:: JORDAN1G semantic presentation
begin
registration
cluster trivial V13() V16( NAT ) Function-like V26() FinSequence-like FinSubsequence-like for set ;
existence
ex b1 being FinSequence st b1 is trivial
proof
take {} ; ::_thesis: {} is trivial
thus {} is trivial ; ::_thesis: verum
end;
end;
theorem Th1: :: JORDAN1G:1
for f being trivial FinSequence holds
( f is empty or ex x being set st f = <*x*> )
proof
let f be trivial FinSequence; ::_thesis: ( f is empty or ex x being set st f = <*x*> )
assume not f is empty ; ::_thesis: ex x being set st f = <*x*>
then consider x being set such that
A1: f = {x} by ZFMISC_1:131;
x in {x} by TARSKI:def_1;
then consider y, z being set such that
A2: x = [y,z] by A1, RELAT_1:def_1;
A3: 1 in dom f by A1, FINSEQ_5:6;
take z ; ::_thesis: f = <*z*>
dom f = {y} by A1, A2, RELAT_1:9;
then 1 = y by A3, TARSKI:def_1;
hence f = <*z*> by A1, A2, FINSEQ_1:def_5; ::_thesis: verum
end;
registration
let p be non trivial FinSequence;
cluster Rev p -> non trivial ;
coherence
not Rev p is trivial
proof
assume A1: Rev p is trivial ; ::_thesis: contradiction
percases ( Rev p is empty or ex x being set st Rev p = <*x*> ) by A1, Th1;
suppose Rev p is empty ; ::_thesis: contradiction
hence contradiction ; ::_thesis: verum
end;
suppose ex x being set st Rev p = <*x*> ; ::_thesis: contradiction
then consider x being set such that
A2: Rev p = <*x*> ;
p = Rev <*x*> by A2
.= <*x*> by FINSEQ_5:60 ;
hence contradiction ; ::_thesis: verum
end;
end;
end;
end;
theorem Th2: :: JORDAN1G:2
for D being non empty set
for f being FinSequence of D
for G being Matrix of D
for p being set st f is_sequence_on G holds
f -: p is_sequence_on G
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for G being Matrix of D
for p being set st f is_sequence_on G holds
f -: p is_sequence_on G
let f be FinSequence of D; ::_thesis: for G being Matrix of D
for p being set st f is_sequence_on G holds
f -: p is_sequence_on G
let G be Matrix of D; ::_thesis: for p being set st f is_sequence_on G holds
f -: p is_sequence_on G
let p be set ; ::_thesis: ( f is_sequence_on G implies f -: p is_sequence_on G )
assume f is_sequence_on G ; ::_thesis: f -: p is_sequence_on G
then f | (p .. f) is_sequence_on G by GOBOARD1:22;
hence f -: p is_sequence_on G by FINSEQ_5:def_1; ::_thesis: verum
end;
theorem Th3: :: JORDAN1G:3
for D being non empty set
for f being FinSequence of D
for G being Matrix of D
for p being Element of D st p in rng f & f is_sequence_on G holds
f :- p is_sequence_on G
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for G being Matrix of D
for p being Element of D st p in rng f & f is_sequence_on G holds
f :- p is_sequence_on G
let f be FinSequence of D; ::_thesis: for G being Matrix of D
for p being Element of D st p in rng f & f is_sequence_on G holds
f :- p is_sequence_on G
let G be Matrix of D; ::_thesis: for p being Element of D st p in rng f & f is_sequence_on G holds
f :- p is_sequence_on G
let p be Element of D; ::_thesis: ( p in rng f & f is_sequence_on G implies f :- p is_sequence_on G )
assume that
A1: p in rng f and
A2: f is_sequence_on G ; ::_thesis: f :- p is_sequence_on G
ex i being Element of NAT st
( i + 1 = p .. f & f :- p = f /^ i ) by A1, FINSEQ_5:49;
hence f :- p is_sequence_on G by A2, JORDAN8:2; ::_thesis: verum
end;
theorem Th4: :: JORDAN1G:4
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_sequence_on Gauge (C,n)
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_sequence_on Gauge (C,n)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Upper_Seq (C,n) is_sequence_on Gauge (C,n)
Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) is_sequence_on Gauge (C,n) by Th2;
hence Upper_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1E:def_1; ::_thesis: verum
end;
theorem Th5: :: JORDAN1G:5
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) is_sequence_on Gauge (C,n)
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) is_sequence_on Gauge (C,n)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Lower_Seq (C,n) is_sequence_on Gauge (C,n)
Cage (C,n) is_sequence_on Gauge (C,n) by JORDAN9:def_1;
then A1: Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))) is_sequence_on Gauge (C,n) by REVROT_1:34;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) is_sequence_on Gauge (C,n) by A1, Th3;
hence Lower_Seq (C,n) is_sequence_on Gauge (C,n) by JORDAN1E:def_2; ::_thesis: verum
end;
registration
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2);
let n be Element of NAT ;
cluster Upper_Seq (C,n) -> standard ;
coherence
Upper_Seq (C,n) is standard
proof
Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4;
hence Upper_Seq (C,n) is standard by JORDAN8:4; ::_thesis: verum
end;
cluster Lower_Seq (C,n) -> standard ;
coherence
Lower_Seq (C,n) is standard
proof
Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5;
hence Lower_Seq (C,n) is standard by JORDAN8:4; ::_thesis: verum
end;
end;
theorem Th6: :: JORDAN1G:6
for G being Y_equal-in-column Y_increasing-in-line Matrix of (TOP-REAL 2)
for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `2 = (G * (i2,j2)) `2 holds
j1 = j2
proof
let G be Y_equal-in-column Y_increasing-in-line Matrix of (TOP-REAL 2); ::_thesis: for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `2 = (G * (i2,j2)) `2 holds
j1 = j2
let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `2 = (G * (i2,j2)) `2 implies j1 = j2 )
assume that
A1: [i1,j1] in Indices G and
A2: [i2,j2] in Indices G and
A3: (G * (i1,j1)) `2 = (G * (i2,j2)) `2 and
A4: j1 <> j2 ; ::_thesis: contradiction
A5: ( 1 <= j1 & j1 <= width G ) by A1, MATRIX_1:38;
A6: ( j1 < j2 or j1 > j2 ) by A4, XXREAL_0:1;
A7: ( 1 <= i2 & i2 <= len G ) by A2, MATRIX_1:38;
A8: ( 1 <= j2 & j2 <= width G ) by A2, MATRIX_1:38;
A9: ( 1 <= i1 & i1 <= len G ) by A1, MATRIX_1:38;
then (G * (i1,j2)) `2 = (G * (1,j2)) `2 by A8, GOBOARD5:1
.= (G * (i2,j2)) `2 by A7, A8, GOBOARD5:1 ;
hence contradiction by A3, A9, A5, A8, A6, GOBOARD5:4; ::_thesis: verum
end;
theorem Th7: :: JORDAN1G:7
for G being X_equal-in-line X_increasing-in-column Matrix of (TOP-REAL 2)
for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `1 = (G * (i2,j2)) `1 holds
i1 = i2
proof
let G be X_equal-in-line X_increasing-in-column Matrix of (TOP-REAL 2); ::_thesis: for i1, i2, j1, j2 being Element of NAT st [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `1 = (G * (i2,j2)) `1 holds
i1 = i2
let i1, i2, j1, j2 be Element of NAT ; ::_thesis: ( [i1,j1] in Indices G & [i2,j2] in Indices G & (G * (i1,j1)) `1 = (G * (i2,j2)) `1 implies i1 = i2 )
assume that
A1: [i1,j1] in Indices G and
A2: [i2,j2] in Indices G and
A3: (G * (i1,j1)) `1 = (G * (i2,j2)) `1 and
A4: i1 <> i2 ; ::_thesis: contradiction
A5: ( 1 <= i1 & i1 <= len G ) by A1, MATRIX_1:38;
A6: ( 1 <= i2 & i2 <= len G ) by A2, MATRIX_1:38;
A7: ( 1 <= j2 & j2 <= width G ) by A2, MATRIX_1:38;
A8: ( i1 < i2 or i1 > i2 ) by A4, XXREAL_0:1;
( 1 <= j1 & j1 <= width G ) by A1, MATRIX_1:38;
then (G * (i1,j1)) `1 = (G * (i1,1)) `1 by A5, GOBOARD5:2
.= (G * (i1,j2)) `1 by A5, A7, GOBOARD5:2 ;
hence contradiction by A3, A5, A6, A7, A8, GOBOARD5:3; ::_thesis: verum
end;
theorem Th8: :: JORDAN1G:8
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) holds
(N-min (L~ f)) `1 < (N-max (L~ f)) `1
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) implies (N-min (L~ f)) `1 < (N-max (L~ f)) `1 )
set p = N-min (L~ f);
set i = (N-min (L~ f)) .. f;
assume A1: ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
A2: len f >= 2 by NAT_D:60;
A3: (N-min (L~ f)) `2 = N-bound (L~ f) by EUCLID:52;
A4: N-min (L~ f) in rng f by SPRECT_2:39;
then A5: (N-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A6: ( 1 <= (N-min (L~ f)) .. f & (N-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A7: N-min (L~ f) = f . ((N-min (L~ f)) .. f) by A4, FINSEQ_4:19
.= f /. ((N-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ;
percases ( (N-min (L~ f)) .. f = 1 or (N-min (L~ f)) .. f = len f or ( 1 < (N-min (L~ f)) .. f & (N-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1;
supposeA8: (N-min (L~ f)) .. f = 1 ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
(N-min (L~ f)) `2 = (N-max (L~ f)) `2 by PSCOMP_1:37;
then A9: (N-min (L~ f)) `1 <> (N-max (L~ f)) `1 by A1, A7, A8, TOPREAL3:6;
(N-min (L~ f)) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:38;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A9, XXREAL_0:1; ::_thesis: verum
end;
supposeA10: (N-min (L~ f)) .. f = len f ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
(N-min (L~ f)) `2 = (N-max (L~ f)) `2 by PSCOMP_1:37;
then A11: (N-min (L~ f)) `1 <> (N-max (L~ f)) `1 by A1, A7, A10, TOPREAL3:6;
(N-min (L~ f)) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:38;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A11, XXREAL_0:1; ::_thesis: verum
end;
supposethat A12: 1 < (N-min (L~ f)) .. f and
A13: (N-min (L~ f)) .. f < len f ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
A14: (((N-min (L~ f)) .. f) -' 1) + 1 = (N-min (L~ f)) .. f by A12, XREAL_1:235;
then A15: ((N-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13;
then A16: f /. (((N-min (L~ f)) .. f) -' 1) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21;
((N-min (L~ f)) .. f) -' 1 <= (N-min (L~ f)) .. f by A14, NAT_1:11;
then ((N-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2;
then A17: ((N-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25;
then A18: f /. (((N-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1;
A19: ((N-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13;
then A20: f /. (((N-min (L~ f)) .. f) + 1) in LSeg (f,((N-min (L~ f)) .. f)) by A12, TOPREAL1:21;
((N-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A21: ((N-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25;
then A22: f /. (((N-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1;
A23: N-min (L~ f) <> f /. (((N-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29;
A24: N-min (L~ f) in LSeg (f,((N-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21;
A25: N-min (L~ f) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21;
A26: N-min (L~ f) <> f /. (((N-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29;
A27: ( not LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is vertical or not LSeg (f,((N-min (L~ f)) .. f)) is vertical )
proof
assume ( LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is vertical & LSeg (f,((N-min (L~ f)) .. f)) is vertical ) ; ::_thesis: contradiction
then A28: ( (N-min (L~ f)) `1 = (f /. (((N-min (L~ f)) .. f) + 1)) `1 & (N-min (L~ f)) `1 = (f /. (((N-min (L~ f)) .. f) -' 1)) `1 ) by A25, A24, A16, A20, SPPOL_1:def_3;
A29: ( (f /. (((N-min (L~ f)) .. f) + 1)) `2 <= (f /. (((N-min (L~ f)) .. f) -' 1)) `2 or (f /. (((N-min (L~ f)) .. f) + 1)) `2 >= (f /. (((N-min (L~ f)) .. f) -' 1)) `2 ) ;
A30: ( (N-min (L~ f)) `2 >= (f /. (((N-min (L~ f)) .. f) + 1)) `2 & (N-min (L~ f)) `2 >= (f /. (((N-min (L~ f)) .. f) -' 1)) `2 ) by A3, A18, A22, PSCOMP_1:24;
( LSeg (f,((N-min (L~ f)) .. f)) = LSeg ((f /. ((N-min (L~ f)) .. f)),(f /. (((N-min (L~ f)) .. f) + 1))) & LSeg (f,(((N-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((N-min (L~ f)) .. f)),(f /. (((N-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3;
then ( f /. (((N-min (L~ f)) .. f) -' 1) in LSeg (f,((N-min (L~ f)) .. f)) or f /. (((N-min (L~ f)) .. f) + 1) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:7;
then ( f /. (((N-min (L~ f)) .. f) -' 1) in (LSeg (f,(((N-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((N-min (L~ f)) .. f))) or f /. (((N-min (L~ f)) .. f) + 1) in (LSeg (f,(((N-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((N-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4;
then ( ((((N-min (L~ f)) .. f) -' 1) + 1) + 1 = (((N-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((N-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((N-min (L~ f)) .. f))) <> {(f /. ((N-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1;
hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum
end;
now__::_thesis:_(N-min_(L~_f))_`1_<_(N-max_(L~_f))_`1
percases ( LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is horizontal or LSeg (f,((N-min (L~ f)) .. f)) is horizontal ) by A27, SPPOL_1:19;
suppose LSeg (f,(((N-min (L~ f)) .. f) -' 1)) is horizontal ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
then A31: (N-min (L~ f)) `2 = (f /. (((N-min (L~ f)) .. f) -' 1)) `2 by A25, A16, SPPOL_1:def_2;
then A32: f /. (((N-min (L~ f)) .. f) -' 1) in N-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:10;
then A33: (f /. (((N-min (L~ f)) .. f) -' 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39;
(f /. (((N-min (L~ f)) .. f) -' 1)) `1 <> (N-min (L~ f)) `1 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6;
then A34: (f /. (((N-min (L~ f)) .. f) -' 1)) `1 > (N-min (L~ f)) `1 by A33, XXREAL_0:1;
(f /. (((N-min (L~ f)) .. f) -' 1)) `1 <= (N-max (L~ f)) `1 by A32, PSCOMP_1:39;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A34, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,((N-min (L~ f)) .. f)) is horizontal ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
then A35: (N-min (L~ f)) `2 = (f /. (((N-min (L~ f)) .. f) + 1)) `2 by A24, A20, SPPOL_1:def_2;
then A36: f /. (((N-min (L~ f)) .. f) + 1) in N-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:10;
then A37: (f /. (((N-min (L~ f)) .. f) + 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39;
(f /. (((N-min (L~ f)) .. f) + 1)) `1 <> (N-min (L~ f)) `1 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6;
then A38: (f /. (((N-min (L~ f)) .. f) + 1)) `1 > (N-min (L~ f)) `1 by A37, XXREAL_0:1;
(f /. (((N-min (L~ f)) .. f) + 1)) `1 <= (N-max (L~ f)) `1 by A36, PSCOMP_1:39;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A38, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 ; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1G:9
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) holds
N-min (L~ f) <> N-max (L~ f)
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) implies N-min (L~ f) <> N-max (L~ f) )
assume ( ( f /. 1 <> N-min (L~ f) & f /. (len f) <> N-min (L~ f) ) or ( f /. 1 <> N-max (L~ f) & f /. (len f) <> N-max (L~ f) ) ) ; ::_thesis: N-min (L~ f) <> N-max (L~ f)
then (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by Th8;
hence N-min (L~ f) <> N-max (L~ f) ; ::_thesis: verum
end;
theorem Th10: :: JORDAN1G:10
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) holds
(S-min (L~ f)) `1 < (S-max (L~ f)) `1
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) implies (S-min (L~ f)) `1 < (S-max (L~ f)) `1 )
set p = S-min (L~ f);
set i = (S-min (L~ f)) .. f;
assume A1: ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
A2: len f >= 2 by NAT_D:60;
A3: (S-min (L~ f)) `2 = S-bound (L~ f) by EUCLID:52;
A4: S-min (L~ f) in rng f by SPRECT_2:41;
then A5: (S-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A6: ( 1 <= (S-min (L~ f)) .. f & (S-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A7: S-min (L~ f) = f . ((S-min (L~ f)) .. f) by A4, FINSEQ_4:19
.= f /. ((S-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ;
percases ( (S-min (L~ f)) .. f = 1 or (S-min (L~ f)) .. f = len f or ( 1 < (S-min (L~ f)) .. f & (S-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1;
supposeA8: (S-min (L~ f)) .. f = 1 ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
(S-min (L~ f)) `2 = (S-max (L~ f)) `2 by PSCOMP_1:53;
then A9: (S-min (L~ f)) `1 <> (S-max (L~ f)) `1 by A1, A7, A8, TOPREAL3:6;
(S-min (L~ f)) `1 <= (S-max (L~ f)) `1 by PSCOMP_1:54;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A9, XXREAL_0:1; ::_thesis: verum
end;
supposeA10: (S-min (L~ f)) .. f = len f ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
(S-min (L~ f)) `2 = (S-max (L~ f)) `2 by PSCOMP_1:53;
then A11: (S-min (L~ f)) `1 <> (S-max (L~ f)) `1 by A1, A7, A10, TOPREAL3:6;
(S-min (L~ f)) `1 <= (S-max (L~ f)) `1 by PSCOMP_1:54;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A11, XXREAL_0:1; ::_thesis: verum
end;
supposethat A12: 1 < (S-min (L~ f)) .. f and
A13: (S-min (L~ f)) .. f < len f ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
A14: (((S-min (L~ f)) .. f) -' 1) + 1 = (S-min (L~ f)) .. f by A12, XREAL_1:235;
then A15: ((S-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13;
then A16: f /. (((S-min (L~ f)) .. f) -' 1) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21;
((S-min (L~ f)) .. f) -' 1 <= (S-min (L~ f)) .. f by A14, NAT_1:11;
then ((S-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2;
then A17: ((S-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25;
then A18: f /. (((S-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1;
A19: ((S-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13;
then A20: f /. (((S-min (L~ f)) .. f) + 1) in LSeg (f,((S-min (L~ f)) .. f)) by A12, TOPREAL1:21;
((S-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A21: ((S-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25;
then A22: f /. (((S-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1;
A23: S-min (L~ f) <> f /. (((S-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29;
A24: S-min (L~ f) in LSeg (f,((S-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21;
A25: S-min (L~ f) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21;
A26: S-min (L~ f) <> f /. (((S-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29;
A27: ( not LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is vertical or not LSeg (f,((S-min (L~ f)) .. f)) is vertical )
proof
assume ( LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is vertical & LSeg (f,((S-min (L~ f)) .. f)) is vertical ) ; ::_thesis: contradiction
then A28: ( (S-min (L~ f)) `1 = (f /. (((S-min (L~ f)) .. f) + 1)) `1 & (S-min (L~ f)) `1 = (f /. (((S-min (L~ f)) .. f) -' 1)) `1 ) by A25, A24, A16, A20, SPPOL_1:def_3;
A29: ( (f /. (((S-min (L~ f)) .. f) + 1)) `2 <= (f /. (((S-min (L~ f)) .. f) -' 1)) `2 or (f /. (((S-min (L~ f)) .. f) + 1)) `2 >= (f /. (((S-min (L~ f)) .. f) -' 1)) `2 ) ;
A30: ( (S-min (L~ f)) `2 <= (f /. (((S-min (L~ f)) .. f) + 1)) `2 & (S-min (L~ f)) `2 <= (f /. (((S-min (L~ f)) .. f) -' 1)) `2 ) by A3, A18, A22, PSCOMP_1:24;
( LSeg (f,((S-min (L~ f)) .. f)) = LSeg ((f /. ((S-min (L~ f)) .. f)),(f /. (((S-min (L~ f)) .. f) + 1))) & LSeg (f,(((S-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((S-min (L~ f)) .. f)),(f /. (((S-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3;
then ( f /. (((S-min (L~ f)) .. f) -' 1) in LSeg (f,((S-min (L~ f)) .. f)) or f /. (((S-min (L~ f)) .. f) + 1) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:7;
then ( f /. (((S-min (L~ f)) .. f) -' 1) in (LSeg (f,(((S-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((S-min (L~ f)) .. f))) or f /. (((S-min (L~ f)) .. f) + 1) in (LSeg (f,(((S-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((S-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4;
then ( ((((S-min (L~ f)) .. f) -' 1) + 1) + 1 = (((S-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((S-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((S-min (L~ f)) .. f))) <> {(f /. ((S-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1;
hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum
end;
now__::_thesis:_(S-min_(L~_f))_`1_<_(S-max_(L~_f))_`1
percases ( LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is horizontal or LSeg (f,((S-min (L~ f)) .. f)) is horizontal ) by A27, SPPOL_1:19;
suppose LSeg (f,(((S-min (L~ f)) .. f) -' 1)) is horizontal ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
then A31: (S-min (L~ f)) `2 = (f /. (((S-min (L~ f)) .. f) -' 1)) `2 by A25, A16, SPPOL_1:def_2;
then A32: f /. (((S-min (L~ f)) .. f) -' 1) in S-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:11;
then A33: (f /. (((S-min (L~ f)) .. f) -' 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55;
(f /. (((S-min (L~ f)) .. f) -' 1)) `1 <> (S-min (L~ f)) `1 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6;
then A34: (f /. (((S-min (L~ f)) .. f) -' 1)) `1 > (S-min (L~ f)) `1 by A33, XXREAL_0:1;
(f /. (((S-min (L~ f)) .. f) -' 1)) `1 <= (S-max (L~ f)) `1 by A32, PSCOMP_1:55;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A34, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,((S-min (L~ f)) .. f)) is horizontal ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
then A35: (S-min (L~ f)) `2 = (f /. (((S-min (L~ f)) .. f) + 1)) `2 by A24, A20, SPPOL_1:def_2;
then A36: f /. (((S-min (L~ f)) .. f) + 1) in S-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:11;
then A37: (f /. (((S-min (L~ f)) .. f) + 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55;
(f /. (((S-min (L~ f)) .. f) + 1)) `1 <> (S-min (L~ f)) `1 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6;
then A38: (f /. (((S-min (L~ f)) .. f) + 1)) `1 > (S-min (L~ f)) `1 by A37, XXREAL_0:1;
(f /. (((S-min (L~ f)) .. f) + 1)) `1 <= (S-max (L~ f)) `1 by A36, PSCOMP_1:55;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A38, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 ; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1G:11
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) holds
S-min (L~ f) <> S-max (L~ f)
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) implies S-min (L~ f) <> S-max (L~ f) )
assume ( ( f /. 1 <> S-min (L~ f) & f /. (len f) <> S-min (L~ f) ) or ( f /. 1 <> S-max (L~ f) & f /. (len f) <> S-max (L~ f) ) ) ; ::_thesis: S-min (L~ f) <> S-max (L~ f)
then (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by Th10;
hence S-min (L~ f) <> S-max (L~ f) ; ::_thesis: verum
end;
theorem Th12: :: JORDAN1G:12
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) holds
(W-min (L~ f)) `2 < (W-max (L~ f)) `2
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) implies (W-min (L~ f)) `2 < (W-max (L~ f)) `2 )
set p = W-min (L~ f);
set i = (W-min (L~ f)) .. f;
assume A1: ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
A2: len f >= 2 by NAT_D:60;
A3: (W-min (L~ f)) `1 = W-bound (L~ f) by EUCLID:52;
A4: W-min (L~ f) in rng f by SPRECT_2:43;
then A5: (W-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A6: ( 1 <= (W-min (L~ f)) .. f & (W-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A7: W-min (L~ f) = f . ((W-min (L~ f)) .. f) by A4, FINSEQ_4:19
.= f /. ((W-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ;
percases ( (W-min (L~ f)) .. f = 1 or (W-min (L~ f)) .. f = len f or ( 1 < (W-min (L~ f)) .. f & (W-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1;
supposeA8: (W-min (L~ f)) .. f = 1 ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
(W-min (L~ f)) `1 = (W-max (L~ f)) `1 by PSCOMP_1:29;
then A9: (W-min (L~ f)) `2 <> (W-max (L~ f)) `2 by A1, A7, A8, TOPREAL3:6;
(W-min (L~ f)) `2 <= (W-max (L~ f)) `2 by PSCOMP_1:30;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A9, XXREAL_0:1; ::_thesis: verum
end;
supposeA10: (W-min (L~ f)) .. f = len f ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
(W-min (L~ f)) `1 = (W-max (L~ f)) `1 by PSCOMP_1:29;
then A11: (W-min (L~ f)) `2 <> (W-max (L~ f)) `2 by A1, A7, A10, TOPREAL3:6;
(W-min (L~ f)) `2 <= (W-max (L~ f)) `2 by PSCOMP_1:30;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A11, XXREAL_0:1; ::_thesis: verum
end;
supposethat A12: 1 < (W-min (L~ f)) .. f and
A13: (W-min (L~ f)) .. f < len f ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
A14: (((W-min (L~ f)) .. f) -' 1) + 1 = (W-min (L~ f)) .. f by A12, XREAL_1:235;
then A15: ((W-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13;
then A16: f /. (((W-min (L~ f)) .. f) -' 1) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21;
((W-min (L~ f)) .. f) -' 1 <= (W-min (L~ f)) .. f by A14, NAT_1:11;
then ((W-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2;
then A17: ((W-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25;
then A18: f /. (((W-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1;
A19: ((W-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13;
then A20: f /. (((W-min (L~ f)) .. f) + 1) in LSeg (f,((W-min (L~ f)) .. f)) by A12, TOPREAL1:21;
((W-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A21: ((W-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25;
then A22: f /. (((W-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1;
A23: W-min (L~ f) <> f /. (((W-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29;
A24: W-min (L~ f) in LSeg (f,((W-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21;
A25: W-min (L~ f) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21;
A26: W-min (L~ f) <> f /. (((W-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29;
A27: ( not LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is horizontal or not LSeg (f,((W-min (L~ f)) .. f)) is horizontal )
proof
assume ( LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is horizontal & LSeg (f,((W-min (L~ f)) .. f)) is horizontal ) ; ::_thesis: contradiction
then A28: ( (W-min (L~ f)) `2 = (f /. (((W-min (L~ f)) .. f) + 1)) `2 & (W-min (L~ f)) `2 = (f /. (((W-min (L~ f)) .. f) -' 1)) `2 ) by A25, A24, A16, A20, SPPOL_1:def_2;
A29: ( (f /. (((W-min (L~ f)) .. f) + 1)) `1 <= (f /. (((W-min (L~ f)) .. f) -' 1)) `1 or (f /. (((W-min (L~ f)) .. f) + 1)) `1 >= (f /. (((W-min (L~ f)) .. f) -' 1)) `1 ) ;
A30: ( (W-min (L~ f)) `1 <= (f /. (((W-min (L~ f)) .. f) + 1)) `1 & (W-min (L~ f)) `1 <= (f /. (((W-min (L~ f)) .. f) -' 1)) `1 ) by A3, A18, A22, PSCOMP_1:24;
( LSeg (f,((W-min (L~ f)) .. f)) = LSeg ((f /. ((W-min (L~ f)) .. f)),(f /. (((W-min (L~ f)) .. f) + 1))) & LSeg (f,(((W-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((W-min (L~ f)) .. f)),(f /. (((W-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3;
then ( f /. (((W-min (L~ f)) .. f) -' 1) in LSeg (f,((W-min (L~ f)) .. f)) or f /. (((W-min (L~ f)) .. f) + 1) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:8;
then ( f /. (((W-min (L~ f)) .. f) -' 1) in (LSeg (f,(((W-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((W-min (L~ f)) .. f))) or f /. (((W-min (L~ f)) .. f) + 1) in (LSeg (f,(((W-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((W-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4;
then ( ((((W-min (L~ f)) .. f) -' 1) + 1) + 1 = (((W-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((W-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((W-min (L~ f)) .. f))) <> {(f /. ((W-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1;
hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum
end;
now__::_thesis:_(W-min_(L~_f))_`2_<_(W-max_(L~_f))_`2
percases ( LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is vertical or LSeg (f,((W-min (L~ f)) .. f)) is vertical ) by A27, SPPOL_1:19;
suppose LSeg (f,(((W-min (L~ f)) .. f) -' 1)) is vertical ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
then A31: (W-min (L~ f)) `1 = (f /. (((W-min (L~ f)) .. f) -' 1)) `1 by A25, A16, SPPOL_1:def_3;
then A32: f /. (((W-min (L~ f)) .. f) -' 1) in W-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:12;
then A33: (f /. (((W-min (L~ f)) .. f) -' 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31;
(f /. (((W-min (L~ f)) .. f) -' 1)) `2 <> (W-min (L~ f)) `2 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6;
then A34: (f /. (((W-min (L~ f)) .. f) -' 1)) `2 > (W-min (L~ f)) `2 by A33, XXREAL_0:1;
(f /. (((W-min (L~ f)) .. f) -' 1)) `2 <= (W-max (L~ f)) `2 by A32, PSCOMP_1:31;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A34, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,((W-min (L~ f)) .. f)) is vertical ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
then A35: (W-min (L~ f)) `1 = (f /. (((W-min (L~ f)) .. f) + 1)) `1 by A24, A20, SPPOL_1:def_3;
then A36: f /. (((W-min (L~ f)) .. f) + 1) in W-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:12;
then A37: (f /. (((W-min (L~ f)) .. f) + 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31;
(f /. (((W-min (L~ f)) .. f) + 1)) `2 <> (W-min (L~ f)) `2 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6;
then A38: (f /. (((W-min (L~ f)) .. f) + 1)) `2 > (W-min (L~ f)) `2 by A37, XXREAL_0:1;
(f /. (((W-min (L~ f)) .. f) + 1)) `2 <= (W-max (L~ f)) `2 by A36, PSCOMP_1:31;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A38, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 ; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1G:13
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) holds
W-min (L~ f) <> W-max (L~ f)
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) implies W-min (L~ f) <> W-max (L~ f) )
assume ( ( f /. 1 <> W-min (L~ f) & f /. (len f) <> W-min (L~ f) ) or ( f /. 1 <> W-max (L~ f) & f /. (len f) <> W-max (L~ f) ) ) ; ::_thesis: W-min (L~ f) <> W-max (L~ f)
then (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by Th12;
hence W-min (L~ f) <> W-max (L~ f) ; ::_thesis: verum
end;
theorem Th14: :: JORDAN1G:14
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) holds
(E-min (L~ f)) `2 < (E-max (L~ f)) `2
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) implies (E-min (L~ f)) `2 < (E-max (L~ f)) `2 )
set p = E-min (L~ f);
set i = (E-min (L~ f)) .. f;
assume A1: ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
A2: len f >= 2 by NAT_D:60;
A3: (E-min (L~ f)) `1 = E-bound (L~ f) by EUCLID:52;
A4: E-min (L~ f) in rng f by SPRECT_2:45;
then A5: (E-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A6: ( 1 <= (E-min (L~ f)) .. f & (E-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A7: E-min (L~ f) = f . ((E-min (L~ f)) .. f) by A4, FINSEQ_4:19
.= f /. ((E-min (L~ f)) .. f) by A5, PARTFUN1:def_6 ;
percases ( (E-min (L~ f)) .. f = 1 or (E-min (L~ f)) .. f = len f or ( 1 < (E-min (L~ f)) .. f & (E-min (L~ f)) .. f < len f ) ) by A6, XXREAL_0:1;
supposeA8: (E-min (L~ f)) .. f = 1 ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
(E-min (L~ f)) `1 = (E-max (L~ f)) `1 by PSCOMP_1:45;
then A9: (E-min (L~ f)) `2 <> (E-max (L~ f)) `2 by A1, A7, A8, TOPREAL3:6;
(E-min (L~ f)) `2 <= (E-max (L~ f)) `2 by PSCOMP_1:46;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A9, XXREAL_0:1; ::_thesis: verum
end;
supposeA10: (E-min (L~ f)) .. f = len f ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
(E-min (L~ f)) `1 = (E-max (L~ f)) `1 by PSCOMP_1:45;
then A11: (E-min (L~ f)) `2 <> (E-max (L~ f)) `2 by A1, A7, A10, TOPREAL3:6;
(E-min (L~ f)) `2 <= (E-max (L~ f)) `2 by PSCOMP_1:46;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A11, XXREAL_0:1; ::_thesis: verum
end;
supposethat A12: 1 < (E-min (L~ f)) .. f and
A13: (E-min (L~ f)) .. f < len f ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
A14: (((E-min (L~ f)) .. f) -' 1) + 1 = (E-min (L~ f)) .. f by A12, XREAL_1:235;
then A15: ((E-min (L~ f)) .. f) -' 1 >= 1 by A12, NAT_1:13;
then A16: f /. (((E-min (L~ f)) .. f) -' 1) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) by A13, A14, TOPREAL1:21;
((E-min (L~ f)) .. f) -' 1 <= (E-min (L~ f)) .. f by A14, NAT_1:11;
then ((E-min (L~ f)) .. f) -' 1 <= len f by A13, XXREAL_0:2;
then A17: ((E-min (L~ f)) .. f) -' 1 in dom f by A15, FINSEQ_3:25;
then A18: f /. (((E-min (L~ f)) .. f) -' 1) in L~ f by A2, GOBOARD1:1;
A19: ((E-min (L~ f)) .. f) + 1 <= len f by A13, NAT_1:13;
then A20: f /. (((E-min (L~ f)) .. f) + 1) in LSeg (f,((E-min (L~ f)) .. f)) by A12, TOPREAL1:21;
((E-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A21: ((E-min (L~ f)) .. f) + 1 in dom f by A19, FINSEQ_3:25;
then A22: f /. (((E-min (L~ f)) .. f) + 1) in L~ f by A2, GOBOARD1:1;
A23: E-min (L~ f) <> f /. (((E-min (L~ f)) .. f) + 1) by A4, A7, A21, FINSEQ_4:20, GOBOARD7:29;
A24: E-min (L~ f) in LSeg (f,((E-min (L~ f)) .. f)) by A7, A12, A19, TOPREAL1:21;
A25: E-min (L~ f) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) by A7, A13, A14, A15, TOPREAL1:21;
A26: E-min (L~ f) <> f /. (((E-min (L~ f)) .. f) -' 1) by A5, A7, A14, A17, GOBOARD7:29;
A27: ( not LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is horizontal or not LSeg (f,((E-min (L~ f)) .. f)) is horizontal )
proof
assume ( LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is horizontal & LSeg (f,((E-min (L~ f)) .. f)) is horizontal ) ; ::_thesis: contradiction
then A28: ( (E-min (L~ f)) `2 = (f /. (((E-min (L~ f)) .. f) + 1)) `2 & (E-min (L~ f)) `2 = (f /. (((E-min (L~ f)) .. f) -' 1)) `2 ) by A25, A24, A16, A20, SPPOL_1:def_2;
A29: ( (f /. (((E-min (L~ f)) .. f) + 1)) `1 <= (f /. (((E-min (L~ f)) .. f) -' 1)) `1 or (f /. (((E-min (L~ f)) .. f) + 1)) `1 >= (f /. (((E-min (L~ f)) .. f) -' 1)) `1 ) ;
A30: ( (E-min (L~ f)) `1 >= (f /. (((E-min (L~ f)) .. f) + 1)) `1 & (E-min (L~ f)) `1 >= (f /. (((E-min (L~ f)) .. f) -' 1)) `1 ) by A3, A18, A22, PSCOMP_1:24;
( LSeg (f,((E-min (L~ f)) .. f)) = LSeg ((f /. ((E-min (L~ f)) .. f)),(f /. (((E-min (L~ f)) .. f) + 1))) & LSeg (f,(((E-min (L~ f)) .. f) -' 1)) = LSeg ((f /. ((E-min (L~ f)) .. f)),(f /. (((E-min (L~ f)) .. f) -' 1))) ) by A12, A13, A14, A15, A19, TOPREAL1:def_3;
then ( f /. (((E-min (L~ f)) .. f) -' 1) in LSeg (f,((E-min (L~ f)) .. f)) or f /. (((E-min (L~ f)) .. f) + 1) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) ) by A7, A28, A30, A29, GOBOARD7:8;
then ( f /. (((E-min (L~ f)) .. f) -' 1) in (LSeg (f,(((E-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((E-min (L~ f)) .. f))) or f /. (((E-min (L~ f)) .. f) + 1) in (LSeg (f,(((E-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((E-min (L~ f)) .. f))) ) by A16, A20, XBOOLE_0:def_4;
then ( ((((E-min (L~ f)) .. f) -' 1) + 1) + 1 = (((E-min (L~ f)) .. f) -' 1) + (1 + 1) & (LSeg (f,(((E-min (L~ f)) .. f) -' 1))) /\ (LSeg (f,((E-min (L~ f)) .. f))) <> {(f /. ((E-min (L~ f)) .. f))} ) by A7, A26, A23, TARSKI:def_1;
hence contradiction by A14, A15, A19, TOPREAL1:def_6; ::_thesis: verum
end;
now__::_thesis:_(E-min_(L~_f))_`2_<_(E-max_(L~_f))_`2
percases ( LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is vertical or LSeg (f,((E-min (L~ f)) .. f)) is vertical ) by A27, SPPOL_1:19;
suppose LSeg (f,(((E-min (L~ f)) .. f) -' 1)) is vertical ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
then A31: (E-min (L~ f)) `1 = (f /. (((E-min (L~ f)) .. f) -' 1)) `1 by A25, A16, SPPOL_1:def_3;
then A32: f /. (((E-min (L~ f)) .. f) -' 1) in E-most (L~ f) by A2, A3, A17, GOBOARD1:1, SPRECT_2:13;
then A33: (f /. (((E-min (L~ f)) .. f) -' 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47;
(f /. (((E-min (L~ f)) .. f) -' 1)) `2 <> (E-min (L~ f)) `2 by A5, A7, A14, A17, A31, GOBOARD7:29, TOPREAL3:6;
then A34: (f /. (((E-min (L~ f)) .. f) -' 1)) `2 > (E-min (L~ f)) `2 by A33, XXREAL_0:1;
(f /. (((E-min (L~ f)) .. f) -' 1)) `2 <= (E-max (L~ f)) `2 by A32, PSCOMP_1:47;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A34, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,((E-min (L~ f)) .. f)) is vertical ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
then A35: (E-min (L~ f)) `1 = (f /. (((E-min (L~ f)) .. f) + 1)) `1 by A24, A20, SPPOL_1:def_3;
then A36: f /. (((E-min (L~ f)) .. f) + 1) in E-most (L~ f) by A2, A3, A21, GOBOARD1:1, SPRECT_2:13;
then A37: (f /. (((E-min (L~ f)) .. f) + 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47;
(f /. (((E-min (L~ f)) .. f) + 1)) `2 <> (E-min (L~ f)) `2 by A5, A7, A21, A35, GOBOARD7:29, TOPREAL3:6;
then A38: (f /. (((E-min (L~ f)) .. f) + 1)) `2 > (E-min (L~ f)) `2 by A37, XXREAL_0:1;
(f /. (((E-min (L~ f)) .. f) + 1)) `2 <= (E-max (L~ f)) `2 by A36, PSCOMP_1:47;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A38, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 ; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1G:15
for f being non trivial special unfolded standard FinSequence of (TOP-REAL 2) st ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) holds
E-min (L~ f) <> E-max (L~ f)
proof
let f be non trivial special unfolded standard FinSequence of (TOP-REAL 2); ::_thesis: ( ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) implies E-min (L~ f) <> E-max (L~ f) )
assume ( ( f /. 1 <> E-min (L~ f) & f /. (len f) <> E-min (L~ f) ) or ( f /. 1 <> E-max (L~ f) & f /. (len f) <> E-max (L~ f) ) ) ; ::_thesis: E-min (L~ f) <> E-max (L~ f)
then (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by Th14;
hence E-min (L~ f) <> E-max (L~ f) ; ::_thesis: verum
end;
theorem Th16: :: JORDAN1G:16
for D being non empty set
for f being FinSequence of D
for p, q being Element of D st p in rng f & q in rng f & q .. f <= p .. f holds
(f -: p) :- q = (f :- q) -: p
proof
let D be non empty set ; ::_thesis: for f being FinSequence of D
for p, q being Element of D st p in rng f & q in rng f & q .. f <= p .. f holds
(f -: p) :- q = (f :- q) -: p
let f be FinSequence of D; ::_thesis: for p, q being Element of D st p in rng f & q in rng f & q .. f <= p .. f holds
(f -: p) :- q = (f :- q) -: p
let p, q be Element of D; ::_thesis: ( p in rng f & q in rng f & q .. f <= p .. f implies (f -: p) :- q = (f :- q) -: p )
assume that
A1: p in rng f and
A2: q in rng f and
A3: q .. f <= p .. f ; ::_thesis: (f -: p) :- q = (f :- q) -: p
A4: ( f -: p = f | (p .. f) & (f :- q) -: p = (f :- q) | (p .. (f :- q)) ) by FINSEQ_5:def_1;
consider i being Element of NAT such that
A5: i + 1 = q .. f and
A6: f :- q = f /^ i by A2, FINSEQ_5:49;
A7: i < p .. f by A3, A5, NAT_1:13;
then p .. f = i + (p .. (f /^ i)) by A1, FINSEQ_6:56;
then A8: p .. (f /^ i) = (p .. f) - i
.= (p .. f) -' i by A7, XREAL_1:233 ;
q in rng (f -: p) by A1, A2, A3, FINSEQ_5:46;
then A9: ex j being Element of NAT st
( j + 1 = q .. (f -: p) & (f -: p) :- q = (f -: p) /^ j ) by FINSEQ_5:49;
q .. (f -: p) = q .. f by A1, A2, A3, SPRECT_5:3;
hence (f -: p) :- q = (f :- q) -: p by A5, A6, A9, A4, A8, FINSEQ_5:80; ::_thesis: verum
end;
theorem Th17: :: JORDAN1G:17
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT holds (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))}
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))}
let n be Element of NAT ; ::_thesis: (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))}
set US = (Cage (C,n)) -: (W-min (L~ (Cage (C,n))));
set LS = (Cage (C,n)) :- (W-min (L~ (Cage (C,n))));
set f = Cage (C,n);
set pW = W-min (L~ (Cage (C,n)));
set pN = N-min (L~ (Cage (C,n)));
set pNa = N-max (L~ (Cage (C,n)));
set pSa = S-max (L~ (Cage (C,n)));
set pSi = S-min (L~ (Cage (C,n)));
set pEa = E-max (L~ (Cage (C,n)));
set pEi = E-min (L~ (Cage (C,n)));
A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A2: (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47;
len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, FINSEQ_5:42;
then ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) = W-min (L~ (Cage (C,n))) by A1, FINSEQ_5:45;
then A3: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A2, REVROT_1:3;
A4: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:71;
then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A4, SPRECT_2:70, XXREAL_0:2;
then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A4, SPRECT_2:72, XXREAL_0:2;
then A5: (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A4, SPRECT_2:73, XXREAL_0:2;
((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A1, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A6: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A2, FINSEQ_6:42;
( N-max (L~ (Cage (C,n))) in rng (Cage (C,n)) & (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) ) by A4, SPRECT_2:40, SPRECT_2:74;
then A7: N-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A1, A5, FINSEQ_5:46, XXREAL_0:2;
{(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A6, A7, A3, ENUMSET1:def_1; ::_thesis: verum
end;
then A8: card {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53;
then A9: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A1, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A10: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A10, A9, TARSKI:def_2; ::_thesis: verum
end;
then A11: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then A12: card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62;
( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
then A13: N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57;
then card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57;
then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A11, A12, XBOOLE_1:1;
then len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A14: rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by SPPOL_2:18;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A1, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A15: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3;
(W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) ;
then A16: ( W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) & W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) ) by A1, FINSEQ_5:46, FINSEQ_6:61;
( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-max (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
then ( N-min (L~ (Cage (C,n))) <> N-max (L~ (Cage (C,n))) & N-max (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) ) by SPRECT_2:52, SPRECT_2:57;
then A17: card {(N-min (L~ (Cage (C,n)))),(N-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 3 by A13, CARD_2:58;
card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by CARD_1:62;
then 3 c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A17, A8, XBOOLE_1:1;
then A18: len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) >= 3 by NAT_1:39;
then A19: rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by SPPOL_2:18, XXREAL_0:2;
thus (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} :: according to XBOOLE_0:def_10 ::_thesis: {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) or x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} )
assume A20: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) ; ::_thesis: x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))}
then reconsider x1 = x as Point of (TOP-REAL 2) ;
assume A21: not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: contradiction
x in L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A20, XBOOLE_0:def_4;
then consider i1 being Element of NAT such that
A22: 1 <= i1 and
A23: i1 + 1 <= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) and
A24: x1 in LSeg (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))),i1) by SPPOL_2:13;
A25: LSeg (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))),i1) = LSeg ((Cage (C,n)),i1) by A23, SPPOL_2:9;
x in L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A20, XBOOLE_0:def_4;
then consider i2 being Element of NAT such that
A26: 1 <= i2 and
A27: i2 + 1 <= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) and
A28: x1 in LSeg (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))),i2) by SPPOL_2:13;
set i3 = i2 -' 1;
A29: (i2 -' 1) + 1 = i2 by A26, XREAL_1:235;
then A30: 1 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= ((i2 -' 1) + 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A26, XREAL_1:7;
A31: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) = ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A1, FINSEQ_5:50;
then i2 < ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A27, NAT_1:13;
then i2 - 1 < (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:19;
then A32: (i2 - 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20;
i2 - 1 >= 1 - 1 by A26, XREAL_1:9;
then A33: (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by A32, XREAL_0:def_2;
A34: LSeg (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))),i2) = LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A1, A29, SPPOL_2:10;
A35: len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, FINSEQ_5:42;
then i1 + 1 < ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 by A23, NAT_1:13;
then i1 + 1 < ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A30, XXREAL_0:2;
then A36: i1 + 1 <= (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by NAT_1:13;
A37: (((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + 1 = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A1, FINSEQ_4:21, XREAL_1:235;
(i2 -' 1) + 1 < ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 by A27, A29, A31, NAT_1:13;
then i2 -' 1 < (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7;
then A38: (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) < len (Cage (C,n)) by XREAL_1:20;
then A39: ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= len (Cage (C,n)) by NAT_1:13;
now__::_thesis:_contradiction
percases ( ( i1 + 1 < (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) or i1 = 1 or i1 + 1 = (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) ) by A22, A36, XXREAL_0:1;
suppose ( i1 + 1 < (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) & i1 > 1 ) ; ::_thesis: contradiction
then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A38, GOBOARD5:def_4;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7;
hence contradiction by A24, A28, A25, A34, XBOOLE_0:def_4; ::_thesis: verum
end;
supposeA40: i1 = 1 ; ::_thesis: contradiction
(i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) >= 0 + 3 by A18, A35, XREAL_1:7;
then A41: i1 + 1 < (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A40, XXREAL_0:2;
now__::_thesis:_contradiction
percases ( ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) or ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ) by A39, XXREAL_0:1;
suppose ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 < len (Cage (C,n)) ; ::_thesis: contradiction
then LSeg ((Cage (C,n)),i1) misses LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) by A41, GOBOARD5:def_4;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {} by XBOOLE_0:def_7;
hence contradiction by A24, A28, A25, A34, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose ((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 = len (Cage (C,n)) ; ::_thesis: contradiction
then (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) - 1 ;
then (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) = (len (Cage (C,n))) -' 1 by XREAL_0:def_2;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. 1)} by A40, GOBOARD7:34, REVROT_1:30;
then x1 in {((Cage (C,n)) /. 1)} by A24, A28, A25, A34, XBOOLE_0:def_4;
then x1 = (Cage (C,n)) /. 1 by TARSKI:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
hence contradiction by A21, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA42: i1 + 1 = (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) ; ::_thesis: contradiction
(i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) >= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by NAT_1:11;
then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A33, XXREAL_0:2;
then ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
then A43: (((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) -' 1) + (1 + 1) <= len (Cage (C,n)) by A37;
0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:7;
then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) = i1 + 1 by A23, A35, A42, XXREAL_0:1;
then (LSeg ((Cage (C,n)),i1)) /\ (LSeg ((Cage (C,n)),((i2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))))) = {((Cage (C,n)) /. ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A22, A37, A42, A43, TOPREAL1:def_6;
then x1 in {((Cage (C,n)) /. ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))} by A24, A28, A25, A34, XBOOLE_0:def_4;
then x1 = (Cage (C,n)) /. ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by TARSKI:def_1
.= W-min (L~ (Cage (C,n))) by A1, FINSEQ_5:38 ;
hence contradiction by A21, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
A44: ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A1, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
not (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) is empty by A17, A8, NAT_1:39;
then A45: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A44, FINSEQ_6:42;
thus {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) )
assume A46: x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))))
percases ( x = N-min (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A46, TARSKI:def_2;
suppose x = N-min (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))))
hence x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A14, A19, A45, A15, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))))
hence x in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A14, A19, A16, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
end;
theorem Th18: :: JORDAN1G:18
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))
set Nmi = N-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Wma = W-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Emi = E-min (L~ (Cage (C,n)));
set Sma = S-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set RotWmi = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
set RotEma = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57;
then A2: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57;
A3: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A4: (Cage (C,n)) -: (W-min (L~ (Cage (C,n)))) <> {} by FINSEQ_5:47;
len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A3, FINSEQ_5:42;
then ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) = W-min (L~ (Cage (C,n))) by A3, FINSEQ_5:45;
then A5: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A4, REVROT_1:3;
((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /. 1 = (Cage (C,n)) /. 1 by A3, FINSEQ_5:44
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A6: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A4, FINSEQ_6:42;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A6, A5, TARSKI:def_2; ::_thesis: verum
end;
then A7: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A2, A7, XBOOLE_1:1;
then len ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A8: rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by SPPOL_2:18;
A9: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:72;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:71, XXREAL_0:2;
then A10: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:73, XXREAL_0:2;
then A11: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:74, XXREAL_0:2;
A12: (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:74;
then A13: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A3, A1, A10, FINSEQ_5:46, XXREAL_0:2;
(N-max (L~ (Cage (C,n)))) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:38;
then ( (N-min (L~ (Cage (C,n)))) `1 < (N-max (L~ (Cage (C,n)))) `1 & (N-max (L~ (Cage (C,n)))) `1 <= E-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_2:51;
then A14: N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by EUCLID:52;
A15: not E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
proof
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53;
then A16: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A3, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A17: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A17, A16, TARSKI:def_2; ::_thesis: verum
end;
then A18: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57;
then A19: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57;
card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A19, A18, XBOOLE_1:1;
then len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A20: rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) c= L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by SPPOL_2:18;
assume E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ; ::_thesis: contradiction
then E-max (L~ (Cage (C,n))) in (L~ ((Cage (C,n)) -: (W-min (L~ (Cage (C,n)))))) /\ (L~ ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A13, A8, A20, XBOOLE_0:def_4;
then E-max (L~ (Cage (C,n))) in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} by Th17;
then E-max (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) by A14, TARSKI:def_2;
hence contradiction by TOPREAL5:19; ::_thesis: verum
end;
A21: (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:70;
A22: (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A9, SPRECT_2:68;
then A23: ( N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) ) by A9, SPRECT_2:39, SPRECT_2:70, XXREAL_0:2;
then A24: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) by A3, A11, FINSEQ_5:46, XXREAL_0:2;
A25: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) <> 1
proof
assume A26: (E-max (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = 1 ; ::_thesis: contradiction
(N-min (L~ (Cage (C,n)))) .. ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) = (N-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A3, A23, A11, SPRECT_5:3, XXREAL_0:2
.= 1 by A9, FINSEQ_6:43 ;
hence contradiction by A22, A21, A13, A24, A26, FINSEQ_5:9; ::_thesis: verum
end;
then E-max (L~ (Cage (C,n))) in rng (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1) by A13, FINSEQ_6:78;
then A27: E-max (L~ (Cage (C,n))) in (rng (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1)) \ (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by A15, XBOOLE_0:def_5;
A28: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A3, A1, A12, A10, FINSEQ_6:62, XXREAL_0:2;
(Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) = (((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1)) :- (E-max (L~ (Cage (C,n)))) by A3, FINSEQ_6:def_2
.= (((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) /^ 1) :- (E-max (L~ (Cage (C,n)))) by A27, FINSEQ_6:65
.= ((Cage (C,n)) -: (W-min (L~ (Cage (C,n))))) :- (E-max (L~ (Cage (C,n)))) by A13, A25, FINSEQ_6:83
.= ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) -: (W-min (L~ (Cage (C,n)))) by A3, A1, A12, A10, Th16, XXREAL_0:2
.= (((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) ^ (((Cage (C,n)) -: (E-max (L~ (Cage (C,n))))) /^ 1)) -: (W-min (L~ (Cage (C,n)))) by A28, FINSEQ_6:66
.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by A1, FINSEQ_6:def_2 ;
hence Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by JORDAN1E:def_2; ::_thesis: verum
end;
theorem Th19: :: JORDAN1G:19
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1
(Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
hence (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 by FINSEQ_6:43; ::_thesis: verum
end;
theorem Th20: :: JORDAN1G:20
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
set Wmi = W-min (L~ (Cage (C,n)));
set Wma = W-max (L~ (Cage (C,n)));
set Nmi = N-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A2: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:43;
A3: W-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:44;
A4: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A5: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
then A6: (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_5:21;
A7: ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A1, A5, JORDAN1E:def_1, SPRECT_5:25;
(N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:24;
then A8: (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:23, XXREAL_0:2;
then (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:25, XXREAL_0:2;
then (W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, A4, A6, SPRECT_5:3, XXREAL_0:2;
hence (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A4, A6, A7, A8, A3, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th21: :: JORDAN1G:21
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
set Wmi = W-min (L~ (Cage (C,n)));
set Wma = W-max (L~ (Cage (C,n)));
set Nmi = N-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A2: W-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:44;
A3: N-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:39;
A4: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A5: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
then A6: (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_5:23;
A7: ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A1, A5, JORDAN1E:def_1, SPRECT_5:25;
A8: (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:24;
then (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, A5, SPRECT_5:25, XXREAL_0:2;
then (W-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (W-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A2, A4, A6, SPRECT_5:3, XXREAL_0:2;
hence (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A4, A6, A8, A7, A3, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th22: :: JORDAN1G:22
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
set Wmi = W-min (L~ (Cage (C,n)));
set Nmi = N-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
then A3: ( (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A1, SPRECT_5:24, SPRECT_5:25;
A4: N-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:40;
N-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:39;
then ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A2, A3, JORDAN1E:def_1, SPRECT_5:3, XXREAL_0:2;
hence (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A2, A3, A4, SPRECT_5:3; ::_thesis: verum
end;
theorem Th23: :: JORDAN1G:23
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))
set Wmi = W-min (L~ (Cage (C,n)));
set Nma = N-max (L~ (Cage (C,n)));
set Ema = E-max (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
A1: L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
then A3: (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_5:25;
N-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, SPRECT_2:40;
then ( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (N-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) = (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A2, A3, JORDAN1E:def_1, SPRECT_5:3;
hence (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by A2, A3, SPRECT_5:3; ::_thesis: verum
end;
theorem Th24: :: JORDAN1G:24
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n))
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n))
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by JORDAN1E:def_1;
then E-max (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_5:46;
then A2: Upper_Seq (C,n) just_once_values E-max (L~ (Cage (C,n))) by FINSEQ_4:8;
(Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
hence (E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by A2, REVROT_1:1; ::_thesis: verum
end;
theorem Th25: :: JORDAN1G:25
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1
(Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) by JORDAN1F:6;
hence (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 by FINSEQ_6:43; ::_thesis: verum
end;
theorem Th26: :: JORDAN1G:26
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
set Ema = E-max (L~ (Cage (C,n)));
set Emi = E-min (L~ (Cage (C,n)));
set Sma = S-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18;
A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A3: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:46;
A4: E-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:45;
A5: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A6: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
then A7: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:37;
A8: (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41;
(S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:40;
then A9: (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:39, XXREAL_0:2;
then (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A5, A7, SPRECT_5:3, XXREAL_0:2;
hence (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A5, A7, A8, A9, A4, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th27: :: JORDAN1G:27
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
set Ema = E-max (L~ (Cage (C,n)));
set Emi = E-min (L~ (Cage (C,n)));
set Sma = S-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18;
A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A3: E-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:45;
A4: S-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:42;
A5: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A6: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
then A7: (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:39;
A8: (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41;
A9: (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:40;
then (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, A6, SPRECT_5:41, XXREAL_0:2;
then (E-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (E-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A5, A7, SPRECT_5:3, XXREAL_0:2;
hence (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A5, A7, A9, A8, A4, SPRECT_5:3, XXREAL_0:2; ::_thesis: verum
end;
theorem Th28: :: JORDAN1G:28
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
set Ema = E-max (L~ (Cage (C,n)));
set Sma = S-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18;
A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A3: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
then A4: ( (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) & (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) ) by A2, SPRECT_5:40, SPRECT_5:41;
A5: S-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:41;
S-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:42;
then (S-max (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A4, SPRECT_5:3, XXREAL_0:2;
hence (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A3, A4, A5, SPRECT_5:3; ::_thesis: verum
end;
theorem Th29: :: JORDAN1G:29
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))
set Ema = E-max (L~ (Cage (C,n)));
set Smi = S-min (L~ (Cage (C,n)));
set Wmi = W-min (L~ (Cage (C,n)));
set Rot = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
A1: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) by Th18;
A2: L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = L~ (Cage (C,n)) by REVROT_1:33;
then A3: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
then A4: (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_5:41;
S-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A2, SPRECT_2:41;
then (S-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) = (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A3, A4, SPRECT_5:3;
hence (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by A1, A3, A4, SPRECT_5:3; ::_thesis: verum
end;
theorem Th30: :: JORDAN1G:30
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n))
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A1: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;
( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) & (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) ) by Th18;
then W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A1, FINSEQ_5:46;
then A2: Lower_Seq (C,n) just_once_values W-min (L~ (Cage (C,n))) by FINSEQ_4:8;
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8;
hence (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) by A2, REVROT_1:1; ::_thesis: verum
end;
theorem Th31: :: JORDAN1G:31
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n)))
set Ca = Cage (C,n);
set US = Upper_Seq (C,n);
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Nmin = N-min (L~ (Cage (C,n)));
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 2 by XXREAL_0:2;
then 2 in Seg (len (Upper_Seq (C,n))) by FINSEQ_1:1;
then A2: 2 in Seg ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by JORDAN1E:8;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_5:53;
then A3: W-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by FINSEQ_6:42;
(Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by SPRECT_2:76;
then A4: ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
( W-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (N-min (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ (Cage (C,n)))) `2 <= (N-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:24;
then N-min (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) by SPRECT_2:57;
then A5: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} = 2 by CARD_2:57;
A6: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A7: 1 <= (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:21;
((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A6, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A8: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A8, A3, TARSKI:def_2; ::_thesis: verum
end;
then A9: card {(N-min (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) :- (W-min (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) by A5, A9, XBOOLE_1:1;
then A10: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A11: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2;
A12: (Upper_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_1
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:44
.= (Cage (C,n)) /. ((1 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A6, A11, REVROT_1:9
.= (Cage (C,n)) /. (0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:232 ;
(Upper_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. 2 by JORDAN1E:def_1
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:43
.= (Cage (C,n)) /. ((2 -' 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A6, A10, REVROT_1:9
.= (Cage (C,n)) /. ((2 - 1) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def_2 ;
hence ((Upper_Seq (C,n)) /. 2) `1 = W-bound (L~ (Cage (C,n))) by A7, A4, A12, JORDAN1E:22, JORDAN1F:5; ::_thesis: verum
end;
theorem :: JORDAN1G:32
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n)))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n)))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n)))
set Ca = Cage (C,n);
set LS = Lower_Seq (C,n);
set Emax = E-max (L~ (Cage (C,n)));
set Emin = E-min (L~ (Cage (C,n)));
set Smax = S-max (L~ (Cage (C,n)));
set Smin = S-min (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
set Nmin = N-min (L~ (Cage (C,n)));
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A1: W-min (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:46;
len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Lower_Seq (C,n)) >= 2 by XXREAL_0:2;
then 2 <= (W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) by Th30;
then 2 <= (W-min (L~ (Cage (C,n)))) .. ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) by Th18;
then 2 <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by A1, FINSEQ_6:72;
then A2: 2 in Seg ((W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n))))))) by FINSEQ_1:1;
((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_5:53;
then A3: E-max (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by FINSEQ_6:42;
( N-max (L~ (Cage (C,n))) in L~ (Cage (C,n)) & (E-max (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:11;
then (N-max (L~ (Cage (C,n)))) `1 <= (E-max (L~ (Cage (C,n)))) `1 by PSCOMP_1:24;
then N-min (L~ (Cage (C,n))) <> E-max (L~ (Cage (C,n))) by SPRECT_2:51;
then A4: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} = 2 by CARD_2:57;
A5: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:71;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:72, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:73, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A5, SPRECT_2:74, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A5, SPRECT_2:76, XXREAL_0:2;
then A6: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
A7: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A8: 1 <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by FINSEQ_4:21;
((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) /. (len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) = (Cage (C,n)) /. (len (Cage (C,n))) by A7, FINSEQ_5:54
.= (Cage (C,n)) /. 1 by FINSEQ_6:def_1
.= N-min (L~ (Cage (C,n))) by JORDAN9:32 ;
then A9: N-min (L~ (Cage (C,n))) in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by REVROT_1:3;
{(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} or x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) )
assume x in {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} ; ::_thesis: x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))
hence x in rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A9, A3, TARSKI:def_2; ::_thesis: verum
end;
then A10: card {(N-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} c= card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_1:11;
card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= card (dom ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) by CARD_2:61;
then card (rng ((Cage (C,n)) :- (E-max (L~ (Cage (C,n)))))) c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by CARD_1:62;
then 2 c= len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) by A4, A10, XBOOLE_1:1;
then A11: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 2 by NAT_1:39;
then A12: len ((Cage (C,n)) :- (E-max (L~ (Cage (C,n))))) >= 1 by XXREAL_0:2;
A13: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 1 by Th18
.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 by A1, FINSEQ_5:44
.= (Cage (C,n)) /. ((1 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A12, REVROT_1:9
.= (Cage (C,n)) /. (0 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_1:232 ;
(Lower_Seq (C,n)) /. 2 = ((Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n))))) /. 2 by Th18
.= (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 2 by A1, A2, FINSEQ_5:43
.= (Cage (C,n)) /. ((2 -' 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by A7, A11, REVROT_1:9
.= (Cage (C,n)) /. ((2 - 1) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) by XREAL_0:def_2 ;
hence ((Lower_Seq (C,n)) /. 2) `1 = E-bound (L~ (Cage (C,n))) by A8, A6, A13, JORDAN1E:20, JORDAN1F:6; ::_thesis: verum
end;
theorem Th33: :: JORDAN1G:33
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound C) + (E-bound C)
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound C) + (E-bound C)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound C) + (E-bound C)
thus (W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n)))) = (W-bound (L~ (Cage (C,n)))) + ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) by JORDAN1A:64
.= ((W-bound C) - (((E-bound C) - (W-bound C)) / (2 |^ n))) + ((E-bound C) + (((E-bound C) - (W-bound C)) / (2 |^ n))) by JORDAN1A:62
.= (W-bound C) + (E-bound C) ; ::_thesis: verum
end;
theorem :: JORDAN1G:34
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C)
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2) holds (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C)
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound C) + (N-bound C)
thus (S-bound (L~ (Cage (C,n)))) + (N-bound (L~ (Cage (C,n)))) = (S-bound (L~ (Cage (C,n)))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN10:6
.= ((S-bound C) - (((N-bound C) - (S-bound C)) / (2 |^ n))) + ((N-bound C) + (((N-bound C) - (S-bound C)) / (2 |^ n))) by JORDAN1A:63
.= (S-bound C) + (N-bound C) ; ::_thesis: verum
end;
theorem Th35: :: JORDAN1G:35
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT
for i being Nat st 1 <= i & i <= width (Gauge (C,n)) & n > 0 holds
((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT
for i being Nat st 1 <= i & i <= width (Gauge (C,n)) & n > 0 holds
((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
let n be Element of NAT ; ::_thesis: for i being Nat st 1 <= i & i <= width (Gauge (C,n)) & n > 0 holds
((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
let i be Nat; ::_thesis: ( 1 <= i & i <= width (Gauge (C,n)) & n > 0 implies ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 )
assume A1: ( 1 <= i & i <= width (Gauge (C,n)) ) ; ::_thesis: ( not n > 0 or ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2 )
reconsider ii = i as Element of NAT by ORDINAL1:def_12;
A2: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
assume A3: n > 0 ; ::_thesis: ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((W-bound C) + (E-bound C)) / 2
len (Gauge (C,1)) >= 4 by JORDAN8:10;
then A4: len (Gauge (C,1)) >= 1 by XXREAL_0:2;
thus ((Gauge (C,n)) * ((Center (Gauge (C,n))),i)) `1 = ((Gauge (C,n)) * ((Center (Gauge (C,n))),ii)) `1
.= ((Gauge (C,1)) * ((Center (Gauge (C,1))),1)) `1 by A1, A2, A4, A3, JORDAN1A:36
.= ((W-bound C) + (E-bound C)) / 2 by A4, JORDAN1A:38 ; ::_thesis: verum
end;
theorem :: JORDAN1G:36
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n, i being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & n > 0 holds
((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n, i being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & n > 0 holds
((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2
let n, i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & n > 0 implies ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 )
assume A1: ( 1 <= i & i <= len (Gauge (C,n)) ) ; ::_thesis: ( not n > 0 or ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2 )
len (Gauge (C,1)) >= 4 by JORDAN8:10;
then A2: len (Gauge (C,1)) >= 1 by XXREAL_0:2;
assume n > 0 ; ::_thesis: ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((S-bound C) + (N-bound C)) / 2
hence ((Gauge (C,n)) * (i,(Center (Gauge (C,n))))) `2 = ((Gauge (C,1)) * (1,(Center (Gauge (C,1))))) `2 by A1, A2, JORDAN1A:37
.= ((S-bound C) + (N-bound C)) / 2 by A2, JORDAN1A:39 ;
::_thesis: verum
end;
theorem Th37: :: JORDAN1G:37
for f being S-Sequence_in_R2
for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. 1 in L~ (mid (f,k1,k2)) & not k1 = 1 holds
k2 = 1
proof
let f be S-Sequence_in_R2; ::_thesis: for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. 1 in L~ (mid (f,k1,k2)) & not k1 = 1 holds
k2 = 1
let k1, k2 be Element of NAT ; ::_thesis: ( 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. 1 in L~ (mid (f,k1,k2)) & not k1 = 1 implies k2 = 1 )
assume that
A1: 1 <= k1 and
A2: k1 <= len f and
A3: 1 <= k2 and
A4: k2 <= len f and
A5: f /. 1 in L~ (mid (f,k1,k2)) ; ::_thesis: ( k1 = 1 or k2 = 1 )
assume that
A6: k1 <> 1 and
A7: k2 <> 1 ; ::_thesis: contradiction
A8: len f >= 2 by TOPREAL1:def_8;
consider j being Element of NAT such that
A9: 1 <= j and
A10: j + 1 <= len (mid (f,k1,k2)) and
A11: f /. 1 in LSeg ((mid (f,k1,k2)),j) by A5, SPPOL_2:13;
percases ( k1 < k2 or k1 > k2 or k1 = k2 ) by XXREAL_0:1;
supposeA12: k1 < k2 ; ::_thesis: contradiction
then len (mid (f,k1,k2)) = (k2 -' k1) + 1 by A1, A2, A3, A4, FINSEQ_6:118;
then j < (k2 -' k1) + 1 by A10, NAT_1:13;
then LSeg ((mid (f,k1,k2)),j) = LSeg (f,((j + k1) -' 1)) by A1, A4, A9, A12, JORDAN4:19;
then A13: (j + k1) -' 1 = 1 by A11, A8, JORDAN5B:30;
j + k1 >= 1 + 1 by A1, A9, XREAL_1:7;
then (j + k1) - 1 >= (1 + 1) - 1 by XREAL_1:9;
then j + (k1 - 1) = 1 by A13, XREAL_0:def_2;
then k1 - 1 = 1 - j ;
then k1 - 1 <= 0 by A9, XREAL_1:47;
then k1 - 1 = 0 by A1, XREAL_1:48;
hence contradiction by A6; ::_thesis: verum
end;
supposeA14: k1 > k2 ; ::_thesis: contradiction
then len (mid (f,k1,k2)) = (k1 -' k2) + 1 by A1, A2, A3, A4, FINSEQ_6:118;
then A15: j < (k1 -' k2) + 1 by A10, NAT_1:13;
k1 - k2 > 0 by A14, XREAL_1:50;
then k1 -' k2 = k1 - k2 by XREAL_0:def_2;
then j - 1 < k1 - k2 by A15, XREAL_1:19;
then (j - 1) + k2 < k1 by XREAL_1:20;
then j + (- (1 - k2)) < k1 ;
then A16: k2 - 1 < k1 - j by XREAL_1:20;
LSeg ((mid (f,k1,k2)),j) = LSeg (f,(k1 -' j)) by A2, A3, A9, A14, A15, JORDAN4:20;
then k1 -' j = 1 by A11, A8, JORDAN5B:30;
then k1 - j = 1 by XREAL_0:def_2;
then k2 < 1 + 1 by A16, XREAL_1:19;
then k2 <= 1 by NAT_1:13;
hence contradiction by A3, A7, XXREAL_0:1; ::_thesis: verum
end;
suppose k1 = k2 ; ::_thesis: contradiction
then mid (f,k1,k2) = <*(f /. k1)*> by A1, A2, JORDAN4:15;
hence contradiction by A5, SPPOL_2:12; ::_thesis: verum
end;
end;
end;
theorem Th38: :: JORDAN1G:38
for f being S-Sequence_in_R2
for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. (len f) in L~ (mid (f,k1,k2)) & not k1 = len f holds
k2 = len f
proof
let f be S-Sequence_in_R2; ::_thesis: for k1, k2 being Element of NAT st 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. (len f) in L~ (mid (f,k1,k2)) & not k1 = len f holds
k2 = len f
let k1, k2 be Element of NAT ; ::_thesis: ( 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & f /. (len f) in L~ (mid (f,k1,k2)) & not k1 = len f implies k2 = len f )
assume that
A1: 1 <= k1 and
A2: k1 <= len f and
A3: 1 <= k2 and
A4: k2 <= len f and
A5: f /. (len f) in L~ (mid (f,k1,k2)) ; ::_thesis: ( k1 = len f or k2 = len f )
assume that
A6: k1 <> len f and
A7: k2 <> len f ; ::_thesis: contradiction
consider j being Element of NAT such that
A8: 1 <= j and
A9: j + 1 <= len (mid (f,k1,k2)) and
A10: f /. (len f) in LSeg ((mid (f,k1,k2)),j) by A5, SPPOL_2:13;
percases ( k1 < k2 or k1 > k2 or k1 = k2 ) by XXREAL_0:1;
supposeA11: k1 < k2 ; ::_thesis: contradiction
then A12: len (mid (f,k1,k2)) = (k2 -' k1) + 1 by A1, A2, A3, A4, FINSEQ_6:118;
then A13: j < (k2 -' k1) + 1 by A9, NAT_1:13;
A14: j + k1 >= 1 + 1 by A1, A8, XREAL_1:7;
then A15: (j + k1) - 1 >= (1 + 1) - 1 by XREAL_1:9;
then A16: (j + k1) -' 1 = (j + k1) - 1 by XREAL_0:def_2;
k2 - k1 > 0 by A11, XREAL_1:50;
then A17: k2 -' k1 = k2 - k1 by XREAL_0:def_2;
then j - 1 < k2 - k1 by A13, XREAL_1:19;
then (j - 1) + k1 < k2 by XREAL_1:20;
then A18: (j + k1) - 1 < len f by A4, XXREAL_0:2;
then A19: (j + k1) -' 1 in dom f by A15, A16, FINSEQ_3:25;
A20: j + k1 >= 1 by A14, XXREAL_0:2;
((j + k1) - 1) + 1 <= len f by A16, A18, NAT_1:13;
then j + k1 in Seg (len f) by A20, FINSEQ_1:1;
then A21: ((j + k1) -' 1) + 1 in dom f by A16, FINSEQ_1:def_3;
LSeg ((mid (f,k1,k2)),j) = LSeg (f,((j + k1) -' 1)) by A1, A4, A8, A11, A13, JORDAN4:19;
then A22: ((j + k1) -' 1) + 1 = len f by A10, A19, A21, GOBOARD2:2;
A23: (j + k1) -' 1 = (j + k1) - 1 by A15, XREAL_0:def_2;
j < (k2 + 1) - k1 by A9, A17, A12, NAT_1:13;
then len f < k2 + 1 by A22, A23, XREAL_1:20;
then len f <= k2 by NAT_1:13;
hence contradiction by A4, A7, XXREAL_0:1; ::_thesis: verum
end;
supposeA24: k1 > k2 ; ::_thesis: contradiction
then len (mid (f,k1,k2)) = (k1 -' k2) + 1 by A1, A2, A3, A4, FINSEQ_6:118;
then A25: j < (k1 -' k2) + 1 by A9, NAT_1:13;
k1 - k2 > 0 by A24, XREAL_1:50;
then k1 -' k2 = k1 - k2 by XREAL_0:def_2;
then j - 1 < k1 - k2 by A25, XREAL_1:19;
then (j - 1) + k2 < k1 by XREAL_1:20;
then A26: j + (- (1 - k2)) < k1 ;
then A27: - (1 - k2) < k1 - j by XREAL_1:20;
A28: k2 - 1 >= 0 by A3, XREAL_1:48;
then A29: (k1 - j) + 1 > 0 + 1 by A27, XREAL_1:6;
k2 - 1 < k1 - j by A26, XREAL_1:20;
then A30: k1 - j > 0 by A3, XREAL_1:48;
then A31: k1 -' j = k1 - j by XREAL_0:def_2;
k1 - j <= k1 - 1 by A8, XREAL_1:10;
then (k1 - j) + 1 <= (k1 - 1) + 1 by XREAL_1:7;
then k1 - j < k1 by A31, NAT_1:13;
then A32: k1 - j < len f by A2, XXREAL_0:2;
then (k1 - j) + 1 <= len f by A31, NAT_1:13;
then A33: (k1 -' j) + 1 in dom f by A31, A29, FINSEQ_3:25;
k1 - j >= 0 + 1 by A27, A28, A31, NAT_1:13;
then A34: k1 -' j in dom f by A31, A32, FINSEQ_3:25;
LSeg ((mid (f,k1,k2)),j) = LSeg (f,(k1 -' j)) by A2, A3, A8, A24, A25, JORDAN4:20;
then (k1 -' j) + 1 = len f by A10, A34, A33, GOBOARD2:2;
then A35: (k1 - j) + 1 = len f by A30, XREAL_0:def_2;
k1 - j <= k1 - 1 by A8, XREAL_1:10;
then len f <= (k1 - 1) + 1 by A35, XREAL_1:7;
hence contradiction by A2, A6, XXREAL_0:1; ::_thesis: verum
end;
suppose k1 = k2 ; ::_thesis: contradiction
then mid (f,k1,k2) = <*(f /. k1)*> by A1, A2, JORDAN4:15;
hence contradiction by A5, SPPOL_2:12; ::_thesis: verum
end;
end;
end;
theorem Th39: :: JORDAN1G:39
for C being compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT holds
( rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) & rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) )
proof
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds
( rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) & rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) )
let n be Element of NAT ; ::_thesis: ( rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) & rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) )
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A1: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
then rng (Upper_Seq (C,n)) c= rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_5:48;
hence rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; ::_thesis: rng (Lower_Seq (C,n)) c= rng (Cage (C,n))
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2;
then rng (Lower_Seq (C,n)) c= rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A1, FINSEQ_5:55;
hence rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by FINSEQ_6:90, SPRECT_2:43; ::_thesis: verum
end;
theorem Th40: :: JORDAN1G:40
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_a_h.c._for Cage (C,n)
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Upper_Seq (C,n) is_a_h.c._for Cage (C,n)
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Upper_Seq (C,n) is_a_h.c._for Cage (C,n)
A1: ((Upper_Seq (C,n)) /. 1) `1 = (W-min (L~ (Cage (C,n)))) `1 by JORDAN1F:5
.= W-bound (L~ (Cage (C,n))) by EUCLID:52 ;
A2: ((Upper_Seq (C,n)) /. (len (Upper_Seq (C,n)))) `1 = (E-max (L~ (Cage (C,n)))) `1 by JORDAN1F:7
.= E-bound (L~ (Cage (C,n))) by EUCLID:52 ;
Upper_Seq (C,n) is_in_the_area_of Cage (C,n) by JORDAN1E:17;
hence Upper_Seq (C,n) is_a_h.c._for Cage (C,n) by A1, A2, SPRECT_2:def_2; ::_thesis: verum
end;
theorem Th41: :: JORDAN1G:41
for n being Element of NAT
for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n)
proof
let n be Element of NAT ; ::_thesis: for C being compact non horizontal non vertical Subset of (TOP-REAL 2) holds Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n)
let C be compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n)
A1: ((Rev (Lower_Seq (C,n))) /. 1) `1 = ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 by FINSEQ_5:65
.= (W-min (L~ (Cage (C,n)))) `1 by JORDAN1F:8
.= W-bound (L~ (Cage (C,n))) by EUCLID:52 ;
A2: ((Rev (Lower_Seq (C,n))) /. (len (Rev (Lower_Seq (C,n))))) `1 = ((Rev (Lower_Seq (C,n))) /. (len (Lower_Seq (C,n)))) `1 by FINSEQ_5:def_3
.= ((Lower_Seq (C,n)) /. 1) `1 by FINSEQ_5:65
.= (E-max (L~ (Cage (C,n)))) `1 by JORDAN1F:6
.= E-bound (L~ (Cage (C,n))) by EUCLID:52 ;
Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51;
hence Rev (Lower_Seq (C,n)) is_a_h.c._for Cage (C,n) by A1, A2, SPRECT_2:def_2; ::_thesis: verum
end;
theorem Th42: :: JORDAN1G:42
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n))
let i be Element of NAT ; ::_thesis: ( 1 < i & i <= len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) )
assume that
A1: ( 1 < i & i <= len (Gauge (C,n)) ) and
A2: (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) ; ::_thesis: contradiction
consider i2 being Nat such that
A3: i2 in dom (Upper_Seq (C,n)) and
A4: (Upper_Seq (C,n)) . i2 = (Gauge (C,n)) * (i,1) by A2, FINSEQ_2:10;
reconsider i2 = i2 as Element of NAT by ORDINAL1:def_12;
A5: ( 1 <= i2 & i2 <= len (Upper_Seq (C,n)) ) by A3, FINSEQ_3:25;
set f = Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))));
set i1 = (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n));
A6: ( E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) & rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = rng (Cage (C,n)) ) by FINSEQ_6:90, SPRECT_2:43, SPRECT_2:46;
W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A7: (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 = W-min (L~ (Cage (C,n))) by FINSEQ_6:92;
L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by REVROT_1:33;
then A8: ( (N-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) < (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) & (N-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by A7, SPRECT_5:24, SPRECT_5:25;
(E-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by Th24;
then (N-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by Th23;
then A9: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) < len (Upper_Seq (C,n)) by Th22, XXREAL_0:2;
3 <= len (Lower_Seq (C,n)) by JORDAN1E:15;
then A10: 2 <= len (Lower_Seq (C,n)) by XXREAL_0:2;
A11: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
4 <= len (Gauge (C,n)) by JORDAN8:10;
then A12: 1 <= len (Gauge (C,n)) by XXREAL_0:2;
( (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = 1 & (W-max (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <= (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) ) by Th19, Th21;
then A13: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) > 1 by Th20, XXREAL_0:2;
then A14: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
( Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) & N-min (L~ (Cage (C,n))) in rng (Cage (C,n)) ) by JORDAN1E:def_1, SPRECT_2:39;
then A15: N-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A6, A8, FINSEQ_5:46, XXREAL_0:2;
then A16: (Upper_Seq (C,n)) /. ((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))) = N-min (L~ (Cage (C,n))) by FINSEQ_5:38;
A17: (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) <> i2
proof
assume (N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) = i2 ; ::_thesis: contradiction
then (Gauge (C,n)) * (i,1) = N-min (L~ (Cage (C,n))) by A4, A14, A16, PARTFUN1:def_6;
then ((Gauge (C,n)) * (i,1)) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
then S-bound (L~ (Cage (C,n))) = N-bound (L~ (Cage (C,n))) by A1, JORDAN1A:72;
hence contradiction by SPRECT_1:16; ::_thesis: verum
end;
then mid ((Upper_Seq (C,n)),((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))),i2) is being_S-Seq by A13, A9, A5, JORDAN3:6;
then reconsider h1 = mid ((Upper_Seq (C,n)),((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))),i2) as one-to-one special FinSequence of (TOP-REAL 2) ;
set h = Rev h1;
A18: len h1 = len (Rev h1) by FINSEQ_5:def_3;
then A19: not h1 is empty by A3, A14, SPRECT_2:5;
then A20: ((Rev h1) /. (len (Rev h1))) `2 = (h1 /. 1) `2 by A18, FINSEQ_5:65
.= ((Upper_Seq (C,n)) /. ((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)))) `2 by A3, A14, SPRECT_2:8
.= (N-min (L~ (Cage (C,n)))) `2 by A15, FINSEQ_5:38
.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;
h1 is_in_the_area_of Cage (C,n) by A3, A14, JORDAN1E:17, SPRECT_2:22;
then A21: Rev h1 is_in_the_area_of Cage (C,n) by SPRECT_3:51;
((Rev h1) /. 1) `2 = (h1 /. (len h1)) `2 by A19, FINSEQ_5:65
.= ((Upper_Seq (C,n)) /. i2) `2 by A3, A14, SPRECT_2:9
.= ((Gauge (C,n)) * (i,1)) `2 by A3, A4, PARTFUN1:def_6
.= S-bound (L~ (Cage (C,n))) by A1, JORDAN1A:72 ;
then A22: ( Rev (Lower_Seq (C,n)) is special & Rev h1 is_a_v.c._for Cage (C,n) ) by A21, A20, SPRECT_2:def_3;
len (Rev h1) >= 1 by A3, A14, A18, SPRECT_2:5;
then len (Rev h1) > 1 by A3, A14, A17, A18, SPRECT_2:6, XXREAL_0:1;
then A23: 1 + 1 <= len (Rev h1) by NAT_1:13;
( len (Lower_Seq (C,n)) = len (Rev (Lower_Seq (C,n))) & Rev h1 is special ) by FINSEQ_5:def_3, SPPOL_2:40;
then ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & L~ (Rev (Lower_Seq (C,n))) meets L~ (Rev h1) ) by A10, A23, A22, Th41, SPPOL_2:22, SPRECT_2:29;
then consider x being set such that
A24: x in L~ (Lower_Seq (C,n)) and
A25: x in L~ (Rev h1) by XBOOLE_0:3;
A26: L~ (Rev h1) = L~ h1 by SPPOL_2:22;
L~ (mid ((Upper_Seq (C,n)),((N-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n))),i2)) c= L~ (Upper_Seq (C,n)) by A13, A9, A5, JORDAN4:35;
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A24, A25, A26, XBOOLE_0:def_4;
then A27: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A27, TARSKI:def_2;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then x = (Upper_Seq (C,n)) /. 1 by JORDAN1F:5;
then i2 = 1 by A13, A9, A5, A25, A26, Th37;
then (Upper_Seq (C,n)) /. 1 = (Gauge (C,n)) * (i,1) by A3, A4, PARTFUN1:def_6;
then W-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,1) by JORDAN1F:5;
then ((Gauge (C,n)) * (i,1)) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * (1,1)) `1 by A12, JORDAN1A:73 ;
hence contradiction by A1, A12, A11, GOBOARD5:3; ::_thesis: verum
end;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then x = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by JORDAN1F:7;
then i2 = len (Upper_Seq (C,n)) by A13, A9, A5, A25, A26, Th38;
then (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = (Gauge (C,n)) * (i,1) by A3, A4, PARTFUN1:def_6;
then A28: E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,1) by JORDAN1F:7;
(SE-corner (L~ (Cage (C,n)))) `2 <= (E-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:46;
then (SE-corner (L~ (Cage (C,n)))) `2 < (E-max (L~ (Cage (C,n)))) `2 by SPRECT_2:53, XXREAL_0:2;
then S-bound (L~ (Cage (C,n))) < ((Gauge (C,n)) * (i,1)) `2 by A28, EUCLID:52;
hence contradiction by A1, JORDAN1A:72; ::_thesis: verum
end;
end;
end;
theorem Th43: :: JORDAN1G:43
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) )
set wi = width (Gauge (C,n));
assume that
A1: ( 1 <= i & i < len (Gauge (C,n)) ) and
A2: (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) ; ::_thesis: contradiction
consider i2 being Nat such that
A3: i2 in dom (Lower_Seq (C,n)) and
A4: (Lower_Seq (C,n)) . i2 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A2, FINSEQ_2:10;
reconsider i2 = i2 as Element of NAT by ORDINAL1:def_12;
A5: ( 1 <= i2 & i2 <= len (Lower_Seq (C,n)) ) by A3, FINSEQ_3:25;
3 <= len (Upper_Seq (C,n)) by JORDAN1E:15;
then A6: 2 <= len (Upper_Seq (C,n)) by XXREAL_0:2;
set f = Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))));
set i1 = (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n));
A7: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A8: (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) /. 1 = E-max (L~ (Cage (C,n))) by FINSEQ_6:92;
L~ (Cage (C,n)) = L~ (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) by REVROT_1:33;
then A9: ( (S-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) < (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) & (S-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) <= (W-min (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) ) by A8, SPRECT_5:40, SPRECT_5:41;
A10: ( W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) & rng (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) = rng (Cage (C,n)) ) by FINSEQ_6:90, SPRECT_2:43, SPRECT_2:46;
(W-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = len (Lower_Seq (C,n)) by Th30;
then (S-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= len (Lower_Seq (C,n)) by Th29;
then A11: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) < len (Lower_Seq (C,n)) by Th28, XXREAL_0:2;
( (E-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = 1 & (E-min (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) ) by Th25, Th27;
then A12: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) > 1 by Th26, XXREAL_0:2;
then A13: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(E-max (L~ (Cage (C,n)))))) -: (W-min (L~ (Cage (C,n)))) & S-max (L~ (Cage (C,n))) in rng (Cage (C,n)) ) by Th18, SPRECT_2:42;
then A14: S-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by A10, A9, FINSEQ_5:46, XXREAL_0:2;
then A15: (Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))) = S-max (L~ (Cage (C,n))) by FINSEQ_5:38;
A16: (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) <> i2
proof
assume (S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)) = i2 ; ::_thesis: contradiction
then (Gauge (C,n)) * (i,(width (Gauge (C,n)))) = S-max (L~ (Cage (C,n))) by A4, A13, A15, PARTFUN1:def_6;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
then N-bound (L~ (Cage (C,n))) = S-bound (L~ (Cage (C,n))) by A1, A7, JORDAN1A:70;
hence contradiction by SPRECT_1:16; ::_thesis: verum
end;
then mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2) is being_S-Seq by A12, A11, A5, JORDAN3:6;
then reconsider h = mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2) as one-to-one special FinSequence of (TOP-REAL 2) ;
A17: (h /. 1) `2 = ((Lower_Seq (C,n)) /. ((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n)))) `2 by A3, A13, SPRECT_2:8
.= (S-max (L~ (Cage (C,n)))) `2 by A14, FINSEQ_5:38
.= S-bound (L~ (Cage (C,n))) by EUCLID:52 ;
len h >= 1 by A3, A13, SPRECT_2:5;
then len h > 1 by A3, A13, A16, SPRECT_2:6, XXREAL_0:1;
then A18: 1 + 1 <= len h by NAT_1:13;
A19: h is_in_the_area_of Cage (C,n) by A3, A13, JORDAN1E:18, SPRECT_2:22;
(h /. (len h)) `2 = ((Lower_Seq (C,n)) /. i2) `2 by A3, A13, SPRECT_2:9
.= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A3, A4, PARTFUN1:def_6
.= N-bound (L~ (Cage (C,n))) by A1, A7, JORDAN1A:70 ;
then h is_a_v.c._for Cage (C,n) by A19, A17, SPRECT_2:def_3;
then L~ (Upper_Seq (C,n)) meets L~ h by A6, A18, Th40, SPRECT_2:29;
then consider x being set such that
A20: x in L~ (Upper_Seq (C,n)) and
A21: x in L~ h by XBOOLE_0:3;
L~ (mid ((Lower_Seq (C,n)),((S-max (L~ (Cage (C,n)))) .. (Lower_Seq (C,n))),i2)) c= L~ (Lower_Seq (C,n)) by A12, A11, A5, JORDAN4:35;
then x in (L~ (Lower_Seq (C,n))) /\ (L~ (Upper_Seq (C,n))) by A20, A21, XBOOLE_0:def_4;
then A22: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
4 <= len (Gauge (C,n)) by JORDAN8:10;
then A23: 1 <= len (Gauge (C,n)) by XXREAL_0:2;
percases ( x = E-max (L~ (Cage (C,n))) or x = W-min (L~ (Cage (C,n))) ) by A22, TARSKI:def_2;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then x = (Lower_Seq (C,n)) /. 1 by JORDAN1F:6;
then i2 = 1 by A12, A11, A5, A21, Th37;
then (Lower_Seq (C,n)) /. 1 = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A3, A4, PARTFUN1:def_6;
then E-max (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by JORDAN1F:6;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52
.= ((Gauge (C,n)) * ((len (Gauge (C,n))),(width (Gauge (C,n))))) `1 by A7, A23, JORDAN1A:71 ;
hence contradiction by A1, A7, A23, GOBOARD5:3; ::_thesis: verum
end;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then x = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by JORDAN1F:8;
then i2 = len (Lower_Seq (C,n)) by A12, A11, A5, A21, Th38;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by A3, A4, PARTFUN1:def_6;
then A24: W-min (L~ (Cage (C,n))) = (Gauge (C,n)) * (i,(width (Gauge (C,n)))) by JORDAN1F:8;
(NW-corner (L~ (Cage (C,n)))) `2 >= (W-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;
then (NW-corner (L~ (Cage (C,n)))) `2 > (W-min (L~ (Cage (C,n)))) `2 by SPRECT_2:57, XXREAL_0:2;
then N-bound (L~ (Cage (C,n))) > ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A24, EUCLID:52;
hence contradiction by A1, A7, JORDAN1A:70; ::_thesis: verum
end;
end;
end;
theorem Th44: :: JORDAN1G:44
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 < i & i <= len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n))
let i be Element of NAT ; ::_thesis: ( 1 < i & i <= len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) )
assume that
A1: ( 1 < i & i <= len (Gauge (C,n)) ) and
A2: (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) ; ::_thesis: contradiction
set Gi1 = (Gauge (C,n)) * (i,1);
consider ii being Element of NAT such that
A3: 1 <= ii and
A4: ii + 1 <= len (Upper_Seq (C,n)) and
A5: (Gauge (C,n)) * (i,1) in LSeg ((Upper_Seq (C,n)),ii) by A2, SPPOL_2:13;
A6: LSeg ((Upper_Seq (C,n)),ii) = LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) by A3, A4, TOPREAL1:def_3;
ii + 1 >= 1 by NAT_1:11;
then A7: ii + 1 in dom (Upper_Seq (C,n)) by A4, FINSEQ_3:25;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then ( len (Gauge (C,n)) = width (Gauge (C,n)) & len (Gauge (C,n)) > 1 ) by JORDAN8:def_1, XXREAL_0:2;
then A8: [i,1] in Indices (Gauge (C,n)) by A1, MATRIX_1:36;
ii < len (Upper_Seq (C,n)) by A4, NAT_1:13;
then A9: ii in dom (Upper_Seq (C,n)) by A3, FINSEQ_3:25;
A10: not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) by A1, Th42;
Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4;
then consider i1, j1, i2, j2 being Element of NAT such that
A11: [i1,j1] in Indices (Gauge (C,n)) and
A12: (Upper_Seq (C,n)) /. ii = (Gauge (C,n)) * (i1,j1) and
A13: [i2,j2] in Indices (Gauge (C,n)) and
A14: (Upper_Seq (C,n)) /. (ii + 1) = (Gauge (C,n)) * (i2,j2) and
A15: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A4, JORDAN8:3;
A16: 1 <= i1 by A11, MATRIX_1:38;
A17: j2 <= width (Gauge (C,n)) by A13, MATRIX_1:38;
A18: 1 <= j1 by A11, MATRIX_1:38;
A19: i1 <= len (Gauge (C,n)) by A11, MATRIX_1:38;
A20: 1 <= j2 by A13, MATRIX_1:38;
A21: i2 <= len (Gauge (C,n)) by A13, MATRIX_1:38;
A22: 1 <= i2 by A13, MATRIX_1:38;
A23: j1 <= width (Gauge (C,n)) by A11, MATRIX_1:38;
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A15;
supposeA24: ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: contradiction
then j1 <= j2 by NAT_1:11;
then ((Gauge (C,n)) * (i1,j1)) `2 <= ((Gauge (C,n)) * (i2,j2)) `2 by A16, A19, A18, A17, A24, SPRECT_3:12;
then A25: ((Gauge (C,n)) * (i1,j1)) `2 <= ((Gauge (C,n)) * (i,1)) `2 by A5, A6, A12, A14, TOPREAL1:4;
((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A24, GOBOARD5:2
.= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ;
then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16;
then ((Gauge (C,n)) * (i,1)) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41;
then A26: i1 = i by A11, A8, Th7;
then ((Gauge (C,n)) * (i,1)) `2 <= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A18, A23, SPRECT_3:12;
then j1 = 1 by A11, A8, A25, Th6, XXREAL_0:1;
hence contradiction by A12, A9, A10, A26, PARTFUN2:2; ::_thesis: verum
end;
supposeA27: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ;
then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15;
then ((Gauge (C,n)) * (i,1)) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40;
then A28: j1 = 1 by A11, A8, Th6;
i2 > 1 by A16, A27, NAT_1:13;
then not (Upper_Seq (C,n)) /. (ii + 1) in rng (Upper_Seq (C,n)) by A14, A21, A27, A28, Th42;
hence contradiction by A7, PARTFUN2:2; ::_thesis: verum
end;
supposeA29: ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ;
then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15;
then ((Gauge (C,n)) * (i,1)) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40;
then A30: j1 = 1 by A11, A8, Th6;
i1 > 1 by A22, A29, NAT_1:13;
then not (Upper_Seq (C,n)) /. ii in rng (Upper_Seq (C,n)) by A12, A19, A30, Th42;
hence contradiction by A9, PARTFUN2:2; ::_thesis: verum
end;
supposeA31: ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: contradiction
then j2 <= j1 by NAT_1:11;
then ((Gauge (C,n)) * (i2,j2)) `2 <= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A23, A20, A31, SPRECT_3:12;
then A32: ((Gauge (C,n)) * (i2,j2)) `2 <= ((Gauge (C,n)) * (i,1)) `2 by A5, A6, A12, A14, TOPREAL1:4;
((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A31, GOBOARD5:2
.= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ;
then LSeg (((Upper_Seq (C,n)) /. ii),((Upper_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16;
then ((Gauge (C,n)) * (i,1)) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41;
then A33: i1 = i by A11, A8, Th7;
then ((Gauge (C,n)) * (i,1)) `2 <= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, A31, SPRECT_3:12;
then j2 = 1 by A13, A8, A32, Th6, XXREAL_0:1;
hence contradiction by A14, A7, A10, A31, A33, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN1G:45
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i being Element of NAT st 1 <= i & i < len (Gauge (C,n)) holds
not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n))
set wi = width (Gauge (C,n));
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len (Gauge (C,n)) implies not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) )
assume that
A1: ( 1 <= i & i < len (Gauge (C,n)) ) and
A2: (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in L~ (Lower_Seq (C,n)) ; ::_thesis: contradiction
set Gi1 = (Gauge (C,n)) * (i,(width (Gauge (C,n))));
consider ii being Element of NAT such that
A3: 1 <= ii and
A4: ii + 1 <= len (Lower_Seq (C,n)) and
A5: (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in LSeg ((Lower_Seq (C,n)),ii) by A2, SPPOL_2:13;
A6: LSeg ((Lower_Seq (C,n)),ii) = LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) by A3, A4, TOPREAL1:def_3;
ii + 1 >= 1 by NAT_1:11;
then A7: ii + 1 in dom (Lower_Seq (C,n)) by A4, FINSEQ_3:25;
len (Gauge (C,n)) >= 4 by JORDAN8:10;
then ( len (Gauge (C,n)) = width (Gauge (C,n)) & len (Gauge (C,n)) > 1 ) by JORDAN8:def_1, XXREAL_0:2;
then A8: [i,(width (Gauge (C,n)))] in Indices (Gauge (C,n)) by A1, MATRIX_1:36;
ii < len (Lower_Seq (C,n)) by A4, NAT_1:13;
then A9: ii in dom (Lower_Seq (C,n)) by A3, FINSEQ_3:25;
A10: not (Gauge (C,n)) * (i,(width (Gauge (C,n)))) in rng (Lower_Seq (C,n)) by A1, Th43;
Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5;
then consider i1, j1, i2, j2 being Element of NAT such that
A11: [i1,j1] in Indices (Gauge (C,n)) and
A12: (Lower_Seq (C,n)) /. ii = (Gauge (C,n)) * (i1,j1) and
A13: [i2,j2] in Indices (Gauge (C,n)) and
A14: (Lower_Seq (C,n)) /. (ii + 1) = (Gauge (C,n)) * (i2,j2) and
A15: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A3, A4, JORDAN8:3;
A16: 1 <= i1 by A11, MATRIX_1:38;
A17: j2 <= width (Gauge (C,n)) by A13, MATRIX_1:38;
A18: 1 <= j1 by A11, MATRIX_1:38;
A19: i1 <= len (Gauge (C,n)) by A11, MATRIX_1:38;
A20: 1 <= j2 by A13, MATRIX_1:38;
A21: i2 <= len (Gauge (C,n)) by A13, MATRIX_1:38;
A22: 1 <= i2 by A13, MATRIX_1:38;
A23: j1 <= width (Gauge (C,n)) by A11, MATRIX_1:38;
percases ( ( i1 = i2 & j2 + 1 = j1 ) or ( i2 + 1 = i1 & j1 = j2 ) or ( i2 = i1 + 1 & j1 = j2 ) or ( i1 = i2 & j2 = j1 + 1 ) ) by A15;
supposeA24: ( i1 = i2 & j2 + 1 = j1 ) ; ::_thesis: contradiction
then j1 >= j2 by NAT_1:11;
then ((Gauge (C,n)) * (i1,j1)) `2 >= ((Gauge (C,n)) * (i2,j2)) `2 by A16, A19, A23, A20, A24, SPRECT_3:12;
then A25: ((Gauge (C,n)) * (i1,j1)) `2 >= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A5, A6, A12, A14, TOPREAL1:4;
((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A24, GOBOARD5:2
.= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ;
then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41;
then A26: i1 = i by A11, A8, Th7;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 >= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A18, A23, SPRECT_3:12;
then j1 = width (Gauge (C,n)) by A11, A8, A25, Th6, XXREAL_0:1;
hence contradiction by A12, A9, A10, A26, PARTFUN2:2; ::_thesis: verum
end;
supposeA27: ( i2 + 1 = i1 & j1 = j2 ) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ;
then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40;
then A28: j1 = width (Gauge (C,n)) by A11, A8, Th6;
i2 < len (Gauge (C,n)) by A19, A27, NAT_1:13;
then not (Lower_Seq (C,n)) /. (ii + 1) in rng (Lower_Seq (C,n)) by A14, A22, A27, A28, Th43;
hence contradiction by A7, PARTFUN2:2; ::_thesis: verum
end;
supposeA29: ( i2 = i1 + 1 & j1 = j2 ) ; ::_thesis: contradiction
then ((Gauge (C,n)) * (i1,j1)) `2 = ((Gauge (C,n)) * (1,j2)) `2 by A16, A19, A18, A23, GOBOARD5:1
.= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, GOBOARD5:1 ;
then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is horizontal by A12, A14, SPPOL_1:15;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 = ((Gauge (C,n)) * (i1,j1)) `2 by A5, A6, A12, SPPOL_1:40;
then A30: j1 = width (Gauge (C,n)) by A11, A8, Th6;
i1 < len (Gauge (C,n)) by A21, A29, NAT_1:13;
then not (Lower_Seq (C,n)) /. ii in rng (Lower_Seq (C,n)) by A12, A16, A30, Th43;
hence contradiction by A9, PARTFUN2:2; ::_thesis: verum
end;
supposeA31: ( i1 = i2 & j2 = j1 + 1 ) ; ::_thesis: contradiction
then j2 >= j1 by NAT_1:11;
then ((Gauge (C,n)) * (i2,j2)) `2 >= ((Gauge (C,n)) * (i1,j1)) `2 by A16, A19, A18, A17, A31, SPRECT_3:12;
then A32: ((Gauge (C,n)) * (i2,j2)) `2 >= ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 by A5, A6, A12, A14, TOPREAL1:4;
((Gauge (C,n)) * (i1,j1)) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A16, A19, A18, A23, A31, GOBOARD5:2
.= ((Gauge (C,n)) * (i2,j2)) `1 by A22, A21, A20, A17, GOBOARD5:2 ;
then LSeg (((Lower_Seq (C,n)) /. ii),((Lower_Seq (C,n)) /. (ii + 1))) is vertical by A12, A14, SPPOL_1:16;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `1 = ((Gauge (C,n)) * (i1,j1)) `1 by A5, A6, A12, SPPOL_1:41;
then A33: i1 = i by A11, A8, Th7;
then ((Gauge (C,n)) * (i,(width (Gauge (C,n))))) `2 >= ((Gauge (C,n)) * (i2,j2)) `2 by A22, A21, A20, A17, A31, SPRECT_3:12;
then j2 = width (Gauge (C,n)) by A13, A8, A32, Th6, XXREAL_0:1;
hence contradiction by A14, A7, A10, A31, A33, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
theorem Th46: :: JORDAN1G:46
for n being Element of NAT
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
proof
let n be Element of NAT ; ::_thesis: for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) implies LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) )
set Gij = (Gauge (C,n)) * (i,j);
assume that
A1: 1 <= i and
A2: i <= len (Gauge (C,n)) and
A3: ( 1 <= j & j <= width (Gauge (C,n)) ) and
A4: (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
A5: Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2;
set Wmi = W-min (L~ (Cage (C,n)));
set h = mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))));
set v1 = L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)));
set NE = NE-corner (L~ (Cage (C,n)));
set Gv1 = <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))));
set v = (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>;
A6: L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
A7: Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
A8: len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then A9: len (Upper_Seq (C,n)) >= 1 by XXREAL_0:2;
A10: len (Lower_Seq (C,n)) >= 3 by JORDAN1E:15;
then A11: ( len (Lower_Seq (C,n)) >= 2 & len (Lower_Seq (C,n)) >= 1 ) by XXREAL_0:2;
A12: len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
A13: ((Gauge (C,n)) * (i,1)) `2 = S-bound (L~ (Cage (C,n))) by A1, A2, JORDAN1A:72;
now__::_thesis:_LSeg_(((Gauge_(C,n))_*_(i,1)),((Gauge_(C,n))_*_(i,j)))_meets_L~_(Lower_Seq_(C,n))
percases ( ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & i = 1 ) or ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) & i > 1 ) or ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) & i > 1 ) or (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) or ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) ) ) by A1, A4, A6, XBOOLE_0:def_3, XXREAL_0:1;
supposeA14: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & i = 1 ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
set G11 = (Gauge (C,n)) * (1,1);
A15: W-min (L~ (Cage (C,n))) in L~ (Cage (C,n)) by SPRECT_1:13;
S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (1,1)) `2 by A2, A14, JORDAN1A:72;
then A16: ( (W-min (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) & ((Gauge (C,n)) * (1,1)) `2 <= (W-min (L~ (Cage (C,n)))) `2 ) by A15, EUCLID:52, PSCOMP_1:24;
A17: rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) by A10, SPPOL_2:18, XXREAL_0:2;
A18: ((Gauge (C,n)) * (i,j)) `1 = W-bound (L~ (Cage (C,n))) by A3, A12, A14, JORDAN1A:73;
then (Gauge (C,n)) * (i,j) in W-most (L~ (Cage (C,n))) by A4, SPRECT_2:12;
then A19: (W-min (L~ (Cage (C,n)))) `2 <= ((Gauge (C,n)) * (i,j)) `2 by PSCOMP_1:31;
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) by JORDAN1F:8;
then A20: W-min (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) by REVROT_1:3;
((Gauge (C,n)) * (1,1)) `1 = W-bound (L~ (Cage (C,n))) by A2, A14, JORDAN1A:73;
then W-min (L~ (Cage (C,n))) in LSeg (((Gauge (C,n)) * (1,1)),((Gauge (C,n)) * (1,j))) by A14, A16, A18, A19, GOBOARD7:7;
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by A14, A17, A20, XBOOLE_0:3; ::_thesis: verum
end;
supposeA21: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) <> NE-corner (L~ (Cage (C,n))) & i > 1 ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
len (Cage (C,n)) > 4 by GOBOARD7:34;
then A22: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;
A23: not NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n))
proof
A24: (NE-corner (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
then ( (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) & (NE-corner (L~ (Cage (C,n)))) `2 >= S-bound (L~ (Cage (C,n))) ) by EUCLID:52, SPRECT_1:22;
then NE-corner (L~ (Cage (C,n))) in { p where p is Point of (TOP-REAL 2) : ( p `1 = E-bound (L~ (Cage (C,n))) & p `2 <= N-bound (L~ (Cage (C,n))) & p `2 >= S-bound (L~ (Cage (C,n))) ) } by A24;
then A25: NE-corner (L~ (Cage (C,n))) in LSeg ((SE-corner (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n))))) by SPRECT_1:23;
assume NE-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) ; ::_thesis: contradiction
then NE-corner (L~ (Cage (C,n))) in (LSeg ((SE-corner (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A22, A25, XBOOLE_0:def_4;
then A26: (NE-corner (L~ (Cage (C,n)))) `2 <= (E-max (L~ (Cage (C,n)))) `2 by PSCOMP_1:47;
A27: (E-max (L~ (Cage (C,n)))) `1 = (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:45;
(E-max (L~ (Cage (C,n)))) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:46;
then (E-max (L~ (Cage (C,n)))) `2 = (NE-corner (L~ (Cage (C,n)))) `2 by A26, XXREAL_0:1;
hence contradiction by A21, A27, TOPREAL3:6; ::_thesis: verum
end;
A28: now__::_thesis:_not_NE-corner_(L~_(Cage_(C,n)))_in_rng_(L_Cut_((Upper_Seq_(C,n)),((Gauge_(C,n))_*_(i,j))))
percases ( (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ) ;
suppose (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = <*((Gauge (C,n)) * (i,j))*> ^ (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by JORDAN3:def_3;
then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:31;
then A29: rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) = {((Gauge (C,n)) * (i,j))} \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:38;
not NE-corner (L~ (Cage (C,n))) in L~ (Cage (C,n))
proof
assume NE-corner (L~ (Cage (C,n))) in L~ (Cage (C,n)) ; ::_thesis: contradiction
then consider i being Element of NAT such that
A30: 1 <= i and
A31: i + 1 <= len (Cage (C,n)) and
A32: NE-corner (L~ (Cage (C,n))) in LSeg (((Cage (C,n)) /. i),((Cage (C,n)) /. (i + 1))) by SPPOL_2:14;
percases ( ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ) by A30, A31, TOPREAL1:def_5;
supposeA33: ((Cage (C,n)) /. i) `1 = ((Cage (C,n)) /. (i + 1)) `1 ; ::_thesis: contradiction
( ((Cage (C,n)) /. i) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or ((Cage (C,n)) /. i) `2 >= ((Cage (C,n)) /. (i + 1)) `2 ) ;
then A34: ( (NE-corner (L~ (Cage (C,n)))) `2 <= ((Cage (C,n)) /. (i + 1)) `2 or (NE-corner (L~ (Cage (C,n)))) `2 <= ((Cage (C,n)) /. i) `2 ) by A32, TOPREAL1:4;
A35: (NE-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. i) `1 by A32, A33, GOBOARD7:5;
A36: 1 <= i + 1 by NAT_1:11;
then A37: i + 1 in dom (Cage (C,n)) by A31, FINSEQ_3:25;
A38: (NE-corner (L~ (Cage (C,n)))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
then A39: ((Cage (C,n)) /. (i + 1)) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by A31, A36, JORDAN5D:11;
A40: i < len (Cage (C,n)) by A31, NAT_1:13;
then ((Cage (C,n)) /. i) `2 <= (NE-corner (L~ (Cage (C,n)))) `2 by A30, A38, JORDAN5D:11;
then ( (NE-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. (i + 1)) `2 or (NE-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. i) `2 ) by A39, A34, XXREAL_0:1;
then A41: ( NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. (i + 1) or NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. i ) by A33, A35, TOPREAL3:6;
i in dom (Cage (C,n)) by A30, A40, FINSEQ_3:25;
hence contradiction by A23, A37, A41, PARTFUN2:2; ::_thesis: verum
end;
supposeA42: ((Cage (C,n)) /. i) `2 = ((Cage (C,n)) /. (i + 1)) `2 ; ::_thesis: contradiction
( ((Cage (C,n)) /. i) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or ((Cage (C,n)) /. i) `1 >= ((Cage (C,n)) /. (i + 1)) `1 ) ;
then A43: ( (NE-corner (L~ (Cage (C,n)))) `1 <= ((Cage (C,n)) /. (i + 1)) `1 or (NE-corner (L~ (Cage (C,n)))) `1 <= ((Cage (C,n)) /. i) `1 ) by A32, TOPREAL1:3;
A44: (NE-corner (L~ (Cage (C,n)))) `2 = ((Cage (C,n)) /. i) `2 by A32, A42, GOBOARD7:6;
A45: 1 <= i + 1 by NAT_1:11;
then A46: i + 1 in dom (Cage (C,n)) by A31, FINSEQ_3:25;
A47: (NE-corner (L~ (Cage (C,n)))) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
then A48: ((Cage (C,n)) /. (i + 1)) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by A31, A45, JORDAN5D:12;
A49: i < len (Cage (C,n)) by A31, NAT_1:13;
then ((Cage (C,n)) /. i) `1 <= (NE-corner (L~ (Cage (C,n)))) `1 by A30, A47, JORDAN5D:12;
then ( (NE-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. (i + 1)) `1 or (NE-corner (L~ (Cage (C,n)))) `1 = ((Cage (C,n)) /. i) `1 ) by A48, A43, XXREAL_0:1;
then A50: ( NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. (i + 1) or NE-corner (L~ (Cage (C,n))) = (Cage (C,n)) /. i ) by A42, A44, TOPREAL3:6;
i in dom (Cage (C,n)) by A30, A49, FINSEQ_3:25;
hence contradiction by A23, A46, A50, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
then A51: not NE-corner (L~ (Cage (C,n))) in {((Gauge (C,n)) * (i,j))} by A4, TARSKI:def_1;
( rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Upper_Seq (C,n)) & rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) ) by Th39, FINSEQ_6:119;
then rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Cage (C,n)) by XBOOLE_1:1;
then not NE-corner (L~ (Cage (C,n))) in rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by A23;
hence not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A29, A51, XBOOLE_0:def_3; ::_thesis: verum
end;
suppose (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))) by JORDAN3:def_3;
then A52: rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119;
rng (Upper_Seq (C,n)) c= rng (Cage (C,n)) by Th39;
then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Cage (C,n)) by A52, XBOOLE_1:1;
hence not NE-corner (L~ (Cage (C,n))) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A23; ::_thesis: verum
end;
end;
end;
S-bound (L~ (Cage (C,n))) < N-bound (L~ (Cage (C,n))) by SPRECT_1:32;
then NE-corner (L~ (Cage (C,n))) <> (Gauge (C,n)) * (i,1) by A13, EUCLID:52;
then not NE-corner (L~ (Cage (C,n))) in {((Gauge (C,n)) * (i,1))} by TARSKI:def_1;
then not NE-corner (L~ (Cage (C,n))) in rng <*((Gauge (C,n)) * (i,1))*> by FINSEQ_1:39;
then not NE-corner (L~ (Cage (C,n))) in (rng <*((Gauge (C,n)) * (i,1))*>) \/ (rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A28, XBOOLE_0:def_3;
then not NE-corner (L~ (Cage (C,n))) in rng (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by FINSEQ_1:31;
then rng (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) misses {(NE-corner (L~ (Cage (C,n))))} by ZFMISC_1:50;
then A53: rng (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) misses rng <*(NE-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;
A54: len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_2:16
.= (1 + (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_5:8 ;
A55: not L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is empty by A21, JORDAN1E:3;
then A56: 0 + 1 <= len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by NAT_1:13;
then 1 in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_3:25;
then A57: (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1 = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A21, JORDAN3:23 ;
then A58: ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1 = (Gauge (C,n)) * (i,j) by A56, BOOLMARK:7;
1 + (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 1 + 1 by A56, XREAL_1:7;
then A59: 2 < len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by A54, NAT_1:13;
A60: L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is being_S-Seq by A21, JORDAN3:34;
(<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> = <*((Gauge (C,n)) * (i,1))*> ^ ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by FINSEQ_1:32;
then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1 = (Gauge (C,n)) * (i,1) by FINSEQ_5:15;
then A61: (((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1) `2 = S-bound (L~ (Cage (C,n))) by A1, A2, JORDAN1A:72;
len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) + 1 by FINSEQ_2:16;
then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. (len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>)) = NE-corner (L~ (Cage (C,n))) by FINSEQ_4:67;
then A62: (((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. (len ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>))) `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
A63: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:70;
then A64: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by A63, SPRECT_2:69, XXREAL_0:2;
(E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:72;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:71, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:73, XXREAL_0:2;
then A65: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A63, SPRECT_2:74, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A63, SPRECT_2:76, XXREAL_0:2;
then A66: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
A67: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then (Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = E-max (L~ (Cage (C,n))) by FINSEQ_5:38;
then A68: ((Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1)) `1 = E-bound (L~ (Cage (C,n))) by A64, A66, JORDAN1E:20;
A69: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A70: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = ((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A67, A65, SPRECT_5:9;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*((Gauge_(C,n))_*_(i,1))*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*((Gauge (C,n)) * (i,1))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
assume A71: m in dom <*((Gauge (C,n)) * (i,1))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then m in Seg 1 by FINSEQ_1:38;
then m = 1 by FINSEQ_1:2, TARSKI:def_1;
then <*((Gauge (C,n)) * (i,1))*> . m = (Gauge (C,n)) * (i,1) by FINSEQ_1:40;
then A72: <*((Gauge (C,n)) * (i,1))*> /. m = (Gauge (C,n)) * (i,1) by A71, PARTFUN1:def_6;
width (Gauge (C,n)) >= 4 by A12, JORDAN8:10;
then A73: 1 <= width (Gauge (C,n)) by XXREAL_0:2;
then ((Gauge (C,n)) * (1,1)) `1 <= ((Gauge (C,n)) * (i,1)) `1 by A1, A2, SPRECT_3:13;
hence W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 by A12, A72, A73, JORDAN1A:73; ::_thesis: ( (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
((Gauge (C,n)) * (i,1)) `1 <= ((Gauge (C,n)) * ((len (Gauge (C,n))),1)) `1 by A1, A2, A73, SPRECT_3:13;
hence (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) by A12, A72, A73, JORDAN1A:71; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
thus S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 by A1, A2, A72, JORDAN1A:72; ::_thesis: (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n)))
S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2 by A1, A2, JORDAN1A:72;
hence (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) by A72, SPRECT_1:22; ::_thesis: verum
end;
then A74: <*((Gauge (C,n)) * (i,1))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
A75: <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:25;
3 <= len (Lower_Seq (C,n)) by JORDAN1E:15;
then 2 <= len (Lower_Seq (C,n)) by XXREAL_0:2;
then A76: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
<*((Gauge (C,n)) * (i,j))*> is_in_the_area_of Cage (C,n) by A21, JORDAN1E:17, SPRECT_3:46;
then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is_in_the_area_of Cage (C,n) by A21, JORDAN1E:17, SPRECT_3:56;
then <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_in_the_area_of Cage (C,n) by A74, SPRECT_2:24;
then (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by A75, SPRECT_2:24;
then A77: (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is_a_v.c._for Cage (C,n) by A61, A62, SPRECT_2:def_3;
A78: ((1 + (((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) - (len (Cage (C,n))) = 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ;
A79: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
then mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A76, JORDAN1E:18, SPRECT_2:22;
then A80: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A65, NAT_1:13;
then (1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 by XREAL_1:20;
then A81: (len (Cage (C,n))) + ((1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) <= (len (Cage (C,n))) + 0 by XREAL_1:6;
A82: len (Lower_Seq (C,n)) >= 2 + 1 by JORDAN1E:15;
then A83: len (Lower_Seq (C,n)) > 2 by NAT_1:13;
(len (Cage (C,n))) + 0 <= (len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:6;
then (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A70, XREAL_1:9;
then ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6;
then A84: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A69, FINSEQ_5:50;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A85: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A86: L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= L~ (Upper_Seq (C,n)) by A21, JORDAN3:42;
A87: len (Lower_Seq (C,n)) > 1 by A82, XXREAL_0:2;
then A88: not mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is empty by A83, JORDAN1B:2;
A89: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A67, FINSEQ_6:90, SPRECT_2:43;
then (Lower_Seq (C,n)) /. (1 + 1) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A5, A76, FINSEQ_5:52
.= (Cage (C,n)) /. (((1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) -' (len (Cage (C,n)))) by A69, A70, A84, A81, REVROT_1:17
.= (Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1) by A70, A78, XREAL_0:def_2 ;
then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. 1) `1 = E-bound (L~ (Cage (C,n))) by A76, A79, A68, SPRECT_2:8;
then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = E-bound (L~ (Cage (C,n))) by A88, FINSEQ_5:65;
then A90: ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))))) `1 = E-bound (L~ (Cage (C,n))) by FINSEQ_5:def_3;
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A5, A89, FINSEQ_5:54
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1
.= W-min (L~ (Cage (C,n))) by A69, FINSEQ_6:92 ;
then ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = W-bound (L~ (Cage (C,n))) by A76, A79, SPRECT_2:9;
then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. 1) `1 = W-bound (L~ (Cage (C,n))) by A88, FINSEQ_5:65;
then A91: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_a_h.c._for Cage (C,n) by A80, A90, SPRECT_2:def_2;
A92: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
set ci = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))));
rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by A8, SPPOL_2:18, XXREAL_0:2;
then A93: not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) by A2, A21, Th44;
not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) by A2, A21, Th44;
then not (Gauge (C,n)) * (i,1) in {((Gauge (C,n)) * (i,j))} by A21, TARSKI:def_1;
then A94: not (Gauge (C,n)) * (i,1) in rng <*((Gauge (C,n)) * (i,j))*> by FINSEQ_1:38;
now__::_thesis:_not_(Gauge_(C,n))_*_(i,1)_in_rng_(L_Cut_((Upper_Seq_(C,n)),((Gauge_(C,n))_*_(i,j))))
percases ( (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ) ;
supposeA95: (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))
rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119;
then not (Gauge (C,n)) * (i,1) in rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by A93;
then not (Gauge (C,n)) * (i,1) in (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by A94, XBOOLE_0:def_3;
then not (Gauge (C,n)) * (i,1) in rng (<*((Gauge (C,n)) * (i,j))*> ^ (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:31;
hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A95, JORDAN3:def_3; ::_thesis: verum
end;
suppose (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))) by JORDAN3:def_3;
then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119;
hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A93; ::_thesis: verum
end;
end;
end;
then {((Gauge (C,n)) * (i,1))} misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by ZFMISC_1:50;
then A96: rng <*((Gauge (C,n)) * (i,1))*> misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_1:38;
A97: <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;
(Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2
.= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ;
then A98: not E-max (L~ (Cage (C,n))) in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) by A83, JORDAN5B:16;
<*((Gauge (C,n)) * (i,1))*> is one-to-one by FINSEQ_3:93;
then <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is one-to-one by A96, A60, FINSEQ_3:91;
then A99: (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is one-to-one by A53, A97, FINSEQ_3:91;
A100: L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) c= L~ (Lower_Seq (C,n)) by A11, JORDAN4:35;
(<*((Gauge (C,n)) * (i,1))*> /. (len <*((Gauge (C,n)) * (i,1))*>)) `1 = (<*((Gauge (C,n)) * (i,1))*> /. 1) `1 by FINSEQ_1:39
.= ((Gauge (C,n)) * (i,1)) `1 by FINSEQ_4:16
.= ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1) `1 by A1, A2, A3, A57, GOBOARD5:2 ;
then A101: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is special by A60, GOBOARD2:8;
len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A56, FINSEQ_3:25;
then A102: (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by PARTFUN1:def_6
.= (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) by A21, JORDAN1B:4
.= (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by A92, PARTFUN1:def_6
.= ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A7, A85, FINSEQ_5:42
.= E-max (L~ (Cage (C,n))) by A85, FINSEQ_5:45 ;
then (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) = E-max (L~ (Cage (C,n))) by A55, SPRECT_3:1;
then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `1 = (NE-corner (L~ (Cage (C,n)))) `1 by PSCOMP_1:45
.= (<*(NE-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_4:16 ;
then A103: (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*> is special by A101, GOBOARD2:8;
mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is S-Sequence_in_R2 by A83, A87, JORDAN3:6;
then A104: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is S-Sequence_in_R2 ;
then 2 <= len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) by TOPREAL1:def_8;
then L~ (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) meets L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by A59, A99, A103, A104, A91, A77, SPRECT_2:29;
then L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) meets L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by SPPOL_2:22;
then consider x being set such that
A105: x in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) and
A106: x in L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) by XBOOLE_0:3;
A107: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
L~ ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) = L~ (<*((Gauge (C,n)) * (i,1))*> ^ ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>)) by FINSEQ_1:32
.= (LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1))) \/ (L~ ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>)) by SPPOL_2:20
.= (LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1))) \/ ((L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) \/ (LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n))))))) by A55, SPPOL_2:19 ;
then A108: ( x in LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1)) or x in (L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) \/ (LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n)))))) ) by A106, XBOOLE_0:def_3;
now__::_thesis:_L~_(Lower_Seq_(C,n))_meets_L~_<*((Gauge_(C,n))_*_(i,1)),((Gauge_(C,n))_*_(i,j))*>
percases ( x in LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1)) or x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) or x in LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n))))) ) by A108, XBOOLE_0:def_3;
suppose x in LSeg (((Gauge (C,n)) * (i,1)),(((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ^ <*(NE-corner (L~ (Cage (C,n))))*>) /. 1)) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*>
then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A58, SPPOL_2:21;
hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A105, A100, XBOOLE_0:3; ::_thesis: verum
end;
supposeA109: x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*>
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A105, A100, A86, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A110: x = W-min (L~ (Cage (C,n))) by A105, A98, TARSKI:def_2;
1 in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
then (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A7, A85, FINSEQ_5:44
.= W-min (L~ (Cage (C,n))) by A107, FINSEQ_6:92 ;
then x = (Gauge (C,n)) * (i,j) by A21, A109, A110, JORDAN1E:7;
then x in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by SPPOL_2:21;
hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A105, A100, XBOOLE_0:3; ::_thesis: verum
end;
supposeA111: x in LSeg (((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))),(NE-corner (L~ (Cage (C,n))))) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*>
x in L~ (Cage (C,n)) by A6, A105, A100, XBOOLE_0:def_3;
then x in (LSeg ((E-max (L~ (Cage (C,n)))),(NE-corner (L~ (Cage (C,n)))))) /\ (L~ (Cage (C,n))) by A102, A111, XBOOLE_0:def_4;
then x in {(E-max (L~ (Cage (C,n))))} by PSCOMP_1:51;
hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A105, A98, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
then L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> meets L~ (Lower_Seq (C,n)) ;
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by SPPOL_2:21; ::_thesis: verum
end;
supposeA112: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) & E-max (L~ (Cage (C,n))) = NE-corner (L~ (Cage (C,n))) & i > 1 ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*((Gauge_(C,n))_*_(i,1))*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_&_(<*((Gauge_(C,n))_*_(i,1))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*((Gauge (C,n)) * (i,1))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
assume A113: m in dom <*((Gauge (C,n)) * (i,1))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 & (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then m in Seg 1 by FINSEQ_1:38;
then m = 1 by FINSEQ_1:2, TARSKI:def_1;
then <*((Gauge (C,n)) * (i,1))*> . m = (Gauge (C,n)) * (i,1) by FINSEQ_1:40;
then A114: <*((Gauge (C,n)) * (i,1))*> /. m = (Gauge (C,n)) * (i,1) by A113, PARTFUN1:def_6;
width (Gauge (C,n)) >= 4 by A12, JORDAN8:10;
then A115: 1 <= width (Gauge (C,n)) by XXREAL_0:2;
then ((Gauge (C,n)) * (1,1)) `1 <= ((Gauge (C,n)) * (i,1)) `1 by A1, A2, SPRECT_3:13;
hence W-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `1 by A12, A114, A115, JORDAN1A:73; ::_thesis: ( (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
((Gauge (C,n)) * (i,1)) `1 <= ((Gauge (C,n)) * ((len (Gauge (C,n))),1)) `1 by A1, A2, A115, SPRECT_3:13;
hence (<*((Gauge (C,n)) * (i,1))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) by A12, A114, A115, JORDAN1A:71; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 & (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
thus S-bound (L~ (Cage (C,n))) <= (<*((Gauge (C,n)) * (i,1))*> /. m) `2 by A1, A2, A114, JORDAN1A:72; ::_thesis: (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n)))
S-bound (L~ (Cage (C,n))) = ((Gauge (C,n)) * (i,1)) `2 by A1, A2, JORDAN1A:72;
hence (<*((Gauge (C,n)) * (i,1))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) by A114, SPRECT_1:22; ::_thesis: verum
end;
then A116: <*((Gauge (C,n)) * (i,1))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
<*((Gauge (C,n)) * (i,j))*> is_in_the_area_of Cage (C,n) by A112, JORDAN1E:17, SPRECT_3:46;
then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is_in_the_area_of Cage (C,n) by A112, JORDAN1E:17, SPRECT_3:56;
then A117: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_in_the_area_of Cage (C,n) by A116, SPRECT_2:24;
A118: len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A119: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A120: not L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is empty by A112, JORDAN1E:3;
then A121: 0 + 1 <= len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by NAT_1:13;
then 1 in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_3:25;
then A122: (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1 = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . 1 by PARTFUN1:def_6
.= (Gauge (C,n)) * (i,j) by A112, JORDAN3:23 ;
len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) in dom (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A121, FINSEQ_3:25;
then (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) . (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by PARTFUN1:def_6
.= (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) by A112, JORDAN1B:4
.= (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by A118, PARTFUN1:def_6
.= ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A7, A119, FINSEQ_5:42
.= E-max (L~ (Cage (C,n))) by A119, FINSEQ_5:45 ;
then (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))))) = E-max (L~ (Cage (C,n))) by A120, SPRECT_3:1;
then A123: ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. (len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))))) `2 = N-bound (L~ (Cage (C,n))) by A112, EUCLID:52;
(<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. 1 = (Gauge (C,n)) * (i,1) by FINSEQ_5:15;
then ((<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) /. 1) `2 = S-bound (L~ (Cage (C,n))) by A1, A2, JORDAN1A:72;
then A124: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is_a_v.c._for Cage (C,n) by A117, A123, SPRECT_2:def_3;
A125: (Cage (C,n)) /. 1 = N-min (L~ (Cage (C,n))) by JORDAN9:32;
then (N-max (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by SPRECT_2:70;
then A126: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) > 1 by A125, SPRECT_2:69, XXREAL_0:2;
(E-min (L~ (Cage (C,n)))) .. (Cage (C,n)) <= (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:72;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-max (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:71, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (S-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:73, XXREAL_0:2;
then A127: (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < (W-min (L~ (Cage (C,n)))) .. (Cage (C,n)) by A125, SPRECT_2:74, XXREAL_0:2;
then (E-max (L~ (Cage (C,n)))) .. (Cage (C,n)) < len (Cage (C,n)) by A125, SPRECT_2:76, XXREAL_0:2;
then A128: ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1 <= len (Cage (C,n)) by NAT_1:13;
A129: E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then (Cage (C,n)) /. ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) = E-max (L~ (Cage (C,n))) by FINSEQ_5:38;
then A130: ((Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1)) `1 = E-bound (L~ (Cage (C,n))) by A126, A128, JORDAN1E:20;
1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A127, NAT_1:13;
then (1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= 0 by XREAL_1:20;
then A131: (len (Cage (C,n))) + ((1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) <= (len (Cage (C,n))) + 0 by XREAL_1:6;
A132: len (Lower_Seq (C,n)) >= 2 + 1 by JORDAN1E:15;
then A133: len (Lower_Seq (C,n)) > 2 by NAT_1:13;
set ci = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))));
rng (Upper_Seq (C,n)) c= L~ (Upper_Seq (C,n)) by A8, SPPOL_2:18, XXREAL_0:2;
then A134: not (Gauge (C,n)) * (i,1) in rng (Upper_Seq (C,n)) by A2, A112, Th44;
not (Gauge (C,n)) * (i,1) in L~ (Upper_Seq (C,n)) by A2, A112, Th44;
then not (Gauge (C,n)) * (i,1) in {((Gauge (C,n)) * (i,j))} by A112, TARSKI:def_1;
then A135: not (Gauge (C,n)) * (i,1) in rng <*((Gauge (C,n)) * (i,j))*> by FINSEQ_1:38;
now__::_thesis:_not_(Gauge_(C,n))_*_(i,1)_in_rng_(L_Cut_((Upper_Seq_(C,n)),((Gauge_(C,n))_*_(i,j))))
percases ( (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) or (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ) ;
supposeA136: (Gauge (C,n)) * (i,j) <> (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))
rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119;
then not (Gauge (C,n)) * (i,1) in rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n))))) by A134;
then not (Gauge (C,n)) * (i,1) in (rng <*((Gauge (C,n)) * (i,j))*>) \/ (rng (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by A135, XBOOLE_0:def_3;
then not (Gauge (C,n)) * (i,1) in rng (<*((Gauge (C,n)) * (i,j))*> ^ (mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))))) by FINSEQ_1:31;
hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A136, JORDAN3:def_3; ::_thesis: verum
end;
suppose (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . ((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1) ; ::_thesis: not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))
then L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) = mid ((Upper_Seq (C,n)),((Index (((Gauge (C,n)) * (i,j)),(Upper_Seq (C,n)))) + 1),(len (Upper_Seq (C,n)))) by JORDAN3:def_3;
then rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= rng (Upper_Seq (C,n)) by FINSEQ_6:119;
hence not (Gauge (C,n)) * (i,1) in rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by A134; ::_thesis: verum
end;
end;
end;
then {((Gauge (C,n)) * (i,1))} misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by ZFMISC_1:50;
then A137: rng <*((Gauge (C,n)) * (i,1))*> misses rng (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) by FINSEQ_1:38;
A138: ((1 + (((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) - (len (Cage (C,n))) = 1 + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) ;
1 + (len (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 1 + 1 by A121, XREAL_1:7;
then A139: len (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) >= 2 by FINSEQ_5:8;
3 <= len (Lower_Seq (C,n)) by JORDAN1E:15;
then 2 <= len (Lower_Seq (C,n)) by XXREAL_0:2;
then A140: 2 in dom (Lower_Seq (C,n)) by FINSEQ_3:25;
(Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2
.= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ;
then A141: not E-max (L~ (Cage (C,n))) in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) by A133, JORDAN5B:16;
A142: L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))) is being_S-Seq by A112, JORDAN3:34;
(<*((Gauge (C,n)) * (i,1))*> /. (len <*((Gauge (C,n)) * (i,1))*>)) `1 = (<*((Gauge (C,n)) * (i,1))*> /. 1) `1 by FINSEQ_1:39
.= ((Gauge (C,n)) * (i,1)) `1 by FINSEQ_4:16
.= ((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1) `1 by A1, A2, A3, A122, GOBOARD5:2 ;
then A143: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is special by A142, GOBOARD2:8;
A144: L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) = (LSeg (((Gauge (C,n)) * (i,1)),((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1))) \/ (L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A120, SPPOL_2:20;
A145: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
then A146: (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) = ((len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n)))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) by A129, A127, SPRECT_5:9;
(len (Cage (C,n))) + 0 <= (len (Cage (C,n))) + ((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) by XREAL_1:6;
then (len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n))) <= (E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A146, XREAL_1:9;
then ((len (Cage (C,n))) - ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) + 1 <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by XREAL_1:6;
then A147: len ((Cage (C,n)) :- (W-min (L~ (Cage (C,n))))) <= 1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A145, FINSEQ_5:50;
A148: len (Lower_Seq (C,n)) > 1 by A132, XXREAL_0:2;
then A149: not mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is empty by A133, JORDAN1B:2;
A150: len (Lower_Seq (C,n)) in dom (Lower_Seq (C,n)) by FINSEQ_5:6;
then mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A140, JORDAN1E:18, SPRECT_2:22;
then A151: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_in_the_area_of Cage (C,n) by SPRECT_3:51;
A152: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by A129, FINSEQ_6:90, SPRECT_2:43;
then (Lower_Seq (C,n)) /. (1 + 1) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) by A5, A140, FINSEQ_5:52
.= (Cage (C,n)) /. (((1 + ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))))) + ((W-min (L~ (Cage (C,n)))) .. (Cage (C,n)))) -' (len (Cage (C,n)))) by A145, A146, A147, A131, REVROT_1:17
.= (Cage (C,n)) /. (((E-max (L~ (Cage (C,n)))) .. (Cage (C,n))) + 1) by A146, A138, XREAL_0:def_2 ;
then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. 1) `1 = E-bound (L~ (Cage (C,n))) by A140, A150, A130, SPRECT_2:8;
then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = E-bound (L~ (Cage (C,n))) by A149, FINSEQ_5:65;
then A153: ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. (len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))))) `1 = E-bound (L~ (Cage (C,n))) by FINSEQ_5:def_3;
(Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A5, A152, FINSEQ_5:54
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1
.= W-min (L~ (Cage (C,n))) by A145, FINSEQ_6:92 ;
then ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
then ((mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) /. (len (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))))) `1 = W-bound (L~ (Cage (C,n))) by A140, A150, SPRECT_2:9;
then ((Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) /. 1) `1 = W-bound (L~ (Cage (C,n))) by A149, FINSEQ_5:65;
then A154: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is_a_h.c._for Cage (C,n) by A151, A153, SPRECT_2:def_2;
<*((Gauge (C,n)) * (i,1))*> is one-to-one by FINSEQ_3:93;
then A155: <*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) is one-to-one by A137, A142, FINSEQ_3:91;
A156: L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) c= L~ (Lower_Seq (C,n)) by A11, JORDAN4:35;
mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))) is S-Sequence_in_R2 by A133, A148, JORDAN3:6;
then A157: Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) is S-Sequence_in_R2 ;
then 2 <= len (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) by TOPREAL1:def_8;
then L~ (Rev (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n)))))) meets L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by A139, A155, A143, A157, A154, A124, SPRECT_2:29;
then L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) meets L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by SPPOL_2:22;
then consider x being set such that
A158: x in L~ (mid ((Lower_Seq (C,n)),2,(len (Lower_Seq (C,n))))) and
A159: x in L~ (<*((Gauge (C,n)) * (i,1))*> ^ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j))))) by XBOOLE_0:3;
A160: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
A161: L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) c= L~ (Upper_Seq (C,n)) by A112, JORDAN3:42;
now__::_thesis:_L~_(Lower_Seq_(C,n))_meets_L~_<*((Gauge_(C,n))_*_(i,1)),((Gauge_(C,n))_*_(i,j))*>
percases ( x in LSeg (((Gauge (C,n)) * (i,1)),((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1)) or x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ) by A159, A144, XBOOLE_0:def_3;
suppose x in LSeg (((Gauge (C,n)) * (i,1)),((L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) /. 1)) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*>
then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A122, SPPOL_2:21;
hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A158, A156, XBOOLE_0:3; ::_thesis: verum
end;
supposeA162: x in L~ (L_Cut ((Upper_Seq (C,n)),((Gauge (C,n)) * (i,j)))) ; ::_thesis: L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*>
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A158, A156, A161, XBOOLE_0:def_4;
then x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then A163: x = W-min (L~ (Cage (C,n))) by A158, A141, TARSKI:def_2;
1 in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
then (Upper_Seq (C,n)) . 1 = (Upper_Seq (C,n)) /. 1 by PARTFUN1:def_6
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A7, A119, FINSEQ_5:44
.= W-min (L~ (Cage (C,n))) by A160, FINSEQ_6:92 ;
then x = (Gauge (C,n)) * (i,j) by A112, A162, A163, JORDAN1E:7;
then x in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
then x in L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by SPPOL_2:21;
hence L~ (Lower_Seq (C,n)) meets L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> by A158, A156, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
then L~ <*((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))*> meets L~ (Lower_Seq (C,n)) ;
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by SPPOL_2:21; ::_thesis: verum
end;
supposeA164: (Gauge (C,n)) * (i,j) in L~ (Lower_Seq (C,n)) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
(Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by A164, XBOOLE_0:3; ::_thesis: verum
end;
supposeA165: ( (Gauge (C,n)) * (i,j) in L~ (Upper_Seq (C,n)) & (Gauge (C,n)) * (i,j) = (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n))
A166: (Gauge (C,n)) * (i,j) in LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) by RLTOPSP1:68;
A167: ( rng (Lower_Seq (C,n)) c= L~ (Lower_Seq (C,n)) & E-max (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) ) by A5, A10, FINSEQ_6:61, SPPOL_2:18, XXREAL_0:2;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A168: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
len (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by A9, FINSEQ_3:25;
then (Upper_Seq (C,n)) . (len (Upper_Seq (C,n))) = (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) by PARTFUN1:def_6
.= ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A7, A168, FINSEQ_5:42
.= E-max (L~ (Cage (C,n))) by A168, FINSEQ_5:45 ;
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) by A165, A167, A166, XBOOLE_0:3; ::_thesis: verum
end;
end;
end;
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets L~ (Lower_Seq (C,n)) ; ::_thesis: verum
end;
theorem Th47: :: JORDAN1G:47
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
let n be Element of NAT ; ::_thesis: ( n > 0 implies First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) )
assume A1: n > 0 ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;
set Ebo = E-bound (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
A2: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;
A3: ( (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) & (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) ) by JORDAN1F:5, JORDAN1F:7;
then A4: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by TOPREAL1:25;
A5: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;
then A6: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;
A7: Center (Gauge (C,n)) <= len (Gauge (C,n)) by JORDAN1B:13;
((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A5, XREAL_1:226;
then A8: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;
then A9: L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A4, A6, JORDAN6:49;
(L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed by A4, A6, A8, JORDAN6:49;
then A10: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A4, A9, JORDAN5C:def_1;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
then consider t being Element of NAT such that
A11: 1 <= t and
A12: t + 1 <= len (Upper_Seq (C,n)) and
A13: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in LSeg ((Upper_Seq (C,n)),t) by SPPOL_2:13;
A14: LSeg ((Upper_Seq (C,n)),t) = LSeg (((Upper_Seq (C,n)) /. t),((Upper_Seq (C,n)) /. (t + 1))) by A11, A12, TOPREAL1:def_3;
t < len (Upper_Seq (C,n)) by A12, NAT_1:13;
then A15: t in dom (Upper_Seq (C,n)) by A11, FINSEQ_3:25;
1 <= t + 1 by A11, NAT_1:13;
then A16: t + 1 in dom (Upper_Seq (C,n)) by A12, FINSEQ_3:25;
First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A10, XBOOLE_0:def_4;
then A17: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;
A18: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = First_Point ((LSeg ((Upper_Seq (C,n)),t)),((Upper_Seq (C,n)) /. t),((Upper_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A3, A9, A11, A12, A13, JORDAN5C:19, JORDAN6:30;
now__::_thesis:_First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Upper_Seq_(C,n))
percases ( LSeg ((Upper_Seq (C,n)),t) is vertical or LSeg ((Upper_Seq (C,n)),t) is horizontal ) by SPPOL_1:19;
supposeA19: LSeg ((Upper_Seq (C,n)),t) is vertical ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
then ((Upper_Seq (C,n)) /. (t + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A14, A17, SPPOL_1:41;
then (Upper_Seq (C,n)) /. (t + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ;
then A20: (Upper_Seq (C,n)) /. (t + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6;
A21: ( LSeg ((Upper_Seq (C,n)),t) is closed & LSeg ((Upper_Seq (C,n)),t) is_an_arc_of (Upper_Seq (C,n)) /. t,(Upper_Seq (C,n)) /. (t + 1) ) by A14, A15, A16, GOBOARD7:29, TOPREAL1:9;
((Upper_Seq (C,n)) /. t) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A14, A17, A19, SPPOL_1:41;
then (Upper_Seq (C,n)) /. t in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ;
then (Upper_Seq (C,n)) /. t in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6;
then LSeg ((Upper_Seq (C,n)),t) c= Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A14, A20, JORDAN1A:13;
then First_Point ((LSeg ((Upper_Seq (C,n)),t)),((Upper_Seq (C,n)) /. t),((Upper_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Upper_Seq (C,n)) /. t by A21, JORDAN5C:7;
hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A18, A15, PARTFUN2:2; ::_thesis: verum
end;
suppose LSeg ((Upper_Seq (C,n)),t) is horizontal ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
then A22: ((Upper_Seq (C,n)) /. t) `2 = ((Upper_Seq (C,n)) /. (t + 1)) `2 by A14, SPPOL_1:15;
then A23: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = ((Upper_Seq (C,n)) /. t) `2 by A13, A14, GOBOARD7:6;
Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4;
then consider i1, j1, i2, j2 being Element of NAT such that
A24: [i1,j1] in Indices (Gauge (C,n)) and
A25: (Upper_Seq (C,n)) /. t = (Gauge (C,n)) * (i1,j1) and
A26: [i2,j2] in Indices (Gauge (C,n)) and
A27: (Upper_Seq (C,n)) /. (t + 1) = (Gauge (C,n)) * (i2,j2) and
A28: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A11, A12, JORDAN8:3;
A29: 1 <= i1 by A24, MATRIX_1:38;
A30: 1 <= i2 by A26, MATRIX_1:38;
A31: i1 <= len (Gauge (C,n)) by A24, MATRIX_1:38;
A32: j1 = j2 by A22, A24, A25, A26, A27, Th6;
A33: i2 <= len (Gauge (C,n)) by A26, MATRIX_1:38;
A34: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A24, MATRIX_1:38;
then A35: ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35
.= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A17, Th33 ;
((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `2 = ((Gauge (C,n)) * (1,j1)) `2 by A2, A7, A34, GOBOARD5:1
.= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A23, A25, A29, A31, A34, GOBOARD5:1 ;
then A36: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Gauge (C,n)) * ((Center (Gauge (C,n))),j1) by A35, TOPREAL3:6;
now__::_thesis:_First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Upper_Seq_(C,n))
percases ( i1 + 1 = i2 or i1 = i2 + 1 ) by A28, A32;
supposeA37: i1 + 1 = i2 ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
i1 < i1 + 1 by NAT_1:13;
then A38: ((Gauge (C,n)) * (i1,j1)) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A29, A34, A33, A37, SPRECT_3:13;
then ((Gauge (C,n)) * (i1,j1)) `1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A13, A14, A25, A27, A32, A37, TOPREAL1:3;
then i1 <= Center (Gauge (C,n)) by A2, A31, A34, A35, GOBOARD5:3;
then ( i1 = Center (Gauge (C,n)) or i1 < Center (Gauge (C,n)) ) by XXREAL_0:1;
then A39: ( i1 = Center (Gauge (C,n)) or i1 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13;
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A13, A14, A25, A27, A32, A37, A38, TOPREAL1:3;
then Center (Gauge (C,n)) <= i1 + 1 by A7, A34, A30, A35, A37, GOBOARD5:3;
then ( i1 = Center (Gauge (C,n)) or i1 + 1 = Center (Gauge (C,n)) ) by A39, XXREAL_0:1;
hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A15, A16, A25, A27, A32, A36, A37, PARTFUN2:2; ::_thesis: verum
end;
supposeA40: i1 = i2 + 1 ; ::_thesis: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n))
i2 < i2 + 1 by NAT_1:13;
then A41: ((Gauge (C,n)) * (i2,j1)) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A31, A34, A30, A40, SPRECT_3:13;
then ((Gauge (C,n)) * (i2,j1)) `1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A13, A14, A25, A27, A32, A40, TOPREAL1:3;
then i2 <= Center (Gauge (C,n)) by A2, A34, A33, A35, GOBOARD5:3;
then ( i2 = Center (Gauge (C,n)) or i2 < Center (Gauge (C,n)) ) by XXREAL_0:1;
then A42: ( i2 = Center (Gauge (C,n)) or i2 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13;
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A13, A14, A25, A27, A32, A40, A41, TOPREAL1:3;
then Center (Gauge (C,n)) <= i2 + 1 by A7, A29, A34, A35, A40, GOBOARD5:3;
then ( i2 = Center (Gauge (C,n)) or i2 + 1 = Center (Gauge (C,n)) ) by A42, XXREAL_0:1;
hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A15, A16, A25, A27, A32, A36, A40, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) ; ::_thesis: verum
end;
end;
end;
hence First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) ; ::_thesis: verum
end;
theorem Th48: :: JORDAN1G:48
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
let n be Element of NAT ; ::_thesis: ( n > 0 implies Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) )
assume A1: n > 0 ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;
set Ebo = E-bound (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
A2: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8;
then A3: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25;
A4: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;
then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1;
then A5: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;
((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A4, JORDAN6:1;
then A6: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;
A7: L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A2, JORDAN5B:14, TOPREAL1:25;
then A8: L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A5, A6, JORDAN6:49;
(L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed by A7, A5, A6, JORDAN6:49;
then A9: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A3, A8, JORDAN5C:def_2;
then Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
then consider t being Element of NAT such that
A10: 1 <= t and
A11: t + 1 <= len (Lower_Seq (C,n)) and
A12: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in LSeg ((Lower_Seq (C,n)),t) by SPPOL_2:13;
A13: LSeg ((Lower_Seq (C,n)),t) = LSeg (((Lower_Seq (C,n)) /. t),((Lower_Seq (C,n)) /. (t + 1))) by A10, A11, TOPREAL1:def_3;
1 <= t + 1 by A10, NAT_1:13;
then A14: t + 1 in dom (Lower_Seq (C,n)) by A11, FINSEQ_3:25;
t < len (Lower_Seq (C,n)) by A11, NAT_1:13;
then A15: t in dom (Lower_Seq (C,n)) by A10, FINSEQ_3:25;
Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A9, XBOOLE_0:def_4;
then A16: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;
A17: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((LSeg ((Lower_Seq (C,n)),t)),((Lower_Seq (C,n)) /. t),((Lower_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A2, A8, A10, A11, A12, JORDAN5C:20, JORDAN6:30;
now__::_thesis:_Last_Point_((L~_(Lower_Seq_(C,n))),(E-max_(L~_(Cage_(C,n)))),(W-min_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Lower_Seq_(C,n))
percases ( LSeg ((Lower_Seq (C,n)),t) is vertical or LSeg ((Lower_Seq (C,n)),t) is horizontal ) by SPPOL_1:19;
supposeA18: LSeg ((Lower_Seq (C,n)),t) is vertical ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
then ((Lower_Seq (C,n)) /. (t + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A12, A13, A16, SPPOL_1:41;
then (Lower_Seq (C,n)) /. (t + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ;
then A19: (Lower_Seq (C,n)) /. (t + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6;
A20: ( LSeg ((Lower_Seq (C,n)),t) is closed & LSeg ((Lower_Seq (C,n)),t) is_an_arc_of (Lower_Seq (C,n)) /. t,(Lower_Seq (C,n)) /. (t + 1) ) by A13, A15, A14, GOBOARD7:29, TOPREAL1:9;
((Lower_Seq (C,n)) /. t) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A12, A13, A16, A18, SPPOL_1:41;
then (Lower_Seq (C,n)) /. t in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ;
then (Lower_Seq (C,n)) /. t in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6;
then LSeg ((Lower_Seq (C,n)),t) c= Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A13, A19, JORDAN1A:13;
then Last_Point ((LSeg ((Lower_Seq (C,n)),t)),((Lower_Seq (C,n)) /. t),((Lower_Seq (C,n)) /. (t + 1)),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Lower_Seq (C,n)) /. (t + 1) by A20, JORDAN5C:7;
hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A17, A14, PARTFUN2:2; ::_thesis: verum
end;
suppose LSeg ((Lower_Seq (C,n)),t) is horizontal ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
then A21: ((Lower_Seq (C,n)) /. t) `2 = ((Lower_Seq (C,n)) /. (t + 1)) `2 by A13, SPPOL_1:15;
then A22: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = ((Lower_Seq (C,n)) /. t) `2 by A12, A13, GOBOARD7:6;
Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5;
then consider i1, j1, i2, j2 being Element of NAT such that
A23: [i1,j1] in Indices (Gauge (C,n)) and
A24: (Lower_Seq (C,n)) /. t = (Gauge (C,n)) * (i1,j1) and
A25: [i2,j2] in Indices (Gauge (C,n)) and
A26: (Lower_Seq (C,n)) /. (t + 1) = (Gauge (C,n)) * (i2,j2) and
A27: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A10, A11, JORDAN8:3;
A28: 1 <= i1 by A23, MATRIX_1:38;
A29: j1 = j2 by A21, A23, A24, A25, A26, Th6;
A30: i2 <= len (Gauge (C,n)) by A25, MATRIX_1:38;
A31: i1 <= len (Gauge (C,n)) by A23, MATRIX_1:38;
A32: 1 <= i2 by A25, MATRIX_1:38;
A33: Center (Gauge (C,n)) <= len (Gauge (C,n)) by JORDAN1B:13;
A34: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A23, MATRIX_1:38;
then A35: ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35
.= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A16, Th33 ;
A36: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;
then ((Gauge (C,n)) * ((Center (Gauge (C,n))),j1)) `2 = ((Gauge (C,n)) * (1,j1)) `2 by A34, A33, GOBOARD5:1
.= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A22, A24, A28, A31, A34, GOBOARD5:1 ;
then A37: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = (Gauge (C,n)) * ((Center (Gauge (C,n))),j1) by A35, TOPREAL3:6;
now__::_thesis:_Last_Point_((L~_(Lower_Seq_(C,n))),(E-max_(L~_(Cage_(C,n)))),(W-min_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)))_in_rng_(Lower_Seq_(C,n))
percases ( i1 + 1 = i2 or i1 = i2 + 1 ) by A27, A29;
supposeA38: i1 + 1 = i2 ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
i1 < i1 + 1 by NAT_1:13;
then A39: ((Gauge (C,n)) * (i1,j1)) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A28, A34, A30, A38, SPRECT_3:13;
then ((Gauge (C,n)) * (i1,j1)) `1 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A12, A13, A24, A26, A29, A38, TOPREAL1:3;
then i1 <= Center (Gauge (C,n)) by A31, A34, A36, A35, GOBOARD5:3;
then ( i1 = Center (Gauge (C,n)) or i1 < Center (Gauge (C,n)) ) by XXREAL_0:1;
then A40: ( i1 = Center (Gauge (C,n)) or i1 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13;
(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i1 + 1),j1)) `1 by A12, A13, A24, A26, A29, A38, A39, TOPREAL1:3;
then Center (Gauge (C,n)) <= i1 + 1 by A34, A32, A33, A35, A38, GOBOARD5:3;
then ( i1 = Center (Gauge (C,n)) or i1 + 1 = Center (Gauge (C,n)) ) by A40, XXREAL_0:1;
hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A15, A14, A24, A26, A29, A37, A38, PARTFUN2:2; ::_thesis: verum
end;
supposeA41: i1 = i2 + 1 ; ::_thesis: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n))
i2 < i2 + 1 by NAT_1:13;
then A42: ((Gauge (C,n)) * (i2,j1)) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A31, A34, A32, A41, SPRECT_3:13;
then ((Gauge (C,n)) * (i2,j1)) `1 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A12, A13, A24, A26, A29, A41, TOPREAL1:3;
then i2 <= Center (Gauge (C,n)) by A34, A30, A36, A35, GOBOARD5:3;
then ( i2 = Center (Gauge (C,n)) or i2 < Center (Gauge (C,n)) ) by XXREAL_0:1;
then A43: ( i2 = Center (Gauge (C,n)) or i2 + 1 <= Center (Gauge (C,n)) ) by NAT_1:13;
(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= ((Gauge (C,n)) * ((i2 + 1),j1)) `1 by A12, A13, A24, A26, A29, A41, A42, TOPREAL1:3;
then Center (Gauge (C,n)) <= i2 + 1 by A28, A34, A33, A35, A41, GOBOARD5:3;
then ( i2 = Center (Gauge (C,n)) or i2 + 1 = Center (Gauge (C,n)) ) by A43, XXREAL_0:1;
hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A15, A14, A24, A26, A29, A37, A41, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) ; ::_thesis: verum
end;
end;
end;
hence Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) ; ::_thesis: verum
end;
theorem Th49: :: JORDAN1G:49
for f being S-Sequence_in_R2
for p being Point of (TOP-REAL 2) st p in rng f holds
R_Cut (f,p) = mid (f,1,(p .. f))
proof
let f be S-Sequence_in_R2; ::_thesis: for p being Point of (TOP-REAL 2) st p in rng f holds
R_Cut (f,p) = mid (f,1,(p .. f))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in rng f implies R_Cut (f,p) = mid (f,1,(p .. f)) )
assume A1: p in rng f ; ::_thesis: R_Cut (f,p) = mid (f,1,(p .. f))
then consider i being Nat such that
A2: i in dom f and
A3: f . i = p by FINSEQ_2:10;
reconsider i = i as Element of NAT by ORDINAL1:def_12;
A4: i <= len f by A2, FINSEQ_3:25;
len f >= 2 by TOPREAL1:def_8;
then A5: rng f c= L~ f by SPPOL_2:18;
then A6: 1 <= Index (p,f) by A1, JORDAN3:8;
A7: Index (p,f) < len f by A1, A5, JORDAN3:8;
A8: 0 + 1 <= i by A2, FINSEQ_3:25;
then A9: i - 1 >= 0 by XREAL_1:19;
percases ( 1 < i or 1 = i ) by A8, XXREAL_0:1;
supposeA10: 1 < i ; ::_thesis: R_Cut (f,p) = mid (f,1,(p .. f))
1 <= len f by A8, A4, XXREAL_0:2;
then 1 in dom f by FINSEQ_3:25;
then p <> f . 1 by A2, A3, A10, FUNCT_1:def_4;
then A11: R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by JORDAN3:def_4;
A12: (Index (p,f)) + 1 = i by A3, A4, A10, JORDAN3:12;
A13: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A6, A7, JORDAN4:8
.= i -' 1 by A1, A5, A12, JORDAN3:8, NAT_D:38 ;
A14: len (mid (f,1,i)) = (i -' 1) + 1 by A8, A4, JORDAN4:8
.= i by A8, XREAL_1:235 ;
then A15: dom (mid (f,1,i)) = Seg i by FINSEQ_1:def_3;
A16: now__::_thesis:_for_j_being_Nat_st_j_in_dom_(mid_(f,1,i))_holds_
(mid_(f,1,i))_._j_=_((mid_(f,1,(Index_(p,f))))_^_<*p*>)_._j
let j be Nat; ::_thesis: ( j in dom (mid (f,1,i)) implies (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j )
reconsider a = j as Element of NAT by ORDINAL1:def_12;
assume A17: j in dom (mid (f,1,i)) ; ::_thesis: (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j
then A18: 1 <= j by A15, FINSEQ_1:1;
A19: j <= i by A15, A17, FINSEQ_1:1;
now__::_thesis:_(mid_(f,1,i))_._j_=_((mid_(f,1,(Index_(p,f))))_^_<*p*>)_._j
percases ( j < i or j = i ) by A19, XXREAL_0:1;
suppose j < i ; ::_thesis: (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j
then A20: j <= Index (p,f) by A12, NAT_1:13;
then j <= i -' 1 by A9, A12, XREAL_0:def_2;
then A21: j in dom (mid (f,1,(Index (p,f)))) by A13, A18, FINSEQ_3:25;
thus (mid (f,1,i)) . j = f . a by A4, A18, A19, FINSEQ_6:123
.= (mid (f,1,(Index (p,f)))) . a by A7, A18, A20, FINSEQ_6:123
.= ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j by A21, FINSEQ_1:def_7 ; ::_thesis: verum
end;
supposeA22: j = i ; ::_thesis: (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j
A23: (i -' 1) + 1 = i by A8, XREAL_1:235;
thus (mid (f,1,i)) . j = f . a by A4, A18, A19, FINSEQ_6:123
.= ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j by A3, A13, A22, A23, FINSEQ_1:42 ; ::_thesis: verum
end;
end;
end;
hence (mid (f,1,i)) . j = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . j ; ::_thesis: verum
end;
len ((mid (f,1,(Index (p,f)))) ^ <*p*>) = (i -' 1) + 1 by A13, FINSEQ_2:16
.= i by A8, XREAL_1:235 ;
then mid (f,1,i) = R_Cut (f,p) by A11, A14, A16, FINSEQ_2:9;
hence R_Cut (f,p) = mid (f,1,(p .. f)) by A2, A3, FINSEQ_5:11; ::_thesis: verum
end;
supposeA24: 1 = i ; ::_thesis: R_Cut (f,p) = mid (f,1,(p .. f))
then A25: R_Cut (f,p) = <*p*> by A3, JORDAN3:def_4;
A26: p = f /. 1 by A2, A3, A24, PARTFUN1:def_6;
then p .. f = 1 by FINSEQ_6:43;
hence R_Cut (f,p) = mid (f,1,(p .. f)) by A4, A24, A25, A26, JORDAN4:15; ::_thesis: verum
end;
end;
end;
theorem Th50: :: JORDAN1G:50
for f being S-Sequence_in_R2
for Q being closed Subset of (TOP-REAL 2) st L~ f meets Q & not f /. 1 in Q & First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f holds
(L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))}
proof
let f be S-Sequence_in_R2; ::_thesis: for Q being closed Subset of (TOP-REAL 2) st L~ f meets Q & not f /. 1 in Q & First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f holds
(L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))}
let Q be closed Subset of (TOP-REAL 2); ::_thesis: ( L~ f meets Q & not f /. 1 in Q & First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f implies (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} )
assume that
A1: ( L~ f meets Q & not f /. 1 in Q ) and
A2: First_Point ((L~ f),(f /. 1),(f /. (len f)),Q) in rng f ; ::_thesis: (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))}
(L~ (R_Cut (f,(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} by A1, SPRECT_4:1;
hence (L~ (mid (f,1,((First_Point ((L~ f),(f /. 1),(f /. (len f)),Q)) .. f)))) /\ Q = {(First_Point ((L~ f),(f /. 1),(f /. (len f)),Q))} by A2, Th49; ::_thesis: verum
end;
theorem Th51: :: JORDAN1G:51
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds
((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds
((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
let n be Element of NAT ; ::_thesis: ( n > 0 implies for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds
((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 )
assume A1: n > 0 ; ::_thesis: for k being Element of NAT st 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) holds
((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
set US = Upper_Seq (C,n);
set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;
set Ebo = E-bound (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
defpred S1[ Nat] means ( 1 <= $1 & $1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) implies ((Upper_Seq (C,n)) /. $1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 );
A2: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then A3: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;
A4: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A2, XREAL_1:226;
A5: for k being non empty Nat st S1[k] holds
S1[k + 1]
proof
set GC1 = (Gauge (C,n)) * ((Center (Gauge (C,n))),1);
let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A6: ( 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) implies ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) ; ::_thesis: S1[k + 1]
4 <= len (Gauge (C,n)) by JORDAN8:10;
then 1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A7: 1 <= width (Gauge (C,n)) by JORDAN8:def_1;
then A8: ((Gauge (C,n)) * ((Center (Gauge (C,n))),1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35
.= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ;
A9: k >= 1 by NAT_1:14;
A10: (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
A11: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A1, Th47;
then A12: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20;
then A13: 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by FINSEQ_3:25;
A14: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;
A15: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
assume that
A16: 1 <= k + 1 and
A17: k + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
A18: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by A12, FINSEQ_3:25;
then A19: k + 1 <= len (Upper_Seq (C,n)) by A17, XXREAL_0:2;
Upper_Seq (C,n) is_sequence_on Gauge (C,n) by Th4;
then consider i1, j1, i2, j2 being Element of NAT such that
A20: [i1,j1] in Indices (Gauge (C,n)) and
A21: (Upper_Seq (C,n)) /. kk = (Gauge (C,n)) * (i1,j1) and
A22: [i2,j2] in Indices (Gauge (C,n)) and
A23: (Upper_Seq (C,n)) /. (kk + 1) = (Gauge (C,n)) * (i2,j2) and
A24: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A9, A19, JORDAN8:3;
A25: 1 <= i1 by A20, MATRIX_1:38;
A26: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A20, MATRIX_1:38;
A27: j2 <= width (Gauge (C,n)) by A22, MATRIX_1:38;
A28: ( 1 <= i2 & 1 <= j2 ) by A22, MATRIX_1:38;
A29: i2 <= len (Gauge (C,n)) by A22, MATRIX_1:38;
A30: i1 <= len (Gauge (C,n)) by A20, MATRIX_1:38;
A31: ( Center (Gauge (C,n)) <= len (Gauge (C,n)) & i1 + 1 >= 1 ) by JORDAN1B:13, NAT_1:11;
now__::_thesis:_((Upper_Seq_(C,n))_/._(k_+_1))_`1_<_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A24;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then ((Upper_Seq (C,n)) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A21, A25, A30, A26, GOBOARD5:2
.= ((Upper_Seq (C,n)) /. (k + 1)) `1 by A23, A29, A28, A27, GOBOARD5:2 ;
hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, A17, NAT_1:13, NAT_1:14; ::_thesis: verum
end;
supposeA32: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
A33: now__::_thesis:_not_((Upper_Seq_(C,n))_/._(k_+_1))_`1_=_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2
A34: k + 1 >= 1 + 1 by A9, XREAL_1:7;
len (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 1) + 1 by A13, A18, JORDAN4:8
.= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by A13, XREAL_1:235 ;
then A35: rng (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A17, A34, SPPOL_2:18, XXREAL_0:2;
A36: (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A11, FINSEQ_5:38;
A37: now__::_thesis:_not_(Upper_Seq_(C,n))_/._1_in_Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)
assume (Upper_Seq (C,n)) /. 1 in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) ; ::_thesis: contradiction
then (W-min (L~ (Cage (C,n)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A15, JORDAN6:31;
hence contradiction by A3, EUCLID:52; ::_thesis: verum
end;
A38: ( (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 & ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 ) by A3, A4, EUCLID:52;
A39: ( Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) is closed & L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) ) by A15, A10, JORDAN6:30, TOPREAL1:25;
First_Point ((L~ (Upper_Seq (C,n))),((Upper_Seq (C,n)) /. 1),((Upper_Seq (C,n)) /. (len (Upper_Seq (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by A1, A15, A10, Th47;
then A40: (L~ (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) = {(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A15, A10, A39, A38, A37, Th50, JORDAN6:49;
A41: ( mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) = (Upper_Seq (C,n)) | ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) & (Upper_Seq (C,n)) | (Seg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) = (Upper_Seq (C,n)) | ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) ) by A13, FINSEQ_1:def_15, FINSEQ_6:116;
assume ((Upper_Seq (C,n)) /. (k + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: contradiction
then (Upper_Seq (C,n)) /. (k + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ;
then A42: (Upper_Seq (C,n)) /. (k + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6;
A43: k + 1 in dom (Upper_Seq (C,n)) by A16, A19, FINSEQ_3:25;
k + 1 in Seg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A16, A17, FINSEQ_1:1;
then (Upper_Seq (C,n)) /. (k + 1) in rng (mid ((Upper_Seq (C,n)),1,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A41, A43, PARTFUN2:18;
then (Upper_Seq (C,n)) /. (k + 1) in {(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A42, A35, A40, XBOOLE_0:def_4;
then (Upper_Seq (C,n)) /. (k + 1) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by TARSKI:def_1;
hence contradiction by A17, A12, A43, A36, PARTFUN2:10; ::_thesis: verum
end;
i1 < Center (Gauge (C,n)) by A6, A17, A21, A30, A26, A14, A7, A8, JORDAN1A:18, NAT_1:13, NAT_1:14;
then i1 + 1 <= Center (Gauge (C,n)) by NAT_1:13;
then ((Upper_Seq (C,n)) /. (k + 1)) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A23, A26, A7, A8, A31, A32, JORDAN1A:18;
hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A33, XXREAL_0:1; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then i2 < i1 by NAT_1:13;
then ((Upper_Seq (C,n)) /. (k + 1)) `1 <= ((Upper_Seq (C,n)) /. k) `1 by A21, A23, A30, A26, A28, A27, JORDAN1A:18;
hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, A17, NAT_1:13, NAT_1:14, XXREAL_0:2; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then ((Upper_Seq (C,n)) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A21, A25, A30, A26, GOBOARD5:2
.= ((Upper_Seq (C,n)) /. (k + 1)) `1 by A23, A29, A28, A27, GOBOARD5:2 ;
hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, A17, NAT_1:13, NAT_1:14; ::_thesis: verum
end;
end;
end;
hence ((Upper_Seq (C,n)) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: verum
end;
A44: S1[1]
proof
assume that
1 <= 1 and
1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: ((Upper_Seq (C,n)) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
(Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
hence ((Upper_Seq (C,n)) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A3, EUCLID:52; ::_thesis: verum
end;
A45: for k being non empty Nat holds S1[k] from NAT_1:sch_10(A44, A5);
let k be Element of NAT ; ::_thesis: ( 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) implies ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 )
assume ( 1 <= k & k < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ) ; ::_thesis: ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
hence ((Upper_Seq (C,n)) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A45; ::_thesis: verum
end;
theorem Th52: :: JORDAN1G:52
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds
((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds
((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
let n be Element of NAT ; ::_thesis: ( n > 0 implies for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds
((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 )
assume A1: n > 0 ; ::_thesis: for k being Nat st 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) holds
((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
set LS = Lower_Seq (C,n);
set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;
set Ebo = E-bound (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wmin = W-min (L~ (Cage (C,n)));
set RLS = Rev (Lower_Seq (C,n));
set FiP = First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
A2: L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) by SPPOL_2:22;
A3: len (Rev (Lower_Seq (C,n))) = len (Lower_Seq (C,n)) by FINSEQ_5:def_3;
defpred S1[ Nat] means ( 1 <= $1 & $1 < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) implies ((Rev (Lower_Seq (C,n))) /. $1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 );
A4: rng (Rev (Lower_Seq (C,n))) = rng (Lower_Seq (C,n)) by FINSEQ_5:57;
A5: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then A6: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;
A7: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A5, XREAL_1:226;
A8: for k being non empty Nat st S1[k] holds
S1[k + 1]
proof
A9: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;
then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1;
then A10: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;
((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A9, JORDAN6:1;
then A11: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;
A12: (Rev (Lower_Seq (C,n))) /. (len (Rev (Lower_Seq (C,n)))) = (Lower_Seq (C,n)) /. 1 by A3, FINSEQ_5:65
.= E-max (L~ (Cage (C,n))) by JORDAN1F:6 ;
set GC1 = (Gauge (C,n)) * ((Center (Gauge (C,n))),1);
let k be non empty Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A13: ( 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) implies ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) ; ::_thesis: S1[k + 1]
4 <= len (Gauge (C,n)) by JORDAN8:10;
then 1 <= len (Gauge (C,n)) by XXREAL_0:2;
then A14: 1 <= width (Gauge (C,n)) by JORDAN8:def_1;
then A15: ((Gauge (C,n)) * ((Center (Gauge (C,n))),1)) `1 = ((W-bound C) + (E-bound C)) / 2 by A1, Th35
.= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ;
A16: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8;
then A17: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25;
A18: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;
A19: (Rev (Lower_Seq (C,n))) /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A16, JORDAN5B:14, TOPREAL1:25;
then ( L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A10, A11, JORDAN6:49;
then A20: First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A2, A17, JORDAN5C:18;
then A21: (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) in dom (Rev (Lower_Seq (C,n))) by A1, A4, Th48, FINSEQ_4:20;
then A22: 1 <= (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by FINSEQ_3:25;
A23: k >= 1 by NAT_1:14;
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
assume that
A24: 1 <= k + 1 and
A25: k + 1 < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
A26: (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) <= len (Rev (Lower_Seq (C,n))) by A21, FINSEQ_3:25;
then A27: k + 1 <= len (Rev (Lower_Seq (C,n))) by A25, XXREAL_0:2;
Lower_Seq (C,n) is_sequence_on Gauge (C,n) by Th5;
then Rev (Lower_Seq (C,n)) is_sequence_on Gauge (C,n) by JORDAN9:5;
then consider i1, j1, i2, j2 being Element of NAT such that
A28: [i1,j1] in Indices (Gauge (C,n)) and
A29: (Rev (Lower_Seq (C,n))) /. kk = (Gauge (C,n)) * (i1,j1) and
A30: [i2,j2] in Indices (Gauge (C,n)) and
A31: (Rev (Lower_Seq (C,n))) /. (kk + 1) = (Gauge (C,n)) * (i2,j2) and
A32: ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A23, A27, JORDAN8:3;
A33: 1 <= i1 by A28, MATRIX_1:38;
A34: ( 1 <= j1 & j1 <= width (Gauge (C,n)) ) by A28, MATRIX_1:38;
A35: i2 <= len (Gauge (C,n)) by A30, MATRIX_1:38;
A36: i1 <= len (Gauge (C,n)) by A28, MATRIX_1:38;
A37: j2 <= width (Gauge (C,n)) by A30, MATRIX_1:38;
A38: ( 1 <= i2 & 1 <= j2 ) by A30, MATRIX_1:38;
A39: ( Center (Gauge (C,n)) <= len (Gauge (C,n)) & i1 + 1 >= 1 ) by JORDAN1B:13, NAT_1:11;
now__::_thesis:_((Rev_(Lower_Seq_(C,n)))_/._(k_+_1))_`1_<_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2
percases ( ( i1 = i2 & j1 + 1 = j2 ) or ( i1 + 1 = i2 & j1 = j2 ) or ( i1 = i2 + 1 & j1 = j2 ) or ( i1 = i2 & j1 = j2 + 1 ) ) by A32;
suppose ( i1 = i2 & j1 + 1 = j2 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then ((Rev (Lower_Seq (C,n))) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A29, A33, A36, A34, GOBOARD5:2
.= ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 by A31, A35, A38, A37, GOBOARD5:2 ;
hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A25, NAT_1:13, NAT_1:14; ::_thesis: verum
end;
supposeA40: ( i1 + 1 = i2 & j1 = j2 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
A41: now__::_thesis:_not_((Rev_(Lower_Seq_(C,n)))_/._(k_+_1))_`1_=_((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2
A42: now__::_thesis:_not_(Rev_(Lower_Seq_(C,n)))_/._1_in_Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2)
assume (Rev (Lower_Seq (C,n))) /. 1 in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) ; ::_thesis: contradiction
then (W-min (L~ (Cage (C,n)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A19, JORDAN6:31;
hence contradiction by A6, EUCLID:52; ::_thesis: verum
end;
assume ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: contradiction
then (Rev (Lower_Seq (C,n))) /. (k + 1) in { p where p is Point of (TOP-REAL 2) : p `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 } ;
then A43: (Rev (Lower_Seq (C,n))) /. (k + 1) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by JORDAN6:def_6;
A44: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by A7, EUCLID:52;
( L~ (Rev (Lower_Seq (C,n))) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) & (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ) by A6, A19, A12, EUCLID:52, TOPREAL1:25;
then A45: L~ (Rev (Lower_Seq (C,n))) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A44, JORDAN6:49;
A46: (Rev (Lower_Seq (C,n))) /. ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) = First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A4, A20, Th48, FINSEQ_5:38;
A47: k + 1 >= 1 + 1 by A23, XREAL_1:7;
len (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) = (((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) -' 1) + 1 by A22, A26, JORDAN4:8
.= (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by A22, XREAL_1:235 ;
then A48: rng (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) c= L~ (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) by A25, A47, SPPOL_2:18, XXREAL_0:2;
A49: k + 1 in dom (Rev (Lower_Seq (C,n))) by A24, A27, FINSEQ_3:25;
( Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) is closed & Rev (Lower_Seq (C,n)) is being_S-Seq ) by JORDAN6:30;
then A50: (L~ (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) = {(First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A1, A4, A20, A19, A12, A45, A42, Th48, Th50;
A51: ( mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) = (Rev (Lower_Seq (C,n))) | ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) & (Rev (Lower_Seq (C,n))) | (Seg ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) = (Rev (Lower_Seq (C,n))) | ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) ) by A22, FINSEQ_1:def_15, FINSEQ_6:116;
k + 1 in Seg ((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A24, A25, FINSEQ_1:1;
then (Rev (Lower_Seq (C,n))) /. (k + 1) in rng (mid ((Rev (Lower_Seq (C,n))),1,((First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))))) by A51, A49, PARTFUN2:18;
then (Rev (Lower_Seq (C,n))) /. (k + 1) in {(First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))} by A43, A48, A50, XBOOLE_0:def_4;
then (Rev (Lower_Seq (C,n))) /. (k + 1) = First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by TARSKI:def_1;
hence contradiction by A25, A21, A49, A46, PARTFUN2:10; ::_thesis: verum
end;
i1 < Center (Gauge (C,n)) by A13, A25, A29, A36, A34, A18, A14, A15, JORDAN1A:18, NAT_1:13, NAT_1:14;
then i1 + 1 <= Center (Gauge (C,n)) by NAT_1:13;
then ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A31, A34, A14, A15, A39, A40, JORDAN1A:18;
hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A41, XXREAL_0:1; ::_thesis: verum
end;
suppose ( i1 = i2 + 1 & j1 = j2 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then i2 < i1 by NAT_1:13;
then ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 <= ((Rev (Lower_Seq (C,n))) /. k) `1 by A29, A31, A36, A34, A38, A37, JORDAN1A:18;
hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A25, NAT_1:13, NAT_1:14, XXREAL_0:2; ::_thesis: verum
end;
suppose ( i1 = i2 & j1 = j2 + 1 ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then ((Rev (Lower_Seq (C,n))) /. k) `1 = ((Gauge (C,n)) * (i2,1)) `1 by A29, A33, A36, A34, GOBOARD5:2
.= ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 by A31, A35, A38, A37, GOBOARD5:2 ;
hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A13, A25, NAT_1:13, NAT_1:14; ::_thesis: verum
end;
end;
end;
hence ((Rev (Lower_Seq (C,n))) /. (k + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 ; ::_thesis: verum
end;
A52: S1[1]
proof
assume that
1 <= 1 and
1 < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
(Rev (Lower_Seq (C,n))) /. 1 = (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
hence ((Rev (Lower_Seq (C,n))) /. 1) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A6, EUCLID:52; ::_thesis: verum
end;
A53: for k being non empty Nat holds S1[k] from NAT_1:sch_10(A52, A8);
let k be Nat; ::_thesis: ( 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) implies ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 )
assume ( 1 <= k & k < (First_Point ((L~ (Rev (Lower_Seq (C,n)))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) ) ; ::_thesis: ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
hence ((Rev (Lower_Seq (C,n))) /. k) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A53; ::_thesis: verum
end;
theorem Th53: :: JORDAN1G:53
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds
q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds
q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
let n be Element of NAT ; ::_thesis: ( n > 0 implies for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds
q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 )
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;
set US = Upper_Seq (C,n);
set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
A1: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
(Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
then A2: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A1, TOPREAL1:25;
assume A3: n > 0 ; ::_thesis: for q being Point of (TOP-REAL 2) st q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) holds
q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then A4: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by Th47;
then A5: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20;
then A6: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by FINSEQ_3:25;
A7: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then A8: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;
((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A7, XREAL_1:226;
then A9: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;
(W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A8, EUCLID:52;
then ( L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A2, A9, JORDAN6:49;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A2, JORDAN5C:def_1;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by XBOOLE_0:def_4;
then A10: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;
A11: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42;
A12: now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))_=_1
assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43
.= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A4, A11, FINSEQ_5:9;
hence contradiction by A8, A10, EUCLID:52; ::_thesis: verum
end;
1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by A5, FINSEQ_3:25;
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1 by A12, XXREAL_0:1;
then A13: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19;
then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def_2;
then A14: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A6, A13, JORDAN4:8;
let q be Point of (TOP-REAL 2); ::_thesis: ( q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) implies q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 )
assume q in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
then consider k being Element of NAT such that
A15: k in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) and
A16: q = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. k by PARTFUN2:2;
k + 2 >= 1 + 1 by NAT_1:11;
then A17: (k + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9;
len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then len (Upper_Seq (C,n)) >= 2 by XXREAL_0:2;
then 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A18: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. k = (Upper_Seq (C,n)) /. ((k + 2) -' 1) by A15, A5, A13, SPRECT_2:3
.= (Upper_Seq (C,n)) /. (k + (2 - 1)) by A17, XREAL_0:def_2 ;
k <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A15, FINSEQ_3:25;
then k < ((((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1) + 1 by A14, NAT_1:13;
then A19: k + 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
percases ( k + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) or k + 1 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ) by A19, XXREAL_0:1;
suppose k + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
hence q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A3, A16, A18, Th51, NAT_1:11; ::_thesis: verum
end;
suppose k + 1 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2
hence q `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A16, A4, A10, A18, FINSEQ_5:38; ::_thesis: verum
end;
end;
end;
theorem Th54: :: JORDAN1G:54
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
(First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2
let n be Element of NAT ; ::_thesis: ( n > 0 implies (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 )
set sr = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2;
set Wmin = W-min (L~ (Cage (C,n)));
set Emax = E-max (L~ (Cage (C,n)));
set Nbo = N-bound (L~ (Cage (C,n)));
set Ebo = E-bound (L~ (Cage (C,n)));
set Sbo = S-bound (L~ (Cage (C,n)));
set Wbo = W-bound (L~ (Cage (C,n)));
set SW = SW-corner (L~ (Cage (C,n)));
set FiP = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
set LaP = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
set g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>;
set h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>;
A1: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by JORDAN1F:5;
A2: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;
then W-bound (L~ (Cage (C,n))) <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:1;
then A3: (W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;
((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= E-bound (L~ (Cage (C,n))) by A2, JORDAN6:1;
then A4: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;
set GCw = (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))));
A5: 1 <= Center (Gauge (C,n)) by JORDAN1B:11;
len (Gauge (C,n)) = width (Gauge (C,n)) by JORDAN8:def_1;
then A6: ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `2 = N-bound (L~ (Cage (C,n))) by A5, JORDAN1A:70, JORDAN1B:13;
A7: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;
A8: |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 = N-bound (L~ (Cage (C,n))) by EUCLID:52;
set RevL = (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))));
A9: ( <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is one-to-one & <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is special ) by FINSEQ_3:93;
A10: rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:48;
A11: ( (Lower_Seq (C,n)) /. 1 = E-max (L~ (Cage (C,n))) & (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = W-min (L~ (Cage (C,n))) ) by JORDAN1F:6, JORDAN1F:8;
then A12: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by TOPREAL1:25;
A13: 4 <= len (Gauge (C,n)) by JORDAN8:10;
then A14: len (Gauge (C,n)) >= 3 by XXREAL_0:2;
A15: W-bound (L~ (Cage (C,n))) < E-bound (L~ (Cage (C,n))) by SPRECT_1:31;
then A16: W-bound (L~ (Cage (C,n))) < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by XREAL_1:226;
L~ (Lower_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A11, JORDAN5B:14, TOPREAL1:25;
then A17: ( L~ (Lower_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A3, A4, JORDAN6:49;
then A18: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Lower_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A12, JORDAN5C:def_2;
then A19: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Lower_Seq (C,n)) by XBOOLE_0:def_4;
then Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def_3;
then A20: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13;
assume A21: n > 0 ; ::_thesis: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2
then A22: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Upper_Seq (C,n)) by Th47;
then A23: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) in dom (Upper_Seq (C,n)) by FINSEQ_4:20;
then A24: 1 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by FINSEQ_3:25;
1 <= len (Gauge (C,n)) by A13, XXREAL_0:2;
then 1 <= width (Gauge (C,n)) by JORDAN8:def_1;
then ((Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n))))) `1 = ((W-bound C) + (E-bound C)) / 2 by A21, Th35
.= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by Th33 ;
then (Gauge (C,n)) * ((Center (Gauge (C,n))),(width (Gauge (C,n)))) = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by A6, EUCLID:53;
then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Lower_Seq (C,n)) by A5, A14, Th43, JORDAN1B:15;
then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;
then A25: not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10;
(SW-corner (L~ (Cage (C,n)))) `2 = S-bound (L~ (Cage (C,n))) by EUCLID:52;
then |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| <> SW-corner (L~ (Cage (C,n))) by A8, SPRECT_1:32;
then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in {(SW-corner (L~ (Cage (C,n))))} by TARSKI:def_1;
then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng <*(SW-corner (L~ (Cage (C,n))))*> by FINSEQ_1:38;
then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in (rng <*(SW-corner (L~ (Cage (C,n))))*>) \/ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A25, XBOOLE_0:def_3;
then not |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| in rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_1:31;
then rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} by ZFMISC_1:50;
then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} = {} by XBOOLE_0:def_7;
then (rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) /\ (rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) = {} by FINSEQ_1:38;
then A26: rng (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) misses rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> by XBOOLE_0:def_7;
Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Lower_Seq (C,n)) by A21, Th48;
then A27: Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;
then A28: not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by FINSEQ_5:47;
A29: len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) by A27, FINSEQ_5:42;
A30: (Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = E-max (L~ (Cage (C,n))) by JORDAN1F:7;
then A31: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A1, TOPREAL1:25;
A32: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 < E-bound (L~ (Cage (C,n))) by A15, XREAL_1:226;
then A33: ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 <= (E-max (L~ (Cage (C,n)))) `1 by EUCLID:52;
(W-min (L~ (Cage (C,n)))) `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A16, EUCLID:52;
then ( L~ (Upper_Seq (C,n)) meets Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) & (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) is closed ) by A31, A33, JORDAN6:49;
then A34: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)) by A31, JORDAN5C:def_1;
then A35: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Upper_Seq (C,n)) by XBOOLE_0:def_4;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by XBOOLE_0:def_3;
then A36: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in L~ (Cage (C,n)) by JORDAN1E:13;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`1_&_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`2_&_(<*|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
assume m in dom <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then m in Seg 1 by FINSEQ_1:38;
then m = 1 by FINSEQ_1:2, TARSKI:def_1;
then A37: <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m = |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| by FINSEQ_4:16;
then (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
(<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A37, EUCLID:52;
hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 & (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by A36, PSCOMP_1:24; ::_thesis: verum
end;
then A38: <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
A39: First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A34, XBOOLE_0:def_4;
then A40: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;
now__::_thesis:_not_(rng_(mid_((Upper_Seq_(C,n)),2,((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))))))_/\_{|[(E-bound_(L~_(Cage_(C,n)))),((First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2)]|}_<>_{}
assume (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} <> {} ; ::_thesis: contradiction
then consider x being set such that
A41: x in (rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) /\ {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} by XBOOLE_0:def_1;
( x in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) & x in {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} ) by A41, XBOOLE_0:def_4;
then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| in rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by TARSKI:def_1;
then |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 <= ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, Th53;
hence contradiction by A32, EUCLID:52; ::_thesis: verum
end;
then rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses {|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|} by XBOOLE_0:def_7;
then A42: rng (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) misses rng <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> by FINSEQ_1:38;
A43: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) <= len (Upper_Seq (C,n)) by A23, FINSEQ_3:25;
Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2) by A18, XBOOLE_0:def_4;
then A44: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by JORDAN6:31;
A45: now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2_=_(Last_Point_((L~_(Lower_Seq_(C,n))),(E-max_(L~_(Cage_(C,n)))),(W-min_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_`2
assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ; ::_thesis: contradiction
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A40, A44, TOPREAL3:6;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A35, A19, XBOOLE_0:def_4;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
then ( First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) or First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = E-max (L~ (Cage (C,n))) ) by TARSKI:def_2;
hence contradiction by A16, A32, A40, EUCLID:52; ::_thesis: verum
end;
len (Upper_Seq (C,n)) >= 3 by JORDAN1E:15;
then A46: len (Upper_Seq (C,n)) > 2 by XXREAL_0:2;
then A47: 2 in dom (Upper_Seq (C,n)) by FINSEQ_3:25;
then A48: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))) `2 = ((Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) `2 by A23, SPRECT_2:9
.= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A22, FINSEQ_5:38
.= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 by EUCLID:52
.= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `2 by FINSEQ_4:16 ;
2 <> (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))
proof
assume 2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) ; ::_thesis: contradiction
then (Upper_Seq (C,n)) /. 2 = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38;
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = W-bound (L~ (Cage (C,n))) by Th31;
then W-bound (L~ (Cage (C,n))) = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A39, JORDAN6:31;
hence contradiction by SPRECT_1:31; ::_thesis: verum
end;
then mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is being_S-Seq by A46, A24, A43, JORDAN3:6;
then reconsider g = (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ^ <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A42, A48, A9, FINSEQ_3:91, GOBOARD2:8;
mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))) is_in_the_area_of Cage (C,n) by A47, A23, JORDAN1E:17, SPRECT_2:22;
then A49: g is_in_the_area_of Cage (C,n) by A38, SPRECT_2:24;
A50: (g /. (len g)) `1 = (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. (len <*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*>)) `1 by SPRECT_3:1
.= (<*|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|*> /. 1) `1 by FINSEQ_1:39
.= |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 by FINSEQ_4:16
.= E-bound (L~ (Cage (C,n))) by EUCLID:52 ;
A51: 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A47, A23, SPRECT_2:5;
then 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by FINSEQ_3:25;
then (g /. 1) `1 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. 1) `1 by FINSEQ_4:68
.= ((Upper_Seq (C,n)) /. 2) `1 by A47, A23, SPRECT_2:8
.= W-bound (L~ (Cage (C,n))) by Th31 ;
then A52: g is_a_h.c._for Cage (C,n) by A49, A50, SPRECT_2:def_2;
assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ; ::_thesis: contradiction
then A53: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 < (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A45, XXREAL_0:1;
A54: rng (Lower_Seq (C,n)) c= rng (Cage (C,n)) by Th39;
now__::_thesis:_contradiction
percases ( SW-corner (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) or SW-corner (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) ) ;
supposeA55: SW-corner (L~ (Cage (C,n))) <> W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
not SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n))
proof
(SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:29;
then A56: (SW-corner (L~ (Cage (C,n)))) `2 <> (W-min (L~ (Cage (C,n)))) `2 by A55, TOPREAL3:6;
assume SW-corner (L~ (Cage (C,n))) in rng (Lower_Seq (C,n)) ; ::_thesis: contradiction
then A57: SW-corner (L~ (Cage (C,n))) in rng (Cage (C,n)) by A54;
len (Cage (C,n)) > 4 by GOBOARD7:34;
then A58: rng (Cage (C,n)) c= L~ (Cage (C,n)) by SPPOL_2:18, XXREAL_0:2;
(SW-corner (L~ (Cage (C,n)))) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
then SW-corner (L~ (Cage (C,n))) in W-most (L~ (Cage (C,n))) by A57, A58, SPRECT_2:12;
then (W-min (L~ (Cage (C,n)))) `2 <= (SW-corner (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;
hence contradiction by A7, A56, XXREAL_0:1; ::_thesis: verum
end;
then not SW-corner (L~ (Cage (C,n))) in rng (Rev (Lower_Seq (C,n))) by FINSEQ_5:57;
then not SW-corner (L~ (Cage (C,n))) in rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A10;
then {(SW-corner (L~ (Cage (C,n))))} misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by ZFMISC_1:50;
then {(SW-corner (L~ (Cage (C,n))))} /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} by XBOOLE_0:def_7;
then (rng <*(SW-corner (L~ (Cage (C,n))))*>) /\ (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = {} by FINSEQ_1:38;
then A59: rng <*(SW-corner (L~ (Cage (C,n))))*> misses rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by XBOOLE_0:def_7;
<*(SW-corner (L~ (Cage (C,n))))*> is one-to-one by FINSEQ_3:93;
then A60: <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is one-to-one by A59, FINSEQ_3:91;
set FiP2 = First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
set midU = mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))));
reconsider RevLS = Rev (Lower_Seq (C,n)) as special FinSequence of (TOP-REAL 2) ;
(<*(SW-corner (L~ (Cage (C,n))))*> /. (len <*(SW-corner (L~ (Cage (C,n))))*>)) `1 = (<*(SW-corner (L~ (Cage (C,n))))*> /. 1) `1 by FINSEQ_1:39
.= (SW-corner (L~ (Cage (C,n)))) `1 by FINSEQ_4:16
.= W-bound (L~ (Cage (C,n))) by EUCLID:52
.= (W-min (L~ (Cage (C,n)))) `1 by EUCLID:52
.= ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `1 by JORDAN1F:8
.= ((Rev (Lower_Seq (C,n))) /. 1) `1 by FINSEQ_5:65
.= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1) `1 by A27, FINSEQ_5:44 ;
then A61: <*(SW-corner (L~ (Cage (C,n))))*> ^ (RevLS -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is special by GOBOARD2:8;
not (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is empty by A27, FINSEQ_5:47;
then A62: ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))))) `1 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) `1 by SPRECT_3:1
.= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) `1 by A27, FINSEQ_5:42
.= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A27, FINSEQ_5:45
.= |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52
.= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. 1) `1 by FINSEQ_4:16 ;
( <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is one-to-one & <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is special ) by FINSEQ_3:93;
then reconsider h = (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A26, A60, A62, A61, FINSEQ_3:91, GOBOARD2:8;
A63: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*(SW-corner_(L~_(Cage_(C,n))))*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`1_&_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`2_&_(<*(SW-corner_(L~_(Cage_(C,n))))*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*(SW-corner (L~ (Cage (C,n))))*> implies ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
assume m in dom <*(SW-corner (L~ (Cage (C,n))))*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then m in Seg 1 by FINSEQ_1:38;
then m = 1 by FINSEQ_1:2, TARSKI:def_1;
then A64: <*(SW-corner (L~ (Cage (C,n))))*> /. m = SW-corner (L~ (Cage (C,n))) by FINSEQ_4:16;
then (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
hence ( W-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by SPRECT_1:21; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
(<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 = S-bound (L~ (Cage (C,n))) by A64, EUCLID:52;
hence ( S-bound (L~ (Cage (C,n))) <= (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 & (<*(SW-corner (L~ (Cage (C,n))))*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; ::_thesis: verum
end;
then A65: <*(SW-corner (L~ (Cage (C,n))))*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
A66: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A27, FINSEQ_5:42
.= Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A27, FINSEQ_5:45 ;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
A67: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;
assume m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then m in Seg 1 by FINSEQ_1:38;
then m = 1 by FINSEQ_1:2, TARSKI:def_1;
then A68: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by FINSEQ_4:16;
then (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;
hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by A67, JORDAN6:1; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
(<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A68, EUCLID:52;
hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; ::_thesis: verum
end;
then A69: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
A70: ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) ) by A12, A17, JORDAN5C:18, SPPOL_2:22;
Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51;
then (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is_in_the_area_of Cage (C,n) by A27, JORDAN1E:1;
then <*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) is_in_the_area_of Cage (C,n) by A65, SPRECT_2:24;
then A71: h is_in_the_area_of Cage (C,n) by A69, SPRECT_2:24;
len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = 1 + (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by FINSEQ_5:8;
then A72: len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) >= 1 by NAT_1:11;
1 in dom h by FINSEQ_5:6;
then h /. 1 = h . 1 by PARTFUN1:def_6;
then A73: (h /. 1) `2 = ((<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /. 1) `2 by A72, FINSEQ_6:109
.= (SW-corner (L~ (Cage (C,n)))) `2 by FINSEQ_5:15
.= S-bound (L~ (Cage (C,n))) by EUCLID:52 ;
A74: len h = (len (<*(SW-corner (L~ (Cage (C,n))))*> ^ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) + 1 by FINSEQ_2:16;
then A75: 1 + 1 <= len h by A72, XREAL_1:7;
L~ (Cage (C,n)) = (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) by JORDAN1E:13;
then A76: L~ (Upper_Seq (C,n)) c= L~ (Cage (C,n)) by XBOOLE_1:7;
A77: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) = (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A47, A23, SPRECT_2:9
.= First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38 ;
A78: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42;
now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))_=_1
assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43
.= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A22, A78, FINSEQ_5:9;
hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum
end;
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1 by A24, XXREAL_0:1;
then A79: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19;
then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def_2;
then A80: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A43, A79, JORDAN4:8
.= ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - (2 - 1) ;
1 in dom ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A28, FINSEQ_5:6;
then A81: (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1 = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1 by FINSEQ_4:68
.= (Rev (Lower_Seq (C,n))) /. 1 by A27, FINSEQ_5:44
.= (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) by FINSEQ_5:65
.= W-min (L~ (Cage (C,n))) by JORDAN1F:8 ;
A82: (SW-corner (L~ (Cage (C,n)))) `2 <= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:30;
len g = (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) + 1 by FINSEQ_2:16;
then A83: 1 + 1 <= len g by A51, XREAL_1:7;
A84: L~ g = (L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) \/ (LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)) by A47, A23, SPPOL_2:19, SPRECT_2:7;
L~ (Rev (Lower_Seq (C,n))) = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (L~ ((Rev (Lower_Seq (C,n))) :- (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A27, SPPOL_2:24;
then L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Rev (Lower_Seq (C,n))) by XBOOLE_1:7;
then A85: L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Lower_Seq (C,n)) by SPPOL_2:22;
A86: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= N-bound (L~ (Cage (C,n))) by A20, PSCOMP_1:24;
A87: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by EUCLID:52;
then A88: LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by SPPOL_1:15;
(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52;
then A89: LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) is vertical by SPPOL_1:16;
A90: L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A47, A23, SPRECT_3:18;
(h /. (len h)) `2 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 by A74, FINSEQ_4:67
.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;
then h is_a_v.c._for Cage (C,n) by A71, A73, SPRECT_2:def_3;
then L~ g meets L~ h by A52, A75, A83, SPRECT_2:29;
then consider x being set such that
A91: x in L~ g and
A92: x in L~ h by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A91;
L~ h = L~ (<*(SW-corner (L~ (Cage (C,n))))*> ^ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by FINSEQ_1:32
.= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ (L~ (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>)) by SPPOL_2:20
.= (LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1))) \/ ((L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|))) by A27, FINSEQ_5:47, SPPOL_2:19 ;
then A93: ( x in LSeg ((SW-corner (L~ (Cage (C,n)))),((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) /. 1)) or x in (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|)) ) by A92, XBOOLE_0:def_3;
A94: (SW-corner (L~ (Cage (C,n)))) `1 = (W-min (L~ (Cage (C,n)))) `1 by PSCOMP_1:29;
then A95: LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) is vertical by SPPOL_1:16;
now__::_thesis:_contradiction
percases ( x in LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) or x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ) by A93, A81, A66, XBOOLE_0:def_3;
supposeA96: x in LSeg ((SW-corner (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n))))) ; ::_thesis: contradiction
then A97: x `2 <= (W-min (L~ (Cage (C,n)))) `2 by A82, TOPREAL1:4;
A98: x `1 = (SW-corner (L~ (Cage (C,n)))) `1 by A95, A96, SPPOL_1:41;
then A99: x `1 = W-bound (L~ (Cage (C,n))) by EUCLID:52;
now__::_thesis:_contradiction
percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A91, A84, A77, XBOOLE_0:def_3;
supposeA100: x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then x in L~ (Upper_Seq (C,n)) by A90;
then x in W-most (L~ (Cage (C,n))) by A76, A98, EUCLID:52, SPRECT_2:12;
then x `2 >= (W-min (L~ (Cage (C,n)))) `2 by PSCOMP_1:31;
then x `2 = (W-min (L~ (Cage (C,n)))) `2 by A97, XXREAL_0:1;
then x = W-min (L~ (Cage (C,n))) by A94, A98, TOPREAL3:6;
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A100, Th37;
then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;
hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum
end;
suppose x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction
hence contradiction by A16, A32, A40, A63, A99, TOPREAL1:3; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA101: x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A91, A84, A77, XBOOLE_0:def_3;
supposeA102: x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A90, A85, A101, XBOOLE_0:def_4;
then A103: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
now__::_thesis:_contradiction
percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A103, TARSKI:def_2;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A102, Th37;
then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;
hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum
end;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by A46, A30, A24, A43, A102, Th38;
then E-max (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A30, A22, FINSEQ_5:38;
hence contradiction by A32, A40, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA104: x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction
LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by A87, SPPOL_1:15;
then A105: x `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A104, SPPOL_1:40;
consider i being Element of NAT such that
A106: 1 <= i and
A107: i + 1 <= len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) and
A108: x in LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) by A101, SPPOL_2:14;
A109: i < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A107, NAT_1:13;
then A110: ((Rev (Lower_Seq (C,n))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A70, A106, Th52;
i in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A106, A109, FINSEQ_1:1;
then A111: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i = (Rev (Lower_Seq (C,n))) /. i by A27, FINSEQ_5:43;
i + 1 >= 1 by NAT_1:11;
then i + 1 in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A107, FINSEQ_1:1;
then A112: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1) = (Rev (Lower_Seq (C,n))) /. (i + 1) by A27, FINSEQ_5:43;
A113: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= x `1 by A32, A40, A63, A104, TOPREAL1:3;
now__::_thesis:_contradiction
percases ( i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ) by A107, XXREAL_0:1;
supposeA114: i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction
( (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) ;
then A115: ( x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) by A108, TOPREAL1:3;
((Rev (Lower_Seq (C,n))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A70, A114, Th52, NAT_1:11;
hence contradiction by A40, A113, A111, A112, A110, A115, XXREAL_0:2; ::_thesis: verum
end;
supposeA116: i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction
then i + 1 <= len (Rev (Lower_Seq (C,n))) by A27, A29, FINSEQ_4:21;
then LSeg (((Rev (Lower_Seq (C,n))) /. i),((Rev (Lower_Seq (C,n))) /. (i + 1))) = LSeg ((Rev (Lower_Seq (C,n))),i) by A106, TOPREAL1:def_3;
then ( LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is vertical or LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is horizontal ) by A111, A112, SPPOL_1:19;
hence contradiction by A44, A45, A66, A105, A108, A111, A110, A116, SPPOL_1:16, SPPOL_1:40; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA117: x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ; ::_thesis: contradiction
then A118: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= x `2 by A8, A86, TOPREAL1:4;
A119: x `1 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A89, A117, SPPOL_1:41;
now__::_thesis:_contradiction
percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A91, A84, A77, XBOOLE_0:def_3;
suppose x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then consider i being Element of NAT such that
A120: 1 <= i and
A121: i + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) and
A122: x in LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) by SPPOL_2:14;
i + 2 >= 1 + 1 by NAT_1:11;
then A123: (i + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9;
i < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A121, NAT_1:13;
then i in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A120, FINSEQ_3:25;
then A124: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i = (Upper_Seq (C,n)) /. ((i + 2) -' 1) by A47, A23, A79, SPRECT_2:3
.= (Upper_Seq (C,n)) /. (i + (2 - 1)) by A123, XREAL_0:def_2 ;
(i + 1) + 2 >= 1 + 1 by NAT_1:11;
then A125: ((i + 1) + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9;
A126: 1 <= i + 1 by NAT_1:11;
then i + 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A121, FINSEQ_3:25;
then A127: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1) = (Upper_Seq (C,n)) /. (((i + 1) + 2) -' 1) by A47, A23, A79, SPRECT_2:3
.= (Upper_Seq (C,n)) /. ((i + 1) + (2 - 1)) by A125, XREAL_0:def_2 ;
A128: (i + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A80, A121, XREAL_1:7;
then i + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then A129: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A124, Th51, NAT_1:11;
(i + 1) + 1 <= len (Upper_Seq (C,n)) by A43, A128, XXREAL_0:2;
then LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) = LSeg ((Upper_Seq (C,n)),(i + 1)) by A124, A126, A127, TOPREAL1:def_3;
then A130: ( LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is vertical or LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is horizontal ) by SPPOL_1:19;
now__::_thesis:_contradiction
percases ( i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ) by A121, XXREAL_0:1;
suppose i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then (i + 1) + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by NAT_1:13;
then ((i + 1) + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A80, XREAL_1:7;
then (i + 1) + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then A131: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A127, Th51, NAT_1:11;
( ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 or ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 ) ;
hence contradiction by A44, A119, A122, A129, A131, TOPREAL1:3; ::_thesis: verum
end;
supposeA132: i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `2 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `2 by A40, A77, A129, A130, SPPOL_1:15, SPPOL_1:16;
hence contradiction by A53, A77, A118, A122, A132, GOBOARD7:6; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
suppose x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction
hence contradiction by A53, A88, A118, SPPOL_1:40; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA133: SW-corner (L~ (Cage (C,n))) = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
reconsider RevLS = Rev (Lower_Seq (C,n)) as special FinSequence of (TOP-REAL 2) ;
set h = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>;
A134: ( <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is one-to-one & RevLS -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is special ) by FINSEQ_3:93;
rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) misses {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} by A25, ZFMISC_1:50;
then (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /\ {|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|} = {} by XBOOLE_0:def_7;
then (rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) /\ (rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*>) = {} by FINSEQ_1:38;
then A135: rng ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) misses rng <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> by XBOOLE_0:def_7;
(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))) `1 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))))) `1 by A27, FINSEQ_5:42
.= (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A27, FINSEQ_5:45
.= |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52
.= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. 1) `1 by FINSEQ_4:16 ;
then reconsider h = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ^ <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> as one-to-one special FinSequence of (TOP-REAL 2) by A135, A134, FINSEQ_3:91, GOBOARD2:8;
now__::_thesis:_for_m_being_Element_of_NAT_st_m_in_dom_<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_holds_
(_W-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`1_<=_E-bound_(L~_(Cage_(C,n)))_&_S-bound_(L~_(Cage_(C,n)))_<=_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_&_(<*|[(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2),(N-bound_(L~_(Cage_(C,n))))]|*>_/._m)_`2_<=_N-bound_(L~_(Cage_(C,n)))_)
let m be Element of NAT ; ::_thesis: ( m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> implies ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) )
A136: W-bound (L~ (Cage (C,n))) <= E-bound (L~ (Cage (C,n))) by SPRECT_1:21;
assume m in dom <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> ; ::_thesis: ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) & S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
then m in Seg 1 by FINSEQ_1:38;
then m = 1 by FINSEQ_1:2, TARSKI:def_1;
then A137: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| by FINSEQ_4:16;
then (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 = ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by EUCLID:52;
hence ( W-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `1 <= E-bound (L~ (Cage (C,n))) ) by A136, JORDAN6:1; ::_thesis: ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) )
(<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 = N-bound (L~ (Cage (C,n))) by A137, EUCLID:52;
hence ( S-bound (L~ (Cage (C,n))) <= (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 & (<*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> /. m) `2 <= N-bound (L~ (Cage (C,n))) ) by SPRECT_1:22; ::_thesis: verum
end;
then A138: <*|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|*> is_in_the_area_of Cage (C,n) by SPRECT_2:def_1;
Rev (Lower_Seq (C,n)) is_in_the_area_of Cage (C,n) by JORDAN1E:18, SPRECT_3:51;
then (Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) is_in_the_area_of Cage (C,n) by A27, JORDAN1E:1;
then A139: h is_in_the_area_of Cage (C,n) by A138, SPRECT_2:24;
A140: len h = (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) + 1 by FINSEQ_2:16;
then A141: (h /. (len h)) `2 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `2 by FINSEQ_4:67
.= N-bound (L~ (Cage (C,n))) by EUCLID:52 ;
L~ (Rev (Lower_Seq (C,n))) = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (L~ ((Rev (Lower_Seq (C,n))) :- (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) by A27, SPPOL_2:24;
then L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Rev (Lower_Seq (C,n))) by XBOOLE_1:7;
then A142: L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) c= L~ (Lower_Seq (C,n)) by SPPOL_2:22;
A143: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= N-bound (L~ (Cage (C,n))) by A20, PSCOMP_1:24;
(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n))) >= 0 + 1 by A28, A29, NAT_1:13;
then A144: len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) >= 1 by A27, FINSEQ_5:42;
1 in dom h by FINSEQ_5:6;
then h /. 1 = h . 1 by PARTFUN1:def_6;
then (h /. 1) `2 = (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. 1) `2 by A144, FINSEQ_6:109
.= ((Rev (Lower_Seq (C,n))) /. 1) `2 by A27, FINSEQ_5:44
.= ((Lower_Seq (C,n)) /. (len (Lower_Seq (C,n)))) `2 by FINSEQ_5:65
.= (W-min (L~ (Cage (C,n)))) `2 by JORDAN1F:8
.= S-bound (L~ (Cage (C,n))) by A133, EUCLID:52 ;
then A145: h is_a_v.c._for Cage (C,n) by A139, A141, SPRECT_2:def_3;
set FiP2 = First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)));
set midU = mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))));
A146: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `1 = E-bound (L~ (Cage (C,n))) by EUCLID:52;
A147: L~ g = (L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) \/ (LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|)) by A47, A23, SPPOL_2:19, SPRECT_2:7;
A148: W-min (L~ (Cage (C,n))) in rng (Upper_Seq (C,n)) by A1, FINSEQ_6:42;
now__::_thesis:_not_(First_Point_((L~_(Upper_Seq_(C,n))),(W-min_(L~_(Cage_(C,n)))),(E-max_(L~_(Cage_(C,n)))),(Vertical_Line_(((W-bound_(L~_(Cage_(C,n))))_+_(E-bound_(L~_(Cage_(C,n)))))_/_2))))_.._(Upper_Seq_(C,n))_=_1
assume (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = ((Upper_Seq (C,n)) /. 1) .. (Upper_Seq (C,n)) by FINSEQ_6:43
.= (W-min (L~ (Cage (C,n)))) .. (Upper_Seq (C,n)) by JORDAN1F:5 ;
then First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = W-min (L~ (Cage (C,n))) by A22, A148, FINSEQ_5:9;
hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum
end;
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) > 1 by A24, XXREAL_0:1;
then A149: (1 + 1) + 0 <= (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 >= 0 by XREAL_1:19;
then ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) -' 2 = ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2 by XREAL_0:def_2;
then A150: len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) = (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 2) + 1 by A43, A149, JORDAN4:8
.= ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - (2 - 1) ;
(Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 = |[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]| `1 by A44, EUCLID:52;
then A151: LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) is vertical by SPPOL_1:16;
len g = (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) + 1 by FINSEQ_2:16;
then A152: 1 + 1 <= len g by A51, XREAL_1:7;
A153: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) = ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A27, FINSEQ_5:42
.= Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A27, FINSEQ_5:45 ;
1 + 1 <= len h by A144, A140, XREAL_1:7;
then L~ g meets L~ h by A52, A145, A152, SPRECT_2:29;
then consider x being set such that
A154: x in L~ g and
A155: x in L~ h by XBOOLE_0:3;
reconsider x = x as Point of (TOP-REAL 2) by A154;
A156: L~ h = (L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))))) \/ (LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|)) by A27, FINSEQ_5:47, SPPOL_2:19;
A157: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)))))) = (Upper_Seq (C,n)) /. ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) by A47, A23, SPRECT_2:9
.= First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A22, FINSEQ_5:38 ;
A158: L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) c= L~ (Upper_Seq (C,n)) by A47, A23, SPRECT_3:18;
A159: ( L~ (Rev (Lower_Seq (C,n))) = L~ (Lower_Seq (C,n)) & First_Point ((L~ (Lower_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) = Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) ) by A12, A17, JORDAN5C:18, SPPOL_2:22;
A160: |[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]| `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by EUCLID:52;
then A161: LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by SPPOL_1:15;
now__::_thesis:_contradiction
percases ( x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ) by A155, A156, A153, XBOOLE_0:def_3;
supposeA162: x in L~ ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A154, A147, A157, XBOOLE_0:def_3;
supposeA163: x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then x in (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) by A158, A142, A162, XBOOLE_0:def_4;
then A164: x in {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} by JORDAN1E:16;
now__::_thesis:_contradiction
percases ( x = W-min (L~ (Cage (C,n))) or x = E-max (L~ (Cage (C,n))) ) by A164, TARSKI:def_2;
suppose x = W-min (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = 1 by A46, A1, A24, A43, A163, Th37;
then W-min (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A1, A22, FINSEQ_5:38;
hence contradiction by A16, A40, EUCLID:52; ::_thesis: verum
end;
suppose x = E-max (L~ (Cage (C,n))) ; ::_thesis: contradiction
then (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) = len (Upper_Seq (C,n)) by A46, A30, A24, A43, A163, Th38;
then E-max (L~ (Cage (C,n))) = First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))) by A30, A22, FINSEQ_5:38;
hence contradiction by A32, A40, EUCLID:52; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA165: x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction
LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) is horizontal by A160, SPPOL_1:15;
then A166: x `2 = (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by A165, SPPOL_1:40;
consider i being Element of NAT such that
A167: 1 <= i and
A168: i + 1 <= len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) and
A169: x in LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) by A162, SPPOL_2:14;
A170: i < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) by A168, NAT_1:13;
then A171: ((Rev (Lower_Seq (C,n))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A159, A167, Th52;
i in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A167, A170, FINSEQ_1:1;
then A172: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i = (Rev (Lower_Seq (C,n))) /. i by A27, FINSEQ_5:43;
i + 1 >= 1 by NAT_1:11;
then i + 1 in Seg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Rev (Lower_Seq (C,n)))) by A29, A168, FINSEQ_1:1;
then A173: ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1) = (Rev (Lower_Seq (C,n))) /. (i + 1) by A27, FINSEQ_5:43;
A174: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 <= x `1 by A32, A40, A146, A165, TOPREAL1:3;
now__::_thesis:_contradiction
percases ( i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) or i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ) by A168, XXREAL_0:1;
supposeA175: i + 1 < len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction
( (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) ;
then A176: ( x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1)) `1 or x `1 <= (((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i) `1 ) by A169, TOPREAL1:3;
((Rev (Lower_Seq (C,n))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A29, A159, A175, Th52, NAT_1:11;
hence contradiction by A40, A174, A172, A173, A171, A176, XXREAL_0:2; ::_thesis: verum
end;
supposeA177: i + 1 = len ((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) ; ::_thesis: contradiction
then i + 1 <= len (Rev (Lower_Seq (C,n))) by A27, A29, FINSEQ_4:21;
then LSeg (((Rev (Lower_Seq (C,n))) /. i),((Rev (Lower_Seq (C,n))) /. (i + 1))) = LSeg ((Rev (Lower_Seq (C,n))),i) by A167, TOPREAL1:def_3;
then ( LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is vertical or LSeg ((((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. i),(((Rev (Lower_Seq (C,n))) -: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2))))) /. (i + 1))) is horizontal ) by A172, A173, SPPOL_1:19;
hence contradiction by A44, A45, A153, A166, A169, A172, A171, A177, SPPOL_1:16, SPPOL_1:40; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA178: x in LSeg ((Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2),(N-bound (L~ (Cage (C,n))))]|) ; ::_thesis: contradiction
then A179: (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 <= x `2 by A8, A143, TOPREAL1:4;
A180: x `1 = (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `1 by A151, A178, SPPOL_1:41;
now__::_thesis:_contradiction
percases ( x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ) by A154, A147, A157, XBOOLE_0:def_3;
suppose x in L~ (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then consider i being Element of NAT such that
A181: 1 <= i and
A182: i + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) and
A183: x in LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) by SPPOL_2:14;
i + 2 >= 1 + 1 by NAT_1:11;
then A184: (i + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9;
i < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A182, NAT_1:13;
then i in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A181, FINSEQ_3:25;
then A185: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i = (Upper_Seq (C,n)) /. ((i + 2) -' 1) by A47, A23, A149, SPRECT_2:3
.= (Upper_Seq (C,n)) /. (i + (2 - 1)) by A184, XREAL_0:def_2 ;
(i + 1) + 2 >= 1 + 1 by NAT_1:11;
then A186: ((i + 1) + 2) - 1 >= (1 + 1) - 1 by XREAL_1:9;
A187: 1 <= i + 1 by NAT_1:11;
then i + 1 in dom (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by A182, FINSEQ_3:25;
then A188: (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1) = (Upper_Seq (C,n)) /. (((i + 1) + 2) -' 1) by A47, A23, A149, SPRECT_2:3
.= (Upper_Seq (C,n)) /. ((i + 1) + (2 - 1)) by A186, XREAL_0:def_2 ;
A189: (i + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A150, A182, XREAL_1:7;
then i + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then A190: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A185, Th51, NAT_1:11;
(i + 1) + 1 <= len (Upper_Seq (C,n)) by A43, A189, XXREAL_0:2;
then LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) = LSeg ((Upper_Seq (C,n)),(i + 1)) by A185, A187, A188, TOPREAL1:def_3;
then A191: ( LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is vertical or LSeg (((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i),((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1))) is horizontal ) by SPPOL_1:19;
now__::_thesis:_contradiction
percases ( i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) or i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ) by A182, XXREAL_0:1;
suppose i + 1 < len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then (i + 1) + 1 <= len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) by NAT_1:13;
then ((i + 1) + 1) + 1 <= (((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))) - 1) + 1 by A150, XREAL_1:7;
then (i + 1) + 1 < (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n)) by NAT_1:13;
then A192: ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 < ((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2 by A21, A188, Th51, NAT_1:11;
( ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 or ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `1 <= ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `1 ) ;
hence contradiction by A44, A180, A183, A190, A192, TOPREAL1:3; ::_thesis: verum
end;
supposeA193: i + 1 = len (mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) ; ::_thesis: contradiction
then ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. i) `2 = ((mid ((Upper_Seq (C,n)),2,((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) .. (Upper_Seq (C,n))))) /. (i + 1)) `2 by A40, A157, A190, A191, SPPOL_1:15, SPPOL_1:16;
hence contradiction by A53, A157, A179, A183, A193, GOBOARD7:6; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
suppose x in LSeg ((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))),|[(E-bound (L~ (Cage (C,n)))),((First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2)]|) ; ::_thesis: contradiction
hence contradiction by A53, A161, A179, SPPOL_1:40; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
theorem Th55: :: JORDAN1G:55
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n)))
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n)))
let n be Element of NAT ; ::_thesis: ( n > 0 implies L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) )
A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then A2: E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) by FINSEQ_6:90, SPRECT_2:43;
A3: Upper_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_1;
then (Upper_Seq (C,n)) /. 1 = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by A2, FINSEQ_5:44;
then A4: (Upper_Seq (C,n)) /. 1 = W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92;
(Upper_Seq (C,n)) /. (len (Upper_Seq (C,n))) = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) -: (E-max (L~ (Cage (C,n))))) /. ((E-max (L~ (Cage (C,n)))) .. (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A3, A2, FINSEQ_5:42
.= E-max (L~ (Cage (C,n))) by A2, FINSEQ_5:45 ;
then A5: L~ (Upper_Seq (C,n)) is_an_arc_of W-min (L~ (Cage (C,n))), E-max (L~ (Cage (C,n))) by A4, TOPREAL1:25;
assume n > 0 ; ::_thesis: L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n)))
then A6: (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 by Th54;
A7: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2
.= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ;
Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) by JORDAN1E:def_2;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by A2, FINSEQ_5:54
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1
.= W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92 ;
then A8: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by A7, TOPREAL1:25;
( (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) = {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} & (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) = L~ (Cage (C,n)) ) by JORDAN1E:13, JORDAN1E:16;
hence L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) by A5, A8, A6, JORDAN6:def_8; ::_thesis: verum
end;
theorem Th56: :: JORDAN1G:56
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n)))
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n)))
let n be Element of NAT ; ::_thesis: ( n > 0 implies L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) )
A1: W-min (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:43;
A2: (Lower_Seq (C,n)) /. 1 = ((Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n))))) /. 1 by JORDAN1E:def_2
.= E-max (L~ (Cage (C,n))) by FINSEQ_5:53 ;
E-max (L~ (Cage (C,n))) in rng (Cage (C,n)) by SPRECT_2:46;
then ( Lower_Seq (C,n) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) :- (E-max (L~ (Cage (C,n)))) & E-max (L~ (Cage (C,n))) in rng (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) ) by FINSEQ_6:90, JORDAN1E:def_2, SPRECT_2:43;
then (Lower_Seq (C,n)) /. (len (Lower_Seq (C,n))) = (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. (len (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n))))))) by FINSEQ_5:54
.= (Rotate ((Cage (C,n)),(W-min (L~ (Cage (C,n)))))) /. 1 by FINSEQ_6:def_1
.= W-min (L~ (Cage (C,n))) by A1, FINSEQ_6:92 ;
then A3: L~ (Lower_Seq (C,n)) is_an_arc_of E-max (L~ (Cage (C,n))), W-min (L~ (Cage (C,n))) by A2, TOPREAL1:25;
assume n > 0 ; ::_thesis: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n)))
then A4: ( L~ (Upper_Seq (C,n)) = Upper_Arc (L~ (Cage (C,n))) & (First_Point ((L~ (Upper_Seq (C,n))),(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 > (Last_Point ((L~ (Lower_Seq (C,n))),(E-max (L~ (Cage (C,n)))),(W-min (L~ (Cage (C,n)))),(Vertical_Line (((W-bound (L~ (Cage (C,n)))) + (E-bound (L~ (Cage (C,n))))) / 2)))) `2 ) by Th54, Th55;
( (L~ (Upper_Seq (C,n))) /\ (L~ (Lower_Seq (C,n))) = {(W-min (L~ (Cage (C,n)))),(E-max (L~ (Cage (C,n))))} & (L~ (Upper_Seq (C,n))) \/ (L~ (Lower_Seq (C,n))) = L~ (Cage (C,n)) ) by JORDAN1E:13, JORDAN1E:16;
hence L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by A3, A4, JORDAN6:def_9; ::_thesis: verum
end;
theorem :: JORDAN1G:57
for C being connected compact non horizontal non vertical Subset of (TOP-REAL 2)
for n being Element of NAT st n > 0 holds
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n)))
proof
let C be connected compact non horizontal non vertical Subset of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n > 0 holds
for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n)))
let n be Element of NAT ; ::_thesis: ( n > 0 implies for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) )
assume n > 0 ; ::_thesis: for i, j being Element of NAT st 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) holds
LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n)))
then A1: L~ (Lower_Seq (C,n)) = Lower_Arc (L~ (Cage (C,n))) by Th56;
let i, j be Element of NAT ; ::_thesis: ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) implies LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) )
assume ( 1 <= i & i <= len (Gauge (C,n)) & 1 <= j & j <= width (Gauge (C,n)) & (Gauge (C,n)) * (i,j) in L~ (Cage (C,n)) ) ; ::_thesis: LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n)))
hence LSeg (((Gauge (C,n)) * (i,1)),((Gauge (C,n)) * (i,j))) meets Lower_Arc (L~ (Cage (C,n))) by A1, Th46; ::_thesis: verum
end;