] I[01] by FUNCT_2:def_1;
then ((((Pcm + Pml) + Plk) + Pkj) + Pjd) .: (dom ((((Pcm + Pml) + Plk) + Pkj) + Pjd)) is compact by WEIERSTR:8;
then A295: rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) is closed by RELAT_1:113;
A296: rng Pml misses V by A19, A29, A289, XBOOLE_1:1, XBOOLE_1:63;
{(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} c= LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} or x in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) )
assume x in {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} ; ::_thesis: x in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))
then ( x = LMP Jc or x = UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) ) by TARSKI:def_2;
hence x in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) by RLTOPSP1:68; ::_thesis: verum
end;
then A297: LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) = ((LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))}) \/ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} by XBOOLE_1:45;
A298: (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} misses V
proof
assume not (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} misses V ; ::_thesis: contradiction
then ex q being set st
( q in (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) \ {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} & q in V ) by XBOOLE_0:3;
then V meets Ux by A10, A82, XBOOLE_0:3;
hence contradiction by A10, A52, A286, A287, A288, CONNSP_1:35; ::_thesis: verum
end;
A299: not LMP Jc in V by A17, A19, A289, XBOOLE_0:3;
not UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd) in V by A18, A41, A289, XBOOLE_0:3;
then {(LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))} misses V by A299, ZFMISC_1:51;
then A300: LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) misses V by A297, A298, XBOOLE_1:70;
A301: rng Pkj misses V by A50, A289, XBOOLE_1:63;
A302: LSeg ((LMP C),|[0,(- 3)]|) misses V by A1, A285, Th90;
LSeg (|[0,3]|,(UMP C)) misses V by A1, A285, Th89;
then ((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) misses V by A296, A300, XBOOLE_1:114;
then A303: rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) misses V by A85, A86, A290, A291, A292, A293, A294, A301, A302, XBOOLE_1:114;
A304: UMP C = |[((UMP C) `1),((UMP C) `2)]| by EUCLID:53;
A305: |[0,3]| = |[(|[0,3]| `1),(|[0,3]| `2)]| by EUCLID:53;
A306: LMP C = |[((LMP C) `1),((LMP C) `2)]| by EUCLID:53;
A307: not |[(- 1),0]| in LSeg (|[0,3]|,(UMP C)) by A12, A13, A21, A23, A24, A304, A305, Lm16, JGRAPH_6:1;
not |[(- 1),0]| in rng Pml by A30, ZFMISC_1:49;
then A308: not |[(- 1),0]| in (LSeg (|[0,3]|,(UMP C))) \/ (rng Pml) by A307, XBOOLE_0:def_3;
not |[(- 1),0]| in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) by A22, A44, A56, A57, A59, Lm16, JGRAPH_6:1;
then A309: not |[(- 1),0]| in ((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) by A308, XBOOLE_0:def_3;
not |[(- 1),0]| in rng Pkj by A49, ZFMISC_1:49;
then A310: not |[(- 1),0]| in (((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))))) \/ (rng Pkj) by A309, XBOOLE_0:def_3;
not |[(- 1),0]| in LSeg ((LMP C),|[0,(- 3)]|) by A34, A35, A58, A306, Lm16, Lm22, JGRAPH_6:1;
then not |[(- 1),0]| in rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A85, A86, A290, A291, A292, A293, A294, A310, XBOOLE_0:def_3;
then consider ra being positive real number such that
A311: Ball (|[(- 1),0]|,ra) misses rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A295, Th25;
A312: not |[1,0]| in LSeg (|[0,3]|,(UMP C)) by A12, A13, A21, A23, A24, A304, A305, Lm17, JGRAPH_6:1;
not |[1,0]| in rng Pml by A30, ZFMISC_1:49;
then A313: not |[1,0]| in (LSeg (|[0,3]|,(UMP C))) \/ (rng Pml) by A312, XBOOLE_0:def_3;
not |[1,0]| in LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))) by A22, A44, A56, A57, A59, Lm17, JGRAPH_6:1;
then A314: not |[1,0]| in ((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) by A313, XBOOLE_0:def_3;
not |[1,0]| in rng Pkj by A49, ZFMISC_1:49;
then A315: not |[1,0]| in (((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))))) \/ (rng Pkj) by A314, XBOOLE_0:def_3;
not |[1,0]| in LSeg ((LMP C),|[0,(- 3)]|) by A34, A35, A58, A306, Lm17, Lm22, JGRAPH_6:1;
then not |[1,0]| in rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A85, A86, A290, A291, A292, A293, A294, A315, XBOOLE_0:def_3;
then consider rb being positive real number such that
A316: Ball (|[1,0]|,rb) misses rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A295, Th25;
set A = Ball (|[(- 1),0]|,ra);
set B = Ball (|[1,0]|,rb);
A317: |[(- 1),0]| in Ball (|[(- 1),0]|,ra) by Th16;
A318: |[1,0]| in Ball (|[1,0]|,rb) by Th16;
not VP is empty ;
then consider t being set such that
A319: t in V by A287, XBOOLE_0:def_1;
V in { W where W is Subset of (TOP-REAL 2) : W is_inside_component_of C } by A285;
then t in BDD C by A319, TARSKI:def_4;
then A320: C = Fr V by A287, A288, Lm15;
then |[(- 1),0]| in Cl V by A14, XBOOLE_0:def_4;
then Ball (|[(- 1),0]|,ra) meets V by A317, PRE_TOPC:def_7;
then consider u being set such that
A321: u in Ball (|[(- 1),0]|,ra) and
A322: u in V by XBOOLE_0:3;
|[1,0]| in Cl V by A15, A320, XBOOLE_0:def_4;
then Ball (|[1,0]|,rb) meets V by A318, PRE_TOPC:def_7;
then consider v being set such that
A323: v in Ball (|[1,0]|,rb) and
A324: v in V by XBOOLE_0:3;
reconsider u = u, v = v as Point of (TOP-REAL 2) by A321, A323;
A325: the carrier of (T2C | VP) = VP by PRE_TOPC:8;
reconsider u1 = u, v1 = v as Point of (T2C | VP) by A287, A322, A324, PRE_TOPC:8;
T2C | VP is pathwise_connected by A288, Th69;
then A326: u1,v1 are_connected by BORSUK_2:def_3;
then consider fuv being Function of I[01],(T2C | VP) such that
A327: fuv is continuous and
A328: fuv . 0 = u1 and
A329: fuv . 1 = v1 by BORSUK_2:def_1;
A330: T2C | VP = (TOP-REAL 2) | V by A287, GOBOARD9:2;
fuv is Path of u1,v1 by A326, A327, A328, A329, BORSUK_2:def_2;
then reconsider uv = fuv as Path of u,v by A326, A330, TOPALG_2:1;
A331: rng fuv c= the carrier of (T2C | VP) ;
then A332: rng uv misses rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A287, A303, A325, XBOOLE_1:63;
consider au being Path of |[(- 1),0]|,u, fau being Function of I[01],((TOP-REAL 2) | (LSeg (|[(- 1),0]|,u))) such that
A333: rng fau = LSeg (|[(- 1),0]|,u) and
A334: au = fau by Th43;
consider vb being Path of v,|[1,0]|, fvb being Function of I[01],((TOP-REAL 2) | (LSeg (v,|[1,0]|))) such that
A335: rng fvb = LSeg (v,|[1,0]|) and
A336: vb = fvb by Th43;
set AB = (au + uv) + vb;
A337: rng ((au + uv) + vb) = ((rng au) \/ (rng uv)) \/ (rng vb) by Th40;
|[(- 1),0]| in Ball (|[(- 1),0]|,ra) by Th16;
then LSeg (|[(- 1),0]|,u) c= Ball (|[(- 1),0]|,ra) by A321, JORDAN1:def_1;
then A338: LSeg (|[(- 1),0]|,u) misses rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A311, XBOOLE_1:63;
|[1,0]| in Ball (|[1,0]|,rb) by Th16;
then LSeg (v,|[1,0]|) c= Ball (|[1,0]|,rb) by A323, JORDAN1:def_1;
then LSeg (v,|[1,0]|) misses rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A316, XBOOLE_1:63;
then A339: rng ((au + uv) + vb) misses rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A332, A333, A334, A335, A336, A337, A338, XBOOLE_1:114;
A340: |[(- 1),0]|,|[1,0]| are_connected by BORSUK_2:def_3;
A341: V c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A1, A285, Th93;
then A342: LSeg (|[(- 1),0]|,u) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A14, A322, JORDAN1:def_1;
A343: LSeg (v,|[1,0]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A15, A324, A341, JORDAN1:def_1;
rng uv c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A287, A325, A331, A341, XBOOLE_1:1;
then (LSeg (|[(- 1),0]|,u)) \/ (rng uv) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A342, XBOOLE_1:8;
then rng ((au + uv) + vb) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A333, A334, A335, A336, A337, A343, XBOOLE_1:8;
then reconsider h = (au + uv) + vb as Path of AR,BR by A340, Th29;
A344: |[0,3]|,|[0,(- 3)]| are_connected by BORSUK_2:def_3;
(LSeg (|[0,3]|,(UMP C))) \/ (rng Pml) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A87, A88, XBOOLE_1:8;
then A345: ((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)))) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A83, XBOOLE_1:8;
rng Pkj c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A50, XBOOLE_1:1;
then A346: (((LSeg (|[0,3]|,(UMP C))) \/ (rng Pml)) \/ (LSeg ((LMP Jc),(UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd))))) \/ (rng Pkj) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A345, XBOOLE_1:8;
LSeg ((LMP C),|[0,(- 3)]|) c= closed_inside_of_rectangle ((- 1),1,(- 3),3) by A11, A32, Lm63, Lm67, JORDAN1:def_1;
then rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) c= the carrier of (Trectangle ((- 1),1,(- 3),3)) by A84, A85, A86, A290, A291, A292, A293, A294, A346, XBOOLE_1:8;
then reconsider v = (((Pcm + Pml) + Plk) + Pkj) + Pjd as Path of CR,DR by A344, Th29;
consider s, t being Point of I[01] such that
A347: h . s = v . t by Lm16, Lm17, Lm21, Lm23, JGRAPH_8:6;
A348: dom h = the carrier of I[01] by FUNCT_2:def_1;
A349: dom v = the carrier of I[01] by FUNCT_2:def_1;
A350: h . s in rng ((au + uv) + vb) by A348, FUNCT_1:def_3;
v . t in rng ((((Pcm + Pml) + Plk) + Pkj) + Pjd) by A349, FUNCT_1:def_3;
hence contradiction by A339, A347, A350, XBOOLE_0:3; ::_thesis: verum
end;
theorem Th96: :: JORDAN:96
for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
BDD C = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `)))
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
BDD C = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: for Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) st Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc holds
BDD C = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `)))
let Jc, Jd be compact with_the_max_arc Subset of (TOP-REAL 2); ::_thesis: ( Jc is_an_arc_of |[(- 1),0]|,|[1,0]| & Jd is_an_arc_of |[(- 1),0]|,|[1,0]| & C = Jc \/ Jd & Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} & UMP C in Jc & LMP C in Jd & W-bound C = W-bound Jc & E-bound C = E-bound Jc implies BDD C = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) )
assume that
A2: Jc is_an_arc_of |[(- 1),0]|,|[1,0]| and
A3: Jd is_an_arc_of |[(- 1),0]|,|[1,0]| and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc ; ::_thesis: BDD C = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `)))
reconsider Ux = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) as Subset of (TOP-REAL 2) by PRE_TOPC:11;
Ux = BDD C
proof
Ux is_inside_component_of C by A1, A2, A3, A4, A5, A6, A7, A8, A9, Th95;
hence Ux c= BDD C by JORDAN2C:22; :: according to XBOOLE_0:def_10 ::_thesis: BDD C c= Ux
set F = { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } ;
let q be set ; :: according to TARSKI:def_3 ::_thesis: ( not q in BDD C or q in Ux )
assume q in BDD C ; ::_thesis: q in Ux
then consider Z being set such that
A10: q in Z and
A11: Z in { B where B is Subset of (TOP-REAL 2) : B is_inside_component_of C } by TARSKI:def_4;
ex B being Subset of (TOP-REAL 2) st
( Z = B & B is_inside_component_of C ) by A11;
hence q in Ux by A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, Th95; ::_thesis: verum
end;
hence BDD C = Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) ; ::_thesis: verum
end;
Lm91: for C being Simple_closed_curve st |[(- 1),0]|,|[1,0]| realize-max-dist-in C holds
C is Jordan
proof
let C be Simple_closed_curve; ::_thesis: ( |[(- 1),0]|,|[1,0]| realize-max-dist-in C implies C is Jordan )
assume A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in C ; ::_thesis: C is Jordan
then consider Jc, Jd being compact with_the_max_arc Subset of (TOP-REAL 2) such that
A2: Jc is_an_arc_of |[(- 1),0]|,|[1,0]| and
A3: Jd is_an_arc_of |[(- 1),0]|,|[1,0]| and
A4: C = Jc \/ Jd and
A5: Jc /\ Jd = {|[(- 1),0]|,|[1,0]|} and
A6: UMP C in Jc and
A7: LMP C in Jd and
A8: W-bound C = W-bound Jc and
A9: E-bound C = E-bound Jc by Lm90;
set l = LMP Jc;
set LJ = (LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd;
set k = UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd);
set x = (1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc));
A10: Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) is a_component by CONNSP_1:40;
A11: Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) = BDD C by A1, A2, A3, A4, A5, A6, A7, A8, A9, Th96;
thus C ` <> {} ; :: according to JORDAN1:def_2 ::_thesis: ex b1, b2 being Element of bool the carrier of (TOP-REAL 2) st
( C ` = b1 \/ b2 & b1 misses b2 & (Cl b1) \ b1 = (Cl b2) \ b2 & ( for b3, b4 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b3 = b1 or not b4 = b2 or ( b3 is a_component & b4 is a_component ) ) ) )
take A1 = UBD C; ::_thesis: ex b1 being Element of bool the carrier of (TOP-REAL 2) st
( C ` = A1 \/ b1 & A1 misses b1 & (Cl A1) \ A1 = (Cl b1) \ b1 & ( for b2, b3 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b2 = A1 or not b3 = b1 or ( b2 is a_component & b3 is a_component ) ) ) )
take A2 = BDD C; ::_thesis: ( C ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for b1, b2 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b1 = A1 or not b2 = A2 or ( b1 is a_component & b2 is a_component ) ) ) )
thus C ` = A1 \/ A2 by JORDAN2C:27; ::_thesis: ( A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for b1, b2 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b1 = A1 or not b2 = A2 or ( b1 is a_component & b2 is a_component ) ) ) )
thus A1 misses A2 by JORDAN2C:24; ::_thesis: ( (Cl A1) \ A1 = (Cl A2) \ A2 & ( for b1, b2 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b1 = A1 or not b2 = A2 or ( b1 is a_component & b2 is a_component ) ) ) )
A12: Component_of (Down (((1 / 2) * ((UMP ((LSeg ((LMP Jc),|[0,(- 3)]|)) /\ Jd)) + (LMP Jc))),(C `))) <> {} ((TOP-REAL 2) | (C `)) by A10, CONNSP_1:32;
A1 is_a_component_of C ` by JORDAN2C:124;
then A13: ex B1 being Subset of ((TOP-REAL 2) | (C `)) st
( B1 = A1 & B1 is a_component ) by CONNSP_1:def_6;
then A14: C = Fr A1 by A11, A12, Lm15
.= (Cl A1) /\ (Cl (A1 `)) ;
A15: C = Fr A2 by A10, A11, A12, Lm15
.= (Cl A2) /\ (Cl (A2 `)) ;
A2 c= C ` by JORDAN2C:25;
then C misses A2 by SUBSET_1:23;
then A16: C c= (Cl A2) \ A2 by A15, XBOOLE_1:17, XBOOLE_1:86;
A17: A1 misses A2 by JORDAN2C:24;
then A2 c= A1 ` by SUBSET_1:23;
then A18: Cl A2 c= A1 ` by TOPS_1:5;
A1 \/ A2 = C ` by JORDAN2C:27;
then A1 \/ A2 misses C by SUBSET_1:23;
then C misses A1 by XBOOLE_1:70;
then A19: A2 \/ C misses A1 by A17, XBOOLE_1:70;
A2 \/ A1 = C ` by JORDAN2C:27;
then (A2 \/ A1) ` misses C ` by SUBSET_1:23;
then ((A2 \/ A1) `) /\ (C `) = {} by XBOOLE_0:def_7;
then ((A2 \/ A1) \/ C) ` = {} by XBOOLE_1:53;
then ((A2 \/ C) \/ A1) ` = {} by XBOOLE_1:4;
then ((A2 \/ C) `) /\ (A1 `) = {} by XBOOLE_1:53;
then (A2 \/ C) ` misses A1 ` by XBOOLE_0:def_7;
then Cl A2 c= A2 \/ C by A18, A19, SUBSET_1:25;
then A20: (Cl A2) \ A2 c= C by XBOOLE_1:43;
A1 c= C ` by JORDAN2C:26;
then C misses A1 by SUBSET_1:23;
then A21: C c= (Cl A1) \ A1 by A14, XBOOLE_1:17, XBOOLE_1:86;
A1 c= A2 ` by A17, SUBSET_1:23;
then A22: Cl A1 c= A2 ` by TOPS_1:5;
A2 \/ A1 = C ` by JORDAN2C:27;
then A2 \/ A1 misses C by SUBSET_1:23;
then C misses A2 by XBOOLE_1:70;
then A23: A1 \/ C misses A2 by A17, XBOOLE_1:70;
A1 \/ A2 = C ` by JORDAN2C:27;
then (A1 \/ A2) ` misses C ` by SUBSET_1:23;
then ((A1 \/ A2) `) /\ (C `) = {} by XBOOLE_0:def_7;
then ((A1 \/ A2) \/ C) ` = {} by XBOOLE_1:53;
then ((A1 \/ C) \/ A2) ` = {} by XBOOLE_1:4;
then ((A1 \/ C) `) /\ (A2 `) = {} by XBOOLE_1:53;
then (A1 \/ C) ` misses A2 ` by XBOOLE_0:def_7;
then Cl A1 c= A1 \/ C by A22, A23, SUBSET_1:25;
then (Cl A1) \ A1 c= C by XBOOLE_1:43;
hence (Cl A1) \ A1 = C by A21, XBOOLE_0:def_10
.= (Cl A2) \ A2 by A16, A20, XBOOLE_0:def_10 ;
::_thesis: for b1, b2 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b1 = A1 or not b2 = A2 or ( b1 is a_component & b2 is a_component ) )
thus for b1, b2 being Element of bool the carrier of ((TOP-REAL 2) | (C `)) holds
( not b1 = A1 or not b2 = A2 or ( b1 is a_component & b2 is a_component ) ) by A11, A13, CONNSP_1:40; ::_thesis: verum
end;
Lm92: for C being Simple_closed_curve holds C is Jordan
proof
let C be Simple_closed_curve; ::_thesis: C is Jordan
consider f being Homeomorphism of TOP-REAL 2 such that
A1: |[(- 1),0]|,|[1,0]| realize-max-dist-in f .: C by JORDAN24:7;
A2: f " is Homeomorphism of TOP-REAL 2 by TOPGRP_1:30;
f .: C is Simple_closed_curve by Th70;
then f .: C is Jordan by A1, Lm91;
then A3: (f ") .: (f .: C) is Jordan by A2, JORDAN24:16;
A4: f " = f " by TOPS_2:def_4;
dom f = the carrier of (TOP-REAL 2) by FUNCT_2:def_1;
hence C is Jordan by A3, A4, FUNCT_1:107; ::_thesis: verum
end;
registration
let C be Simple_closed_curve;
cluster BDD C -> non empty ;
coherence
not BDD C is empty
proof
C is Jordan by Lm92;
then BDD C is_inside_component_of C by JORDAN2C:108;
then BDD C is_a_component_of C ` by JORDAN2C:def_2;
then ex B1 being Subset of ((TOP-REAL 2) | (C `)) st
( B1 = BDD C & B1 is a_component ) by CONNSP_1:def_6;
then BDD C <> {} ((TOP-REAL 2) | (C `)) by CONNSP_1:32;
hence not BDD C is empty ; ::_thesis: verum
end;
end;
theorem :: JORDAN:97
for C being Simple_closed_curve
for P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `)) st U = P & U is a_component holds
C = Fr P
proof
let C be Simple_closed_curve; ::_thesis: for P being Subset of (TOP-REAL 2)
for U being Subset of ((TOP-REAL 2) | (C `)) st U = P & U is a_component holds
C = Fr P
let P be Subset of (TOP-REAL 2); ::_thesis: for U being Subset of ((TOP-REAL 2) | (C `)) st U = P & U is a_component holds
C = Fr P
let U be Subset of ((TOP-REAL 2) | (C `)); ::_thesis: ( U = P & U is a_component implies C = Fr P )
not BDD C is empty ;
hence ( U = P & U is a_component implies C = Fr P ) by Lm15; ::_thesis: verum
end;
theorem :: JORDAN:98
for C being Simple_closed_curve ex A1, A2 being Subset of (TOP-REAL 2) st
( C ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (C `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) )
proof
let C be Simple_closed_curve; ::_thesis: ex A1, A2 being Subset of (TOP-REAL 2) st
( C ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (C `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) )
C is Jordan by Lm92;
hence ex A1, A2 being Subset of (TOP-REAL 2) st
( C ` = A1 \/ A2 & A1 misses A2 & (Cl A1) \ A1 = (Cl A2) \ A2 & ( for C1, C2 being Subset of ((TOP-REAL 2) | (C `)) st C1 = A1 & C2 = A2 holds
( C1 is a_component & C2 is a_component ) ) ) by JORDAN1:def_2; ::_thesis: verum
end;
theorem :: JORDAN:99
for C being Simple_closed_curve holds C is Jordan by Lm92;