:: SPRECT_2 semantic presentation
begin
theorem Th1: :: SPRECT_2:1
for i, j, k being Element of NAT
for D being non empty set
for f being FinSequence of D st i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(k + i) -' 1 in dom f
proof
let i, j, k be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(k + i) -' 1 in dom f
let D be non empty set ; ::_thesis: for f being FinSequence of D st i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(k + i) -' 1 in dom f
let f be FinSequence of D; ::_thesis: ( i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) implies (k + i) -' 1 in dom f )
assume that
A1: i <= j and
A2: i in dom f and
A3: j in dom f ; ::_thesis: ( not k in dom (mid (f,i,j)) or (k + i) -' 1 in dom f )
A4: j <= len f by A3, FINSEQ_3:25;
A5: 1 + 0 <= i by A2, FINSEQ_3:25;
then i - 1 >= 0 by XREAL_1:19;
then A6: k + 0 <= k + (i - 1) by XREAL_1:6;
assume A7: k in dom (mid (f,i,j)) ; ::_thesis: (k + i) -' 1 in dom f
then A8: k <= len (mid (f,i,j)) by FINSEQ_3:25;
( i <= len f & 1 <= j ) by A2, A3, FINSEQ_3:25;
then len (mid (f,i,j)) = (j -' i) + 1 by A1, A5, A4, FINSEQ_6:118;
then k <= (j - i) + 1 by A1, A8, XREAL_1:233;
then k <= (j + 1) - i ;
then k + i <= j + 1 by XREAL_1:19;
then ( k + i >= i & (k + i) - 1 <= j ) by NAT_1:11, XREAL_1:20;
then (k + i) -' 1 <= j by A5, XREAL_1:233, XXREAL_0:2;
then A9: (k + i) -' 1 <= len f by A4, XXREAL_0:2;
1 <= k by A7, FINSEQ_3:25;
then 1 <= (k + i) - 1 by A6, XXREAL_0:2;
then 1 <= (k + i) -' 1 by NAT_D:39;
hence (k + i) -' 1 in dom f by A9, FINSEQ_3:25; ::_thesis: verum
end;
theorem Th2: :: SPRECT_2:2
for i, j, k being Element of NAT
for D being non empty set
for f being FinSequence of D st i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(i -' k) + 1 in dom f
proof
let i, j, k be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(i -' k) + 1 in dom f
let D be non empty set ; ::_thesis: for f being FinSequence of D st i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(i -' k) + 1 in dom f
let f be FinSequence of D; ::_thesis: ( i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) implies (i -' k) + 1 in dom f )
assume that
A1: i > j and
A2: i in dom f and
A3: j in dom f ; ::_thesis: ( not k in dom (mid (f,i,j)) or (i -' k) + 1 in dom f )
A4: i <= len f by A2, FINSEQ_3:25;
A5: 1 + 0 <= j by A3, FINSEQ_3:25;
then 1 - j <= 0 by XREAL_1:47;
then A6: i + (1 - j) <= i + 0 by XREAL_1:6;
assume A7: k in dom (mid (f,i,j)) ; ::_thesis: (i -' k) + 1 in dom f
then A8: k <= len (mid (f,i,j)) by FINSEQ_3:25;
k >= 1 by A7, FINSEQ_3:25;
then 1 - k <= 0 by XREAL_1:47;
then i + (1 - k) <= i + 0 by XREAL_1:6;
then A9: (i - k) + 1 <= i ;
( 1 + 0 <= i & j <= len f ) by A2, A3, FINSEQ_3:25;
then len (mid (f,i,j)) = (i -' j) + 1 by A1, A4, A5, FINSEQ_6:118;
then k <= (i - j) + 1 by A1, A8, XREAL_1:233;
then (i -' k) + 1 <= i by A6, A9, XREAL_1:233, XXREAL_0:2;
then A10: (i -' k) + 1 <= len f by A4, XXREAL_0:2;
1 <= (i -' k) + 1 by NAT_1:11;
hence (i -' k) + 1 in dom f by A10, FINSEQ_3:25; ::_thesis: verum
end;
theorem Th3: :: SPRECT_2:3
for i, j, k being Element of NAT
for D being non empty set
for f being FinSequence of D st i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(mid (f,i,j)) /. k = f /. ((k + i) -' 1)
proof
let i, j, k be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(mid (f,i,j)) /. k = f /. ((k + i) -' 1)
let D be non empty set ; ::_thesis: for f being FinSequence of D st i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(mid (f,i,j)) /. k = f /. ((k + i) -' 1)
let f be FinSequence of D; ::_thesis: ( i <= j & i in dom f & j in dom f & k in dom (mid (f,i,j)) implies (mid (f,i,j)) /. k = f /. ((k + i) -' 1) )
assume that
A1: i <= j and
A2: i in dom f and
A3: j in dom f and
A4: k in dom (mid (f,i,j)) ; ::_thesis: (mid (f,i,j)) /. k = f /. ((k + i) -' 1)
A5: ( 1 <= i & i <= len f ) by A2, FINSEQ_3:25;
A6: ( 1 <= k & k <= len (mid (f,i,j)) ) by A4, FINSEQ_3:25;
A7: ( 1 <= j & j <= len f ) by A3, FINSEQ_3:25;
thus (mid (f,i,j)) /. k = (mid (f,i,j)) . k by A4, PARTFUN1:def_6
.= f . ((k + i) -' 1) by A1, A5, A7, A6, FINSEQ_6:118
.= f /. ((k + i) -' 1) by A1, A2, A3, A4, Th1, PARTFUN1:def_6 ; ::_thesis: verum
end;
theorem Th4: :: SPRECT_2:4
for i, j, k being Element of NAT
for D being non empty set
for f being FinSequence of D st i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(mid (f,i,j)) /. k = f /. ((i -' k) + 1)
proof
let i, j, k be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(mid (f,i,j)) /. k = f /. ((i -' k) + 1)
let D be non empty set ; ::_thesis: for f being FinSequence of D st i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) holds
(mid (f,i,j)) /. k = f /. ((i -' k) + 1)
let f be FinSequence of D; ::_thesis: ( i > j & i in dom f & j in dom f & k in dom (mid (f,i,j)) implies (mid (f,i,j)) /. k = f /. ((i -' k) + 1) )
assume that
A1: i > j and
A2: i in dom f and
A3: j in dom f and
A4: k in dom (mid (f,i,j)) ; ::_thesis: (mid (f,i,j)) /. k = f /. ((i -' k) + 1)
A5: ( 1 <= i & i <= len f ) by A2, FINSEQ_3:25;
A6: ( 1 <= k & k <= len (mid (f,i,j)) ) by A4, FINSEQ_3:25;
A7: ( 1 <= j & j <= len f ) by A3, FINSEQ_3:25;
thus (mid (f,i,j)) /. k = (mid (f,i,j)) . k by A4, PARTFUN1:def_6
.= f . ((i -' k) + 1) by A1, A5, A7, A6, FINSEQ_6:118
.= f /. ((i -' k) + 1) by A1, A2, A3, A4, Th2, PARTFUN1:def_6 ; ::_thesis: verum
end;
theorem Th5: :: SPRECT_2:5
for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
len (mid (f,i,j)) >= 1
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
len (mid (f,i,j)) >= 1
let D be non empty set ; ::_thesis: for f being FinSequence of D st i in dom f & j in dom f holds
len (mid (f,i,j)) >= 1
let f be FinSequence of D; ::_thesis: ( i in dom f & j in dom f implies len (mid (f,i,j)) >= 1 )
A1: ( i <= j or j < i ) ;
assume i in dom f ; ::_thesis: ( not j in dom f or len (mid (f,i,j)) >= 1 )
then A2: ( 1 <= i & i <= len f ) by FINSEQ_3:25;
assume j in dom f ; ::_thesis: len (mid (f,i,j)) >= 1
then ( 1 <= j & j <= len f ) by FINSEQ_3:25;
then ( len (mid (f,i,j)) = (i -' j) + 1 or len (mid (f,i,j)) = (j -' i) + 1 ) by A2, A1, FINSEQ_6:118;
hence len (mid (f,i,j)) >= 1 by NAT_1:11; ::_thesis: verum
end;
theorem Th6: :: SPRECT_2:6
for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f & len (mid (f,i,j)) = 1 holds
i = j
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f & len (mid (f,i,j)) = 1 holds
i = j
let D be non empty set ; ::_thesis: for f being FinSequence of D st i in dom f & j in dom f & len (mid (f,i,j)) = 1 holds
i = j
let f be FinSequence of D; ::_thesis: ( i in dom f & j in dom f & len (mid (f,i,j)) = 1 implies i = j )
assume A1: i in dom f ; ::_thesis: ( not j in dom f or not len (mid (f,i,j)) = 1 or i = j )
then A2: 1 <= i by FINSEQ_3:25;
A3: i <= len f by A1, FINSEQ_3:25;
assume A4: j in dom f ; ::_thesis: ( not len (mid (f,i,j)) = 1 or i = j )
then A5: 1 <= j by FINSEQ_3:25;
A6: j <= len f by A4, FINSEQ_3:25;
assume A7: len (mid (f,i,j)) = 1 ; ::_thesis: i = j
percases ( i <= j or i >= j ) ;
supposeA8: i <= j ; ::_thesis: i = j
then 0 + 1 = (j -' i) + 1 by A2, A6, A7, JORDAN4:8;
then 0 = j - i by A8, XREAL_1:233;
hence i = j ; ::_thesis: verum
end;
supposeA9: i >= j ; ::_thesis: i = j
then 0 + 1 = (i -' j) + 1 by A3, A5, A7, JORDAN4:9;
then 0 = i - j by A9, XREAL_1:233;
hence i = j ; ::_thesis: verum
end;
end;
end;
theorem Th7: :: SPRECT_2:7
for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
not mid (f,i,j) is empty
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
not mid (f,i,j) is empty
let D be non empty set ; ::_thesis: for f being FinSequence of D st i in dom f & j in dom f holds
not mid (f,i,j) is empty
let f be FinSequence of D; ::_thesis: ( i in dom f & j in dom f implies not mid (f,i,j) is empty )
assume ( i in dom f & j in dom f ) ; ::_thesis: not mid (f,i,j) is empty
then len (mid (f,i,j)) >= 1 by Th5;
hence not mid (f,i,j) is empty by FINSEQ_3:25, RELAT_1:38; ::_thesis: verum
end;
theorem Th8: :: SPRECT_2:8
for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. 1 = f /. i
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. 1 = f /. i
let D be non empty set ; ::_thesis: for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. 1 = f /. i
let f be FinSequence of D; ::_thesis: ( i in dom f & j in dom f implies (mid (f,i,j)) /. 1 = f /. i )
assume A1: i in dom f ; ::_thesis: ( not j in dom f or (mid (f,i,j)) /. 1 = f /. i )
then A2: ( 1 <= i & i <= len f ) by FINSEQ_3:25;
assume A3: j in dom f ; ::_thesis: (mid (f,i,j)) /. 1 = f /. i
then A4: ( 1 <= j & j <= len f ) by FINSEQ_3:25;
not mid (f,i,j) is empty by A1, A3, Th7;
then 1 in dom (mid (f,i,j)) by FINSEQ_5:6;
hence (mid (f,i,j)) /. 1 = (mid (f,i,j)) . 1 by PARTFUN1:def_6
.= f . i by A2, A4, FINSEQ_6:118
.= f /. i by A1, PARTFUN1:def_6 ;
::_thesis: verum
end;
theorem Th9: :: SPRECT_2:9
for i, j being Element of NAT
for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
proof
let i, j be Element of NAT ; ::_thesis: for D being non empty set
for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
let D be non empty set ; ::_thesis: for f being FinSequence of D st i in dom f & j in dom f holds
(mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
let f be FinSequence of D; ::_thesis: ( i in dom f & j in dom f implies (mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j )
assume A1: i in dom f ; ::_thesis: ( not j in dom f or (mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j )
then A2: ( 1 <= i & i <= len f ) by FINSEQ_3:25;
assume A3: j in dom f ; ::_thesis: (mid (f,i,j)) /. (len (mid (f,i,j))) = f /. j
then A4: ( 1 <= j & j <= len f ) by FINSEQ_3:25;
not mid (f,i,j) is empty by A1, A3, Th7;
then len (mid (f,i,j)) in dom (mid (f,i,j)) by FINSEQ_5:6;
hence (mid (f,i,j)) /. (len (mid (f,i,j))) = (mid (f,i,j)) . (len (mid (f,i,j))) by PARTFUN1:def_6
.= f . j by A2, A4, JORDAN4:11
.= f /. j by A3, PARTFUN1:def_6 ;
::_thesis: verum
end;
begin
theorem Th10: :: SPRECT_2:10
for X being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in X & p `2 = N-bound X holds
p in N-most X
proof
let X be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in X & p `2 = N-bound X holds
p in N-most X
let p be Point of (TOP-REAL 2); ::_thesis: ( p in X & p `2 = N-bound X implies p in N-most X )
assume that
A1: p in X and
A2: p `2 = N-bound X ; ::_thesis: p in N-most X
A3: ( (NW-corner X) `2 = N-bound X & (NE-corner X) `2 = N-bound X ) by EUCLID:52;
A4: ( (NW-corner X) `1 = W-bound X & (NE-corner X) `1 = E-bound X ) by EUCLID:52;
( W-bound X <= p `1 & p `1 <= E-bound X ) by A1, PSCOMP_1:24;
then p in LSeg ((NW-corner X),(NE-corner X)) by A2, A3, A4, GOBOARD7:8;
hence p in N-most X by A1, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th11: :: SPRECT_2:11
for X being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in X & p `2 = S-bound X holds
p in S-most X
proof
let X be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in X & p `2 = S-bound X holds
p in S-most X
let p be Point of (TOP-REAL 2); ::_thesis: ( p in X & p `2 = S-bound X implies p in S-most X )
assume that
A1: p in X and
A2: p `2 = S-bound X ; ::_thesis: p in S-most X
A3: ( (SW-corner X) `2 = S-bound X & (SE-corner X) `2 = S-bound X ) by EUCLID:52;
A4: ( (SW-corner X) `1 = W-bound X & (SE-corner X) `1 = E-bound X ) by EUCLID:52;
( W-bound X <= p `1 & p `1 <= E-bound X ) by A1, PSCOMP_1:24;
then p in LSeg ((SW-corner X),(SE-corner X)) by A2, A3, A4, GOBOARD7:8;
hence p in S-most X by A1, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th12: :: SPRECT_2:12
for X being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in X & p `1 = W-bound X holds
p in W-most X
proof
let X be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in X & p `1 = W-bound X holds
p in W-most X
let p be Point of (TOP-REAL 2); ::_thesis: ( p in X & p `1 = W-bound X implies p in W-most X )
assume that
A1: p in X and
A2: p `1 = W-bound X ; ::_thesis: p in W-most X
A3: ( (SW-corner X) `1 = W-bound X & (NW-corner X) `1 = W-bound X ) by EUCLID:52;
A4: ( (SW-corner X) `2 = S-bound X & (NW-corner X) `2 = N-bound X ) by EUCLID:52;
( S-bound X <= p `2 & p `2 <= N-bound X ) by A1, PSCOMP_1:24;
then p in LSeg ((SW-corner X),(NW-corner X)) by A2, A3, A4, GOBOARD7:7;
hence p in W-most X by A1, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th13: :: SPRECT_2:13
for X being compact Subset of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in X & p `1 = E-bound X holds
p in E-most X
proof
let X be compact Subset of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in X & p `1 = E-bound X holds
p in E-most X
let p be Point of (TOP-REAL 2); ::_thesis: ( p in X & p `1 = E-bound X implies p in E-most X )
assume that
A1: p in X and
A2: p `1 = E-bound X ; ::_thesis: p in E-most X
A3: ( (SE-corner X) `1 = E-bound X & (NE-corner X) `1 = E-bound X ) by EUCLID:52;
A4: ( (SE-corner X) `2 = S-bound X & (NE-corner X) `2 = N-bound X ) by EUCLID:52;
( S-bound X <= p `2 & p `2 <= N-bound X ) by A1, PSCOMP_1:24;
then p in LSeg ((SE-corner X),(NE-corner X)) by A2, A3, A4, GOBOARD7:7;
hence p in E-most X by A1, XBOOLE_0:def_4; ::_thesis: verum
end;
begin
theorem Th14: :: SPRECT_2:14
for i, j being Element of NAT
for f being FinSequence of (TOP-REAL 2) st 1 <= i & i <= j & j <= len f holds
L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) }
proof
let i, j be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL 2) st 1 <= i & i <= j & j <= len f holds
L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) }
let f be FinSequence of (TOP-REAL 2); ::_thesis: ( 1 <= i & i <= j & j <= len f implies L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } )
assume that
A1: 1 <= i and
A2: i <= j and
A3: j <= len f ; ::_thesis: L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) }
set A = { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) } ;
set B = { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } ;
percases ( i = j or i < j ) by A2, XXREAL_0:1;
supposeA4: i = j ; ::_thesis: L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) }
A5: { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } = {}
proof
assume { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } <> {} ; ::_thesis: contradiction
then consider z being set such that
A6: z in { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } by XBOOLE_0:def_1;
ex l being Element of NAT st
( z = LSeg (f,l) & i <= l & l < j ) by A6;
hence contradiction by A4; ::_thesis: verum
end;
mid (f,i,j) = <*(f /. i)*> by A1, A3, A4, JORDAN4:15;
hence L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } by A5, SPPOL_2:12, ZFMISC_1:2; ::_thesis: verum
end;
supposeA7: i < j ; ::_thesis: L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) }
{ (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) } = { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) }
proof
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } c= { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) }
let x be set ; ::_thesis: ( x in { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) } implies x in { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } )
assume x in { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) } ; ::_thesis: x in { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) }
then consider m being Element of NAT such that
A8: x = LSeg ((mid (f,i,j)),m) and
A9: 0 + 1 <= m and
A10: m + 1 <= len (mid (f,i,j)) ;
i < m + i by A9, XREAL_1:29;
then A11: i <= (m + i) -' 1 by NAT_D:49;
len (mid (f,i,j)) = (j -' i) + 1 by A1, A3, A7, JORDAN4:8;
then A12: m < (j -' i) + 1 by A10, NAT_1:13;
then m <= j -' i by NAT_1:13;
then m <= j - i by A7, XREAL_1:233;
then ( m + i >= m & m + i <= j ) by NAT_1:11, XREAL_1:19;
then ((m + i) -' 1) + 1 <= j by A9, XREAL_1:235, XXREAL_0:2;
then A13: (m + i) -' 1 < j by NAT_1:13;
x = LSeg (f,((m + i) -' 1)) by A1, A3, A7, A8, A9, A12, JORDAN4:19;
hence x in { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } by A13, A11; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } or x in { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) } )
assume x in { (LSeg (f,l)) where l is Element of NAT : ( i <= l & l < j ) } ; ::_thesis: x in { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) }
then consider l being Element of NAT such that
A14: x = LSeg (f,l) and
A15: i <= l and
A16: l < j ;
set m = (l -' i) + 1;
A17: l - i < j - i by A16, XREAL_1:9;
( l -' i = l - i & j -' i = j - i ) by A15, A16, XREAL_1:233, XXREAL_0:2;
then A18: (l -' i) + 1 < (j -' i) + 1 by A17, XREAL_1:6;
len (mid (f,i,j)) = (j -' i) + 1 by A1, A3, A7, JORDAN4:8;
then A19: ((l -' i) + 1) + 1 <= len (mid (f,i,j)) by A18, NAT_1:13;
A20: 1 <= (l -' i) + 1 by NAT_1:11;
(((l -' i) + 1) + i) -' 1 = (((l -' i) + i) + 1) -' 1
.= (l + 1) -' 1 by A15, XREAL_1:235
.= l by NAT_D:34 ;
then x = LSeg ((mid (f,i,j)),((l -' i) + 1)) by A1, A3, A7, A14, A20, A18, JORDAN4:19;
hence x in { (LSeg ((mid (f,i,j)),m)) where m is Element of NAT : ( 1 <= m & m + 1 <= len (mid (f,i,j)) ) } by A20, A19; ::_thesis: verum
end;
hence L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } ; ::_thesis: verum
end;
end;
end;
theorem Th15: :: SPRECT_2:15
for f being FinSequence of (TOP-REAL 2) holds dom (X_axis f) = dom f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: dom (X_axis f) = dom f
len (X_axis f) = len f by GOBOARD1:def_1;
hence dom (X_axis f) = dom f by FINSEQ_3:29; ::_thesis: verum
end;
theorem Th16: :: SPRECT_2:16
for f being FinSequence of (TOP-REAL 2) holds dom (Y_axis f) = dom f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: dom (Y_axis f) = dom f
len (Y_axis f) = len f by GOBOARD1:def_2;
hence dom (Y_axis f) = dom f by FINSEQ_3:29; ::_thesis: verum
end;
theorem Th17: :: SPRECT_2:17
for a, b, c being Point of (TOP-REAL 2) st b in LSeg (a,c) & a `1 <= b `1 & c `1 <= b `1 & not a = b & not b = c holds
( a `1 = b `1 & c `1 = b `1 )
proof
let a, b, c be Point of (TOP-REAL 2); ::_thesis: ( b in LSeg (a,c) & a `1 <= b `1 & c `1 <= b `1 & not a = b & not b = c implies ( a `1 = b `1 & c `1 = b `1 ) )
assume that
A1: b in LSeg (a,c) and
A2: ( a `1 <= b `1 & c `1 <= b `1 ) ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
consider r being Real such that
A3: b = ((1 - r) * a) + (r * c) and
0 <= r and
r <= 1 by A1;
percases ( ( a `1 = b `1 & c `1 < b `1 ) or ( a `1 < b `1 & c `1 = b `1 ) or ( a `1 < b `1 & c `1 < b `1 ) or ( a `1 = b `1 & c `1 = b `1 ) ) by A2, XXREAL_0:1;
supposethat A4: a `1 = b `1 and
A5: c `1 < b `1 ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
(b `1) + 0 = (((1 - r) * a) `1) + ((r * c) `1) by A3, TOPREAL3:2
.= (((1 - r) * a) `1) + (r * (c `1)) by TOPREAL3:4
.= ((1 - r) * (b `1)) + (r * (c `1)) by A4, TOPREAL3:4
.= (b `1) + ((r * (c `1)) - (r * (b `1))) ;
then A6: 0 = r * ((c `1) - (b `1)) ;
(c `1) - (b `1) < 0 by A5, XREAL_1:49;
then r = 0 by A6, XCMPLX_1:6;
then b = (1 * a) + (0. (TOP-REAL 2)) by A3, EUCLID:29
.= 1 * a by EUCLID:27
.= a by EUCLID:29 ;
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) ; ::_thesis: verum
end;
supposethat A7: a `1 < b `1 and
A8: c `1 = b `1 ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
b `1 = (((1 - r) * a) `1) + ((r * c) `1) by A3, TOPREAL3:2
.= (((1 - r) * a) `1) + (r * (c `1)) by TOPREAL3:4
.= ((1 - r) * (a `1)) + (r * (b `1)) by A8, TOPREAL3:4 ;
then A9: 0 = (1 - r) * ((a `1) - (b `1)) ;
(a `1) - (b `1) < 0 by A7, XREAL_1:49;
then 1 - r = 0 by A9, XCMPLX_1:6;
then b = (0. (TOP-REAL 2)) + (1 * c) by A3, EUCLID:29
.= 1 * c by EUCLID:27
.= c by EUCLID:29 ;
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) ; ::_thesis: verum
end;
supposeA10: ( a `1 < b `1 & c `1 < b `1 ) ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
( a `1 <= c `1 or c `1 <= a `1 ) ;
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) by A1, A10, TOPREAL1:3; ::_thesis: verum
end;
suppose ( a `1 = b `1 & c `1 = b `1 ) ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) ; ::_thesis: verum
end;
end;
end;
theorem Th18: :: SPRECT_2:18
for a, b, c being Point of (TOP-REAL 2) st b in LSeg (a,c) & a `2 <= b `2 & c `2 <= b `2 & not a = b & not b = c holds
( a `2 = b `2 & c `2 = b `2 )
proof
let a, b, c be Point of (TOP-REAL 2); ::_thesis: ( b in LSeg (a,c) & a `2 <= b `2 & c `2 <= b `2 & not a = b & not b = c implies ( a `2 = b `2 & c `2 = b `2 ) )
assume that
A1: b in LSeg (a,c) and
A2: ( a `2 <= b `2 & c `2 <= b `2 ) ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
consider r being Real such that
A3: b = ((1 - r) * a) + (r * c) and
0 <= r and
r <= 1 by A1;
percases ( ( a `2 = b `2 & c `2 < b `2 ) or ( a `2 < b `2 & c `2 = b `2 ) or ( a `2 < b `2 & c `2 < b `2 ) or ( a `2 = b `2 & c `2 = b `2 ) ) by A2, XXREAL_0:1;
supposethat A4: a `2 = b `2 and
A5: c `2 < b `2 ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
(b `2) + 0 = (((1 - r) * a) `2) + ((r * c) `2) by A3, TOPREAL3:2
.= (((1 - r) * a) `2) + (r * (c `2)) by TOPREAL3:4
.= ((1 - r) * (b `2)) + (r * (c `2)) by A4, TOPREAL3:4
.= (b `2) + ((r * (c `2)) - (r * (b `2))) ;
then A6: 0 = r * ((c `2) - (b `2)) ;
(c `2) - (b `2) < 0 by A5, XREAL_1:49;
then r = 0 by A6, XCMPLX_1:6;
then b = (1 * a) + (0. (TOP-REAL 2)) by A3, EUCLID:29
.= 1 * a by EUCLID:27
.= a by EUCLID:29 ;
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) ; ::_thesis: verum
end;
supposethat A7: a `2 < b `2 and
A8: c `2 = b `2 ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
b `2 = (((1 - r) * a) `2) + ((r * c) `2) by A3, TOPREAL3:2
.= (((1 - r) * a) `2) + (r * (c `2)) by TOPREAL3:4
.= ((1 - r) * (a `2)) + (r * (b `2)) by A8, TOPREAL3:4 ;
then A9: 0 = (1 - r) * ((a `2) - (b `2)) ;
(a `2) - (b `2) < 0 by A7, XREAL_1:49;
then 1 - r = 0 by A9, XCMPLX_1:6;
then b = (0. (TOP-REAL 2)) + (1 * c) by A3, EUCLID:29
.= 1 * c by EUCLID:27
.= c by EUCLID:29 ;
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) ; ::_thesis: verum
end;
supposeA10: ( a `2 < b `2 & c `2 < b `2 ) ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
( a `2 <= c `2 or c `2 <= a `2 ) ;
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) by A1, A10, TOPREAL1:4; ::_thesis: verum
end;
suppose ( a `2 = b `2 & c `2 = b `2 ) ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) ; ::_thesis: verum
end;
end;
end;
theorem Th19: :: SPRECT_2:19
for a, b, c being Point of (TOP-REAL 2) st b in LSeg (a,c) & a `1 >= b `1 & c `1 >= b `1 & not a = b & not b = c holds
( a `1 = b `1 & c `1 = b `1 )
proof
let a, b, c be Point of (TOP-REAL 2); ::_thesis: ( b in LSeg (a,c) & a `1 >= b `1 & c `1 >= b `1 & not a = b & not b = c implies ( a `1 = b `1 & c `1 = b `1 ) )
assume that
A1: b in LSeg (a,c) and
A2: ( a `1 >= b `1 & c `1 >= b `1 ) ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
consider r being Real such that
A3: b = ((1 - r) * a) + (r * c) and
0 <= r and
r <= 1 by A1;
percases ( ( a `1 = b `1 & c `1 > b `1 ) or ( a `1 > b `1 & c `1 = b `1 ) or ( a `1 > b `1 & c `1 > b `1 ) or ( a `1 = b `1 & c `1 = b `1 ) ) by A2, XXREAL_0:1;
supposethat A4: a `1 = b `1 and
A5: c `1 > b `1 ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
(b `1) + 0 = (((1 - r) * a) `1) + ((r * c) `1) by A3, TOPREAL3:2
.= (((1 - r) * a) `1) + (r * (c `1)) by TOPREAL3:4
.= ((1 - r) * (b `1)) + (r * (c `1)) by A4, TOPREAL3:4
.= (b `1) + ((r * (c `1)) - (r * (b `1))) ;
then A6: 0 = r * ((c `1) - (b `1)) ;
(c `1) - (b `1) > 0 by A5, XREAL_1:50;
then r = 0 by A6, XCMPLX_1:6;
then b = (1 * a) + (0. (TOP-REAL 2)) by A3, EUCLID:29
.= 1 * a by EUCLID:27
.= a by EUCLID:29 ;
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) ; ::_thesis: verum
end;
supposethat A7: a `1 > b `1 and
A8: c `1 = b `1 ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
b `1 = (((1 - r) * a) `1) + ((r * c) `1) by A3, TOPREAL3:2
.= (((1 - r) * a) `1) + (r * (c `1)) by TOPREAL3:4
.= ((1 - r) * (a `1)) + (r * (b `1)) by A8, TOPREAL3:4 ;
then A9: 0 = (1 - r) * ((a `1) - (b `1)) ;
(a `1) - (b `1) > 0 by A7, XREAL_1:50;
then 1 - r = 0 by A9, XCMPLX_1:6;
then b = (0. (TOP-REAL 2)) + (1 * c) by A3, EUCLID:29
.= 1 * c by EUCLID:27
.= c by EUCLID:29 ;
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) ; ::_thesis: verum
end;
supposeA10: ( a `1 > b `1 & c `1 > b `1 ) ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
( a `1 >= c `1 or c `1 >= a `1 ) ;
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) by A1, A10, TOPREAL1:3; ::_thesis: verum
end;
suppose ( a `1 = b `1 & c `1 = b `1 ) ; ::_thesis: ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) )
hence ( a = b or b = c or ( a `1 = b `1 & c `1 = b `1 ) ) ; ::_thesis: verum
end;
end;
end;
theorem Th20: :: SPRECT_2:20
for a, b, c being Point of (TOP-REAL 2) st b in LSeg (a,c) & a `2 >= b `2 & c `2 >= b `2 & not a = b & not b = c holds
( a `2 = b `2 & c `2 = b `2 )
proof
let a, b, c be Point of (TOP-REAL 2); ::_thesis: ( b in LSeg (a,c) & a `2 >= b `2 & c `2 >= b `2 & not a = b & not b = c implies ( a `2 = b `2 & c `2 = b `2 ) )
assume that
A1: b in LSeg (a,c) and
A2: ( a `2 >= b `2 & c `2 >= b `2 ) ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
consider r being Real such that
A3: b = ((1 - r) * a) + (r * c) and
0 <= r and
r <= 1 by A1;
percases ( ( a `2 = b `2 & c `2 > b `2 ) or ( a `2 > b `2 & c `2 = b `2 ) or ( a `2 > b `2 & c `2 > b `2 ) or ( a `2 = b `2 & c `2 = b `2 ) ) by A2, XXREAL_0:1;
supposethat A4: a `2 = b `2 and
A5: c `2 > b `2 ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
(b `2) + 0 = (((1 - r) * a) `2) + ((r * c) `2) by A3, TOPREAL3:2
.= (((1 - r) * a) `2) + (r * (c `2)) by TOPREAL3:4
.= ((1 - r) * (b `2)) + (r * (c `2)) by A4, TOPREAL3:4
.= (b `2) + ((r * (c `2)) - (r * (b `2))) ;
then A6: 0 = r * ((c `2) - (b `2)) ;
(c `2) - (b `2) > 0 by A5, XREAL_1:50;
then r = 0 by A6, XCMPLX_1:6;
then b = (1 * a) + (0. (TOP-REAL 2)) by A3, EUCLID:29
.= 1 * a by EUCLID:27
.= a by EUCLID:29 ;
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) ; ::_thesis: verum
end;
supposethat A7: a `2 > b `2 and
A8: c `2 = b `2 ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
b `2 = (((1 - r) * a) `2) + ((r * c) `2) by A3, TOPREAL3:2
.= (((1 - r) * a) `2) + (r * (c `2)) by TOPREAL3:4
.= ((1 - r) * (a `2)) + (r * (b `2)) by A8, TOPREAL3:4 ;
then A9: 0 = (1 - r) * ((a `2) - (b `2)) ;
(a `2) - (b `2) > 0 by A7, XREAL_1:50;
then 1 - r = 0 by A9, XCMPLX_1:6;
then b = (0. (TOP-REAL 2)) + (1 * c) by A3, EUCLID:29
.= 1 * c by EUCLID:27
.= c by EUCLID:29 ;
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) ; ::_thesis: verum
end;
supposeA10: ( a `2 > b `2 & c `2 > b `2 ) ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
( a `2 >= c `2 or c `2 >= a `2 ) ;
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) by A1, A10, TOPREAL1:4; ::_thesis: verum
end;
suppose ( a `2 = b `2 & c `2 = b `2 ) ; ::_thesis: ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) )
hence ( a = b or b = c or ( a `2 = b `2 & c `2 = b `2 ) ) ; ::_thesis: verum
end;
end;
end;
begin
definition
let f, g be FinSequence of (TOP-REAL 2);
predg is_in_the_area_of f means :Def1: :: SPRECT_2:def 1
for n being Element of NAT st n in dom g holds
( W-bound (L~ f) <= (g /. n) `1 & (g /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (g /. n) `2 & (g /. n) `2 <= N-bound (L~ f) );
end;
:: deftheorem Def1 defines is_in_the_area_of SPRECT_2:def_1_:_
for f, g being FinSequence of (TOP-REAL 2) holds
( g is_in_the_area_of f iff for n being Element of NAT st n in dom g holds
( W-bound (L~ f) <= (g /. n) `1 & (g /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (g /. n) `2 & (g /. n) `2 <= N-bound (L~ f) ) );
theorem Th21: :: SPRECT_2:21
for f being non trivial FinSequence of (TOP-REAL 2) holds f is_in_the_area_of f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: f is_in_the_area_of f
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom f implies ( W-bound (L~ f) <= (f /. n) `1 & (f /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (f /. n) `2 & (f /. n) `2 <= N-bound (L~ f) ) )
assume A1: n in dom f ; ::_thesis: ( W-bound (L~ f) <= (f /. n) `1 & (f /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (f /. n) `2 & (f /. n) `2 <= N-bound (L~ f) )
len f >= 2 by NAT_D:60;
then f /. n in L~ f by A1, GOBOARD1:1;
hence ( W-bound (L~ f) <= (f /. n) `1 & (f /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (f /. n) `2 & (f /. n) `2 <= N-bound (L~ f) ) by PSCOMP_1:24; ::_thesis: verum
end;
theorem Th22: :: SPRECT_2:22
for f, g being FinSequence of (TOP-REAL 2) st g is_in_the_area_of f holds
for i, j being Element of NAT st i in dom g & j in dom g holds
mid (g,i,j) is_in_the_area_of f
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( g is_in_the_area_of f implies for i, j being Element of NAT st i in dom g & j in dom g holds
mid (g,i,j) is_in_the_area_of f )
assume A1: for n being Element of NAT st n in dom g holds
( W-bound (L~ f) <= (g /. n) `1 & (g /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (g /. n) `2 & (g /. n) `2 <= N-bound (L~ f) ) ; :: according to SPRECT_2:def_1 ::_thesis: for i, j being Element of NAT st i in dom g & j in dom g holds
mid (g,i,j) is_in_the_area_of f
let i, j be Element of NAT ; ::_thesis: ( i in dom g & j in dom g implies mid (g,i,j) is_in_the_area_of f )
assume A2: ( i in dom g & j in dom g ) ; ::_thesis: mid (g,i,j) is_in_the_area_of f
set h = mid (g,i,j);
percases ( i <= j or i > j ) ;
supposeA3: i <= j ; ::_thesis: mid (g,i,j) is_in_the_area_of f
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom (mid (g,i,j)) implies ( W-bound (L~ f) <= ((mid (g,i,j)) /. n) `1 & ((mid (g,i,j)) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((mid (g,i,j)) /. n) `2 & ((mid (g,i,j)) /. n) `2 <= N-bound (L~ f) ) )
assume n in dom (mid (g,i,j)) ; ::_thesis: ( W-bound (L~ f) <= ((mid (g,i,j)) /. n) `1 & ((mid (g,i,j)) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((mid (g,i,j)) /. n) `2 & ((mid (g,i,j)) /. n) `2 <= N-bound (L~ f) )
then ( (n + i) -' 1 in dom g & (mid (g,i,j)) /. n = g /. ((n + i) -' 1) ) by A2, A3, Th1, Th3;
hence ( W-bound (L~ f) <= ((mid (g,i,j)) /. n) `1 & ((mid (g,i,j)) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((mid (g,i,j)) /. n) `2 & ((mid (g,i,j)) /. n) `2 <= N-bound (L~ f) ) by A1; ::_thesis: verum
end;
supposeA4: i > j ; ::_thesis: mid (g,i,j) is_in_the_area_of f
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom (mid (g,i,j)) implies ( W-bound (L~ f) <= ((mid (g,i,j)) /. n) `1 & ((mid (g,i,j)) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((mid (g,i,j)) /. n) `2 & ((mid (g,i,j)) /. n) `2 <= N-bound (L~ f) ) )
assume n in dom (mid (g,i,j)) ; ::_thesis: ( W-bound (L~ f) <= ((mid (g,i,j)) /. n) `1 & ((mid (g,i,j)) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((mid (g,i,j)) /. n) `2 & ((mid (g,i,j)) /. n) `2 <= N-bound (L~ f) )
then ( (i -' n) + 1 in dom g & (mid (g,i,j)) /. n = g /. ((i -' n) + 1) ) by A2, A4, Th2, Th4;
hence ( W-bound (L~ f) <= ((mid (g,i,j)) /. n) `1 & ((mid (g,i,j)) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((mid (g,i,j)) /. n) `2 & ((mid (g,i,j)) /. n) `2 <= N-bound (L~ f) ) by A1; ::_thesis: verum
end;
end;
end;
theorem :: SPRECT_2:23
for f being non trivial FinSequence of (TOP-REAL 2)
for i, j being Element of NAT st i in dom f & j in dom f holds
mid (f,i,j) is_in_the_area_of f by Th21, Th22;
theorem Th24: :: SPRECT_2:24
for f, g, h being FinSequence of (TOP-REAL 2) st g is_in_the_area_of f & h is_in_the_area_of f holds
g ^ h is_in_the_area_of f
proof
let f, g, h be FinSequence of (TOP-REAL 2); ::_thesis: ( g is_in_the_area_of f & h is_in_the_area_of f implies g ^ h is_in_the_area_of f )
assume that
A1: g is_in_the_area_of f and
A2: h is_in_the_area_of f ; ::_thesis: g ^ h is_in_the_area_of f
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom (g ^ h) implies ( W-bound (L~ f) <= ((g ^ h) /. n) `1 & ((g ^ h) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((g ^ h) /. n) `2 & ((g ^ h) /. n) `2 <= N-bound (L~ f) ) )
assume A3: n in dom (g ^ h) ; ::_thesis: ( W-bound (L~ f) <= ((g ^ h) /. n) `1 & ((g ^ h) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((g ^ h) /. n) `2 & ((g ^ h) /. n) `2 <= N-bound (L~ f) )
percases ( n in dom g or ex i being Nat st
( i in dom h & n = (len g) + i ) ) by A3, FINSEQ_1:25;
supposeA4: n in dom g ; ::_thesis: ( W-bound (L~ f) <= ((g ^ h) /. n) `1 & ((g ^ h) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((g ^ h) /. n) `2 & ((g ^ h) /. n) `2 <= N-bound (L~ f) )
then (g ^ h) /. n = g /. n by FINSEQ_4:68;
hence ( W-bound (L~ f) <= ((g ^ h) /. n) `1 & ((g ^ h) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((g ^ h) /. n) `2 & ((g ^ h) /. n) `2 <= N-bound (L~ f) ) by A1, A4, Def1; ::_thesis: verum
end;
suppose ex i being Nat st
( i in dom h & n = (len g) + i ) ; ::_thesis: ( W-bound (L~ f) <= ((g ^ h) /. n) `1 & ((g ^ h) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((g ^ h) /. n) `2 & ((g ^ h) /. n) `2 <= N-bound (L~ f) )
then consider i being Nat such that
A5: i in dom h and
A6: n = (len g) + i ;
(g ^ h) /. n = h /. i by A5, A6, FINSEQ_4:69;
hence ( W-bound (L~ f) <= ((g ^ h) /. n) `1 & ((g ^ h) /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= ((g ^ h) /. n) `2 & ((g ^ h) /. n) `2 <= N-bound (L~ f) ) by A2, A5, Def1; ::_thesis: verum
end;
end;
end;
theorem Th25: :: SPRECT_2:25
for f being non trivial FinSequence of (TOP-REAL 2) holds <*(NE-corner (L~ f))*> is_in_the_area_of f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: <*(NE-corner (L~ f))*> is_in_the_area_of f
set g = <*(NE-corner (L~ f))*>;
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom <*(NE-corner (L~ f))*> implies ( W-bound (L~ f) <= (<*(NE-corner (L~ f))*> /. n) `1 & (<*(NE-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(NE-corner (L~ f))*> /. n) `2 & (<*(NE-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) )
assume A1: n in dom <*(NE-corner (L~ f))*> ; ::_thesis: ( W-bound (L~ f) <= (<*(NE-corner (L~ f))*> /. n) `1 & (<*(NE-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(NE-corner (L~ f))*> /. n) `2 & (<*(NE-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) )
dom <*(NE-corner (L~ f))*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then n = 1 by A1, TARSKI:def_1;
then <*(NE-corner (L~ f))*> /. n = |[(E-bound (L~ f)),(N-bound (L~ f))]| by FINSEQ_4:16;
then ( (<*(NE-corner (L~ f))*> /. n) `1 = E-bound (L~ f) & (<*(NE-corner (L~ f))*> /. n) `2 = N-bound (L~ f) ) by EUCLID:52;
hence ( W-bound (L~ f) <= (<*(NE-corner (L~ f))*> /. n) `1 & (<*(NE-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(NE-corner (L~ f))*> /. n) `2 & (<*(NE-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) by SPRECT_1:21, SPRECT_1:22; ::_thesis: verum
end;
theorem Th26: :: SPRECT_2:26
for f being non trivial FinSequence of (TOP-REAL 2) holds <*(NW-corner (L~ f))*> is_in_the_area_of f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: <*(NW-corner (L~ f))*> is_in_the_area_of f
set g = <*(NW-corner (L~ f))*>;
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom <*(NW-corner (L~ f))*> implies ( W-bound (L~ f) <= (<*(NW-corner (L~ f))*> /. n) `1 & (<*(NW-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(NW-corner (L~ f))*> /. n) `2 & (<*(NW-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) )
assume A1: n in dom <*(NW-corner (L~ f))*> ; ::_thesis: ( W-bound (L~ f) <= (<*(NW-corner (L~ f))*> /. n) `1 & (<*(NW-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(NW-corner (L~ f))*> /. n) `2 & (<*(NW-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) )
dom <*(NW-corner (L~ f))*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then n = 1 by A1, TARSKI:def_1;
then <*(NW-corner (L~ f))*> /. n = |[(W-bound (L~ f)),(N-bound (L~ f))]| by FINSEQ_4:16;
then ( (<*(NW-corner (L~ f))*> /. n) `1 = W-bound (L~ f) & (<*(NW-corner (L~ f))*> /. n) `2 = N-bound (L~ f) ) by EUCLID:52;
hence ( W-bound (L~ f) <= (<*(NW-corner (L~ f))*> /. n) `1 & (<*(NW-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(NW-corner (L~ f))*> /. n) `2 & (<*(NW-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) by SPRECT_1:21, SPRECT_1:22; ::_thesis: verum
end;
theorem Th27: :: SPRECT_2:27
for f being non trivial FinSequence of (TOP-REAL 2) holds <*(SE-corner (L~ f))*> is_in_the_area_of f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: <*(SE-corner (L~ f))*> is_in_the_area_of f
set g = <*(SE-corner (L~ f))*>;
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom <*(SE-corner (L~ f))*> implies ( W-bound (L~ f) <= (<*(SE-corner (L~ f))*> /. n) `1 & (<*(SE-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(SE-corner (L~ f))*> /. n) `2 & (<*(SE-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) )
assume A1: n in dom <*(SE-corner (L~ f))*> ; ::_thesis: ( W-bound (L~ f) <= (<*(SE-corner (L~ f))*> /. n) `1 & (<*(SE-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(SE-corner (L~ f))*> /. n) `2 & (<*(SE-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) )
dom <*(SE-corner (L~ f))*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then n = 1 by A1, TARSKI:def_1;
then <*(SE-corner (L~ f))*> /. n = |[(E-bound (L~ f)),(S-bound (L~ f))]| by FINSEQ_4:16;
then ( (<*(SE-corner (L~ f))*> /. n) `1 = E-bound (L~ f) & (<*(SE-corner (L~ f))*> /. n) `2 = S-bound (L~ f) ) by EUCLID:52;
hence ( W-bound (L~ f) <= (<*(SE-corner (L~ f))*> /. n) `1 & (<*(SE-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(SE-corner (L~ f))*> /. n) `2 & (<*(SE-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) by SPRECT_1:21, SPRECT_1:22; ::_thesis: verum
end;
theorem Th28: :: SPRECT_2:28
for f being non trivial FinSequence of (TOP-REAL 2) holds <*(SW-corner (L~ f))*> is_in_the_area_of f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: <*(SW-corner (L~ f))*> is_in_the_area_of f
set g = <*(SW-corner (L~ f))*>;
let n be Element of NAT ; :: according to SPRECT_2:def_1 ::_thesis: ( n in dom <*(SW-corner (L~ f))*> implies ( W-bound (L~ f) <= (<*(SW-corner (L~ f))*> /. n) `1 & (<*(SW-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(SW-corner (L~ f))*> /. n) `2 & (<*(SW-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) )
assume A1: n in dom <*(SW-corner (L~ f))*> ; ::_thesis: ( W-bound (L~ f) <= (<*(SW-corner (L~ f))*> /. n) `1 & (<*(SW-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(SW-corner (L~ f))*> /. n) `2 & (<*(SW-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) )
dom <*(SW-corner (L~ f))*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then n = 1 by A1, TARSKI:def_1;
then <*(SW-corner (L~ f))*> /. n = |[(W-bound (L~ f)),(S-bound (L~ f))]| by FINSEQ_4:16;
then ( (<*(SW-corner (L~ f))*> /. n) `1 = W-bound (L~ f) & (<*(SW-corner (L~ f))*> /. n) `2 = S-bound (L~ f) ) by EUCLID:52;
hence ( W-bound (L~ f) <= (<*(SW-corner (L~ f))*> /. n) `1 & (<*(SW-corner (L~ f))*> /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (<*(SW-corner (L~ f))*> /. n) `2 & (<*(SW-corner (L~ f))*> /. n) `2 <= N-bound (L~ f) ) by SPRECT_1:21, SPRECT_1:22; ::_thesis: verum
end;
begin
definition
let f, g be FinSequence of (TOP-REAL 2);
predg is_a_h.c._for f means :Def2: :: SPRECT_2:def 2
( g is_in_the_area_of f & (g /. 1) `1 = W-bound (L~ f) & (g /. (len g)) `1 = E-bound (L~ f) );
predg is_a_v.c._for f means :Def3: :: SPRECT_2:def 3
( g is_in_the_area_of f & (g /. 1) `2 = S-bound (L~ f) & (g /. (len g)) `2 = N-bound (L~ f) );
end;
:: deftheorem Def2 defines is_a_h.c._for SPRECT_2:def_2_:_
for f, g being FinSequence of (TOP-REAL 2) holds
( g is_a_h.c._for f iff ( g is_in_the_area_of f & (g /. 1) `1 = W-bound (L~ f) & (g /. (len g)) `1 = E-bound (L~ f) ) );
:: deftheorem Def3 defines is_a_v.c._for SPRECT_2:def_3_:_
for f, g being FinSequence of (TOP-REAL 2) holds
( g is_a_v.c._for f iff ( g is_in_the_area_of f & (g /. 1) `2 = S-bound (L~ f) & (g /. (len g)) `2 = N-bound (L~ f) ) );
theorem Th29: :: SPRECT_2:29
for f being FinSequence of (TOP-REAL 2)
for g, h being one-to-one special FinSequence of (TOP-REAL 2) st 2 <= len g & 2 <= len h & g is_a_h.c._for f & h is_a_v.c._for f holds
L~ g meets L~ h
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for g, h being one-to-one special FinSequence of (TOP-REAL 2) st 2 <= len g & 2 <= len h & g is_a_h.c._for f & h is_a_v.c._for f holds
L~ g meets L~ h
let g, h be one-to-one special FinSequence of (TOP-REAL 2); ::_thesis: ( 2 <= len g & 2 <= len h & g is_a_h.c._for f & h is_a_v.c._for f implies L~ g meets L~ h )
assume that
A1: ( 2 <= len g & 2 <= len h ) and
A2: for n being Element of NAT st n in dom g holds
( W-bound (L~ f) <= (g /. n) `1 & (g /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (g /. n) `2 & (g /. n) `2 <= N-bound (L~ f) ) and
A3: (g /. 1) `1 = W-bound (L~ f) and
A4: (g /. (len g)) `1 = E-bound (L~ f) and
A5: for n being Element of NAT st n in dom h holds
( W-bound (L~ f) <= (h /. n) `1 & (h /. n) `1 <= E-bound (L~ f) & S-bound (L~ f) <= (h /. n) `2 & (h /. n) `2 <= N-bound (L~ f) ) and
A6: (h /. 1) `2 = S-bound (L~ f) and
A7: (h /. (len h)) `2 = N-bound (L~ f) ; :: according to SPRECT_2:def_1,SPRECT_2:def_2,SPRECT_2:def_3 ::_thesis: L~ g meets L~ h
reconsider g = g, h = h as non empty one-to-one special FinSequence of (TOP-REAL 2) by A1, CARD_1:27;
A8: X_axis h lies_between (X_axis g) . 1,(X_axis g) . (len g)
proof
let n be Element of NAT ; :: according to GOBOARD4:def_2 ::_thesis: ( not n in dom (X_axis h) or ( (X_axis g) . 1 <= (X_axis h) . n & (X_axis h) . n <= (X_axis g) . (len g) ) )
set F = X_axis g;
set r = (X_axis g) . 1;
set s = (X_axis g) . (len g);
set H = X_axis h;
assume n in dom (X_axis h) ; ::_thesis: ( (X_axis g) . 1 <= (X_axis h) . n & (X_axis h) . n <= (X_axis g) . (len g) )
then A9: ( n in dom h & (X_axis h) . n = (h /. n) `1 ) by Th15, GOBOARD1:def_1;
1 in dom (X_axis g) by FINSEQ_5:6;
then (X_axis g) . 1 = W-bound (L~ f) by A3, GOBOARD1:def_1;
hence (X_axis g) . 1 <= (X_axis h) . n by A5, A9; ::_thesis: (X_axis h) . n <= (X_axis g) . (len g)
( len (X_axis g) = len g & len (X_axis g) in dom (X_axis g) ) by FINSEQ_5:6, GOBOARD1:def_1;
then (X_axis g) . (len g) = E-bound (L~ f) by A4, GOBOARD1:def_1;
hence (X_axis h) . n <= (X_axis g) . (len g) by A5, A9; ::_thesis: verum
end;
A10: Y_axis h lies_between (Y_axis h) . 1,(Y_axis h) . (len h)
proof
let n be Element of NAT ; :: according to GOBOARD4:def_2 ::_thesis: ( not n in dom (Y_axis h) or ( (Y_axis h) . 1 <= (Y_axis h) . n & (Y_axis h) . n <= (Y_axis h) . (len h) ) )
set F = Y_axis h;
set r = (Y_axis h) . 1;
set s = (Y_axis h) . (len h);
assume n in dom (Y_axis h) ; ::_thesis: ( (Y_axis h) . 1 <= (Y_axis h) . n & (Y_axis h) . n <= (Y_axis h) . (len h) )
then A11: ( n in dom h & (Y_axis h) . n = (h /. n) `2 ) by Th16, GOBOARD1:def_2;
1 in dom (Y_axis h) by FINSEQ_5:6;
then (Y_axis h) . 1 = S-bound (L~ f) by A6, GOBOARD1:def_2;
hence (Y_axis h) . 1 <= (Y_axis h) . n by A5, A11; ::_thesis: (Y_axis h) . n <= (Y_axis h) . (len h)
( len (Y_axis h) = len h & len (Y_axis h) in dom (Y_axis h) ) by FINSEQ_5:6, GOBOARD1:def_2;
then (Y_axis h) . (len h) = N-bound (L~ f) by A7, GOBOARD1:def_2;
hence (Y_axis h) . n <= (Y_axis h) . (len h) by A5, A11; ::_thesis: verum
end;
A12: Y_axis g lies_between (Y_axis h) . 1,(Y_axis h) . (len h)
proof
let n be Element of NAT ; :: according to GOBOARD4:def_2 ::_thesis: ( not n in dom (Y_axis g) or ( (Y_axis h) . 1 <= (Y_axis g) . n & (Y_axis g) . n <= (Y_axis h) . (len h) ) )
set F = Y_axis h;
set r = (Y_axis h) . 1;
set s = (Y_axis h) . (len h);
set G = Y_axis g;
assume n in dom (Y_axis g) ; ::_thesis: ( (Y_axis h) . 1 <= (Y_axis g) . n & (Y_axis g) . n <= (Y_axis h) . (len h) )
then A13: ( n in dom g & (Y_axis g) . n = (g /. n) `2 ) by Th16, GOBOARD1:def_2;
1 in dom (Y_axis h) by FINSEQ_5:6;
then (Y_axis h) . 1 = S-bound (L~ f) by A6, GOBOARD1:def_2;
hence (Y_axis h) . 1 <= (Y_axis g) . n by A2, A13; ::_thesis: (Y_axis g) . n <= (Y_axis h) . (len h)
( len (Y_axis h) = len h & len (Y_axis h) in dom (Y_axis h) ) by FINSEQ_5:6, GOBOARD1:def_2;
then (Y_axis h) . (len h) = N-bound (L~ f) by A7, GOBOARD1:def_2;
hence (Y_axis g) . n <= (Y_axis h) . (len h) by A2, A13; ::_thesis: verum
end;
X_axis g lies_between (X_axis g) . 1,(X_axis g) . (len g)
proof
let n be Element of NAT ; :: according to GOBOARD4:def_2 ::_thesis: ( not n in dom (X_axis g) or ( (X_axis g) . 1 <= (X_axis g) . n & (X_axis g) . n <= (X_axis g) . (len g) ) )
set F = X_axis g;
set r = (X_axis g) . 1;
set s = (X_axis g) . (len g);
assume n in dom (X_axis g) ; ::_thesis: ( (X_axis g) . 1 <= (X_axis g) . n & (X_axis g) . n <= (X_axis g) . (len g) )
then A14: ( n in dom g & (X_axis g) . n = (g /. n) `1 ) by Th15, GOBOARD1:def_1;
1 in dom (X_axis g) by FINSEQ_5:6;
then (X_axis g) . 1 = W-bound (L~ f) by A3, GOBOARD1:def_1;
hence (X_axis g) . 1 <= (X_axis g) . n by A2, A14; ::_thesis: (X_axis g) . n <= (X_axis g) . (len g)
( len (X_axis g) = len g & len (X_axis g) in dom (X_axis g) ) by FINSEQ_5:6, GOBOARD1:def_1;
then (X_axis g) . (len g) = E-bound (L~ f) by A4, GOBOARD1:def_1;
hence (X_axis g) . n <= (X_axis g) . (len g) by A2, A14; ::_thesis: verum
end;
hence L~ g meets L~ h by A1, A8, A12, A10, GOBOARD4:5; ::_thesis: verum
end;
begin
definition
let f be FinSequence of (TOP-REAL 2);
attrf is clockwise_oriented means :Def4: :: SPRECT_2:def 4
(Rotate (f,(N-min (L~ f)))) /. 2 in N-most (L~ f);
end;
:: deftheorem Def4 defines clockwise_oriented SPRECT_2:def_4_:_
for f being FinSequence of (TOP-REAL 2) holds
( f is clockwise_oriented iff (Rotate (f,(N-min (L~ f)))) /. 2 in N-most (L~ f) );
theorem Th30: :: SPRECT_2:30
for f being non constant standard special_circular_sequence st f /. 1 = N-min (L~ f) holds
( f is clockwise_oriented iff f /. 2 in N-most (L~ f) )
proof
let f be non constant standard special_circular_sequence; ::_thesis: ( f /. 1 = N-min (L~ f) implies ( f is clockwise_oriented iff f /. 2 in N-most (L~ f) ) )
assume f /. 1 = N-min (L~ f) ; ::_thesis: ( f is clockwise_oriented iff f /. 2 in N-most (L~ f) )
then Rotate (f,(N-min (L~ f))) = f by FINSEQ_6:89;
hence ( f is clockwise_oriented iff f /. 2 in N-most (L~ f) ) by Def4; ::_thesis: verum
end;
registration
cluster R^2-unit_square -> compact ;
coherence
R^2-unit_square is compact by TOPREAL2:2;
end;
theorem Th31: :: SPRECT_2:31
N-bound R^2-unit_square = 1
proof
set X = R^2-unit_square ;
reconsider Z = (proj2 | R^2-unit_square) .: the carrier of ((TOP-REAL 2) | R^2-unit_square) as Subset of REAL ;
A1: R^2-unit_square = [#] ((TOP-REAL 2) | R^2-unit_square) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | R^2-unit_square) ;
A2: for q being real number st ( for p being real number st p in Z holds
p <= q ) holds
1 <= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p <= q ) implies 1 <= q )
assume A3: for p being real number st p in Z holds
p <= q ; ::_thesis: 1 <= q
|[1,1]| in LSeg (|[1,0]|,|[1,1]|) by RLTOPSP1:68;
then |[1,1]| in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) by XBOOLE_0:def_3;
then A4: |[1,1]| in R^2-unit_square by XBOOLE_0:def_3;
then (proj2 | R^2-unit_square) . |[1,1]| = |[1,1]| `2 by PSCOMP_1:23
.= 1 by EUCLID:52 ;
hence 1 <= q by A1, A3, A4, FUNCT_2:35; ::_thesis: verum
end;
for p being real number st p in Z holds
p <= 1
proof
let p be real number ; ::_thesis: ( p in Z implies p <= 1 )
assume p in Z ; ::_thesis: p <= 1
then consider p0 being set such that
A5: p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
A6: p = (proj2 | R^2-unit_square) . p0 by FUNCT_2:64;
reconsider p0 = p0 as Point of (TOP-REAL 2) by A1, A5;
ex q being Point of (TOP-REAL 2) st
( p0 = q & ( ( q `1 = 0 & q `2 <= 1 & q `2 >= 0 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 0 ) or ( q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) ) ) by A1, A5, TOPREAL1:14;
hence p <= 1 by A1, A5, A6, PSCOMP_1:23; ::_thesis: verum
end;
hence N-bound R^2-unit_square = 1 by A2, SEQ_4:46; ::_thesis: verum
end;
theorem Th32: :: SPRECT_2:32
W-bound R^2-unit_square = 0
proof
set X = R^2-unit_square ;
reconsider Z = (proj1 | R^2-unit_square) .: the carrier of ((TOP-REAL 2) | R^2-unit_square) as Subset of REAL ;
A1: R^2-unit_square = [#] ((TOP-REAL 2) | R^2-unit_square) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | R^2-unit_square) ;
A2: for q being real number st ( for p being real number st p in Z holds
p >= q ) holds
0 >= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p >= q ) implies 0 >= q )
assume A3: for p being real number st p in Z holds
p >= q ; ::_thesis: 0 >= q
|[0,0]| in LSeg (|[0,0]|,|[1,0]|) by RLTOPSP1:68;
then |[0,0]| in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) by XBOOLE_0:def_3;
then A4: |[0,0]| in R^2-unit_square by XBOOLE_0:def_3;
then (proj1 | R^2-unit_square) . |[0,0]| = |[0,0]| `1 by PSCOMP_1:22
.= 0 by EUCLID:52 ;
hence 0 >= q by A1, A3, A4, FUNCT_2:35; ::_thesis: verum
end;
for p being real number st p in Z holds
p >= 0
proof
let p be real number ; ::_thesis: ( p in Z implies p >= 0 )
assume p in Z ; ::_thesis: p >= 0
then consider p0 being set such that
A5: p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
A6: p = (proj1 | R^2-unit_square) . p0 by FUNCT_2:64;
reconsider p0 = p0 as Point of (TOP-REAL 2) by A1, A5;
ex q being Point of (TOP-REAL 2) st
( p0 = q & ( ( q `1 = 0 & q `2 <= 1 & q `2 >= 0 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 0 ) or ( q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) ) ) by A1, A5, TOPREAL1:14;
hence p >= 0 by A1, A5, A6, PSCOMP_1:22; ::_thesis: verum
end;
hence W-bound R^2-unit_square = 0 by A2, SEQ_4:44; ::_thesis: verum
end;
theorem Th33: :: SPRECT_2:33
E-bound R^2-unit_square = 1
proof
set X = R^2-unit_square ;
reconsider Z = (proj1 | R^2-unit_square) .: the carrier of ((TOP-REAL 2) | R^2-unit_square) as Subset of REAL ;
A1: R^2-unit_square = [#] ((TOP-REAL 2) | R^2-unit_square) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | R^2-unit_square) ;
A2: for q being real number st ( for p being real number st p in Z holds
p <= q ) holds
1 <= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p <= q ) implies 1 <= q )
assume A3: for p being real number st p in Z holds
p <= q ; ::_thesis: 1 <= q
|[1,1]| in LSeg (|[1,0]|,|[1,1]|) by RLTOPSP1:68;
then |[1,1]| in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) by XBOOLE_0:def_3;
then A4: |[1,1]| in R^2-unit_square by XBOOLE_0:def_3;
then (proj1 | R^2-unit_square) . |[1,1]| = |[1,1]| `1 by PSCOMP_1:22
.= 1 by EUCLID:52 ;
hence 1 <= q by A1, A3, A4, FUNCT_2:35; ::_thesis: verum
end;
for p being real number st p in Z holds
p <= 1
proof
let p be real number ; ::_thesis: ( p in Z implies p <= 1 )
assume p in Z ; ::_thesis: p <= 1
then consider p0 being set such that
A5: p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
A6: p = (proj1 | R^2-unit_square) . p0 by FUNCT_2:64;
reconsider p0 = p0 as Point of (TOP-REAL 2) by A1, A5;
ex q being Point of (TOP-REAL 2) st
( p0 = q & ( ( q `1 = 0 & q `2 <= 1 & q `2 >= 0 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 0 ) or ( q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) ) ) by A1, A5, TOPREAL1:14;
hence p <= 1 by A1, A5, A6, PSCOMP_1:22; ::_thesis: verum
end;
hence E-bound R^2-unit_square = 1 by A2, SEQ_4:46; ::_thesis: verum
end;
theorem :: SPRECT_2:34
S-bound R^2-unit_square = 0
proof
set X = R^2-unit_square ;
reconsider Z = (proj2 | R^2-unit_square) .: the carrier of ((TOP-REAL 2) | R^2-unit_square) as Subset of REAL ;
A1: R^2-unit_square = [#] ((TOP-REAL 2) | R^2-unit_square) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | R^2-unit_square) ;
A2: for q being real number st ( for p being real number st p in Z holds
p >= q ) holds
0 >= q
proof
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p >= q ) implies 0 >= q )
assume A3: for p being real number st p in Z holds
p >= q ; ::_thesis: 0 >= q
|[1,0]| in LSeg (|[1,0]|,|[1,1]|) by RLTOPSP1:68;
then |[1,0]| in (LSeg (|[0,0]|,|[1,0]|)) \/ (LSeg (|[1,0]|,|[1,1]|)) by XBOOLE_0:def_3;
then A4: |[1,0]| in R^2-unit_square by XBOOLE_0:def_3;
then (proj2 | R^2-unit_square) . |[1,0]| = |[1,0]| `2 by PSCOMP_1:23
.= 0 by EUCLID:52 ;
hence 0 >= q by A1, A3, A4, FUNCT_2:35; ::_thesis: verum
end;
for p being real number st p in Z holds
p >= 0
proof
let p be real number ; ::_thesis: ( p in Z implies p >= 0 )
assume p in Z ; ::_thesis: p >= 0
then consider p0 being set such that
A5: p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
p0 in the carrier of ((TOP-REAL 2) | R^2-unit_square) and
A6: p = (proj2 | R^2-unit_square) . p0 by FUNCT_2:64;
reconsider p0 = p0 as Point of (TOP-REAL 2) by A1, A5;
ex q being Point of (TOP-REAL 2) st
( p0 = q & ( ( q `1 = 0 & q `2 <= 1 & q `2 >= 0 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 1 ) or ( q `1 <= 1 & q `1 >= 0 & q `2 = 0 ) or ( q `1 = 1 & q `2 <= 1 & q `2 >= 0 ) ) ) by A1, A5, TOPREAL1:14;
hence p >= 0 by A1, A5, A6, PSCOMP_1:23; ::_thesis: verum
end;
hence S-bound R^2-unit_square = 0 by A2, SEQ_4:44; ::_thesis: verum
end;
theorem Th35: :: SPRECT_2:35
N-most R^2-unit_square = LSeg (|[0,1]|,|[1,1]|)
proof
set X = R^2-unit_square ;
( (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) c= R^2-unit_square & LSeg (|[0,1]|,|[1,1]|) c= (LSeg (|[0,0]|,|[0,1]|)) \/ (LSeg (|[0,1]|,|[1,1]|)) ) by XBOOLE_1:7;
hence N-most R^2-unit_square = LSeg (|[0,1]|,|[1,1]|) by Th31, Th32, Th33, XBOOLE_1:1, XBOOLE_1:28; ::_thesis: verum
end;
theorem :: SPRECT_2:36
N-min R^2-unit_square = |[0,1]|
proof
lower_bound (proj1 | (LSeg (|[0,1]|,|[1,1]|))) = 0
proof
set X = LSeg (|[0,1]|,|[1,1]|);
reconsider Z = (proj1 | (LSeg (|[0,1]|,|[1,1]|))) .: the carrier of ((TOP-REAL 2) | (LSeg (|[0,1]|,|[1,1]|))) as Subset of REAL ;
A1: LSeg (|[0,1]|,|[1,1]|) = [#] ((TOP-REAL 2) | (LSeg (|[0,1]|,|[1,1]|))) by PRE_TOPC:def_5
.= the carrier of ((TOP-REAL 2) | (LSeg (|[0,1]|,|[1,1]|))) ;
A2: for p being real number st p in Z holds
p >= 0
proof
let p be real number ; ::_thesis: ( p in Z implies p >= 0 )
assume p in Z ; ::_thesis: p >= 0
then consider p0 being set such that
A3: p0 in the carrier of ((TOP-REAL 2) | (LSeg (|[0,1]|,|[1,1]|))) and
p0 in the carrier of ((TOP-REAL 2) | (LSeg (|[0,1]|,|[1,1]|))) and
A4: p = (proj1 | (LSeg (|[0,1]|,|[1,1]|))) . p0 by FUNCT_2:64;
reconsider p0 = p0 as Point of (TOP-REAL 2) by A1, A3;
( |[0,1]| `1 = 0 & |[1,1]| `1 = 1 ) by EUCLID:52;
then p0 `1 >= 0 by A1, A3, TOPREAL1:3;
hence p >= 0 by A1, A3, A4, PSCOMP_1:22; ::_thesis: verum
end;
for q being real number st ( for p being real number st p in Z holds
p >= q ) holds
0 >= q
proof
A5: (proj1 | (LSeg (|[0,1]|,|[1,1]|))) . |[0,1]| = |[0,1]| `1 by PSCOMP_1:22, RLTOPSP1:68
.= 0 by EUCLID:52 ;
A6: |[0,1]| in LSeg (|[0,1]|,|[1,1]|) by RLTOPSP1:68;
let q be real number ; ::_thesis: ( ( for p being real number st p in Z holds
p >= q ) implies 0 >= q )
assume for p being real number st p in Z holds
p >= q ; ::_thesis: 0 >= q
hence 0 >= q by A1, A6, A5, FUNCT_2:35; ::_thesis: verum
end;
hence lower_bound (proj1 | (LSeg (|[0,1]|,|[1,1]|))) = 0 by A2, SEQ_4:44; ::_thesis: verum
end;
hence N-min R^2-unit_square = |[0,1]| by Th31, Th35; ::_thesis: verum
end;
registration
let X be non empty compact non horizontal non vertical Subset of (TOP-REAL 2);
cluster SpStSeq X -> clockwise_oriented ;
coherence
SpStSeq X is clockwise_oriented
proof
set f = SpStSeq X;
(SpStSeq X) /. 2 = N-max (L~ (SpStSeq X)) by SPRECT_1:84;
then ( (SpStSeq X) /. 1 = N-min (L~ (SpStSeq X)) & (SpStSeq X) /. 2 in N-most (L~ (SpStSeq X)) ) by PSCOMP_1:42, SPRECT_1:83;
hence SpStSeq X is clockwise_oriented by Th30; ::_thesis: verum
end;
end;
registration
cluster non empty non trivial V13() V16( NAT ) V17( the carrier of (TOP-REAL 2)) Function-like non constant V26() FinSequence-like FinSubsequence-like circular special unfolded s.c.c. standard clockwise_oriented for FinSequence of the carrier of (TOP-REAL 2);
existence
ex b1 being non constant standard special_circular_sequence st b1 is clockwise_oriented
proof
set X = the non empty compact non horizontal non vertical Subset of (TOP-REAL 2);
take SpStSeq the non empty compact non horizontal non vertical Subset of (TOP-REAL 2) ; ::_thesis: SpStSeq the non empty compact non horizontal non vertical Subset of (TOP-REAL 2) is clockwise_oriented
thus SpStSeq the non empty compact non horizontal non vertical Subset of (TOP-REAL 2) is clockwise_oriented ; ::_thesis: verum
end;
end;
theorem Th37: :: SPRECT_2:37
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i > j & ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) holds
mid (f,i,j) is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i > j & ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) holds
mid (f,i,j) is S-Sequence_in_R2
let i, j be Element of NAT ; ::_thesis: ( i > j & ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) implies mid (f,i,j) is S-Sequence_in_R2 )
assume that
A1: i > j and
A2: ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) ; ::_thesis: mid (f,i,j) is S-Sequence_in_R2
A3: Rev (mid (f,j,i)) = mid (f,i,j) by JORDAN4:18;
percases ( ( 1 < j & i <= len f ) or ( 1 <= j & i < len f ) ) by A2;
suppose ( 1 < j & i <= len f ) ; ::_thesis: mid (f,i,j) is S-Sequence_in_R2
then mid (f,j,i) is S-Sequence_in_R2 by A1, JORDAN4:40;
hence mid (f,i,j) is S-Sequence_in_R2 by A3; ::_thesis: verum
end;
supposeA4: ( 1 <= j & i < len f ) ; ::_thesis: mid (f,i,j) is S-Sequence_in_R2
then i + 1 <= len f by NAT_1:13;
then mid (f,j,i) is S-Sequence_in_R2 by A1, A4, JORDAN4:39;
hence mid (f,i,j) is S-Sequence_in_R2 by A3; ::_thesis: verum
end;
end;
end;
theorem Th38: :: SPRECT_2:38
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i < j & ( ( 1 < i & j <= len f ) or ( 1 <= i & j < len f ) ) holds
mid (f,i,j) is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i < j & ( ( 1 < i & j <= len f ) or ( 1 <= i & j < len f ) ) holds
mid (f,i,j) is S-Sequence_in_R2
let i, j be Element of NAT ; ::_thesis: ( i < j & ( ( 1 < i & j <= len f ) or ( 1 <= i & j < len f ) ) implies mid (f,i,j) is S-Sequence_in_R2 )
assume ( i < j & ( ( 1 < i & j <= len f ) or ( 1 <= i & j < len f ) ) ) ; ::_thesis: mid (f,i,j) is S-Sequence_in_R2
then mid (f,j,i) is S-Sequence_in_R2 by Th37;
then ( Rev (Rev (mid (f,i,j))) = mid (f,i,j) & Rev (mid (f,i,j)) is S-Sequence_in_R2 ) by JORDAN4:18;
hence mid (f,i,j) is S-Sequence_in_R2 ; ::_thesis: verum
end;
theorem Th39: :: SPRECT_2:39
for f being non trivial FinSequence of (TOP-REAL 2) holds N-min (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: N-min (L~ f) in rng f
set p = N-min (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: N-min (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:11;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `2 <= N-bound (L~ f) by PSCOMP_1:24;
A7: (N-min (L~ f)) `2 = N-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `2 <= N-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_N-min_(L~_f)_in_rng_f
percases ( N-min (L~ f) = f /. i or N-min (L~ f) = f /. (i + 1) or ( (N-min (L~ f)) `2 = (f /. i) `2 & (N-min (L~ f)) `2 = (f /. (i + 1)) `2 ) ) by A4, A9, A6, A7, Th18;
suppose N-min (L~ f) = f /. i ; ::_thesis: N-min (L~ f) in rng f
hence N-min (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose N-min (L~ f) = f /. (i + 1) ; ::_thesis: N-min (L~ f) in rng f
hence N-min (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (N-min (L~ f)) `2 = (f /. i) `2 & (N-min (L~ f)) `2 = (f /. (i + 1)) `2 ) ; ::_thesis: N-min (L~ f) in rng f
then f /. (i + 1) in N-most (L~ f) by A1, A5, A7, Th10, GOBOARD1:1;
then A11: (f /. (i + 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39;
( (f /. i) `1 <= (f /. (i + 1)) `1 or (f /. (i + 1)) `1 <= (f /. i) `1 ) ;
then A12: ( (f /. i) `1 <= (N-min (L~ f)) `1 or (f /. (i + 1)) `1 <= (N-min (L~ f)) `1 ) by A4, TOPREAL1:3;
f /. i in N-most (L~ f) by A1, A8, A7, A10, Th10, GOBOARD1:1;
then (f /. i) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39;
then ( (N-min (L~ f)) `1 = (f /. i) `1 or (N-min (L~ f)) `1 = (f /. (i + 1)) `1 ) by A11, A12, XXREAL_0:1;
then ( N-min (L~ f) = f /. i or N-min (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence N-min (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence N-min (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th40: :: SPRECT_2:40
for f being non trivial FinSequence of (TOP-REAL 2) holds N-max (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: N-max (L~ f) in rng f
set p = N-max (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: N-max (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:11;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `2 <= N-bound (L~ f) by PSCOMP_1:24;
A7: (N-max (L~ f)) `2 = N-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `2 <= N-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_N-max_(L~_f)_in_rng_f
percases ( N-max (L~ f) = f /. i or N-max (L~ f) = f /. (i + 1) or ( (N-max (L~ f)) `2 = (f /. i) `2 & (N-max (L~ f)) `2 = (f /. (i + 1)) `2 ) ) by A4, A9, A6, A7, Th18;
suppose N-max (L~ f) = f /. i ; ::_thesis: N-max (L~ f) in rng f
hence N-max (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose N-max (L~ f) = f /. (i + 1) ; ::_thesis: N-max (L~ f) in rng f
hence N-max (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (N-max (L~ f)) `2 = (f /. i) `2 & (N-max (L~ f)) `2 = (f /. (i + 1)) `2 ) ; ::_thesis: N-max (L~ f) in rng f
then f /. (i + 1) in N-most (L~ f) by A1, A5, A7, Th10, GOBOARD1:1;
then A11: (f /. (i + 1)) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:39;
( (f /. i) `1 >= (f /. (i + 1)) `1 or (f /. (i + 1)) `1 >= (f /. i) `1 ) ;
then A12: ( (f /. i) `1 >= (N-max (L~ f)) `1 or (f /. (i + 1)) `1 >= (N-max (L~ f)) `1 ) by A4, TOPREAL1:3;
f /. i in N-most (L~ f) by A1, A8, A7, A10, Th10, GOBOARD1:1;
then (f /. i) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:39;
then ( (N-max (L~ f)) `1 = (f /. i) `1 or (N-max (L~ f)) `1 = (f /. (i + 1)) `1 ) by A11, A12, XXREAL_0:1;
then ( N-max (L~ f) = f /. i or N-max (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence N-max (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence N-max (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th41: :: SPRECT_2:41
for f being non trivial FinSequence of (TOP-REAL 2) holds S-min (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: S-min (L~ f) in rng f
set p = S-min (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: S-min (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:12;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `2 >= S-bound (L~ f) by PSCOMP_1:24;
A7: (S-min (L~ f)) `2 = S-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `2 >= S-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_S-min_(L~_f)_in_rng_f
percases ( S-min (L~ f) = f /. i or S-min (L~ f) = f /. (i + 1) or ( (S-min (L~ f)) `2 = (f /. i) `2 & (S-min (L~ f)) `2 = (f /. (i + 1)) `2 ) ) by A4, A9, A6, A7, Th20;
suppose S-min (L~ f) = f /. i ; ::_thesis: S-min (L~ f) in rng f
hence S-min (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose S-min (L~ f) = f /. (i + 1) ; ::_thesis: S-min (L~ f) in rng f
hence S-min (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (S-min (L~ f)) `2 = (f /. i) `2 & (S-min (L~ f)) `2 = (f /. (i + 1)) `2 ) ; ::_thesis: S-min (L~ f) in rng f
then f /. (i + 1) in S-most (L~ f) by A1, A5, A7, Th11, GOBOARD1:1;
then A11: (f /. (i + 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55;
( (f /. i) `1 <= (f /. (i + 1)) `1 or (f /. (i + 1)) `1 <= (f /. i) `1 ) ;
then A12: ( (f /. i) `1 <= (S-min (L~ f)) `1 or (f /. (i + 1)) `1 <= (S-min (L~ f)) `1 ) by A4, TOPREAL1:3;
f /. i in S-most (L~ f) by A1, A8, A7, A10, Th11, GOBOARD1:1;
then (f /. i) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55;
then ( (S-min (L~ f)) `1 = (f /. i) `1 or (S-min (L~ f)) `1 = (f /. (i + 1)) `1 ) by A11, A12, XXREAL_0:1;
then ( S-min (L~ f) = f /. i or S-min (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence S-min (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence S-min (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th42: :: SPRECT_2:42
for f being non trivial FinSequence of (TOP-REAL 2) holds S-max (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: S-max (L~ f) in rng f
set p = S-max (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: S-max (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:12;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `2 >= S-bound (L~ f) by PSCOMP_1:24;
A7: (S-max (L~ f)) `2 = S-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `2 >= S-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_S-max_(L~_f)_in_rng_f
percases ( S-max (L~ f) = f /. i or S-max (L~ f) = f /. (i + 1) or ( (S-max (L~ f)) `2 = (f /. i) `2 & (S-max (L~ f)) `2 = (f /. (i + 1)) `2 ) ) by A4, A9, A6, A7, Th20;
suppose S-max (L~ f) = f /. i ; ::_thesis: S-max (L~ f) in rng f
hence S-max (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose S-max (L~ f) = f /. (i + 1) ; ::_thesis: S-max (L~ f) in rng f
hence S-max (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (S-max (L~ f)) `2 = (f /. i) `2 & (S-max (L~ f)) `2 = (f /. (i + 1)) `2 ) ; ::_thesis: S-max (L~ f) in rng f
then f /. (i + 1) in S-most (L~ f) by A1, A5, A7, Th11, GOBOARD1:1;
then A11: (f /. (i + 1)) `1 <= (S-max (L~ f)) `1 by PSCOMP_1:55;
( (f /. i) `1 >= (f /. (i + 1)) `1 or (f /. (i + 1)) `1 >= (f /. i) `1 ) ;
then A12: ( (f /. i) `1 >= (S-max (L~ f)) `1 or (f /. (i + 1)) `1 >= (S-max (L~ f)) `1 ) by A4, TOPREAL1:3;
f /. i in S-most (L~ f) by A1, A8, A7, A10, Th11, GOBOARD1:1;
then (f /. i) `1 <= (S-max (L~ f)) `1 by PSCOMP_1:55;
then ( (S-max (L~ f)) `1 = (f /. i) `1 or (S-max (L~ f)) `1 = (f /. (i + 1)) `1 ) by A11, A12, XXREAL_0:1;
then ( S-max (L~ f) = f /. i or S-max (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence S-max (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence S-max (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th43: :: SPRECT_2:43
for f being non trivial FinSequence of (TOP-REAL 2) holds W-min (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: W-min (L~ f) in rng f
set p = W-min (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: W-min (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:13;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `1 >= W-bound (L~ f) by PSCOMP_1:24;
A7: (W-min (L~ f)) `1 = W-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `1 >= W-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_W-min_(L~_f)_in_rng_f
percases ( W-min (L~ f) = f /. i or W-min (L~ f) = f /. (i + 1) or ( (W-min (L~ f)) `1 = (f /. i) `1 & (W-min (L~ f)) `1 = (f /. (i + 1)) `1 ) ) by A4, A9, A6, A7, Th19;
suppose W-min (L~ f) = f /. i ; ::_thesis: W-min (L~ f) in rng f
hence W-min (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose W-min (L~ f) = f /. (i + 1) ; ::_thesis: W-min (L~ f) in rng f
hence W-min (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (W-min (L~ f)) `1 = (f /. i) `1 & (W-min (L~ f)) `1 = (f /. (i + 1)) `1 ) ; ::_thesis: W-min (L~ f) in rng f
then f /. (i + 1) in W-most (L~ f) by A1, A5, A7, Th12, GOBOARD1:1;
then A11: (f /. (i + 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31;
( (f /. i) `2 <= (f /. (i + 1)) `2 or (f /. (i + 1)) `2 <= (f /. i) `2 ) ;
then A12: ( (f /. i) `2 <= (W-min (L~ f)) `2 or (f /. (i + 1)) `2 <= (W-min (L~ f)) `2 ) by A4, TOPREAL1:4;
f /. i in W-most (L~ f) by A1, A8, A7, A10, Th12, GOBOARD1:1;
then (f /. i) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31;
then ( (W-min (L~ f)) `2 = (f /. i) `2 or (W-min (L~ f)) `2 = (f /. (i + 1)) `2 ) by A11, A12, XXREAL_0:1;
then ( W-min (L~ f) = f /. i or W-min (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence W-min (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence W-min (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th44: :: SPRECT_2:44
for f being non trivial FinSequence of (TOP-REAL 2) holds W-max (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: W-max (L~ f) in rng f
set p = W-max (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: W-max (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:13;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `1 >= W-bound (L~ f) by PSCOMP_1:24;
A7: (W-max (L~ f)) `1 = W-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `1 >= W-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_W-max_(L~_f)_in_rng_f
percases ( W-max (L~ f) = f /. i or W-max (L~ f) = f /. (i + 1) or ( (W-max (L~ f)) `1 = (f /. i) `1 & (W-max (L~ f)) `1 = (f /. (i + 1)) `1 ) ) by A4, A9, A6, A7, Th19;
suppose W-max (L~ f) = f /. i ; ::_thesis: W-max (L~ f) in rng f
hence W-max (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose W-max (L~ f) = f /. (i + 1) ; ::_thesis: W-max (L~ f) in rng f
hence W-max (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (W-max (L~ f)) `1 = (f /. i) `1 & (W-max (L~ f)) `1 = (f /. (i + 1)) `1 ) ; ::_thesis: W-max (L~ f) in rng f
then f /. (i + 1) in W-most (L~ f) by A1, A5, A7, Th12, GOBOARD1:1;
then A11: (f /. (i + 1)) `2 <= (W-max (L~ f)) `2 by PSCOMP_1:31;
( (f /. i) `2 >= (f /. (i + 1)) `2 or (f /. (i + 1)) `2 >= (f /. i) `2 ) ;
then A12: ( (f /. i) `2 >= (W-max (L~ f)) `2 or (f /. (i + 1)) `2 >= (W-max (L~ f)) `2 ) by A4, TOPREAL1:4;
f /. i in W-most (L~ f) by A1, A8, A7, A10, Th12, GOBOARD1:1;
then (f /. i) `2 <= (W-max (L~ f)) `2 by PSCOMP_1:31;
then ( (W-max (L~ f)) `2 = (f /. i) `2 or (W-max (L~ f)) `2 = (f /. (i + 1)) `2 ) by A11, A12, XXREAL_0:1;
then ( W-max (L~ f) = f /. i or W-max (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence W-max (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence W-max (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th45: :: SPRECT_2:45
for f being non trivial FinSequence of (TOP-REAL 2) holds E-min (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: E-min (L~ f) in rng f
set p = E-min (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: E-min (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:14;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `1 <= E-bound (L~ f) by PSCOMP_1:24;
A7: (E-min (L~ f)) `1 = E-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `1 <= E-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_E-min_(L~_f)_in_rng_f
percases ( E-min (L~ f) = f /. i or E-min (L~ f) = f /. (i + 1) or ( (E-min (L~ f)) `1 = (f /. i) `1 & (E-min (L~ f)) `1 = (f /. (i + 1)) `1 ) ) by A4, A9, A6, A7, Th17;
suppose E-min (L~ f) = f /. i ; ::_thesis: E-min (L~ f) in rng f
hence E-min (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose E-min (L~ f) = f /. (i + 1) ; ::_thesis: E-min (L~ f) in rng f
hence E-min (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (E-min (L~ f)) `1 = (f /. i) `1 & (E-min (L~ f)) `1 = (f /. (i + 1)) `1 ) ; ::_thesis: E-min (L~ f) in rng f
then f /. (i + 1) in E-most (L~ f) by A1, A5, A7, Th13, GOBOARD1:1;
then A11: (f /. (i + 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47;
( (f /. i) `2 <= (f /. (i + 1)) `2 or (f /. (i + 1)) `2 <= (f /. i) `2 ) ;
then A12: ( (f /. i) `2 <= (E-min (L~ f)) `2 or (f /. (i + 1)) `2 <= (E-min (L~ f)) `2 ) by A4, TOPREAL1:4;
f /. i in E-most (L~ f) by A1, A8, A7, A10, Th13, GOBOARD1:1;
then (f /. i) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47;
then ( (E-min (L~ f)) `2 = (f /. i) `2 or (E-min (L~ f)) `2 = (f /. (i + 1)) `2 ) by A11, A12, XXREAL_0:1;
then ( E-min (L~ f) = f /. i or E-min (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence E-min (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence E-min (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th46: :: SPRECT_2:46
for f being non trivial FinSequence of (TOP-REAL 2) holds E-max (L~ f) in rng f
proof
let f be non trivial FinSequence of (TOP-REAL 2); ::_thesis: E-max (L~ f) in rng f
set p = E-max (L~ f);
A1: len f >= 2 by NAT_D:60;
consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
A4: E-max (L~ f) in LSeg ((f /. i),(f /. (i + 1))) by SPPOL_2:14, SPRECT_1:14;
i + 1 >= 1 by NAT_1:11;
then A5: i + 1 in dom f by A3, FINSEQ_3:25;
then f /. (i + 1) in L~ f by A1, GOBOARD1:1;
then A6: (f /. (i + 1)) `1 <= E-bound (L~ f) by PSCOMP_1:24;
A7: (E-max (L~ f)) `1 = E-bound (L~ f) by EUCLID:52;
i <= i + 1 by NAT_1:11;
then i <= len f by A3, XXREAL_0:2;
then A8: i in dom f by A2, FINSEQ_3:25;
then f /. i in L~ f by A1, GOBOARD1:1;
then A9: (f /. i) `1 <= E-bound (L~ f) by PSCOMP_1:24;
now__::_thesis:_E-max_(L~_f)_in_rng_f
percases ( E-max (L~ f) = f /. i or E-max (L~ f) = f /. (i + 1) or ( (E-max (L~ f)) `1 = (f /. i) `1 & (E-max (L~ f)) `1 = (f /. (i + 1)) `1 ) ) by A4, A9, A6, A7, Th17;
suppose E-max (L~ f) = f /. i ; ::_thesis: E-max (L~ f) in rng f
hence E-max (L~ f) in rng f by A8, PARTFUN2:2; ::_thesis: verum
end;
suppose E-max (L~ f) = f /. (i + 1) ; ::_thesis: E-max (L~ f) in rng f
hence E-max (L~ f) in rng f by A5, PARTFUN2:2; ::_thesis: verum
end;
supposeA10: ( (E-max (L~ f)) `1 = (f /. i) `1 & (E-max (L~ f)) `1 = (f /. (i + 1)) `1 ) ; ::_thesis: E-max (L~ f) in rng f
then f /. (i + 1) in E-most (L~ f) by A1, A5, A7, Th13, GOBOARD1:1;
then A11: (f /. (i + 1)) `2 <= (E-max (L~ f)) `2 by PSCOMP_1:47;
( (f /. i) `2 >= (f /. (i + 1)) `2 or (f /. (i + 1)) `2 >= (f /. i) `2 ) ;
then A12: ( (f /. i) `2 >= (E-max (L~ f)) `2 or (f /. (i + 1)) `2 >= (E-max (L~ f)) `2 ) by A4, TOPREAL1:4;
f /. i in E-most (L~ f) by A1, A8, A7, A10, Th13, GOBOARD1:1;
then (f /. i) `2 <= (E-max (L~ f)) `2 by PSCOMP_1:47;
then ( (E-max (L~ f)) `2 = (f /. i) `2 or (E-max (L~ f)) `2 = (f /. (i + 1)) `2 ) by A11, A12, XXREAL_0:1;
then ( E-max (L~ f) = f /. i or E-max (L~ f) = f /. (i + 1) ) by A10, TOPREAL3:6;
hence E-max (L~ f) in rng f by A8, A5, PARTFUN2:2; ::_thesis: verum
end;
end;
end;
hence E-max (L~ f) in rng f ; ::_thesis: verum
end;
theorem Th47: :: SPRECT_2:47
for i, j, m, n being Element of NAT
for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,i,j)) misses L~ (mid (f,m,n))
proof
let i, j, m, n be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,i,j)) misses L~ (mid (f,m,n))
let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,i,j)) misses L~ (mid (f,m,n)) )
assume that
A1: ( 1 <= i & i <= j ) and
A2: j < m and
A3: m <= n and
A4: n <= len f and
A5: ( 1 < i or n < len f ) ; ::_thesis: L~ (mid (f,i,j)) misses L~ (mid (f,m,n))
set A = { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } ;
set B = { (LSeg (f,l)) where l is Element of NAT : ( m <= l & l < n ) } ;
1 <= j by A1, XXREAL_0:2;
then 1 < m by A2, XXREAL_0:2;
then A6: L~ (mid (f,m,n)) = union { (LSeg (f,l)) where l is Element of NAT : ( m <= l & l < n ) } by A3, A4, Th14;
A7: for x, y being set st x in { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } & y in { (LSeg (f,l)) where l is Element of NAT : ( m <= l & l < n ) } holds
x misses y
proof
let x, y be set ; ::_thesis: ( x in { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } & y in { (LSeg (f,l)) where l is Element of NAT : ( m <= l & l < n ) } implies x misses y )
assume x in { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } ; ::_thesis: ( not y in { (LSeg (f,l)) where l is Element of NAT : ( m <= l & l < n ) } or x misses y )
then consider k being Element of NAT such that
A8: x = LSeg (f,k) and
A9: i <= k and
A10: k < j ;
assume y in { (LSeg (f,l)) where l is Element of NAT : ( m <= l & l < n ) } ; ::_thesis: x misses y
then consider l being Element of NAT such that
A11: y = LSeg (f,l) and
A12: m <= l and
A13: l < n ;
A14: l < len f by A4, A13, XXREAL_0:2;
l + 1 <= n by A13, NAT_1:13;
then A15: ( k > 1 or l + 1 < len f ) by A5, A9, XXREAL_0:2;
k + 1 <= j by A10, NAT_1:13;
then k + 1 < m by A2, XXREAL_0:2;
then k + 1 < l by A12, XXREAL_0:2;
hence x misses y by A8, A11, A14, A15, GOBOARD5:def_4; ::_thesis: verum
end;
m <= len f by A3, A4, XXREAL_0:2;
then j < len f by A2, XXREAL_0:2;
then L~ (mid (f,i,j)) = union { (LSeg (f,k)) where k is Element of NAT : ( i <= k & k < j ) } by A1, Th14;
hence L~ (mid (f,i,j)) misses L~ (mid (f,m,n)) by A6, A7, ZFMISC_1:126; ::_thesis: verum
end;
theorem Th48: :: SPRECT_2:48
for i, j, m, n being Element of NAT
for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,i,j)) misses L~ (mid (f,n,m))
proof
let i, j, m, n be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,i,j)) misses L~ (mid (f,n,m))
let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,i,j)) misses L~ (mid (f,n,m)) )
mid (f,n,m) = Rev (mid (f,m,n)) by JORDAN4:18;
then L~ (mid (f,n,m)) = L~ (mid (f,m,n)) by SPPOL_2:22;
hence ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,i,j)) misses L~ (mid (f,n,m)) ) by Th47; ::_thesis: verum
end;
theorem Th49: :: SPRECT_2:49
for i, j, m, n being Element of NAT
for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,j,i)) misses L~ (mid (f,n,m))
proof
let i, j, m, n be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,j,i)) misses L~ (mid (f,n,m))
let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,j,i)) misses L~ (mid (f,n,m)) )
mid (f,i,j) = Rev (mid (f,j,i)) by JORDAN4:18;
then L~ (mid (f,i,j)) = L~ (mid (f,j,i)) by SPPOL_2:22;
hence ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,j,i)) misses L~ (mid (f,n,m)) ) by Th48; ::_thesis: verum
end;
theorem Th50: :: SPRECT_2:50
for i, j, m, n being Element of NAT
for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,j,i)) misses L~ (mid (f,m,n))
proof
let i, j, m, n be Element of NAT ; ::_thesis: for f being non constant standard special_circular_sequence st 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) holds
L~ (mid (f,j,i)) misses L~ (mid (f,m,n))
let f be non constant standard special_circular_sequence; ::_thesis: ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,j,i)) misses L~ (mid (f,m,n)) )
mid (f,i,j) = Rev (mid (f,j,i)) by JORDAN4:18;
then L~ (mid (f,i,j)) = L~ (mid (f,j,i)) by SPPOL_2:22;
hence ( 1 <= i & i <= j & j < m & m <= n & n <= len f & ( 1 < i or n < len f ) implies L~ (mid (f,j,i)) misses L~ (mid (f,m,n)) ) by Th47; ::_thesis: verum
end;
theorem Th51: :: SPRECT_2:51
for f being non constant standard special_circular_sequence holds (N-min (L~ f)) `1 < (N-max (L~ f)) `1
proof
let f be non constant standard special_circular_sequence; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
set p = N-min (L~ f);
set i = (N-min (L~ f)) .. f;
A1: len f > 3 + 1 by GOBOARD7:34;
A2: len f >= 1 + 1 by GOBOARD7:34, XXREAL_0:2;
A3: N-min (L~ f) in rng f by Th39;
then A4: (N-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A5: ( 1 <= (N-min (L~ f)) .. f & (N-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A6: N-min (L~ f) = f . ((N-min (L~ f)) .. f) by A3, FINSEQ_4:19
.= f /. ((N-min (L~ f)) .. f) by A4, PARTFUN1:def_6 ;
A7: (N-min (L~ f)) `2 = N-bound (L~ f) by EUCLID:52;
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 A5, XXREAL_0:1;
supposeA8: ( (N-min (L~ f)) .. f = 1 or (N-min (L~ f)) .. f = len f ) ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
then N-min (L~ f) = f /. 1 by A6, FINSEQ_6:def_1;
then A9: N-min (L~ f) in LSeg (f,1) by A2, TOPREAL1:21;
A10: 1 + 1 in dom f by A2, FINSEQ_3:25;
then A11: f /. (1 + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A12: f /. (1 + 1) in LSeg (f,1) by A2, TOPREAL1:21;
A13: ((len f) -' 1) + 1 = len f by A1, XREAL_1:235, XXREAL_0:2;
then (len f) -' 1 > 3 by A1, XREAL_1:6;
then A14: (len f) -' 1 > 1 by XXREAL_0:2;
then A15: f /. ((len f) -' 1) in LSeg (f,((len f) -' 1)) by A13, TOPREAL1:21;
(len f) -' 1 <= len f by A13, NAT_1:11;
then A16: (len f) -' 1 in dom f by A14, FINSEQ_3:25;
then A17: f /. ((len f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A18: f /. 1 = f /. (len f) by FINSEQ_6:def_1;
then A19: N-min (L~ f) in LSeg (f,((len f) -' 1)) by A6, A8, A13, A14, TOPREAL1:21;
A20: 1 in dom f by FINSEQ_5:6;
then A21: N-min (L~ f) <> f /. (1 + 1) by A6, A8, A18, A10, GOBOARD7:29;
A22: len f in dom f by FINSEQ_5:6;
then A23: N-min (L~ f) <> f /. ((len f) -' 1) by A6, A8, A18, A13, A16, GOBOARD7:29;
A24: ( not LSeg (f,((len f) -' 1)) is vertical or not LSeg (f,1) is vertical )
proof
assume ( LSeg (f,((len f) -' 1)) is vertical & LSeg (f,1) is vertical ) ; ::_thesis: contradiction
then A25: ( (N-min (L~ f)) `1 = (f /. (1 + 1)) `1 & (N-min (L~ f)) `1 = (f /. ((len f) -' 1)) `1 ) by A19, A9, A15, A12, SPPOL_1:def_3;
A26: ( (f /. (1 + 1)) `2 <= (f /. ((len f) -' 1)) `2 or (f /. (1 + 1)) `2 >= (f /. ((len f) -' 1)) `2 ) ;
A27: ( (N-min (L~ f)) `2 >= (f /. (1 + 1)) `2 & (N-min (L~ f)) `2 >= (f /. ((len f) -' 1)) `2 ) by A7, A17, A11, PSCOMP_1:24;
( LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) & LSeg (f,((len f) -' 1)) = LSeg ((f /. 1),(f /. ((len f) -' 1))) ) by A2, A18, A13, A14, TOPREAL1:def_3;
then ( f /. ((len f) -' 1) in LSeg (f,1) or f /. (1 + 1) in LSeg (f,((len f) -' 1)) ) by A6, A8, A18, A25, A27, A26, GOBOARD7:7;
then ( f /. ((len f) -' 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) or f /. (1 + 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) ) by A15, A12, XBOOLE_0:def_4;
then A28: (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) <> {(f /. 1)} by A6, A8, A18, A23, A21, TARSKI:def_1;
f . 1 = f /. 1 by A20, PARTFUN1:def_6;
hence contradiction by A28, JORDAN4:42; ::_thesis: verum
end;
now__::_thesis:_(N-min_(L~_f))_`1_<_(N-max_(L~_f))_`1
percases ( LSeg (f,((len f) -' 1)) is horizontal or LSeg (f,1) is horizontal ) by A24, SPPOL_1:19;
suppose LSeg (f,((len f) -' 1)) is horizontal ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
then A29: (N-min (L~ f)) `2 = (f /. ((len f) -' 1)) `2 by A19, A15, SPPOL_1:def_2;
then A30: f /. ((len f) -' 1) in N-most (L~ f) by A2, A7, A16, Th10, GOBOARD1:1;
then A31: (f /. ((len f) -' 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39;
(f /. ((len f) -' 1)) `1 <> (N-min (L~ f)) `1 by A6, A8, A22, A18, A13, A16, A29, GOBOARD7:29, TOPREAL3:6;
then A32: (f /. ((len f) -' 1)) `1 > (N-min (L~ f)) `1 by A31, XXREAL_0:1;
(f /. ((len f) -' 1)) `1 <= (N-max (L~ f)) `1 by A30, PSCOMP_1:39;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A32, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,1) is horizontal ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
then A33: (N-min (L~ f)) `2 = (f /. (1 + 1)) `2 by A9, A12, SPPOL_1:def_2;
then A34: f /. (1 + 1) in N-most (L~ f) by A2, A7, A10, Th10, GOBOARD1:1;
then A35: (f /. (1 + 1)) `1 >= (N-min (L~ f)) `1 by PSCOMP_1:39;
(f /. (1 + 1)) `1 <> (N-min (L~ f)) `1 by A6, A8, A20, A18, A10, A33, GOBOARD7:29, TOPREAL3:6;
then A36: (f /. (1 + 1)) `1 > (N-min (L~ f)) `1 by A35, XXREAL_0:1;
(f /. (1 + 1)) `1 <= (N-max (L~ f)) `1 by A34, PSCOMP_1:39;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A36, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 ; ::_thesis: verum
end;
supposethat A37: 1 < (N-min (L~ f)) .. f and
A38: (N-min (L~ f)) .. f < len f ; ::_thesis: (N-min (L~ f)) `1 < (N-max (L~ f)) `1
A39: (((N-min (L~ f)) .. f) -' 1) + 1 = (N-min (L~ f)) .. f by A37, XREAL_1:235;
then A40: ((N-min (L~ f)) .. f) -' 1 >= 1 by A37, NAT_1:13;
then A41: f /. (((N-min (L~ f)) .. f) -' 1) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) by A38, A39, TOPREAL1:21;
((N-min (L~ f)) .. f) -' 1 <= (N-min (L~ f)) .. f by A39, NAT_1:11;
then ((N-min (L~ f)) .. f) -' 1 <= len f by A38, XXREAL_0:2;
then A42: ((N-min (L~ f)) .. f) -' 1 in dom f by A40, FINSEQ_3:25;
then A43: f /. (((N-min (L~ f)) .. f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A44: ((N-min (L~ f)) .. f) + 1 <= len f by A38, NAT_1:13;
then A45: f /. (((N-min (L~ f)) .. f) + 1) in LSeg (f,((N-min (L~ f)) .. f)) by A37, TOPREAL1:21;
((N-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A46: ((N-min (L~ f)) .. f) + 1 in dom f by A44, FINSEQ_3:25;
then A47: f /. (((N-min (L~ f)) .. f) + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A48: N-min (L~ f) <> f /. (((N-min (L~ f)) .. f) + 1) by A3, A6, A46, FINSEQ_4:20, GOBOARD7:29;
A49: N-min (L~ f) in LSeg (f,((N-min (L~ f)) .. f)) by A6, A37, A44, TOPREAL1:21;
A50: N-min (L~ f) in LSeg (f,(((N-min (L~ f)) .. f) -' 1)) by A6, A38, A39, A40, TOPREAL1:21;
A51: N-min (L~ f) <> f /. (((N-min (L~ f)) .. f) -' 1) by A4, A6, A39, A42, GOBOARD7:29;
A52: ( 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 A53: ( (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 A50, A49, A41, A45, SPPOL_1:def_3;
A54: ( (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 ) ;
A55: ( (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 A7, A43, A47, 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 A37, A38, A39, A40, A44, 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 A6, A53, A55, A54, 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 A41, A45, 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 A6, A51, A48, TARSKI:def_1;
hence contradiction by A39, A40, A44, 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 A52, 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 A56: (N-min (L~ f)) `2 = (f /. (((N-min (L~ f)) .. f) -' 1)) `2 by A50, A41, SPPOL_1:def_2;
then A57: f /. (((N-min (L~ f)) .. f) -' 1) in N-most (L~ f) by A2, A7, A42, Th10, GOBOARD1:1;
then A58: (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 A4, A6, A39, A42, A56, GOBOARD7:29, TOPREAL3:6;
then A59: (f /. (((N-min (L~ f)) .. f) -' 1)) `1 > (N-min (L~ f)) `1 by A58, XXREAL_0:1;
(f /. (((N-min (L~ f)) .. f) -' 1)) `1 <= (N-max (L~ f)) `1 by A57, PSCOMP_1:39;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A59, 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 A60: (N-min (L~ f)) `2 = (f /. (((N-min (L~ f)) .. f) + 1)) `2 by A49, A45, SPPOL_1:def_2;
then A61: f /. (((N-min (L~ f)) .. f) + 1) in N-most (L~ f) by A2, A7, A46, Th10, GOBOARD1:1;
then A62: (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 A4, A6, A46, A60, GOBOARD7:29, TOPREAL3:6;
then A63: (f /. (((N-min (L~ f)) .. f) + 1)) `1 > (N-min (L~ f)) `1 by A62, XXREAL_0:1;
(f /. (((N-min (L~ f)) .. f) + 1)) `1 <= (N-max (L~ f)) `1 by A61, PSCOMP_1:39;
hence (N-min (L~ f)) `1 < (N-max (L~ f)) `1 by A63, 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 Th52: :: SPRECT_2:52
for f being non constant standard special_circular_sequence holds N-min (L~ f) <> N-max (L~ f)
proof
let f be non constant standard special_circular_sequence; ::_thesis: N-min (L~ f) <> N-max (L~ f)
(N-min (L~ f)) `1 < (N-max (L~ f)) `1 by Th51;
hence N-min (L~ f) <> N-max (L~ f) ; ::_thesis: verum
end;
theorem Th53: :: SPRECT_2:53
for f being non constant standard special_circular_sequence holds (E-min (L~ f)) `2 < (E-max (L~ f)) `2
proof
let f be non constant standard special_circular_sequence; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
set p = E-min (L~ f);
set i = (E-min (L~ f)) .. f;
A1: len f > 3 + 1 by GOBOARD7:34;
A2: len f >= 1 + 1 by GOBOARD7:34, XXREAL_0:2;
A3: E-min (L~ f) in rng f by Th45;
then A4: (E-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A5: ( 1 <= (E-min (L~ f)) .. f & (E-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A6: E-min (L~ f) = f . ((E-min (L~ f)) .. f) by A3, FINSEQ_4:19
.= f /. ((E-min (L~ f)) .. f) by A4, PARTFUN1:def_6 ;
A7: (E-min (L~ f)) `1 = E-bound (L~ f) by EUCLID:52;
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 A5, XXREAL_0:1;
supposeA8: ( (E-min (L~ f)) .. f = 1 or (E-min (L~ f)) .. f = len f ) ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
then E-min (L~ f) = f /. 1 by A6, FINSEQ_6:def_1;
then A9: E-min (L~ f) in LSeg (f,1) by A2, TOPREAL1:21;
A10: 1 + 1 in dom f by A2, FINSEQ_3:25;
then A11: f /. (1 + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A12: f /. (1 + 1) in LSeg (f,1) by A2, TOPREAL1:21;
A13: ((len f) -' 1) + 1 = len f by A1, XREAL_1:235, XXREAL_0:2;
then (len f) -' 1 > 3 by A1, XREAL_1:6;
then A14: (len f) -' 1 > 1 by XXREAL_0:2;
then A15: f /. ((len f) -' 1) in LSeg (f,((len f) -' 1)) by A13, TOPREAL1:21;
(len f) -' 1 <= len f by A13, NAT_1:11;
then A16: (len f) -' 1 in dom f by A14, FINSEQ_3:25;
then A17: f /. ((len f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A18: f /. 1 = f /. (len f) by FINSEQ_6:def_1;
then A19: E-min (L~ f) in LSeg (f,((len f) -' 1)) by A6, A8, A13, A14, TOPREAL1:21;
A20: 1 in dom f by FINSEQ_5:6;
then A21: E-min (L~ f) <> f /. (1 + 1) by A6, A8, A18, A10, GOBOARD7:29;
A22: len f in dom f by FINSEQ_5:6;
then A23: E-min (L~ f) <> f /. ((len f) -' 1) by A6, A8, A18, A13, A16, GOBOARD7:29;
A24: ( not LSeg (f,((len f) -' 1)) is horizontal or not LSeg (f,1) is horizontal )
proof
assume ( LSeg (f,((len f) -' 1)) is horizontal & LSeg (f,1) is horizontal ) ; ::_thesis: contradiction
then A25: ( (E-min (L~ f)) `2 = (f /. (1 + 1)) `2 & (E-min (L~ f)) `2 = (f /. ((len f) -' 1)) `2 ) by A19, A9, A15, A12, SPPOL_1:def_2;
A26: ( (f /. (1 + 1)) `1 <= (f /. ((len f) -' 1)) `1 or (f /. (1 + 1)) `1 >= (f /. ((len f) -' 1)) `1 ) ;
A27: ( (E-min (L~ f)) `1 >= (f /. (1 + 1)) `1 & (E-min (L~ f)) `1 >= (f /. ((len f) -' 1)) `1 ) by A7, A17, A11, PSCOMP_1:24;
( LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) & LSeg (f,((len f) -' 1)) = LSeg ((f /. 1),(f /. ((len f) -' 1))) ) by A2, A18, A13, A14, TOPREAL1:def_3;
then ( f /. ((len f) -' 1) in LSeg (f,1) or f /. (1 + 1) in LSeg (f,((len f) -' 1)) ) by A6, A8, A18, A25, A27, A26, GOBOARD7:8;
then ( f /. ((len f) -' 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) or f /. (1 + 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) ) by A15, A12, XBOOLE_0:def_4;
then A28: (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) <> {(f /. 1)} by A6, A8, A18, A23, A21, TARSKI:def_1;
f . 1 = f /. 1 by A20, PARTFUN1:def_6;
hence contradiction by A28, JORDAN4:42; ::_thesis: verum
end;
now__::_thesis:_(E-min_(L~_f))_`2_<_(E-max_(L~_f))_`2
percases ( LSeg (f,((len f) -' 1)) is vertical or LSeg (f,1) is vertical ) by A24, SPPOL_1:19;
suppose LSeg (f,((len f) -' 1)) is vertical ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
then A29: (E-min (L~ f)) `1 = (f /. ((len f) -' 1)) `1 by A19, A15, SPPOL_1:def_3;
then A30: f /. ((len f) -' 1) in E-most (L~ f) by A2, A7, A16, Th13, GOBOARD1:1;
then A31: (f /. ((len f) -' 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47;
(f /. ((len f) -' 1)) `2 <> (E-min (L~ f)) `2 by A6, A8, A22, A18, A13, A16, A29, GOBOARD7:29, TOPREAL3:6;
then A32: (f /. ((len f) -' 1)) `2 > (E-min (L~ f)) `2 by A31, XXREAL_0:1;
(f /. ((len f) -' 1)) `2 <= (E-max (L~ f)) `2 by A30, PSCOMP_1:47;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A32, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,1) is vertical ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
then A33: (E-min (L~ f)) `1 = (f /. (1 + 1)) `1 by A9, A12, SPPOL_1:def_3;
then A34: f /. (1 + 1) in E-most (L~ f) by A2, A7, A10, Th13, GOBOARD1:1;
then A35: (f /. (1 + 1)) `2 >= (E-min (L~ f)) `2 by PSCOMP_1:47;
(f /. (1 + 1)) `2 <> (E-min (L~ f)) `2 by A6, A8, A20, A18, A10, A33, GOBOARD7:29, TOPREAL3:6;
then A36: (f /. (1 + 1)) `2 > (E-min (L~ f)) `2 by A35, XXREAL_0:1;
(f /. (1 + 1)) `2 <= (E-max (L~ f)) `2 by A34, PSCOMP_1:47;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A36, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 ; ::_thesis: verum
end;
supposethat A37: 1 < (E-min (L~ f)) .. f and
A38: (E-min (L~ f)) .. f < len f ; ::_thesis: (E-min (L~ f)) `2 < (E-max (L~ f)) `2
A39: (((E-min (L~ f)) .. f) -' 1) + 1 = (E-min (L~ f)) .. f by A37, XREAL_1:235;
then A40: ((E-min (L~ f)) .. f) -' 1 >= 1 by A37, NAT_1:13;
then A41: f /. (((E-min (L~ f)) .. f) -' 1) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) by A38, A39, TOPREAL1:21;
((E-min (L~ f)) .. f) -' 1 <= (E-min (L~ f)) .. f by A39, NAT_1:11;
then ((E-min (L~ f)) .. f) -' 1 <= len f by A38, XXREAL_0:2;
then A42: ((E-min (L~ f)) .. f) -' 1 in dom f by A40, FINSEQ_3:25;
then A43: f /. (((E-min (L~ f)) .. f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A44: ((E-min (L~ f)) .. f) + 1 <= len f by A38, NAT_1:13;
then A45: f /. (((E-min (L~ f)) .. f) + 1) in LSeg (f,((E-min (L~ f)) .. f)) by A37, TOPREAL1:21;
((E-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A46: ((E-min (L~ f)) .. f) + 1 in dom f by A44, FINSEQ_3:25;
then A47: f /. (((E-min (L~ f)) .. f) + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A48: E-min (L~ f) <> f /. (((E-min (L~ f)) .. f) + 1) by A3, A6, A46, FINSEQ_4:20, GOBOARD7:29;
A49: E-min (L~ f) in LSeg (f,((E-min (L~ f)) .. f)) by A6, A37, A44, TOPREAL1:21;
A50: E-min (L~ f) in LSeg (f,(((E-min (L~ f)) .. f) -' 1)) by A6, A38, A39, A40, TOPREAL1:21;
A51: E-min (L~ f) <> f /. (((E-min (L~ f)) .. f) -' 1) by A4, A6, A39, A42, GOBOARD7:29;
A52: ( 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 A53: ( (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 A50, A49, A41, A45, SPPOL_1:def_2;
A54: ( (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 ) ;
A55: ( (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 A7, A43, A47, 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 A37, A38, A39, A40, A44, 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 A6, A53, A55, A54, 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 A41, A45, 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 A6, A51, A48, TARSKI:def_1;
hence contradiction by A39, A40, A44, 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 A52, 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 A56: (E-min (L~ f)) `1 = (f /. (((E-min (L~ f)) .. f) -' 1)) `1 by A50, A41, SPPOL_1:def_3;
then A57: f /. (((E-min (L~ f)) .. f) -' 1) in E-most (L~ f) by A2, A7, A42, Th13, GOBOARD1:1;
then A58: (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 A4, A6, A39, A42, A56, GOBOARD7:29, TOPREAL3:6;
then A59: (f /. (((E-min (L~ f)) .. f) -' 1)) `2 > (E-min (L~ f)) `2 by A58, XXREAL_0:1;
(f /. (((E-min (L~ f)) .. f) -' 1)) `2 <= (E-max (L~ f)) `2 by A57, PSCOMP_1:47;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A59, 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 A60: (E-min (L~ f)) `1 = (f /. (((E-min (L~ f)) .. f) + 1)) `1 by A49, A45, SPPOL_1:def_3;
then A61: f /. (((E-min (L~ f)) .. f) + 1) in E-most (L~ f) by A2, A7, A46, Th13, GOBOARD1:1;
then A62: (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 A4, A6, A46, A60, GOBOARD7:29, TOPREAL3:6;
then A63: (f /. (((E-min (L~ f)) .. f) + 1)) `2 > (E-min (L~ f)) `2 by A62, XXREAL_0:1;
(f /. (((E-min (L~ f)) .. f) + 1)) `2 <= (E-max (L~ f)) `2 by A61, PSCOMP_1:47;
hence (E-min (L~ f)) `2 < (E-max (L~ f)) `2 by A63, 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 :: SPRECT_2:54
for f being non constant standard special_circular_sequence holds E-min (L~ f) <> E-max (L~ f)
proof
let f be non constant standard special_circular_sequence; ::_thesis: E-min (L~ f) <> E-max (L~ f)
(E-min (L~ f)) `2 < (E-max (L~ f)) `2 by Th53;
hence E-min (L~ f) <> E-max (L~ f) ; ::_thesis: verum
end;
theorem Th55: :: SPRECT_2:55
for f being non constant standard special_circular_sequence holds (S-min (L~ f)) `1 < (S-max (L~ f)) `1
proof
let f be non constant standard special_circular_sequence; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
set p = S-min (L~ f);
set i = (S-min (L~ f)) .. f;
A1: len f > 3 + 1 by GOBOARD7:34;
A2: len f >= 1 + 1 by GOBOARD7:34, XXREAL_0:2;
A3: S-min (L~ f) in rng f by Th41;
then A4: (S-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A5: ( 1 <= (S-min (L~ f)) .. f & (S-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A6: S-min (L~ f) = f . ((S-min (L~ f)) .. f) by A3, FINSEQ_4:19
.= f /. ((S-min (L~ f)) .. f) by A4, PARTFUN1:def_6 ;
A7: (S-min (L~ f)) `2 = S-bound (L~ f) by EUCLID:52;
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 A5, XXREAL_0:1;
supposeA8: ( (S-min (L~ f)) .. f = 1 or (S-min (L~ f)) .. f = len f ) ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
then S-min (L~ f) = f /. 1 by A6, FINSEQ_6:def_1;
then A9: S-min (L~ f) in LSeg (f,1) by A2, TOPREAL1:21;
A10: 1 + 1 in dom f by A2, FINSEQ_3:25;
then A11: f /. (1 + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A12: f /. (1 + 1) in LSeg (f,1) by A2, TOPREAL1:21;
A13: ((len f) -' 1) + 1 = len f by A1, XREAL_1:235, XXREAL_0:2;
then (len f) -' 1 > 3 by A1, XREAL_1:6;
then A14: (len f) -' 1 > 1 by XXREAL_0:2;
then A15: f /. ((len f) -' 1) in LSeg (f,((len f) -' 1)) by A13, TOPREAL1:21;
(len f) -' 1 <= len f by A13, NAT_1:11;
then A16: (len f) -' 1 in dom f by A14, FINSEQ_3:25;
then A17: f /. ((len f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A18: f /. 1 = f /. (len f) by FINSEQ_6:def_1;
then A19: S-min (L~ f) in LSeg (f,((len f) -' 1)) by A6, A8, A13, A14, TOPREAL1:21;
A20: 1 in dom f by FINSEQ_5:6;
then A21: S-min (L~ f) <> f /. (1 + 1) by A6, A8, A18, A10, GOBOARD7:29;
A22: len f in dom f by FINSEQ_5:6;
then A23: S-min (L~ f) <> f /. ((len f) -' 1) by A6, A8, A18, A13, A16, GOBOARD7:29;
A24: ( not LSeg (f,((len f) -' 1)) is vertical or not LSeg (f,1) is vertical )
proof
assume ( LSeg (f,((len f) -' 1)) is vertical & LSeg (f,1) is vertical ) ; ::_thesis: contradiction
then A25: ( (S-min (L~ f)) `1 = (f /. (1 + 1)) `1 & (S-min (L~ f)) `1 = (f /. ((len f) -' 1)) `1 ) by A19, A9, A15, A12, SPPOL_1:def_3;
A26: ( (f /. (1 + 1)) `2 <= (f /. ((len f) -' 1)) `2 or (f /. (1 + 1)) `2 >= (f /. ((len f) -' 1)) `2 ) ;
A27: ( (S-min (L~ f)) `2 <= (f /. (1 + 1)) `2 & (S-min (L~ f)) `2 <= (f /. ((len f) -' 1)) `2 ) by A7, A17, A11, PSCOMP_1:24;
( LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) & LSeg (f,((len f) -' 1)) = LSeg ((f /. 1),(f /. ((len f) -' 1))) ) by A2, A18, A13, A14, TOPREAL1:def_3;
then ( f /. ((len f) -' 1) in LSeg (f,1) or f /. (1 + 1) in LSeg (f,((len f) -' 1)) ) by A6, A8, A18, A25, A27, A26, GOBOARD7:7;
then ( f /. ((len f) -' 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) or f /. (1 + 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) ) by A15, A12, XBOOLE_0:def_4;
then A28: (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) <> {(f /. 1)} by A6, A8, A18, A23, A21, TARSKI:def_1;
f . 1 = f /. 1 by A20, PARTFUN1:def_6;
hence contradiction by A28, JORDAN4:42; ::_thesis: verum
end;
now__::_thesis:_(S-min_(L~_f))_`1_<_(S-max_(L~_f))_`1
percases ( LSeg (f,((len f) -' 1)) is horizontal or LSeg (f,1) is horizontal ) by A24, SPPOL_1:19;
suppose LSeg (f,((len f) -' 1)) is horizontal ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
then A29: (S-min (L~ f)) `2 = (f /. ((len f) -' 1)) `2 by A19, A15, SPPOL_1:def_2;
then A30: f /. ((len f) -' 1) in S-most (L~ f) by A2, A7, A16, Th11, GOBOARD1:1;
then A31: (f /. ((len f) -' 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55;
(f /. ((len f) -' 1)) `1 <> (S-min (L~ f)) `1 by A6, A8, A22, A18, A13, A16, A29, GOBOARD7:29, TOPREAL3:6;
then A32: (f /. ((len f) -' 1)) `1 > (S-min (L~ f)) `1 by A31, XXREAL_0:1;
(f /. ((len f) -' 1)) `1 <= (S-max (L~ f)) `1 by A30, PSCOMP_1:55;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A32, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,1) is horizontal ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
then A33: (S-min (L~ f)) `2 = (f /. (1 + 1)) `2 by A9, A12, SPPOL_1:def_2;
then A34: f /. (1 + 1) in S-most (L~ f) by A2, A7, A10, Th11, GOBOARD1:1;
then A35: (f /. (1 + 1)) `1 >= (S-min (L~ f)) `1 by PSCOMP_1:55;
(f /. (1 + 1)) `1 <> (S-min (L~ f)) `1 by A6, A8, A20, A18, A10, A33, GOBOARD7:29, TOPREAL3:6;
then A36: (f /. (1 + 1)) `1 > (S-min (L~ f)) `1 by A35, XXREAL_0:1;
(f /. (1 + 1)) `1 <= (S-max (L~ f)) `1 by A34, PSCOMP_1:55;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A36, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 ; ::_thesis: verum
end;
supposethat A37: 1 < (S-min (L~ f)) .. f and
A38: (S-min (L~ f)) .. f < len f ; ::_thesis: (S-min (L~ f)) `1 < (S-max (L~ f)) `1
A39: (((S-min (L~ f)) .. f) -' 1) + 1 = (S-min (L~ f)) .. f by A37, XREAL_1:235;
then A40: ((S-min (L~ f)) .. f) -' 1 >= 1 by A37, NAT_1:13;
then A41: f /. (((S-min (L~ f)) .. f) -' 1) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) by A38, A39, TOPREAL1:21;
((S-min (L~ f)) .. f) -' 1 <= (S-min (L~ f)) .. f by A39, NAT_1:11;
then ((S-min (L~ f)) .. f) -' 1 <= len f by A38, XXREAL_0:2;
then A42: ((S-min (L~ f)) .. f) -' 1 in dom f by A40, FINSEQ_3:25;
then A43: f /. (((S-min (L~ f)) .. f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A44: ((S-min (L~ f)) .. f) + 1 <= len f by A38, NAT_1:13;
then A45: f /. (((S-min (L~ f)) .. f) + 1) in LSeg (f,((S-min (L~ f)) .. f)) by A37, TOPREAL1:21;
((S-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A46: ((S-min (L~ f)) .. f) + 1 in dom f by A44, FINSEQ_3:25;
then A47: f /. (((S-min (L~ f)) .. f) + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A48: S-min (L~ f) <> f /. (((S-min (L~ f)) .. f) + 1) by A3, A6, A46, FINSEQ_4:20, GOBOARD7:29;
A49: S-min (L~ f) in LSeg (f,((S-min (L~ f)) .. f)) by A6, A37, A44, TOPREAL1:21;
A50: S-min (L~ f) in LSeg (f,(((S-min (L~ f)) .. f) -' 1)) by A6, A38, A39, A40, TOPREAL1:21;
A51: S-min (L~ f) <> f /. (((S-min (L~ f)) .. f) -' 1) by A4, A6, A39, A42, GOBOARD7:29;
A52: ( 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 A53: ( (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 A50, A49, A41, A45, SPPOL_1:def_3;
A54: ( (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 ) ;
A55: ( (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 A7, A43, A47, 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 A37, A38, A39, A40, A44, 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 A6, A53, A55, A54, 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 A41, A45, 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 A6, A51, A48, TARSKI:def_1;
hence contradiction by A39, A40, A44, 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 A52, 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 A56: (S-min (L~ f)) `2 = (f /. (((S-min (L~ f)) .. f) -' 1)) `2 by A50, A41, SPPOL_1:def_2;
then A57: f /. (((S-min (L~ f)) .. f) -' 1) in S-most (L~ f) by A2, A7, A42, Th11, GOBOARD1:1;
then A58: (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 A4, A6, A39, A42, A56, GOBOARD7:29, TOPREAL3:6;
then A59: (f /. (((S-min (L~ f)) .. f) -' 1)) `1 > (S-min (L~ f)) `1 by A58, XXREAL_0:1;
(f /. (((S-min (L~ f)) .. f) -' 1)) `1 <= (S-max (L~ f)) `1 by A57, PSCOMP_1:55;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A59, 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 A60: (S-min (L~ f)) `2 = (f /. (((S-min (L~ f)) .. f) + 1)) `2 by A49, A45, SPPOL_1:def_2;
then A61: f /. (((S-min (L~ f)) .. f) + 1) in S-most (L~ f) by A2, A7, A46, Th11, GOBOARD1:1;
then A62: (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 A4, A6, A46, A60, GOBOARD7:29, TOPREAL3:6;
then A63: (f /. (((S-min (L~ f)) .. f) + 1)) `1 > (S-min (L~ f)) `1 by A62, XXREAL_0:1;
(f /. (((S-min (L~ f)) .. f) + 1)) `1 <= (S-max (L~ f)) `1 by A61, PSCOMP_1:55;
hence (S-min (L~ f)) `1 < (S-max (L~ f)) `1 by A63, 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 Th56: :: SPRECT_2:56
for f being non constant standard special_circular_sequence holds S-min (L~ f) <> S-max (L~ f)
proof
let f be non constant standard special_circular_sequence; ::_thesis: S-min (L~ f) <> S-max (L~ f)
(S-min (L~ f)) `1 < (S-max (L~ f)) `1 by Th55;
hence S-min (L~ f) <> S-max (L~ f) ; ::_thesis: verum
end;
theorem Th57: :: SPRECT_2:57
for f being non constant standard special_circular_sequence holds (W-min (L~ f)) `2 < (W-max (L~ f)) `2
proof
let f be non constant standard special_circular_sequence; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
set p = W-min (L~ f);
set i = (W-min (L~ f)) .. f;
A1: len f > 3 + 1 by GOBOARD7:34;
A2: len f >= 1 + 1 by GOBOARD7:34, XXREAL_0:2;
A3: W-min (L~ f) in rng f by Th43;
then A4: (W-min (L~ f)) .. f in dom f by FINSEQ_4:20;
then A5: ( 1 <= (W-min (L~ f)) .. f & (W-min (L~ f)) .. f <= len f ) by FINSEQ_3:25;
A6: W-min (L~ f) = f . ((W-min (L~ f)) .. f) by A3, FINSEQ_4:19
.= f /. ((W-min (L~ f)) .. f) by A4, PARTFUN1:def_6 ;
A7: (W-min (L~ f)) `1 = W-bound (L~ f) by EUCLID:52;
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 A5, XXREAL_0:1;
supposeA8: ( (W-min (L~ f)) .. f = 1 or (W-min (L~ f)) .. f = len f ) ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
then W-min (L~ f) = f /. 1 by A6, FINSEQ_6:def_1;
then A9: W-min (L~ f) in LSeg (f,1) by A2, TOPREAL1:21;
A10: 1 + 1 in dom f by A2, FINSEQ_3:25;
then A11: f /. (1 + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A12: f /. (1 + 1) in LSeg (f,1) by A2, TOPREAL1:21;
A13: ((len f) -' 1) + 1 = len f by A1, XREAL_1:235, XXREAL_0:2;
then (len f) -' 1 > 3 by A1, XREAL_1:6;
then A14: (len f) -' 1 > 1 by XXREAL_0:2;
then A15: f /. ((len f) -' 1) in LSeg (f,((len f) -' 1)) by A13, TOPREAL1:21;
(len f) -' 1 <= len f by A13, NAT_1:11;
then A16: (len f) -' 1 in dom f by A14, FINSEQ_3:25;
then A17: f /. ((len f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A18: f /. 1 = f /. (len f) by FINSEQ_6:def_1;
then A19: W-min (L~ f) in LSeg (f,((len f) -' 1)) by A6, A8, A13, A14, TOPREAL1:21;
A20: 1 in dom f by FINSEQ_5:6;
then A21: W-min (L~ f) <> f /. (1 + 1) by A6, A8, A18, A10, GOBOARD7:29;
A22: len f in dom f by FINSEQ_5:6;
then A23: W-min (L~ f) <> f /. ((len f) -' 1) by A6, A8, A18, A13, A16, GOBOARD7:29;
A24: ( not LSeg (f,((len f) -' 1)) is horizontal or not LSeg (f,1) is horizontal )
proof
assume ( LSeg (f,((len f) -' 1)) is horizontal & LSeg (f,1) is horizontal ) ; ::_thesis: contradiction
then A25: ( (W-min (L~ f)) `2 = (f /. (1 + 1)) `2 & (W-min (L~ f)) `2 = (f /. ((len f) -' 1)) `2 ) by A19, A9, A15, A12, SPPOL_1:def_2;
A26: ( (f /. (1 + 1)) `1 <= (f /. ((len f) -' 1)) `1 or (f /. (1 + 1)) `1 >= (f /. ((len f) -' 1)) `1 ) ;
A27: ( (W-min (L~ f)) `1 <= (f /. (1 + 1)) `1 & (W-min (L~ f)) `1 <= (f /. ((len f) -' 1)) `1 ) by A7, A17, A11, PSCOMP_1:24;
( LSeg (f,1) = LSeg ((f /. 1),(f /. (1 + 1))) & LSeg (f,((len f) -' 1)) = LSeg ((f /. 1),(f /. ((len f) -' 1))) ) by A2, A18, A13, A14, TOPREAL1:def_3;
then ( f /. ((len f) -' 1) in LSeg (f,1) or f /. (1 + 1) in LSeg (f,((len f) -' 1)) ) by A6, A8, A18, A25, A27, A26, GOBOARD7:8;
then ( f /. ((len f) -' 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) or f /. (1 + 1) in (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) ) by A15, A12, XBOOLE_0:def_4;
then A28: (LSeg (f,((len f) -' 1))) /\ (LSeg (f,1)) <> {(f /. 1)} by A6, A8, A18, A23, A21, TARSKI:def_1;
f . 1 = f /. 1 by A20, PARTFUN1:def_6;
hence contradiction by A28, JORDAN4:42; ::_thesis: verum
end;
now__::_thesis:_(W-min_(L~_f))_`2_<_(W-max_(L~_f))_`2
percases ( LSeg (f,((len f) -' 1)) is vertical or LSeg (f,1) is vertical ) by A24, SPPOL_1:19;
suppose LSeg (f,((len f) -' 1)) is vertical ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
then A29: (W-min (L~ f)) `1 = (f /. ((len f) -' 1)) `1 by A19, A15, SPPOL_1:def_3;
then A30: f /. ((len f) -' 1) in W-most (L~ f) by A2, A7, A16, Th12, GOBOARD1:1;
then A31: (f /. ((len f) -' 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31;
(f /. ((len f) -' 1)) `2 <> (W-min (L~ f)) `2 by A6, A8, A22, A18, A13, A16, A29, GOBOARD7:29, TOPREAL3:6;
then A32: (f /. ((len f) -' 1)) `2 > (W-min (L~ f)) `2 by A31, XXREAL_0:1;
(f /. ((len f) -' 1)) `2 <= (W-max (L~ f)) `2 by A30, PSCOMP_1:31;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A32, XXREAL_0:2; ::_thesis: verum
end;
suppose LSeg (f,1) is vertical ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
then A33: (W-min (L~ f)) `1 = (f /. (1 + 1)) `1 by A9, A12, SPPOL_1:def_3;
then A34: f /. (1 + 1) in W-most (L~ f) by A2, A7, A10, Th12, GOBOARD1:1;
then A35: (f /. (1 + 1)) `2 >= (W-min (L~ f)) `2 by PSCOMP_1:31;
(f /. (1 + 1)) `2 <> (W-min (L~ f)) `2 by A6, A8, A20, A18, A10, A33, GOBOARD7:29, TOPREAL3:6;
then A36: (f /. (1 + 1)) `2 > (W-min (L~ f)) `2 by A35, XXREAL_0:1;
(f /. (1 + 1)) `2 <= (W-max (L~ f)) `2 by A34, PSCOMP_1:31;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A36, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 ; ::_thesis: verum
end;
supposethat A37: 1 < (W-min (L~ f)) .. f and
A38: (W-min (L~ f)) .. f < len f ; ::_thesis: (W-min (L~ f)) `2 < (W-max (L~ f)) `2
A39: (((W-min (L~ f)) .. f) -' 1) + 1 = (W-min (L~ f)) .. f by A37, XREAL_1:235;
then A40: ((W-min (L~ f)) .. f) -' 1 >= 1 by A37, NAT_1:13;
then A41: f /. (((W-min (L~ f)) .. f) -' 1) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) by A38, A39, TOPREAL1:21;
((W-min (L~ f)) .. f) -' 1 <= (W-min (L~ f)) .. f by A39, NAT_1:11;
then ((W-min (L~ f)) .. f) -' 1 <= len f by A38, XXREAL_0:2;
then A42: ((W-min (L~ f)) .. f) -' 1 in dom f by A40, FINSEQ_3:25;
then A43: f /. (((W-min (L~ f)) .. f) -' 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A44: ((W-min (L~ f)) .. f) + 1 <= len f by A38, NAT_1:13;
then A45: f /. (((W-min (L~ f)) .. f) + 1) in LSeg (f,((W-min (L~ f)) .. f)) by A37, TOPREAL1:21;
((W-min (L~ f)) .. f) + 1 >= 1 by NAT_1:11;
then A46: ((W-min (L~ f)) .. f) + 1 in dom f by A44, FINSEQ_3:25;
then A47: f /. (((W-min (L~ f)) .. f) + 1) in L~ f by A1, GOBOARD1:1, XXREAL_0:2;
A48: W-min (L~ f) <> f /. (((W-min (L~ f)) .. f) + 1) by A3, A6, A46, FINSEQ_4:20, GOBOARD7:29;
A49: W-min (L~ f) in LSeg (f,((W-min (L~ f)) .. f)) by A6, A37, A44, TOPREAL1:21;
A50: W-min (L~ f) in LSeg (f,(((W-min (L~ f)) .. f) -' 1)) by A6, A38, A39, A40, TOPREAL1:21;
A51: W-min (L~ f) <> f /. (((W-min (L~ f)) .. f) -' 1) by A4, A6, A39, A42, GOBOARD7:29;
A52: ( 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 A53: ( (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 A50, A49, A41, A45, SPPOL_1:def_2;
A54: ( (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 ) ;
A55: ( (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 A7, A43, A47, 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 A37, A38, A39, A40, A44, 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 A6, A53, A55, A54, 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 A41, A45, 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 A6, A51, A48, TARSKI:def_1;
hence contradiction by A39, A40, A44, 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 A52, 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 A56: (W-min (L~ f)) `1 = (f /. (((W-min (L~ f)) .. f) -' 1)) `1 by A50, A41, SPPOL_1:def_3;
then A57: f /. (((W-min (L~ f)) .. f) -' 1) in W-most (L~ f) by A2, A7, A42, Th12, GOBOARD1:1;
then A58: (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 A4, A6, A39, A42, A56, GOBOARD7:29, TOPREAL3:6;
then A59: (f /. (((W-min (L~ f)) .. f) -' 1)) `2 > (W-min (L~ f)) `2 by A58, XXREAL_0:1;
(f /. (((W-min (L~ f)) .. f) -' 1)) `2 <= (W-max (L~ f)) `2 by A57, PSCOMP_1:31;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A59, 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 A60: (W-min (L~ f)) `1 = (f /. (((W-min (L~ f)) .. f) + 1)) `1 by A49, A45, SPPOL_1:def_3;
then A61: f /. (((W-min (L~ f)) .. f) + 1) in W-most (L~ f) by A2, A7, A46, Th12, GOBOARD1:1;
then A62: (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 A4, A6, A46, A60, GOBOARD7:29, TOPREAL3:6;
then A63: (f /. (((W-min (L~ f)) .. f) + 1)) `2 > (W-min (L~ f)) `2 by A62, XXREAL_0:1;
(f /. (((W-min (L~ f)) .. f) + 1)) `2 <= (W-max (L~ f)) `2 by A61, PSCOMP_1:31;
hence (W-min (L~ f)) `2 < (W-max (L~ f)) `2 by A63, 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 Th58: :: SPRECT_2:58
for f being non constant standard special_circular_sequence holds W-min (L~ f) <> W-max (L~ f)
proof
let f be non constant standard special_circular_sequence; ::_thesis: W-min (L~ f) <> W-max (L~ f)
(W-min (L~ f)) `2 < (W-max (L~ f)) `2 by Th57;
hence W-min (L~ f) <> W-max (L~ f) ; ::_thesis: verum
end;
theorem Th59: :: SPRECT_2:59
for f being non constant standard special_circular_sequence holds LSeg ((NW-corner (L~ f)),(N-min (L~ f))) misses LSeg ((N-max (L~ f)),(NE-corner (L~ f)))
proof
let f be non constant standard special_circular_sequence; ::_thesis: LSeg ((NW-corner (L~ f)),(N-min (L~ f))) misses LSeg ((N-max (L~ f)),(NE-corner (L~ f)))
A1: (N-min (L~ f)) `2 = (N-max (L~ f)) `2 by PSCOMP_1:37;
assume LSeg ((NW-corner (L~ f)),(N-min (L~ f))) meets LSeg ((N-max (L~ f)),(NE-corner (L~ f))) ; ::_thesis: contradiction
then consider p being set such that
A2: p in LSeg ((NW-corner (L~ f)),(N-min (L~ f))) and
A3: p in LSeg ((N-max (L~ f)),(NE-corner (L~ f))) by XBOOLE_0:3;
reconsider p = p as Point of (TOP-REAL 2) by A2;
(N-max (L~ f)) `1 <= (NE-corner (L~ f)) `1 by PSCOMP_1:38;
then A4: (N-max (L~ f)) `1 <= p `1 by A3, TOPREAL1:3;
(NW-corner (L~ f)) `1 <= (N-min (L~ f)) `1 by PSCOMP_1:38;
then p `1 <= (N-min (L~ f)) `1 by A2, TOPREAL1:3;
then A5: (N-min (L~ f)) `1 >= (N-max (L~ f)) `1 by A4, XXREAL_0:2;
(N-min (L~ f)) `1 <= (N-max (L~ f)) `1 by PSCOMP_1:38;
then (N-min (L~ f)) `1 = (N-max (L~ f)) `1 by A5, XXREAL_0:1;
hence contradiction by A1, Th52, TOPREAL3:6; ::_thesis: verum
end;
theorem Th60: :: SPRECT_2:60
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p <> f /. 1 & ( p `1 = (f /. 1) `1 or p `2 = (f /. 1) `2 ) & (LSeg (p,(f /. 1))) /\ (L~ f) = {(f /. 1)} holds
<*p*> ^ f is S-Sequence_in_R2
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p <> f /. 1 & ( p `1 = (f /. 1) `1 or p `2 = (f /. 1) `2 ) & (LSeg (p,(f /. 1))) /\ (L~ f) = {(f /. 1)} holds
<*p*> ^ f is S-Sequence_in_R2
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p <> f /. 1 & ( p `1 = (f /. 1) `1 or p `2 = (f /. 1) `2 ) & (LSeg (p,(f /. 1))) /\ (L~ f) = {(f /. 1)} implies <*p*> ^ f is S-Sequence_in_R2 )
assume that
A1: f is being_S-Seq and
A2: p <> f /. 1 and
A3: ( p `1 = (f /. 1) `1 or p `2 = (f /. 1) `2 ) and
A4: (LSeg (p,(f /. 1))) /\ (L~ f) = {(f /. 1)} ; ::_thesis: <*p*> ^ f is S-Sequence_in_R2
reconsider f = f as S-Sequence_in_R2 by A1;
A5: len f >= 1 + 1 by TOPREAL1:def_8;
then A6: f /. 1 in LSeg (f,1) by TOPREAL1:21;
set g = <*p*> ^ f;
len (<*p*> ^ f) = (len <*p*>) + (len f) by FINSEQ_1:22;
then len (<*p*> ^ f) >= len f by NAT_1:11;
then A7: len (<*p*> ^ f) >= 2 by A5, XXREAL_0:2;
now__::_thesis:_not_p_in_rng_f
assume A8: p in rng f ; ::_thesis: contradiction
( rng f c= L~ f & p in LSeg (p,(f /. 1)) ) by A5, RLTOPSP1:68, SPPOL_2:18;
then p in {(f /. 1)} by A4, A8, XBOOLE_0:def_4;
hence contradiction by A2, TARSKI:def_1; ::_thesis: verum
end;
then {p} misses rng f by ZFMISC_1:50;
then ( <*p*> is one-to-one & rng <*p*> misses rng f ) by FINSEQ_1:39, FINSEQ_3:93;
then A9: <*p*> ^ f is one-to-one by FINSEQ_3:91;
L~ <*p*> = {} by SPPOL_2:12;
then (L~ <*p*>) /\ (L~ f) = {} ;
then A10: L~ <*p*> misses L~ f by XBOOLE_0:def_7;
A11: 1 in dom f by FINSEQ_5:6;
A12: now__::_thesis:_for_i_being_Element_of_NAT_st_1_+_1_<=_i_&_i_+_1_<=_len_f_holds_
LSeg_(f,i)_misses_LSeg_(p,(f_/._1))
let i be Element of NAT ; ::_thesis: ( 1 + 1 <= i & i + 1 <= len f implies LSeg (f,i) misses LSeg (p,(f /. 1)) )
assume that
A13: 1 + 1 <= i and
A14: i + 1 <= len f ; ::_thesis: LSeg (f,i) misses LSeg (p,(f /. 1))
A15: 2 in dom f by A5, FINSEQ_3:25;
now__::_thesis:_not_f_/._1_in_LSeg_(f,i)
assume f /. 1 in LSeg (f,i) ; ::_thesis: contradiction
then A16: f /. 1 in (LSeg (f,1)) /\ (LSeg (f,i)) by A6, XBOOLE_0:def_4;
then A17: LSeg (f,1) meets LSeg (f,i) by XBOOLE_0:4;
now__::_thesis:_(_(_i_=_1_+_1_&_f_/._1_=_f_/._2_)_or_(_i_>_1_+_1_&_contradiction_)_)
percases ( i = 1 + 1 or i > 1 + 1 ) by A13, XXREAL_0:1;
caseA18: i = 1 + 1 ; ::_thesis: f /. 1 = f /. 2
then (LSeg (f,1)) /\ (LSeg (f,(1 + 1))) = {(f /. 2)} by A14, TOPREAL1:def_6;
hence f /. 1 = f /. 2 by A16, A18, TARSKI:def_1; ::_thesis: verum
end;
case i > 1 + 1 ; ::_thesis: contradiction
hence contradiction by A17, TOPREAL1:def_7; ::_thesis: verum
end;
end;
end;
then f . 1 = f /. 2 by A11, PARTFUN1:def_6
.= f . 2 by A15, PARTFUN1:def_6 ;
hence contradiction by A11, A15, FUNCT_1:def_4; ::_thesis: verum
end;
then not f /. 1 in (LSeg (f,i)) /\ (LSeg (p,(f /. 1))) by XBOOLE_0:def_4;
then A19: (LSeg (f,i)) /\ (LSeg (p,(f /. 1))) <> {(f /. 1)} by TARSKI:def_1;
(LSeg (f,i)) /\ (LSeg (p,(f /. 1))) c= {(f /. 1)} by A4, TOPREAL3:19, XBOOLE_1:26;
then (LSeg (f,i)) /\ (LSeg (p,(f /. 1))) = {} by A19, ZFMISC_1:33;
hence LSeg (f,i) misses LSeg (p,(f /. 1)) by XBOOLE_0:def_7; ::_thesis: verum
end;
A20: len <*p*> = 1 by FINSEQ_1:39;
then A21: ( <*p*> is s.n.c. & <*p*> /. (len <*p*>) = p ) by FINSEQ_4:16, SPPOL_2:33;
A22: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_+_2_<=_len_<*p*>_holds_
LSeg_(<*p*>,i)_misses_LSeg_(p,(f_/._1))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i + 2 <= len <*p*> implies LSeg (<*p*>,i) misses LSeg (p,(f /. 1)) )
assume 1 <= i ; ::_thesis: ( i + 2 <= len <*p*> implies LSeg (<*p*>,i) misses LSeg (p,(f /. 1)) )
A23: 2 <= i + 2 by NAT_1:11;
assume i + 2 <= len <*p*> ; ::_thesis: LSeg (<*p*>,i) misses LSeg (p,(f /. 1))
hence LSeg (<*p*>,i) misses LSeg (p,(f /. 1)) by A20, A23, XXREAL_0:2; ::_thesis: verum
end;
(LSeg (p,(f /. 1))) /\ (LSeg (f,1)) = {(f /. 1)} by A4, A6, TOPREAL3:19, ZFMISC_1:124;
then ( <*p*> ^ f is unfolded & <*p*> ^ f is s.n.c. & <*p*> ^ f is special ) by A3, A21, A10, A22, A12, GOBOARD2:8, SPPOL_2:29, SPPOL_2:36;
hence <*p*> ^ f is S-Sequence_in_R2 by A9, A7, TOPREAL1:def_8; ::_thesis: verum
end;
theorem Th61: :: SPRECT_2:61
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p <> f /. (len f) & ( p `1 = (f /. (len f)) `1 or p `2 = (f /. (len f)) `2 ) & (LSeg (p,(f /. (len f)))) /\ (L~ f) = {(f /. (len f))} holds
f ^ <*p*> is S-Sequence_in_R2
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p <> f /. (len f) & ( p `1 = (f /. (len f)) `1 or p `2 = (f /. (len f)) `2 ) & (LSeg (p,(f /. (len f)))) /\ (L~ f) = {(f /. (len f))} holds
f ^ <*p*> is S-Sequence_in_R2
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p <> f /. (len f) & ( p `1 = (f /. (len f)) `1 or p `2 = (f /. (len f)) `2 ) & (LSeg (p,(f /. (len f)))) /\ (L~ f) = {(f /. (len f))} implies f ^ <*p*> is S-Sequence_in_R2 )
assume that
A1: f is being_S-Seq and
A2: ( p <> f /. (len f) & ( p `1 = (f /. (len f)) `1 or p `2 = (f /. (len f)) `2 ) ) and
A3: (LSeg (p,(f /. (len f)))) /\ (L~ f) = {(f /. (len f))} ; ::_thesis: f ^ <*p*> is S-Sequence_in_R2
set g = <*(f /. (len f)),p*>;
A4: <*(f /. (len f)),p*> is being_S-Seq by A2, SPPOL_2:43;
len <*(f /. (len f)),p*> = 1 + 1 by FINSEQ_1:44;
then A5: mid (<*(f /. (len f)),p*>,2,(len <*(f /. (len f)),p*>)) = <*(<*(f /. (len f)),p*> /. 2)*> by JORDAN4:15
.= <*p*> by FINSEQ_4:17 ;
reconsider f9 = f as S-Sequence_in_R2 by A1;
A6: len f9 in dom f9 by FINSEQ_5:6;
A7: <*(f /. (len f)),p*> . 1 = f /. (len f) by FINSEQ_1:44
.= f . (len f) by A6, PARTFUN1:def_6 ;
(L~ f) /\ (L~ <*(f /. (len f)),p*>) = {(f /. (len f))} by A3, SPPOL_2:21
.= {(f . (len f))} by A6, PARTFUN1:def_6 ;
hence f ^ <*p*> is S-Sequence_in_R2 by A1, A7, A4, A5, JORDAN3:38; ::_thesis: verum
end;
begin
theorem Th62: :: SPRECT_2:62
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = N-max (L~ f) & N-max (L~ f) <> NE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = N-max (L~ f) & N-max (L~ f) <> NE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
set p = NE-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = N-max (L~ f) & N-max (L~ f) <> NE-corner (L~ f) implies (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. j = N-max (L~ f) and
A5: N-max (L~ f) <> NE-corner (L~ f) ; ::_thesis: (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
A7: (mid (f,i,j)) /. (len (mid (f,i,j))) = N-max (L~ f) by A1, A2, A4, Th9;
then A8: (NE-corner (L~ f)) `2 = ((mid (f,i,j)) /. (len (mid (f,i,j)))) `2 by PSCOMP_1:37;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then ( (LSeg ((NE-corner (L~ f)),(N-max (L~ f)))) /\ (L~ f) = {(N-max (L~ f))} & N-max (L~ f) in L~ (mid (f,i,j)) ) by A7, JORDAN3:1, PSCOMP_1:43;
then (LSeg ((NE-corner (L~ f)),((mid (f,i,j)) /. (len (mid (f,i,j)))))) /\ (L~ (mid (f,i,j))) = {((mid (f,i,j)) /. (len (mid (f,i,j))))} by A7, A6, A9, JORDAN4:35, ZFMISC_1:124;
hence (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 by A3, A5, A7, A8, Th61; ::_thesis: verum
end;
theorem :: SPRECT_2:63
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = E-max (L~ f) & E-max (L~ f) <> NE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = E-max (L~ f) & E-max (L~ f) <> NE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
set p = NE-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = E-max (L~ f) & E-max (L~ f) <> NE-corner (L~ f) implies (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. j = E-max (L~ f) and
A5: E-max (L~ f) <> NE-corner (L~ f) ; ::_thesis: (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
A7: (mid (f,i,j)) /. (len (mid (f,i,j))) = E-max (L~ f) by A1, A2, A4, Th9;
then A8: (NE-corner (L~ f)) `1 = ((mid (f,i,j)) /. (len (mid (f,i,j)))) `1 by PSCOMP_1:45;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then ( (LSeg ((NE-corner (L~ f)),(E-max (L~ f)))) /\ (L~ f) = {(E-max (L~ f))} & E-max (L~ f) in L~ (mid (f,i,j)) ) by A7, JORDAN3:1, PSCOMP_1:51;
then (LSeg ((NE-corner (L~ f)),((mid (f,i,j)) /. (len (mid (f,i,j)))))) /\ (L~ (mid (f,i,j))) = {((mid (f,i,j)) /. (len (mid (f,i,j))))} by A7, A6, A9, JORDAN4:35, ZFMISC_1:124;
hence (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 by A3, A5, A7, A8, Th61; ::_thesis: verum
end;
theorem Th64: :: SPRECT_2:64
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = S-max (L~ f) & S-max (L~ f) <> SE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = S-max (L~ f) & S-max (L~ f) <> SE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2
set p = SE-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = S-max (L~ f) & S-max (L~ f) <> SE-corner (L~ f) implies (mid (f,i,j)) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. j = S-max (L~ f) and
A5: S-max (L~ f) <> SE-corner (L~ f) ; ::_thesis: (mid (f,i,j)) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
A7: (mid (f,i,j)) /. (len (mid (f,i,j))) = S-max (L~ f) by A1, A2, A4, Th9;
then A8: (SE-corner (L~ f)) `2 = ((mid (f,i,j)) /. (len (mid (f,i,j)))) `2 by PSCOMP_1:53;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then ( (LSeg ((SE-corner (L~ f)),(S-max (L~ f)))) /\ (L~ f) = {(S-max (L~ f))} & S-max (L~ f) in L~ (mid (f,i,j)) ) by A7, JORDAN3:1, PSCOMP_1:59;
then (LSeg ((SE-corner (L~ f)),((mid (f,i,j)) /. (len (mid (f,i,j)))))) /\ (L~ (mid (f,i,j))) = {((mid (f,i,j)) /. (len (mid (f,i,j))))} by A7, A6, A9, JORDAN4:35, ZFMISC_1:124;
hence (mid (f,i,j)) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2 by A3, A5, A7, A8, Th61; ::_thesis: verum
end;
theorem Th65: :: SPRECT_2:65
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = E-max (L~ f) & E-max (L~ f) <> NE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = E-max (L~ f) & E-max (L~ f) <> NE-corner (L~ f) holds
(mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
set p = NE-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. j = E-max (L~ f) & E-max (L~ f) <> NE-corner (L~ f) implies (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. j = E-max (L~ f) and
A5: E-max (L~ f) <> NE-corner (L~ f) ; ::_thesis: (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
A7: (mid (f,i,j)) /. (len (mid (f,i,j))) = E-max (L~ f) by A1, A2, A4, Th9;
then A8: (NE-corner (L~ f)) `1 = ((mid (f,i,j)) /. (len (mid (f,i,j)))) `1 by PSCOMP_1:45;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then ( (LSeg ((NE-corner (L~ f)),(E-max (L~ f)))) /\ (L~ f) = {(E-max (L~ f))} & E-max (L~ f) in L~ (mid (f,i,j)) ) by A7, JORDAN3:1, PSCOMP_1:51;
then (LSeg ((NE-corner (L~ f)),((mid (f,i,j)) /. (len (mid (f,i,j)))))) /\ (L~ (mid (f,i,j))) = {((mid (f,i,j)) /. (len (mid (f,i,j))))} by A7, A6, A9, JORDAN4:35, ZFMISC_1:124;
hence (mid (f,i,j)) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 by A3, A5, A7, A8, Th61; ::_thesis: verum
end;
theorem Th66: :: SPRECT_2:66
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) holds
<*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) holds
<*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2
set p = NW-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) implies <*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. i = N-min (L~ f) and
A5: N-min (L~ f) <> NW-corner (L~ f) ; ::_thesis: <*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
A7: (mid (f,i,j)) /. 1 = N-min (L~ f) by A1, A2, A4, Th8;
then A8: (NW-corner (L~ f)) `2 = ((mid (f,i,j)) /. 1) `2 by PSCOMP_1:37;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then ( (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) /\ (L~ f) = {(N-min (L~ f))} & N-min (L~ f) in L~ (mid (f,i,j)) ) by A7, JORDAN3:1, PSCOMP_1:43;
then (LSeg ((NW-corner (L~ f)),((mid (f,i,j)) /. 1))) /\ (L~ (mid (f,i,j))) = {((mid (f,i,j)) /. 1)} by A7, A6, A9, JORDAN4:35, ZFMISC_1:124;
hence <*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2 by A3, A5, A7, A8, Th60; ::_thesis: verum
end;
theorem Th67: :: SPRECT_2:67
for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = W-min (L~ f) & W-min (L~ f) <> SW-corner (L~ f) holds
<*(SW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = W-min (L~ f) & W-min (L~ f) <> SW-corner (L~ f) holds
<*(SW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2
set p = SW-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = W-min (L~ f) & W-min (L~ f) <> SW-corner (L~ f) implies <*(SW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. i = W-min (L~ f) and
A5: W-min (L~ f) <> SW-corner (L~ f) ; ::_thesis: <*(SW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2
A6: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
A7: (mid (f,i,j)) /. 1 = W-min (L~ f) by A1, A2, A4, Th8;
then A8: (SW-corner (L~ f)) `1 = ((mid (f,i,j)) /. 1) `1 by PSCOMP_1:29;
A9: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then ( (LSeg ((SW-corner (L~ f)),(W-min (L~ f)))) /\ (L~ f) = {(W-min (L~ f))} & W-min (L~ f) in L~ (mid (f,i,j)) ) by A7, JORDAN3:1, PSCOMP_1:35;
then (LSeg ((SW-corner (L~ f)),((mid (f,i,j)) /. 1))) /\ (L~ (mid (f,i,j))) = {((mid (f,i,j)) /. 1)} by A7, A6, A9, JORDAN4:35, ZFMISC_1:124;
hence <*(SW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2 by A3, A5, A7, A8, Th60; ::_thesis: verum
end;
Lm1: for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) & f /. j = N-max (L~ f) & N-max (L~ f) <> NE-corner (L~ f) holds
(<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) & f /. j = N-max (L~ f) & N-max (L~ f) <> NE-corner (L~ f) holds
(<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
set p = NW-corner (L~ f);
set q = NE-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) & f /. j = N-max (L~ f) & N-max (L~ f) <> NE-corner (L~ f) implies (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. i = N-min (L~ f) and
A5: N-min (L~ f) <> NW-corner (L~ f) and
A6: f /. j = N-max (L~ f) and
A7: N-max (L~ f) <> NE-corner (L~ f) ; ::_thesis: (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2
set g = <*(NW-corner (L~ f))*> ^ (mid (f,i,j));
A8: <*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2 by A1, A2, A3, A4, A5, Th66;
( len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) = (len <*(NW-corner (L~ f))*>) + (len (mid (f,i,j))) & len (mid (f,i,j)) in dom (mid (f,i,j)) ) by A3, FINSEQ_1:22, FINSEQ_5:6;
then A9: (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))) = (mid (f,i,j)) /. (len (mid (f,i,j))) by FINSEQ_4:69;
then A10: (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))) = N-max (L~ f) by A1, A2, A6, Th9;
then A11: (NE-corner (L~ f)) `2 = ((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))) `2 by PSCOMP_1:37;
(mid (f,i,j)) /. 1 = f /. i by A1, A2, Th8;
then A12: L~ (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) = (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) \/ (L~ (mid (f,i,j))) by A3, A4, SPPOL_2:20;
A13: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
A14: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then A15: ( (LSeg ((NE-corner (L~ f)),(N-max (L~ f)))) /\ (L~ f) = {(N-max (L~ f))} & N-max (L~ f) in L~ (mid (f,i,j)) ) by A9, A10, JORDAN3:1, PSCOMP_1:43;
LSeg (((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))),(NE-corner (L~ f))) misses LSeg ((NW-corner (L~ f)),(N-min (L~ f))) by A10, Th59;
then (LSeg ((NE-corner (L~ f)),((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))))) /\ (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) = {} by XBOOLE_0:def_7;
then (LSeg ((NE-corner (L~ f)),((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))))) /\ (L~ (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))) = ((LSeg ((NE-corner (L~ f)),((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))))) /\ (L~ (mid (f,i,j)))) \/ {} by A12, XBOOLE_1:23
.= {((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))))} by A10, A15, A14, A13, JORDAN4:35, ZFMISC_1:124 ;
hence (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(NE-corner (L~ f))*> is S-Sequence_in_R2 by A7, A8, A10, A11, Th61; ::_thesis: verum
end;
registration
let f be non constant standard special_circular_sequence;
cluster L~ f -> being_simple_closed_curve ;
coherence
L~ f is being_simple_closed_curve by JORDAN4:51;
end;
Lm2: for f being non constant standard special_circular_sequence holds LSeg ((S-max (L~ f)),(SE-corner (L~ f))) misses LSeg ((NW-corner (L~ f)),(N-min (L~ f)))
proof
let f be non constant standard special_circular_sequence; ::_thesis: LSeg ((S-max (L~ f)),(SE-corner (L~ f))) misses LSeg ((NW-corner (L~ f)),(N-min (L~ f)))
A1: ( (NW-corner (L~ f)) `2 = N-bound (L~ f) & (N-min (L~ f)) `2 = N-bound (L~ f) ) by EUCLID:52;
assume LSeg ((S-max (L~ f)),(SE-corner (L~ f))) meets LSeg ((NW-corner (L~ f)),(N-min (L~ f))) ; ::_thesis: contradiction
then (LSeg ((S-max (L~ f)),(SE-corner (L~ f)))) /\ (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) <> {} by XBOOLE_0:def_7;
then consider x being set such that
A2: x in (LSeg ((S-max (L~ f)),(SE-corner (L~ f)))) /\ (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) by XBOOLE_0:def_1;
reconsider p = x as Point of (TOP-REAL 2) by A2;
p in LSeg ((NW-corner (L~ f)),(N-min (L~ f))) by A2, XBOOLE_0:def_4;
then ( N-bound (L~ f) <= p `2 & p `2 <= N-bound (L~ f) ) by A1, TOPREAL1:4;
then A3: p `2 = N-bound (L~ f) by XXREAL_0:1;
A4: ( (SE-corner (L~ f)) `2 = S-bound (L~ f) & (S-max (L~ f)) `2 = S-bound (L~ f) ) by EUCLID:52;
x in LSeg ((S-max (L~ f)),(SE-corner (L~ f))) by A2, XBOOLE_0:def_4;
then p `2 <= S-bound (L~ f) by A4, TOPREAL1:4;
hence contradiction by A3, TOPREAL5:16; ::_thesis: verum
end;
Lm3: for f being non constant standard special_circular_sequence
for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) & f /. j = S-max (L~ f) & S-max (L~ f) <> SE-corner (L~ f) holds
(<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2
proof
let f be non constant standard special_circular_sequence; ::_thesis: for i, j being Element of NAT st i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) & f /. j = S-max (L~ f) & S-max (L~ f) <> SE-corner (L~ f) holds
(<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2
set p = NW-corner (L~ f);
set q = SE-corner (L~ f);
let i, j be Element of NAT ; ::_thesis: ( i in dom f & j in dom f & mid (f,i,j) is S-Sequence_in_R2 & f /. i = N-min (L~ f) & N-min (L~ f) <> NW-corner (L~ f) & f /. j = S-max (L~ f) & S-max (L~ f) <> SE-corner (L~ f) implies (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2 )
assume that
A1: i in dom f and
A2: j in dom f and
A3: mid (f,i,j) is S-Sequence_in_R2 and
A4: f /. i = N-min (L~ f) and
A5: N-min (L~ f) <> NW-corner (L~ f) and
A6: f /. j = S-max (L~ f) and
A7: S-max (L~ f) <> SE-corner (L~ f) ; ::_thesis: (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2
set g = <*(NW-corner (L~ f))*> ^ (mid (f,i,j));
A8: <*(NW-corner (L~ f))*> ^ (mid (f,i,j)) is S-Sequence_in_R2 by A1, A2, A3, A4, A5, Th66;
( len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) = (len <*(NW-corner (L~ f))*>) + (len (mid (f,i,j))) & len (mid (f,i,j)) in dom (mid (f,i,j)) ) by A3, FINSEQ_1:22, FINSEQ_5:6;
then A9: (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))) = (mid (f,i,j)) /. (len (mid (f,i,j))) by FINSEQ_4:69;
then A10: (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))) = S-max (L~ f) by A1, A2, A6, Th9;
then A11: (SE-corner (L~ f)) `2 = ((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))) `2 by PSCOMP_1:53;
(mid (f,i,j)) /. 1 = f /. i by A1, A2, Th8;
then A12: L~ (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) = (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) \/ (L~ (mid (f,i,j))) by A3, A4, SPPOL_2:20;
A13: ( 1 <= j & j <= len f ) by A2, FINSEQ_3:25;
A14: ( 1 <= i & i <= len f ) by A1, FINSEQ_3:25;
len (mid (f,i,j)) >= 2 by A3, TOPREAL1:def_8;
then A15: ( (LSeg ((SE-corner (L~ f)),(S-max (L~ f)))) /\ (L~ f) = {(S-max (L~ f))} & S-max (L~ f) in L~ (mid (f,i,j)) ) by A9, A10, JORDAN3:1, PSCOMP_1:59;
LSeg (((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))),(SE-corner (L~ f))) misses LSeg ((NW-corner (L~ f)),(N-min (L~ f))) by A10, Lm2;
then (LSeg ((SE-corner (L~ f)),((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))))) /\ (LSeg ((NW-corner (L~ f)),(N-min (L~ f)))) = {} by XBOOLE_0:def_7;
then (LSeg ((SE-corner (L~ f)),((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))))) /\ (L~ (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))) = ((LSeg ((SE-corner (L~ f)),((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))))))) /\ (L~ (mid (f,i,j)))) \/ {} by A12, XBOOLE_1:23
.= {((<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) /. (len (<*(NW-corner (L~ f))*> ^ (mid (f,i,j)))))} by A10, A15, A14, A13, JORDAN4:35, ZFMISC_1:124 ;
hence (<*(NW-corner (L~ f))*> ^ (mid (f,i,j))) ^ <*(SE-corner (L~ f))*> is S-Sequence_in_R2 by A7, A8, A10, A11, Th61; ::_thesis: verum
end;
begin
theorem Th68: :: SPRECT_2:68
for f being non constant standard special_circular_sequence st f /. 1 = N-min (L~ f) holds
(N-min (L~ f)) .. f < (N-max (L~ f)) .. f
proof
let f be non constant standard special_circular_sequence; ::_thesis: ( f /. 1 = N-min (L~ f) implies (N-min (L~ f)) .. f < (N-max (L~ f)) .. f )
assume f /. 1 = N-min (L~ f) ; ::_thesis: (N-min (L~ f)) .. f < (N-max (L~ f)) .. f
then A1: (N-min (L~ f)) .. f = 1 by FINSEQ_6:43;
A2: N-max (L~ f) in rng f by Th40;
then (N-max (L~ f)) .. f in dom f by FINSEQ_4:20;
then A3: (N-max (L~ f)) .. f >= 1 by FINSEQ_3:25;
N-min (L~ f) in rng f by Th39;
then (N-min (L~ f)) .. f <> (N-max (L~ f)) .. f by A2, Th52, FINSEQ_5:9;
hence (N-min (L~ f)) .. f < (N-max (L~ f)) .. f by A3, A1, XXREAL_0:1; ::_thesis: verum
end;
theorem :: SPRECT_2:69
for f being non constant standard special_circular_sequence st f /. 1 = N-min (L~ f) holds
(N-max (L~ f)) .. f > 1
proof
let f be non constant standard special_circular_sequence; ::_thesis: ( f /. 1 = N-min (L~ f) implies (N-max (L~ f)) .. f > 1 )
assume A1: f /. 1 = N-min (L~ f) ; ::_thesis: (N-max (L~ f)) .. f > 1
then (N-min (L~ f)) .. f = 1 by FINSEQ_6:43;
hence (N-max (L~ f)) .. f > 1 by A1, Th68; ::_thesis: verum
end;
Lm4: for f being non constant standard special_circular_sequence st f /. 1 = N-min (L~ f) holds
(N-min (L~ f)) .. f < (E-max (L~ f)) .. f
proof
let f be non constant standard special_circular_sequence; ::_thesis: ( f /. 1 = N-min (L~ f) implies (N-min (L~ f)) .. f < (E-max (L~ f)) .. f )
A1: N-min (L~ f) in rng f by Th39;
assume f /. 1 = N-min (L~ f) ; ::_thesis: (N-min (L~ f)) .. f < (E-max (L~ f)) .. f
then A2: (N-min (L~ f)) .. f = 1 by FINSEQ_6:43;
(N-max (L~ f)) `1 <= (NE-corner (L~ f)) `1 by PSCOMP_1:38;
then (N-max (L~ f)) `1 <= E-bound (L~ f) by EUCLID:52;
then (N-min (L~ f)) `1 < E-bound (L~ f) by Th51, XXREAL_0:2;
then A3: (N-min (L~ f)) `1 < (E-max (L~ f)) `1 by EUCLID:52;
A4: E-max (L~ f) in rng f by Th46;
then (E-max (L~ f)) .. f >= 1 by FINSEQ_4:21;
hence (N-min (L~ f)) .. f < (E-max (L~ f)) .. f by A4, A1, A3, A2, FINSEQ_5:9, XXREAL_0:1; ::_thesis: verum
end;
Lm5: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(N-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (N-max (L~ z)) .. z < (S-max (L~ z)) .. z )
set i1 = (N-max (L~ z)) .. z;
set i2 = (S-max (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: (N-max (L~ z)) .. z >= (S-max (L~ z)) .. z ; ::_thesis: contradiction
A3: N-min (L~ z) <> N-max (L~ z) by Th52;
A4: S-max (L~ z) in rng z by Th42;
then A5: (S-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A6: (S-max (L~ z)) .. z <= len z by FINSEQ_3:25;
A7: z /. ((S-max (L~ z)) .. z) = z . ((S-max (L~ z)) .. z) by A5, PARTFUN1:def_6
.= S-max (L~ z) by A4, FINSEQ_4:19 ;
then A8: (z /. ((S-max (L~ z)) .. z)) `2 = S-bound (L~ z) by EUCLID:52;
A9: 1 <= (S-max (L~ z)) .. z by A5, FINSEQ_3:25;
A10: (S-max (L~ z)) .. z <> 0 by A5, FINSEQ_3:25;
(z /. 1) `2 = N-bound (L~ z) by A1, EUCLID:52;
then A11: (S-max (L~ z)) .. z <> 1 by A8, TOPREAL5:16;
z /. 2 in N-most (L~ z) by A1, Th30;
then A12: (z /. 2) `2 = (N-min (L~ z)) `2 by PSCOMP_1:39
.= N-bound (L~ z) by EUCLID:52 ;
then (S-max (L~ z)) .. z <> 2 by A8, TOPREAL5:16;
then A13: (S-max (L~ z)) .. z > 2 by A10, A11, NAT_1:26;
then reconsider h = mid (z,((S-max (L~ z)) .. z),2) as S-Sequence_in_R2 by A6, Th37;
A14: 2 <= len z by NAT_D:60;
then A15: 2 in dom z by FINSEQ_3:25;
then A16: (h /. (len h)) `2 = N-bound (L~ z) by A5, A12, Th9;
h /. 1 = S-max (L~ z) by A5, A7, A15, Th8;
then A17: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
h is_in_the_area_of z by A5, A15, Th21, Th22;
then A18: h is_a_v.c._for z by A17, A16, Def3;
A19: N-max (L~ z) in rng z by Th40;
then A20: (N-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A21: z /. ((N-max (L~ z)) .. z) = z . ((N-max (L~ z)) .. z) by PARTFUN1:def_6
.= N-max (L~ z) by A19, FINSEQ_4:19 ;
A22: (N-max (L~ z)) .. z <= len z by A20, FINSEQ_3:25;
z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
then A23: (N-max (L~ z)) .. z < len z by A22, A21, A3, XXREAL_0:1;
then ((N-max (L~ z)) .. z) + 1 <= len z by NAT_1:13;
then (len z) - ((N-max (L~ z)) .. z) >= 1 by XREAL_1:19;
then (len z) -' ((N-max (L~ z)) .. z) >= 1 by NAT_D:39;
then A24: ((len z) -' ((N-max (L~ z)) .. z)) + 1 >= 1 + 1 by XREAL_1:6;
( (N-max (L~ z)) `2 = N-bound (L~ z) & (S-max (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then z /. ((N-max (L~ z)) .. z) <> z /. ((S-max (L~ z)) .. z) by A7, A21, TOPREAL5:16;
then A25: (N-max (L~ z)) .. z > (S-max (L~ z)) .. z by A2, XXREAL_0:1;
then (N-max (L~ z)) .. z > 1 by A9, XXREAL_0:2;
then reconsider M = mid (z,(len z),((N-max (L~ z)) .. z)) as S-Sequence_in_R2 by A23, Th37;
A26: 1 in dom M by FINSEQ_5:6;
A27: len z in dom z by FINSEQ_5:6;
then A28: M /. (len M) = z /. ((N-max (L~ z)) .. z) by A20, Th9
.= N-max (L~ z) by A19, FINSEQ_5:38 ;
A29: L~ M misses L~ h by A22, A25, A13, Th49;
A30: 2 <= len h by TOPREAL1:def_8;
1 <= (N-max (L~ z)) .. z by A20, FINSEQ_3:25;
then A31: len M = ((len z) -' ((N-max (L~ z)) .. z)) + 1 by A22, JORDAN4:9;
then A32: M /. (len M) in L~ M by A24, JORDAN3:1;
A33: z /. 1 = z /. (len z) by FINSEQ_6:def_1;
then A34: M /. 1 = z /. 1 by A20, A27, Th8;
percases ( ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) = N-max (L~ z) ) or ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) = N-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) ) ;
supposethat A35: NW-corner (L~ z) = N-min (L~ z) and
A36: NE-corner (L~ z) = N-max (L~ z) ; ::_thesis: contradiction
A37: (M /. (len M)) `1 = E-bound (L~ z) by A28, A36, EUCLID:52;
M /. 1 = z /. (len z) by A20, A27, Th8;
then A38: (M /. 1) `1 = W-bound (L~ z) by A1, A33, A35, EUCLID:52;
M is_in_the_area_of z by A20, A27, Th21, Th22;
then M is_a_h.c._for z by A38, A37, Def2;
hence contradiction by A18, A29, A31, A24, A30, Th29; ::_thesis: verum
end;
supposethat A39: NW-corner (L~ z) = N-min (L~ z) and
A40: NE-corner (L~ z) <> N-max (L~ z) ; ::_thesis: contradiction
reconsider g = M ^ <*(NE-corner (L~ z))*> as S-Sequence_in_R2 by A20, A21, A27, A40, Th62;
A41: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((M /. (len M)),(NE-corner (L~ z)))) ) by SPPOL_2:19, TOPREAL1:def_8;
len g = (len M) + (len <*(NE-corner (L~ z))*>) by FINSEQ_1:22
.= (len M) + 1 by FINSEQ_1:39 ;
then g /. (len g) = NE-corner (L~ z) by FINSEQ_4:67;
then A42: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M is_in_the_area_of z & <*(NE-corner (L~ z))*> is_in_the_area_of z ) by A20, A27, Th21, Th22, Th25;
then A43: g is_in_the_area_of z by Th24;
(LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ z) by A9, A6, A14, JORDAN4:35, XBOOLE_1:26;
then A44: (LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ h) c= {(M /. (len M))} by A28, PSCOMP_1:43;
g /. 1 = M /. 1 by A26, FINSEQ_4:68
.= z /. 1 by A20, A27, A33, Th8 ;
then (g /. 1) `1 = W-bound (L~ z) by A1, A39, EUCLID:52;
then g is_a_h.c._for z by A43, A42, Def2;
hence contradiction by A18, A29, A32, A30, A41, A44, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposethat A45: NW-corner (L~ z) <> N-min (L~ z) and
A46: NE-corner (L~ z) = N-max (L~ z) ; ::_thesis: contradiction
reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A20, A27, A33, A45, Th66;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((N-max (L~ z)) .. z) by A20, A27, Th9
.= N-max (L~ z) by A19, FINSEQ_5:38 ;
then A47: (g /. (len g)) `1 = E-bound (L~ z) by A46, EUCLID:52;
A48: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A49: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A9, A6, A14, JORDAN4:35, XBOOLE_1:26;
then A50: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A34, PSCOMP_1:43;
A51: M /. 1 in L~ M by A31, A24, JORDAN3:1;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A20, A27, Th21, Th22, Th26;
then g is_in_the_area_of z by Th24;
then g is_a_h.c._for z by A49, A47, Def2;
hence contradiction by A18, A29, A30, A48, A50, A51, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposeA52: ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) ; ::_thesis: contradiction
set K = <*(NW-corner (L~ z))*> ^ M;
reconsider g = (<*(NW-corner (L~ z))*> ^ M) ^ <*(NE-corner (L~ z))*> as S-Sequence_in_R2 by A1, A20, A21, A27, A33, A52, Lm1;
1 in dom (<*(NW-corner (L~ z))*> ^ M) by FINSEQ_5:6;
then g /. 1 = (<*(NW-corner (L~ z))*> ^ M) /. 1 by FINSEQ_4:68
.= NW-corner (L~ z) by FINSEQ_5:15 ;
then A53: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
len g = (len (<*(NW-corner (L~ z))*> ^ M)) + (len <*(NE-corner (L~ z))*>) by FINSEQ_1:22
.= (len (<*(NW-corner (L~ z))*> ^ M)) + 1 by FINSEQ_1:39 ;
then g /. (len g) = NE-corner (L~ z) by FINSEQ_4:67;
then A54: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A20, A27, Th21, Th22, Th26;
then A55: <*(NW-corner (L~ z))*> ^ M is_in_the_area_of z by Th24;
<*(NE-corner (L~ z))*> is_in_the_area_of z by Th25;
then g is_in_the_area_of z by A55, Th24;
then A56: g is_a_h.c._for z by A53, A54, Def2;
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) by FINSEQ_1:22;
then len (<*(NW-corner (L~ z))*> ^ M) >= len M by NAT_1:11;
then len (<*(NW-corner (L~ z))*> ^ M) >= 2 by A31, A24, XXREAL_0:2;
then A57: (<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) in L~ (<*(NW-corner (L~ z))*> ^ M) by JORDAN3:1;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A9, A6, A14, JORDAN4:35, XBOOLE_1:26;
then A58: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A34, PSCOMP_1:43;
( L~ (<*(NW-corner (L~ z))*> ^ M) = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) & M /. 1 in L~ M ) by A31, A24, JORDAN3:1, SPPOL_2:20;
then A59: L~ (<*(NW-corner (L~ z))*> ^ M) misses L~ h by A29, A58, ZFMISC_1:125;
( len M in dom M & len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then A60: (<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) = M /. (len M) by FINSEQ_4:69
.= z /. ((N-max (L~ z)) .. z) by A20, A27, Th9
.= N-max (L~ z) by A19, FINSEQ_5:38 ;
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ h) c= (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ z) by A9, A6, A14, JORDAN4:35, XBOOLE_1:26;
then A61: (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ h) c= {((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)))} by A60, PSCOMP_1:43;
( len g >= 2 & L~ g = (L~ (<*(NW-corner (L~ z))*> ^ M)) \/ (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) ) by SPPOL_2:19, TOPREAL1:def_8;
hence contradiction by A18, A30, A56, A59, A57, A61, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
Lm6: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(N-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (N-max (L~ z)) .. z < (S-min (L~ z)) .. z )
set i1 = (N-max (L~ z)) .. z;
set i2 = (S-min (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: (N-max (L~ z)) .. z >= (S-min (L~ z)) .. z ; ::_thesis: contradiction
A3: N-min (L~ z) <> N-max (L~ z) by Th52;
z /. 2 in N-most (L~ z) by A1, Th30;
then A4: (z /. 2) `2 = (N-min (L~ z)) `2 by PSCOMP_1:39
.= N-bound (L~ z) by EUCLID:52 ;
A5: S-bound (L~ z) < N-bound (L~ z) by TOPREAL5:16;
A6: S-min (L~ z) in rng z by Th41;
then A7: (S-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A8: (S-min (L~ z)) .. z <= len z by FINSEQ_3:25;
A9: z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A7, PARTFUN1:def_6
.= S-min (L~ z) by A6, FINSEQ_4:19 ;
then A10: (z /. ((S-min (L~ z)) .. z)) `2 = S-bound (L~ z) by EUCLID:52;
A11: 1 <= (S-min (L~ z)) .. z by A7, FINSEQ_3:25;
A12: (S-min (L~ z)) .. z <> 0 by A7, FINSEQ_3:25;
(z /. 1) `2 = N-bound (L~ z) by A1, EUCLID:52;
then A13: (S-min (L~ z)) .. z > 2 by A4, A12, A10, A5, NAT_1:26;
then reconsider h = mid (z,((S-min (L~ z)) .. z),2) as S-Sequence_in_R2 by A8, Th37;
A14: 2 <= len z by NAT_D:60;
then A15: 2 in dom z by FINSEQ_3:25;
then h /. 1 = S-min (L~ z) by A7, A9, Th8;
then A16: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. 2 ) by A7, A15, Th9, Th21, Th22;
then A17: ( len h >= 2 & h is_a_v.c._for z ) by A4, A16, Def3, TOPREAL1:def_8;
A18: N-max (L~ z) in rng z by Th40;
then A19: (N-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A20: z /. ((N-max (L~ z)) .. z) = z . ((N-max (L~ z)) .. z) by PARTFUN1:def_6
.= N-max (L~ z) by A18, FINSEQ_4:19 ;
A21: (N-max (L~ z)) .. z <= len z by A19, FINSEQ_3:25;
z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
then A22: (N-max (L~ z)) .. z < len z by A21, A20, A3, XXREAL_0:1;
then ((N-max (L~ z)) .. z) + 1 <= len z by NAT_1:13;
then (len z) - ((N-max (L~ z)) .. z) >= 1 by XREAL_1:19;
then (len z) -' ((N-max (L~ z)) .. z) >= 1 by NAT_D:39;
then A23: ((len z) -' ((N-max (L~ z)) .. z)) + 1 >= 1 + 1 by XREAL_1:6;
( (N-max (L~ z)) `2 = N-bound (L~ z) & (S-min (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then z /. ((N-max (L~ z)) .. z) <> z /. ((S-min (L~ z)) .. z) by A9, A20, TOPREAL5:16;
then A24: (N-max (L~ z)) .. z > (S-min (L~ z)) .. z by A2, XXREAL_0:1;
then (N-max (L~ z)) .. z > 1 by A11, XXREAL_0:2;
then reconsider M = mid (z,(len z),((N-max (L~ z)) .. z)) as S-Sequence_in_R2 by A22, Th37;
A25: 1 in dom M by FINSEQ_5:6;
A26: len z in dom z by FINSEQ_5:6;
then A27: M /. (len M) = z /. ((N-max (L~ z)) .. z) by A19, Th9
.= N-max (L~ z) by A18, FINSEQ_5:38 ;
A28: L~ M misses L~ h by A21, A24, A13, Th49;
1 <= (N-max (L~ z)) .. z by A19, FINSEQ_3:25;
then A29: len M = ((len z) -' ((N-max (L~ z)) .. z)) + 1 by A21, JORDAN4:9;
then A30: M /. (len M) in L~ M by A23, JORDAN3:1;
A31: z /. 1 = z /. (len z) by FINSEQ_6:def_1;
then A32: M /. 1 = z /. 1 by A19, A26, Th8;
percases ( ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) = N-max (L~ z) ) or ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) = N-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) ) ;
supposethat A33: NW-corner (L~ z) = N-min (L~ z) and
A34: NE-corner (L~ z) = N-max (L~ z) ; ::_thesis: contradiction
A35: (M /. (len M)) `1 = E-bound (L~ z) by A27, A34, EUCLID:52;
M /. 1 = z /. (len z) by A19, A26, Th8;
then A36: (M /. 1) `1 = W-bound (L~ z) by A1, A31, A33, EUCLID:52;
M is_in_the_area_of z by A19, A26, Th21, Th22;
then M is_a_h.c._for z by A36, A35, Def2;
hence contradiction by A17, A28, A29, A23, Th29; ::_thesis: verum
end;
supposethat A37: NW-corner (L~ z) = N-min (L~ z) and
A38: NE-corner (L~ z) <> N-max (L~ z) ; ::_thesis: contradiction
reconsider g = M ^ <*(NE-corner (L~ z))*> as S-Sequence_in_R2 by A19, A20, A26, A38, Th62;
A39: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((M /. (len M)),(NE-corner (L~ z)))) ) by SPPOL_2:19, TOPREAL1:def_8;
len g = (len M) + (len <*(NE-corner (L~ z))*>) by FINSEQ_1:22
.= (len M) + 1 by FINSEQ_1:39 ;
then g /. (len g) = NE-corner (L~ z) by FINSEQ_4:67;
then A40: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M is_in_the_area_of z & <*(NE-corner (L~ z))*> is_in_the_area_of z ) by A19, A26, Th21, Th22, Th25;
then A41: g is_in_the_area_of z by Th24;
(LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ z) by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A42: (LSeg ((M /. (len M)),(NE-corner (L~ z)))) /\ (L~ h) c= {(M /. (len M))} by A27, PSCOMP_1:43;
g /. 1 = M /. 1 by A25, FINSEQ_4:68
.= z /. 1 by A19, A26, A31, Th8 ;
then (g /. 1) `1 = W-bound (L~ z) by A1, A37, EUCLID:52;
then g is_a_h.c._for z by A41, A40, Def2;
hence contradiction by A17, A28, A30, A39, A42, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposethat A43: NW-corner (L~ z) <> N-min (L~ z) and
A44: NE-corner (L~ z) = N-max (L~ z) ; ::_thesis: contradiction
reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A19, A26, A31, A43, Th66;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((N-max (L~ z)) .. z) by A19, A26, Th9
.= N-max (L~ z) by A18, FINSEQ_5:38 ;
then A45: (g /. (len g)) `1 = E-bound (L~ z) by A44, EUCLID:52;
A46: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A47: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A48: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A32, PSCOMP_1:43;
A49: M /. 1 in L~ M by A29, A23, JORDAN3:1;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A19, A26, Th21, Th22, Th26;
then g is_in_the_area_of z by Th24;
then g is_a_h.c._for z by A47, A45, Def2;
hence contradiction by A17, A28, A46, A48, A49, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposeA50: ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) <> N-max (L~ z) ) ; ::_thesis: contradiction
set K = <*(NW-corner (L~ z))*> ^ M;
reconsider g = (<*(NW-corner (L~ z))*> ^ M) ^ <*(NE-corner (L~ z))*> as S-Sequence_in_R2 by A1, A19, A20, A26, A31, A50, Lm1;
1 in dom (<*(NW-corner (L~ z))*> ^ M) by FINSEQ_5:6;
then g /. 1 = (<*(NW-corner (L~ z))*> ^ M) /. 1 by FINSEQ_4:68
.= NW-corner (L~ z) by FINSEQ_5:15 ;
then A51: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
len g = (len (<*(NW-corner (L~ z))*> ^ M)) + (len <*(NE-corner (L~ z))*>) by FINSEQ_1:22
.= (len (<*(NW-corner (L~ z))*> ^ M)) + 1 by FINSEQ_1:39 ;
then g /. (len g) = NE-corner (L~ z) by FINSEQ_4:67;
then A52: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A19, A26, Th21, Th22, Th26;
then A53: <*(NW-corner (L~ z))*> ^ M is_in_the_area_of z by Th24;
<*(NE-corner (L~ z))*> is_in_the_area_of z by Th25;
then g is_in_the_area_of z by A53, Th24;
then A54: g is_a_h.c._for z by A51, A52, Def2;
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) by FINSEQ_1:22;
then len (<*(NW-corner (L~ z))*> ^ M) >= len M by NAT_1:11;
then len (<*(NW-corner (L~ z))*> ^ M) >= 2 by A29, A23, XXREAL_0:2;
then A55: (<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) in L~ (<*(NW-corner (L~ z))*> ^ M) by JORDAN3:1;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A56: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A32, PSCOMP_1:43;
( L~ (<*(NW-corner (L~ z))*> ^ M) = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) & M /. 1 in L~ M ) by A29, A23, JORDAN3:1, SPPOL_2:20;
then A57: L~ (<*(NW-corner (L~ z))*> ^ M) misses L~ h by A28, A56, ZFMISC_1:125;
( len M in dom M & len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then A58: (<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) = M /. (len M) by FINSEQ_4:69
.= z /. ((N-max (L~ z)) .. z) by A19, A26, Th9
.= N-max (L~ z) by A18, FINSEQ_5:38 ;
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ h) c= (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ z) by A11, A8, A14, JORDAN4:35, XBOOLE_1:26;
then A59: (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) /\ (L~ h) c= {((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)))} by A58, PSCOMP_1:43;
( len g >= 2 & L~ g = (L~ (<*(NW-corner (L~ z))*> ^ M)) \/ (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(NE-corner (L~ z)))) ) by SPPOL_2:19, TOPREAL1:def_8;
hence contradiction by A17, A54, A57, A55, A59, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
theorem :: SPRECT_2:70
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) & N-max (L~ z) <> E-max (L~ z) holds
(N-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) & N-max (L~ z) <> E-max (L~ z) implies (N-max (L~ z)) .. z < (E-max (L~ z)) .. z )
set i1 = (N-max (L~ z)) .. z;
set i2 = (E-max (L~ z)) .. z;
set j = (S-max (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: ( N-max (L~ z) <> E-max (L~ z) & (N-max (L~ z)) .. z >= (E-max (L~ z)) .. z ) ; ::_thesis: contradiction
(N-min (L~ z)) .. z = 1 by A1, FINSEQ_6:43;
then A3: 1 < (E-max (L~ z)) .. z by A1, Lm4;
( (N-min (L~ z)) `2 = N-bound (L~ z) & (S-max (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then A4: N-min (L~ z) <> S-max (L~ z) by TOPREAL5:16;
A5: S-max (L~ z) in rng z by Th42;
then A6: (S-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A7: z /. ((S-max (L~ z)) .. z) = z . ((S-max (L~ z)) .. z) by PARTFUN1:def_6
.= S-max (L~ z) by A5, FINSEQ_4:19 ;
A8: (S-max (L~ z)) .. z <= len z by A6, FINSEQ_3:25;
z /. (len z) = z /. 1 by FINSEQ_6:def_1;
then A9: (S-max (L~ z)) .. z < len z by A1, A8, A7, A4, XXREAL_0:1;
A10: N-max (L~ z) in rng z by Th40;
then A11: (N-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A12: 1 <= (N-max (L~ z)) .. z by FINSEQ_3:25;
A13: z /. ((N-max (L~ z)) .. z) = z . ((N-max (L~ z)) .. z) by A11, PARTFUN1:def_6
.= N-max (L~ z) by A10, FINSEQ_4:19 ;
A14: (S-max (L~ z)) .. z > (N-max (L~ z)) .. z by A1, Lm5;
then reconsider h = mid (z,((S-max (L~ z)) .. z),((N-max (L~ z)) .. z)) as S-Sequence_in_R2 by A12, A9, Th37;
h /. 1 = S-max (L~ z) by A11, A6, A7, Th8;
then A15: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
h /. (len h) = z /. ((N-max (L~ z)) .. z) by A11, A6, Th9;
then A16: (h /. (len h)) `2 = N-bound (L~ z) by A13, EUCLID:52;
h is_in_the_area_of z by A11, A6, Th21, Th22;
then A17: h is_a_v.c._for z by A15, A16, Def3;
A18: 1 <= (S-max (L~ z)) .. z by A6, FINSEQ_3:25;
A19: (N-max (L~ z)) .. z <= len z by A11, FINSEQ_3:25;
A20: E-max (L~ z) in rng z by Th46;
then A21: (E-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A22: ( 1 <= (E-max (L~ z)) .. z & (E-max (L~ z)) .. z <= len z ) by FINSEQ_3:25;
z /. ((E-max (L~ z)) .. z) = z . ((E-max (L~ z)) .. z) by A21, PARTFUN1:def_6
.= E-max (L~ z) by A20, FINSEQ_4:19 ;
then A23: (N-max (L~ z)) .. z > (E-max (L~ z)) .. z by A2, A13, XXREAL_0:1;
then (E-max (L~ z)) .. z < len z by A19, XXREAL_0:2;
then reconsider M = mid (z,1,((E-max (L~ z)) .. z)) as S-Sequence_in_R2 by A3, Th38;
A24: len M >= 2 by TOPREAL1:def_8;
A25: 1 in dom z by FINSEQ_5:6;
then A26: M /. (len M) = z /. ((E-max (L~ z)) .. z) by A21, Th9
.= E-max (L~ z) by A20, FINSEQ_5:38 ;
A27: ( len h >= 2 & L~ M misses L~ h ) by A14, A9, A3, A23, Th48, TOPREAL1:def_8;
percases ( NW-corner (L~ z) = N-min (L~ z) or NW-corner (L~ z) <> N-min (L~ z) ) ;
supposeA28: NW-corner (L~ z) = N-min (L~ z) ; ::_thesis: contradiction
M /. 1 = z /. 1 by A25, A21, Th8;
then A29: (M /. 1) `1 = W-bound (L~ z) by A1, A28, EUCLID:52;
( M is_in_the_area_of z & (M /. (len M)) `1 = E-bound (L~ z) ) by A25, A21, A26, Th21, Th22, EUCLID:52;
then M is_a_h.c._for z by A29, Def2;
hence contradiction by A17, A24, A27, Th29; ::_thesis: verum
end;
suppose NW-corner (L~ z) <> N-min (L~ z) ; ::_thesis: contradiction
then reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A25, A21, Th66;
A30: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A31: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
len M = (((E-max (L~ z)) .. z) -' 1) + 1 by A22, JORDAN4:8
.= (E-max (L~ z)) .. z by A3, XREAL_1:235 ;
then len M >= 1 + 1 by A3, NAT_1:13;
then A32: M /. 1 in L~ M by JORDAN3:1;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((E-max (L~ z)) .. z) by A25, A21, Th9
.= E-max (L~ z) by A20, FINSEQ_5:38 ;
then A33: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M /. 1 = z /. 1 & (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) ) by A25, A12, A19, A18, A8, A21, Th8, JORDAN4:35, XBOOLE_1:26;
then A34: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, PSCOMP_1:43;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A25, A21, Th21, Th22, Th26;
then g is_in_the_area_of z by Th24;
then g is_a_h.c._for z by A31, A33, Def2;
hence contradiction by A17, A27, A30, A34, A32, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
Lm7: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (E-max (L~ z)) .. z < (S-max (L~ z)) .. z )
set i1 = (E-max (L~ z)) .. z;
set i2 = (S-max (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: (E-max (L~ z)) .. z >= (S-max (L~ z)) .. z ; ::_thesis: contradiction
A3: (N-min (L~ z)) `1 < (N-max (L~ z)) `1 by Th51;
z /. 2 in N-most (L~ z) by A1, Th30;
then A4: (z /. 2) `2 = (N-min (L~ z)) `2 by PSCOMP_1:39
.= N-bound (L~ z) by EUCLID:52 ;
E-min (L~ z) in L~ z by SPRECT_1:14;
then A5: S-bound (L~ z) <= (E-min (L~ z)) `2 by PSCOMP_1:24;
A6: S-bound (L~ z) < N-bound (L~ z) by TOPREAL5:16;
A7: S-max (L~ z) in rng z by Th42;
then A8: (S-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A9: (S-max (L~ z)) .. z <= len z by FINSEQ_3:25;
A10: (S-max (L~ z)) .. z <> 0 by A8, FINSEQ_3:25;
A11: z /. ((S-max (L~ z)) .. z) = z . ((S-max (L~ z)) .. z) by A8, PARTFUN1:def_6
.= S-max (L~ z) by A7, FINSEQ_4:19 ;
then A12: (z /. ((S-max (L~ z)) .. z)) `2 = S-bound (L~ z) by EUCLID:52;
A13: 1 <= (S-max (L~ z)) .. z by A8, FINSEQ_3:25;
(z /. 1) `2 = N-bound (L~ z) by A1, EUCLID:52;
then A14: (S-max (L~ z)) .. z > 2 by A4, A10, A12, A6, NAT_1:26;
then reconsider h = mid (z,((S-max (L~ z)) .. z),2) as S-Sequence_in_R2 by A9, Th37;
A15: 2 <= len z by NAT_D:60;
then A16: 2 in dom z by FINSEQ_3:25;
then h /. 1 = S-max (L~ z) by A8, A11, Th8;
then A17: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. 2 ) by A8, A16, Th9, Th21, Th22;
then A18: ( len h >= 2 & h is_a_v.c._for z ) by A4, A17, Def3, TOPREAL1:def_8;
N-max (L~ z) in L~ z by SPRECT_1:11;
then A19: (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
A20: E-max (L~ z) in rng z by Th46;
then A21: (E-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A22: z /. ((E-max (L~ z)) .. z) = z . ((E-max (L~ z)) .. z) by PARTFUN1:def_6
.= E-max (L~ z) by A20, FINSEQ_4:19 ;
A23: (E-max (L~ z)) .. z <= len z by A21, FINSEQ_3:25;
z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
then (E-max (L~ z)) .. z <> len z by A22, A3, A19, EUCLID:52;
then A24: (E-max (L~ z)) .. z < len z by A23, XXREAL_0:1;
then ((E-max (L~ z)) .. z) + 1 <= len z by NAT_1:13;
then (len z) - ((E-max (L~ z)) .. z) >= 1 by XREAL_1:19;
then (len z) -' ((E-max (L~ z)) .. z) >= 1 by NAT_D:39;
then A25: ((len z) -' ((E-max (L~ z)) .. z)) + 1 >= 1 + 1 by XREAL_1:6;
(E-min (L~ z)) `2 < (E-max (L~ z)) `2 by Th53;
then E-max (L~ z) <> S-max (L~ z) by A5, EUCLID:52;
then A26: (E-max (L~ z)) .. z > (S-max (L~ z)) .. z by A2, A11, A22, XXREAL_0:1;
then (E-max (L~ z)) .. z > 1 by A13, XXREAL_0:2;
then reconsider M = mid (z,(len z),((E-max (L~ z)) .. z)) as S-Sequence_in_R2 by A24, Th37;
A27: len M >= 2 by TOPREAL1:def_8;
1 <= (E-max (L~ z)) .. z by A21, FINSEQ_3:25;
then A28: len M = ((len z) -' ((E-max (L~ z)) .. z)) + 1 by A23, JORDAN4:9;
A29: len z in dom z by FINSEQ_5:6;
then A30: M /. (len M) = z /. ((E-max (L~ z)) .. z) by A21, Th9
.= E-max (L~ z) by A20, FINSEQ_5:38 ;
A31: L~ M misses L~ h by A23, A26, A14, Th49;
A32: z /. 1 = z /. (len z) by FINSEQ_6:def_1;
then A33: M /. 1 = z /. 1 by A21, A29, Th8;
percases ( NW-corner (L~ z) = N-min (L~ z) or NW-corner (L~ z) <> N-min (L~ z) ) ;
supposeA34: NW-corner (L~ z) = N-min (L~ z) ; ::_thesis: contradiction
M /. 1 = z /. (len z) by A21, A29, Th8;
then A35: (M /. 1) `1 = W-bound (L~ z) by A1, A32, A34, EUCLID:52;
( M is_in_the_area_of z & (M /. (len M)) `1 = E-bound (L~ z) ) by A21, A29, A30, Th21, Th22, EUCLID:52;
then M is_a_h.c._for z by A35, Def2;
hence contradiction by A18, A27, A31, Th29; ::_thesis: verum
end;
suppose NW-corner (L~ z) <> N-min (L~ z) ; ::_thesis: contradiction
then reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A21, A29, A32, Th66;
A36: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A37: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A13, A9, A15, JORDAN4:35, XBOOLE_1:26;
then A38: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A33, PSCOMP_1:43;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((E-max (L~ z)) .. z) by A21, A29, Th9
.= E-max (L~ z) by A20, FINSEQ_5:38 ;
then A39: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
A40: M /. 1 in L~ M by A28, A25, JORDAN3:1;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A21, A29, Th21, Th22, Th26;
then g is_in_the_area_of z by Th24;
then g is_a_h.c._for z by A37, A39, Def2;
hence contradiction by A18, A31, A36, A38, A40, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
Lm8: for f being non constant standard special_circular_sequence holds (LSeg ((N-min (L~ f)),(NW-corner (L~ f)))) /\ (LSeg ((NE-corner (L~ f)),(E-max (L~ f)))) = {}
proof
let f be non constant standard special_circular_sequence; ::_thesis: (LSeg ((N-min (L~ f)),(NW-corner (L~ f)))) /\ (LSeg ((NE-corner (L~ f)),(E-max (L~ f)))) = {}
A1: ( (NE-corner (L~ f)) `1 = E-bound (L~ f) & (E-max (L~ f)) `1 = E-bound (L~ f) ) by EUCLID:52;
assume (LSeg ((N-min (L~ f)),(NW-corner (L~ f)))) /\ (LSeg ((NE-corner (L~ f)),(E-max (L~ f)))) <> {} ; ::_thesis: contradiction
then consider x being set such that
A2: x in (LSeg ((N-min (L~ f)),(NW-corner (L~ f)))) /\ (LSeg ((NE-corner (L~ f)),(E-max (L~ f)))) by XBOOLE_0:def_1;
reconsider p = x as Point of (TOP-REAL 2) by A2;
p in LSeg ((NE-corner (L~ f)),(E-max (L~ f))) by A2, XBOOLE_0:def_4;
then ( E-bound (L~ f) <= p `1 & p `1 <= E-bound (L~ f) ) by A1, TOPREAL1:3;
then A3: p `1 = E-bound (L~ f) by XXREAL_0:1;
A4: (NW-corner (L~ f)) `1 <= (N-min (L~ f)) `1 by PSCOMP_1:38;
(N-max (L~ f)) `1 <= (NE-corner (L~ f)) `1 by PSCOMP_1:38;
then A5: (N-min (L~ f)) `1 < (NE-corner (L~ f)) `1 by Th51, XXREAL_0:2;
x in LSeg ((N-min (L~ f)),(NW-corner (L~ f))) by A2, XBOOLE_0:def_4;
then p `1 <= (N-min (L~ f)) `1 by A4, TOPREAL1:3;
hence contradiction by A3, A5, EUCLID:52; ::_thesis: verum
end;
theorem :: SPRECT_2:71
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (E-max (L~ z)) .. z < (E-min (L~ z)) .. z )
set i1 = (E-max (L~ z)) .. z;
set i2 = (E-min (L~ z)) .. z;
set j = (S-max (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: (E-max (L~ z)) .. z >= (E-min (L~ z)) .. z ; ::_thesis: contradiction
A3: (E-max (L~ z)) .. z < (S-max (L~ z)) .. z by A1, Lm7;
A4: E-min (L~ z) in rng z by Th45;
then A5: (E-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A6: 1 <= (E-min (L~ z)) .. z by FINSEQ_3:25;
A7: z /. ((E-min (L~ z)) .. z) = z . ((E-min (L~ z)) .. z) by A5, PARTFUN1:def_6
.= E-min (L~ z) by A4, FINSEQ_4:19 ;
N-max (L~ z) in L~ z by SPRECT_1:11;
then (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
then (N-min (L~ z)) `1 < E-bound (L~ z) by Th51, XXREAL_0:2;
then (N-min (L~ z)) `1 < (E-min (L~ z)) `1 by EUCLID:52;
then A8: (E-min (L~ z)) .. z > 1 by A1, A6, A7, XXREAL_0:1;
A9: (E-min (L~ z)) .. z <= len z by A5, FINSEQ_3:25;
then A10: 1 <= len z by A6, XXREAL_0:2;
A11: (S-max (L~ z)) `2 = S-bound (L~ z) by EUCLID:52;
A12: ( S-bound (L~ z) < N-bound (L~ z) & (N-min (L~ z)) `2 = N-bound (L~ z) ) by EUCLID:52, TOPREAL5:16;
A13: S-max (L~ z) in rng z by Th42;
then A14: (S-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A15: (S-max (L~ z)) .. z <= len z by FINSEQ_3:25;
A16: 1 <= (S-max (L~ z)) .. z by A14, FINSEQ_3:25;
A17: E-max (L~ z) in rng z by Th46;
then A18: (E-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A19: z /. ((E-max (L~ z)) .. z) = z . ((E-max (L~ z)) .. z) by PARTFUN1:def_6
.= E-max (L~ z) by A17, FINSEQ_4:19 ;
A20: 1 <= (E-max (L~ z)) .. z by A18, FINSEQ_3:25;
A21: (E-max (L~ z)) .. z <= len z by A18, FINSEQ_3:25;
(E-min (L~ z)) `2 < (E-max (L~ z)) `2 by Th53;
then A22: (E-max (L~ z)) .. z > (E-min (L~ z)) .. z by A2, A7, A19, XXREAL_0:1;
then (E-min (L~ z)) .. z < len z by A21, XXREAL_0:2;
then reconsider M = mid (z,1,((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A8, Th38;
A23: 1 in dom z by FINSEQ_5:6;
then A24: M /. 1 = z /. 1 by A5, Th8;
(E-max (L~ z)) .. z > 1 by A6, A22, XXREAL_0:2;
then reconsider h = mid (z,((S-max (L~ z)) .. z),((E-max (L~ z)) .. z)) as S-Sequence_in_R2 by A15, A3, Th37;
A25: h /. (len h) = z /. ((E-max (L~ z)) .. z) by A18, A14, Th9;
A26: z /. ((S-max (L~ z)) .. z) = z . ((S-max (L~ z)) .. z) by A14, PARTFUN1:def_6
.= S-max (L~ z) by A13, FINSEQ_4:19 ;
then h /. 1 = S-max (L~ z) by A18, A14, Th8;
then A27: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
M /. (len M) = z /. ((E-min (L~ z)) .. z) by A23, A5, Th9
.= E-min (L~ z) by A4, FINSEQ_5:38 ;
then A28: (M /. (len M)) `1 = E-bound (L~ z) by EUCLID:52;
A29: M is_in_the_area_of z by A23, A5, Th21, Th22;
len h >= 1 by A18, A14, Th5;
then len h > 1 by A18, A14, A3, Th6, XXREAL_0:1;
then A30: len h >= 1 + 1 by NAT_1:13;
len M = (((E-min (L~ z)) .. z) -' 1) + 1 by A6, A9, JORDAN4:8
.= (E-min (L~ z)) .. z by A6, XREAL_1:235 ;
then A31: len M >= 1 + 1 by A8, NAT_1:13;
A32: h is_in_the_area_of z by A18, A14, Th21, Th22;
z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
then (S-max (L~ z)) .. z < len z by A15, A26, A12, A11, XXREAL_0:1;
then A33: L~ M misses L~ h by A6, A22, A3, Th48;
percases ( ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) = E-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) = E-max (L~ z) ) or ( NW-corner (L~ z) = N-min (L~ z) & NE-corner (L~ z) <> E-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & NE-corner (L~ z) <> E-max (L~ z) ) ) ;
supposethat A34: NW-corner (L~ z) = N-min (L~ z) and
A35: NE-corner (L~ z) = E-max (L~ z) ; ::_thesis: contradiction
(M /. 1) `1 = W-bound (L~ z) by A1, A24, A34, EUCLID:52;
then A36: M is_a_h.c._for z by A29, A28, Def2;
(h /. (len h)) `2 = N-bound (L~ z) by A19, A25, A35, EUCLID:52;
then h is_a_v.c._for z by A32, A27, Def3;
hence contradiction by A33, A31, A30, A36, Th29; ::_thesis: verum
end;
supposethat A37: NW-corner (L~ z) <> N-min (L~ z) and
A38: NE-corner (L~ z) = E-max (L~ z) ; ::_thesis: contradiction
reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A23, A5, A37, Th66;
A39: ( 2 <= len g & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
(h /. (len h)) `2 = N-bound (L~ z) by A19, A25, A38, EUCLID:52;
then A40: h is_a_v.c._for z by A32, A27, Def3;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A41: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((E-min (L~ z)) .. z) by A23, A5, Th9
.= E-min (L~ z) by A4, FINSEQ_5:38 ;
then A42: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
<*(NW-corner (L~ z))*> is_in_the_area_of z by Th26;
then g is_in_the_area_of z by A29, Th24;
then A43: g is_a_h.c._for z by A41, A42, Def2;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A20, A21, A16, A15, JORDAN4:35, XBOOLE_1:26;
then A44: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A24, PSCOMP_1:43;
M /. 1 in L~ M by A31, JORDAN3:1;
hence contradiction by A33, A30, A40, A43, A39, A44, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposethat A45: NW-corner (L~ z) = N-min (L~ z) and
A46: NE-corner (L~ z) <> E-max (L~ z) ; ::_thesis: contradiction
reconsider N = h ^ <*(NE-corner (L~ z))*> as S-Sequence_in_R2 by A18, A19, A14, A46, Th65;
A47: ( len M >= 2 & len N >= 2 ) by TOPREAL1:def_8;
(LSeg ((h /. (len h)),(NE-corner (L~ z)))) /\ (L~ M) c= (LSeg ((h /. (len h)),(NE-corner (L~ z)))) /\ (L~ z) by A6, A9, A10, JORDAN4:35, XBOOLE_1:26;
then A48: (LSeg ((h /. (len h)),(NE-corner (L~ z)))) /\ (L~ M) c= {(h /. (len h))} by A19, A25, PSCOMP_1:51;
( L~ N = (L~ h) \/ (LSeg ((NE-corner (L~ z)),(h /. (len h)))) & h /. (len h) in L~ h ) by A30, JORDAN3:1, SPPOL_2:19;
then A49: L~ M misses L~ N by A33, A48, ZFMISC_1:125;
len N = (len h) + (len <*(NE-corner (L~ z))*>) by FINSEQ_1:22
.= (len h) + 1 by FINSEQ_1:39 ;
then N /. (len N) = NE-corner (L~ z) by FINSEQ_4:67;
then A50: (N /. (len N)) `2 = N-bound (L~ z) by EUCLID:52;
M /. 1 = z /. 1 by A23, A5, Th8;
then (M /. 1) `1 = W-bound (L~ z) by A1, A45, EUCLID:52;
then A51: M is_a_h.c._for z by A29, A28, Def2;
1 in dom h by FINSEQ_5:6;
then A52: (N /. 1) `2 = S-bound (L~ z) by A27, FINSEQ_4:68;
<*(NE-corner (L~ z))*> is_in_the_area_of z by Th25;
then N is_in_the_area_of z by A32, Th24;
then N is_a_v.c._for z by A52, A50, Def3;
hence contradiction by A51, A47, A49, Th29; ::_thesis: verum
end;
supposethat A53: NW-corner (L~ z) <> N-min (L~ z) and
A54: NE-corner (L~ z) <> E-max (L~ z) ; ::_thesis: contradiction
reconsider N = h ^ <*(NE-corner (L~ z))*> as S-Sequence_in_R2 by A18, A19, A14, A54, Th65;
reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A23, A5, A53, Th66;
A55: ( len g >= 2 & len N >= 2 ) by TOPREAL1:def_8;
A56: L~ N = (L~ h) \/ (LSeg ((NE-corner (L~ z)),(h /. (len h)))) by SPPOL_2:19;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (LSeg ((NE-corner (L~ z)),(h /. (len h)))) = {} by A1, A19, A25, A24, Lm8;
then (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ N) = ((LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h)) \/ {} by A56, XBOOLE_1:23
.= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) ;
then (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ N) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A20, A21, A16, A15, JORDAN4:35, XBOOLE_1:26;
then A57: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ N) c= {(M /. 1)} by A1, A24, PSCOMP_1:43;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A58: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((E-min (L~ z)) .. z) by A23, A5, Th9
.= E-min (L~ z) by A4, FINSEQ_5:38 ;
then A59: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
len N = (len h) + (len <*(NE-corner (L~ z))*>) by FINSEQ_1:22
.= (len h) + 1 by FINSEQ_1:39 ;
then N /. (len N) = NE-corner (L~ z) by FINSEQ_4:67;
then A60: (N /. (len N)) `2 = N-bound (L~ z) by EUCLID:52;
(LSeg ((h /. (len h)),(NE-corner (L~ z)))) /\ (L~ M) c= (LSeg ((h /. (len h)),(NE-corner (L~ z)))) /\ (L~ z) by A6, A9, A10, JORDAN4:35, XBOOLE_1:26;
then A61: (LSeg ((h /. (len h)),(NE-corner (L~ z)))) /\ (L~ M) c= {(h /. (len h))} by A19, A25, PSCOMP_1:51;
h /. (len h) in L~ h by A30, JORDAN3:1;
then A62: L~ M misses L~ N by A33, A56, A61, ZFMISC_1:125;
1 in dom h by FINSEQ_5:6;
then A63: (N /. 1) `2 = S-bound (L~ z) by A27, FINSEQ_4:68;
<*(NE-corner (L~ z))*> is_in_the_area_of z by Th25;
then N is_in_the_area_of z by A32, Th24;
then A64: N is_a_v.c._for z by A63, A60, Def3;
<*(NW-corner (L~ z))*> is_in_the_area_of z by Th26;
then g is_in_the_area_of z by A29, Th24;
then A65: g is_a_h.c._for z by A58, A59, Def2;
( L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) & M /. 1 in L~ M ) by A31, JORDAN3:1, SPPOL_2:20;
hence contradiction by A65, A55, A64, A62, A57, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
theorem Th72: :: SPRECT_2:72
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) & E-min (L~ z) <> S-max (L~ z) holds
(E-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) & E-min (L~ z) <> S-max (L~ z) implies (E-min (L~ z)) .. z < (S-max (L~ z)) .. z )
set i1 = (E-min (L~ z)) .. z;
set i2 = (S-max (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: ( E-min (L~ z) <> S-max (L~ z) & (E-min (L~ z)) .. z >= (S-max (L~ z)) .. z ) ; ::_thesis: contradiction
A3: S-bound (L~ z) < N-bound (L~ z) by TOPREAL5:16;
z /. 2 in N-most (L~ z) by A1, Th30;
then A4: (z /. 2) `2 = (N-min (L~ z)) `2 by PSCOMP_1:39
.= N-bound (L~ z) by EUCLID:52 ;
A5: S-max (L~ z) in rng z by Th42;
then A6: (S-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A7: (S-max (L~ z)) .. z <= len z by FINSEQ_3:25;
A8: z /. ((S-max (L~ z)) .. z) = z . ((S-max (L~ z)) .. z) by A6, PARTFUN1:def_6
.= S-max (L~ z) by A5, FINSEQ_4:19 ;
then A9: (z /. ((S-max (L~ z)) .. z)) `2 = S-bound (L~ z) by EUCLID:52;
A10: 1 <= (S-max (L~ z)) .. z by A6, FINSEQ_3:25;
A11: (S-max (L~ z)) .. z <> 0 by A6, FINSEQ_3:25;
(z /. 1) `2 = N-bound (L~ z) by A1, EUCLID:52;
then A12: (S-max (L~ z)) .. z > 2 by A4, A11, A9, A3, NAT_1:26;
then reconsider h = mid (z,((S-max (L~ z)) .. z),2) as S-Sequence_in_R2 by A7, Th37;
A13: 2 <= len z by NAT_D:60;
then A14: 2 in dom z by FINSEQ_3:25;
then h /. 1 = S-max (L~ z) by A6, A8, Th8;
then A15: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. 2 ) by A6, A14, Th9, Th21, Th22;
then A16: ( len h >= 2 & h is_a_v.c._for z ) by A4, A15, Def3, TOPREAL1:def_8;
N-max (L~ z) in L~ z by SPRECT_1:11;
then (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
then (N-min (L~ z)) `1 < E-bound (L~ z) by Th51, XXREAL_0:2;
then A17: (N-min (L~ z)) `1 < (E-min (L~ z)) `1 by EUCLID:52;
A18: E-min (L~ z) in rng z by Th45;
then A19: (E-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A20: z /. ((E-min (L~ z)) .. z) = z . ((E-min (L~ z)) .. z) by PARTFUN1:def_6
.= E-min (L~ z) by A18, FINSEQ_4:19 ;
A21: (E-min (L~ z)) .. z <= len z by A19, FINSEQ_3:25;
z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
then A22: (E-min (L~ z)) .. z < len z by A21, A20, A17, XXREAL_0:1;
then ((E-min (L~ z)) .. z) + 1 <= len z by NAT_1:13;
then (len z) - ((E-min (L~ z)) .. z) >= 1 by XREAL_1:19;
then (len z) -' ((E-min (L~ z)) .. z) >= 1 by NAT_D:39;
then A23: ((len z) -' ((E-min (L~ z)) .. z)) + 1 >= 1 + 1 by XREAL_1:6;
A24: (E-min (L~ z)) .. z > (S-max (L~ z)) .. z by A2, A8, A20, XXREAL_0:1;
then (E-min (L~ z)) .. z > 1 by A10, XXREAL_0:2;
then reconsider M = mid (z,(len z),((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A22, Th37;
A25: len z in dom z by FINSEQ_5:6;
then A26: M /. (len M) = z /. ((E-min (L~ z)) .. z) by A19, Th9
.= E-min (L~ z) by A18, FINSEQ_5:38 ;
1 <= (E-min (L~ z)) .. z by A19, FINSEQ_3:25;
then A27: len M = ((len z) -' ((E-min (L~ z)) .. z)) + 1 by A21, JORDAN4:9;
A28: L~ M misses L~ h by A21, A24, A12, Th49;
A29: z /. 1 = z /. (len z) by FINSEQ_6:def_1;
then A30: M /. 1 = z /. 1 by A19, A25, Th8;
percases ( NW-corner (L~ z) = N-min (L~ z) or NW-corner (L~ z) <> N-min (L~ z) ) ;
supposeA31: NW-corner (L~ z) = N-min (L~ z) ; ::_thesis: contradiction
M /. 1 = z /. (len z) by A19, A25, Th8;
then A32: (M /. 1) `1 = W-bound (L~ z) by A1, A29, A31, EUCLID:52;
( M is_in_the_area_of z & (M /. (len M)) `1 = E-bound (L~ z) ) by A19, A25, A26, Th21, Th22, EUCLID:52;
then M is_a_h.c._for z by A32, Def2;
hence contradiction by A16, A28, A27, A23, Th29; ::_thesis: verum
end;
suppose NW-corner (L~ z) <> N-min (L~ z) ; ::_thesis: contradiction
then reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A19, A25, A29, Th66;
A33: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A34: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A10, A7, A13, JORDAN4:35, XBOOLE_1:26;
then A35: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A30, PSCOMP_1:43;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= z /. ((E-min (L~ z)) .. z) by A19, A25, Th9
.= E-min (L~ z) by A18, FINSEQ_5:38 ;
then A36: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
A37: M /. 1 in L~ M by A27, A23, JORDAN3:1;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A19, A25, Th21, Th22, Th26;
then g is_in_the_area_of z by Th24;
then g is_a_h.c._for z by A34, A36, Def2;
hence contradiction by A16, A28, A33, A35, A37, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
theorem Th73: :: SPRECT_2:73
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(S-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (S-max (L~ z)) .. z < (S-min (L~ z)) .. z )
set i1 = (S-max (L~ z)) .. z;
set i2 = (S-min (L~ z)) .. z;
set j = (N-max (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: (S-max (L~ z)) .. z >= (S-min (L~ z)) .. z ; ::_thesis: contradiction
A3: z /. 1 = z /. (len z) by FINSEQ_6:def_1;
A4: S-min (L~ z) in rng z by Th41;
then A5: (S-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A6: (S-min (L~ z)) .. z <= len z by FINSEQ_3:25;
A7: 1 <= (S-min (L~ z)) .. z by A5, FINSEQ_3:25;
A8: S-max (L~ z) in rng z by Th42;
then A9: (S-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A10: z /. ((S-max (L~ z)) .. z) = z . ((S-max (L~ z)) .. z) by PARTFUN1:def_6
.= S-max (L~ z) by A8, FINSEQ_4:19 ;
A11: (S-max (L~ z)) .. z <= len z by A9, FINSEQ_3:25;
( (N-min (L~ z)) `2 = N-bound (L~ z) & (S-max (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then N-min (L~ z) <> S-max (L~ z) by TOPREAL5:16;
then A12: (S-max (L~ z)) .. z < len z by A1, A11, A10, A3, XXREAL_0:1;
then ((S-max (L~ z)) .. z) + 1 <= len z by NAT_1:13;
then (len z) - ((S-max (L~ z)) .. z) >= 1 by XREAL_1:19;
then (len z) -' ((S-max (L~ z)) .. z) >= 1 by NAT_D:39;
then A13: ((len z) -' ((S-max (L~ z)) .. z)) + 1 >= 1 + 1 by XREAL_1:6;
A14: N-max (L~ z) in rng z by Th40;
then A15: (N-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A16: 1 <= (N-max (L~ z)) .. z by FINSEQ_3:25;
then (S-max (L~ z)) .. z > 1 by A1, Lm5, XXREAL_0:2;
then reconsider M = mid (z,(len z),((S-max (L~ z)) .. z)) as S-Sequence_in_R2 by A12, Th37;
A17: z /. ((N-max (L~ z)) .. z) = z . ((N-max (L~ z)) .. z) by A15, PARTFUN1:def_6
.= N-max (L~ z) by A14, FINSEQ_4:19 ;
then A18: (z /. ((N-max (L~ z)) .. z)) `2 = N-bound (L~ z) by EUCLID:52;
N-min (L~ z) <> N-max (L~ z) by Th52;
then A19: 1 < (N-max (L~ z)) .. z by A1, A16, A17, XXREAL_0:1;
A20: len z in dom z by FINSEQ_5:6;
then A21: M /. 1 = z /. (len z) by A9, Th8;
1 <= (S-max (L~ z)) .. z by A9, FINSEQ_3:25;
then A22: len M = ((len z) -' ((S-max (L~ z)) .. z)) + 1 by A11, JORDAN4:9;
then A23: M /. (len M) in L~ M by A13, JORDAN3:1;
A24: 1 in dom M by FINSEQ_5:6;
A25: (N-max (L~ z)) .. z <= len z by A15, FINSEQ_3:25;
A26: (S-min (L~ z)) .. z > (N-max (L~ z)) .. z by A1, Lm6;
then reconsider h = mid (z,((S-min (L~ z)) .. z),((N-max (L~ z)) .. z)) as S-Sequence_in_R2 by A6, A19, Th37;
A27: z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A5, PARTFUN1:def_6
.= S-min (L~ z) by A4, FINSEQ_4:19 ;
then h /. 1 = S-min (L~ z) by A5, A15, Th8;
then A28: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. ((N-max (L~ z)) .. z) ) by A5, A15, Th9, Th21, Th22;
then A29: ( len h >= 2 & h is_a_v.c._for z ) by A18, A28, Def3, TOPREAL1:def_8;
S-min (L~ z) <> S-max (L~ z) by Th56;
then (S-max (L~ z)) .. z > (S-min (L~ z)) .. z by A2, A27, A10, XXREAL_0:1;
then A30: L~ h misses L~ M by A11, A26, A19, Th49;
A31: M /. (len M) = S-max (L~ z) by A9, A10, A20, Th9;
percases ( ( NW-corner (L~ z) = N-min (L~ z) & SE-corner (L~ z) = S-max (L~ z) ) or ( NW-corner (L~ z) = N-min (L~ z) & SE-corner (L~ z) <> S-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & SE-corner (L~ z) = S-max (L~ z) ) or ( NW-corner (L~ z) <> N-min (L~ z) & SE-corner (L~ z) <> S-max (L~ z) ) ) ;
supposeA32: ( NW-corner (L~ z) = N-min (L~ z) & SE-corner (L~ z) = S-max (L~ z) ) ; ::_thesis: contradiction
A33: M is_in_the_area_of z by A9, A20, Th21, Th22;
( (M /. 1) `1 = W-bound (L~ z) & (M /. (len M)) `1 = E-bound (L~ z) ) by A1, A3, A31, A21, A32, EUCLID:52;
then M is_a_h.c._for z by A33, Def2;
hence contradiction by A29, A30, A22, A13, Th29; ::_thesis: verum
end;
supposethat A34: NW-corner (L~ z) = N-min (L~ z) and
A35: SE-corner (L~ z) <> S-max (L~ z) ; ::_thesis: contradiction
reconsider g = M ^ <*(SE-corner (L~ z))*> as S-Sequence_in_R2 by A9, A10, A20, A35, Th64;
A36: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((M /. (len M)),(SE-corner (L~ z)))) ) by SPPOL_2:19, TOPREAL1:def_8;
len g = (len M) + (len <*(SE-corner (L~ z))*>) by FINSEQ_1:22
.= (len M) + 1 by FINSEQ_1:39 ;
then g /. (len g) = SE-corner (L~ z) by FINSEQ_4:67;
then A37: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M is_in_the_area_of z & <*(SE-corner (L~ z))*> is_in_the_area_of z ) by A9, A20, Th21, Th22, Th27;
then A38: g is_in_the_area_of z by Th24;
(LSeg ((M /. (len M)),(SE-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. (len M)),(SE-corner (L~ z)))) /\ (L~ z) by A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A39: (LSeg ((M /. (len M)),(SE-corner (L~ z)))) /\ (L~ h) c= {(M /. (len M))} by A31, PSCOMP_1:59;
g /. 1 = M /. 1 by A24, FINSEQ_4:68
.= z /. 1 by A9, A3, A20, Th8 ;
then (g /. 1) `1 = W-bound (L~ z) by A1, A34, EUCLID:52;
then g is_a_h.c._for z by A38, A37, Def2;
hence contradiction by A29, A30, A23, A36, A39, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposethat A40: NW-corner (L~ z) <> N-min (L~ z) and
A41: SE-corner (L~ z) = S-max (L~ z) ; ::_thesis: contradiction
reconsider g = <*(NW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A1, A9, A3, A20, A40, Th66;
( len M in dom M & len g = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69
.= S-max (L~ z) by A9, A10, A20, Th9 ;
then A42: (g /. (len g)) `1 = E-bound (L~ z) by A41, EUCLID:52;
A43: ( len g >= 2 & L~ g = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) ) by SPPOL_2:20, TOPREAL1:def_8;
g /. 1 = NW-corner (L~ z) by FINSEQ_5:15;
then A44: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A45: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A3, A21, PSCOMP_1:43;
A46: M /. 1 in L~ M by A22, A13, JORDAN3:1;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A9, A20, Th21, Th22, Th26;
then g is_in_the_area_of z by Th24;
then g is_a_h.c._for z by A44, A42, Def2;
hence contradiction by A29, A30, A43, A45, A46, Th29, ZFMISC_1:125; ::_thesis: verum
end;
supposeA47: ( NW-corner (L~ z) <> N-min (L~ z) & SE-corner (L~ z) <> S-max (L~ z) ) ; ::_thesis: contradiction
set K = <*(NW-corner (L~ z))*> ^ M;
reconsider g = (<*(NW-corner (L~ z))*> ^ M) ^ <*(SE-corner (L~ z))*> as S-Sequence_in_R2 by A1, A9, A10, A3, A20, A47, Lm3;
1 in dom (<*(NW-corner (L~ z))*> ^ M) by FINSEQ_5:6;
then g /. 1 = (<*(NW-corner (L~ z))*> ^ M) /. 1 by FINSEQ_4:68
.= NW-corner (L~ z) by FINSEQ_5:15 ;
then A48: (g /. 1) `1 = W-bound (L~ z) by EUCLID:52;
len g = (len (<*(NW-corner (L~ z))*> ^ M)) + (len <*(SE-corner (L~ z))*>) by FINSEQ_1:22
.= (len (<*(NW-corner (L~ z))*> ^ M)) + 1 by FINSEQ_1:39 ;
then g /. (len g) = SE-corner (L~ z) by FINSEQ_4:67;
then A49: (g /. (len g)) `1 = E-bound (L~ z) by EUCLID:52;
( M is_in_the_area_of z & <*(NW-corner (L~ z))*> is_in_the_area_of z ) by A9, A20, Th21, Th22, Th26;
then A50: <*(NW-corner (L~ z))*> ^ M is_in_the_area_of z by Th24;
<*(SE-corner (L~ z))*> is_in_the_area_of z by Th27;
then g is_in_the_area_of z by A50, Th24;
then A51: g is_a_h.c._for z by A48, A49, Def2;
len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) by FINSEQ_1:22;
then len (<*(NW-corner (L~ z))*> ^ M) >= len M by NAT_1:11;
then len (<*(NW-corner (L~ z))*> ^ M) >= 2 by A22, A13, XXREAL_0:2;
then A52: (<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) in L~ (<*(NW-corner (L~ z))*> ^ M) by JORDAN3:1;
(LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ z) by A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A53: (LSeg ((M /. 1),(NW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A1, A3, A21, PSCOMP_1:43;
( L~ (<*(NW-corner (L~ z))*> ^ M) = (L~ M) \/ (LSeg ((NW-corner (L~ z)),(M /. 1))) & M /. 1 in L~ M ) by A22, A13, JORDAN3:1, SPPOL_2:20;
then A54: L~ (<*(NW-corner (L~ z))*> ^ M) misses L~ h by A30, A53, ZFMISC_1:125;
( len M in dom M & len (<*(NW-corner (L~ z))*> ^ M) = (len M) + (len <*(NW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then A55: (<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)) = M /. (len M) by FINSEQ_4:69
.= z /. ((S-max (L~ z)) .. z) by A9, A20, Th9
.= S-max (L~ z) by A8, FINSEQ_5:38 ;
(LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) /\ (L~ h) c= (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) /\ (L~ z) by A7, A6, A16, A25, JORDAN4:35, XBOOLE_1:26;
then A56: (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) /\ (L~ h) c= {((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M)))} by A55, PSCOMP_1:59;
( len g >= 2 & L~ g = (L~ (<*(NW-corner (L~ z))*> ^ M)) \/ (LSeg (((<*(NW-corner (L~ z))*> ^ M) /. (len (<*(NW-corner (L~ z))*> ^ M))),(SE-corner (L~ z)))) ) by SPPOL_2:19, TOPREAL1:def_8;
hence contradiction by A29, A51, A54, A52, A56, Th29, ZFMISC_1:125; ::_thesis: verum
end;
end;
end;
Lm9: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-min (L~ z)) .. z < (S-min (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (E-min (L~ z)) .. z < (S-min (L~ z)) .. z )
assume A1: z /. 1 = N-min (L~ z) ; ::_thesis: (E-min (L~ z)) .. z < (S-min (L~ z)) .. z
then (E-min (L~ z)) .. z <= (S-max (L~ z)) .. z by Th72;
hence (E-min (L~ z)) .. z < (S-min (L~ z)) .. z by A1, Th73, XXREAL_0:2; ::_thesis: verum
end;
Lm10: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) & N-min (L~ z) <> W-max (L~ z) holds
(E-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) & N-min (L~ z) <> W-max (L~ z) implies (E-min (L~ z)) .. z < (W-max (L~ z)) .. z )
set i1 = (E-min (L~ z)) .. z;
set i2 = (W-max (L~ z)) .. z;
set j = (S-min (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: N-min (L~ z) <> W-max (L~ z) and
A3: (E-min (L~ z)) .. z >= (W-max (L~ z)) .. z ; ::_thesis: contradiction
A4: z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
N-max (L~ z) in L~ z by SPRECT_1:11;
then (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
then (N-min (L~ z)) `1 < E-bound (L~ z) by Th51, XXREAL_0:2;
then A5: (N-min (L~ z)) `1 < (E-min (L~ z)) `1 by EUCLID:52;
( (N-min (L~ z)) `2 = N-bound (L~ z) & (S-min (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then A6: N-min (L~ z) <> S-min (L~ z) by TOPREAL5:16;
A7: S-min (L~ z) in rng z by Th41;
then A8: (S-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A9: (S-min (L~ z)) .. z <= len z by FINSEQ_3:25;
A10: E-min (L~ z) in rng z by Th45;
then A11: (E-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A12: z /. ((E-min (L~ z)) .. z) = z . ((E-min (L~ z)) .. z) by PARTFUN1:def_6
.= E-min (L~ z) by A10, FINSEQ_4:19 ;
A13: W-max (L~ z) in rng z by Th44;
then A14: (W-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A15: z /. ((W-max (L~ z)) .. z) = z . ((W-max (L~ z)) .. z) by PARTFUN1:def_6
.= W-max (L~ z) by A13, FINSEQ_4:19 ;
A16: 1 <= (W-max (L~ z)) .. z by A14, FINSEQ_3:25;
( (W-max (L~ z)) `1 = W-bound (L~ z) & (E-min (L~ z)) `1 = E-bound (L~ z) ) by EUCLID:52;
then (W-max (L~ z)) `1 < (E-min (L~ z)) `1 by TOPREAL5:17;
then A17: (E-min (L~ z)) .. z > (W-max (L~ z)) .. z by A3, A15, A12, XXREAL_0:1;
then (E-min (L~ z)) .. z > 1 by A16, XXREAL_0:2;
then A18: (S-min (L~ z)) .. z > 1 by A1, Lm9, XXREAL_0:2;
z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A8, PARTFUN1:def_6
.= S-min (L~ z) by A7, FINSEQ_4:19 ;
then (S-min (L~ z)) .. z < len z by A4, A9, A6, XXREAL_0:1;
then reconsider h = mid (z,((S-min (L~ z)) .. z),(len z)) as S-Sequence_in_R2 by A18, Th38;
A19: (E-min (L~ z)) .. z < (S-min (L~ z)) .. z by A1, Lm9;
A20: len z in dom z by FINSEQ_5:6;
then h /. (len h) = z /. (len z) by A8, Th9;
then A21: (h /. (len h)) `2 = N-bound (L~ z) by A4, EUCLID:52;
(E-min (L~ z)) .. z <= len z by A11, FINSEQ_3:25;
then (E-min (L~ z)) .. z < len z by A4, A12, A5, XXREAL_0:1;
then reconsider M = mid (z,((W-max (L~ z)) .. z),((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A16, A17, Th38;
M /. (len M) = z /. ((E-min (L~ z)) .. z) by A11, A14, Th9
.= E-min (L~ z) by A10, FINSEQ_5:38 ;
then A22: (M /. (len M)) `1 = E-bound (L~ z) by EUCLID:52;
M /. 1 = W-max (L~ z) by A11, A14, A15, Th8;
then A23: (M /. 1) `1 = W-bound (L~ z) by EUCLID:52;
M is_in_the_area_of z by A11, A14, Th21, Th22;
then A24: M is_a_h.c._for z by A23, A22, Def2;
z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A8, PARTFUN1:def_6
.= S-min (L~ z) by A7, FINSEQ_4:19 ;
then h /. 1 = S-min (L~ z) by A20, A8, Th8;
then A25: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
h is_in_the_area_of z by A20, A8, Th21, Th22;
then A26: h is_a_v.c._for z by A25, A21, Def3;
(W-max (L~ z)) .. z > 1 by A1, A2, A16, A15, XXREAL_0:1;
then A27: L~ M misses L~ h by A3, A9, A19, Th47;
( len h >= 2 & len M >= 2 ) by TOPREAL1:def_8;
hence contradiction by A26, A24, A27, Th29; ::_thesis: verum
end;
Lm11: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (E-min (L~ z)) .. z < (W-min (L~ z)) .. z )
set i1 = (E-min (L~ z)) .. z;
set i2 = (W-min (L~ z)) .. z;
set j = (S-min (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: (E-min (L~ z)) .. z >= (W-min (L~ z)) .. z ; ::_thesis: contradiction
A3: z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
N-max (L~ z) in L~ z by SPRECT_1:11;
then (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
then (N-min (L~ z)) `1 < E-bound (L~ z) by Th51, XXREAL_0:2;
then A4: (N-min (L~ z)) `1 < (E-min (L~ z)) `1 by EUCLID:52;
( (N-min (L~ z)) `2 = N-bound (L~ z) & (S-min (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then A5: N-min (L~ z) <> S-min (L~ z) by TOPREAL5:16;
A6: S-min (L~ z) in rng z by Th41;
then A7: (S-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A8: (S-min (L~ z)) .. z <= len z by FINSEQ_3:25;
A9: E-min (L~ z) in rng z by Th45;
then A10: (E-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A11: z /. ((E-min (L~ z)) .. z) = z . ((E-min (L~ z)) .. z) by PARTFUN1:def_6
.= E-min (L~ z) by A9, FINSEQ_4:19 ;
A12: W-min (L~ z) in rng z by Th43;
then A13: (W-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A14: z /. ((W-min (L~ z)) .. z) = z . ((W-min (L~ z)) .. z) by PARTFUN1:def_6
.= W-min (L~ z) by A12, FINSEQ_4:19 ;
A15: 1 <= (W-min (L~ z)) .. z by A13, FINSEQ_3:25;
( (W-min (L~ z)) `1 = W-bound (L~ z) & (E-min (L~ z)) `1 = E-bound (L~ z) ) by EUCLID:52;
then z /. ((E-min (L~ z)) .. z) <> z /. ((W-min (L~ z)) .. z) by A14, A11, TOPREAL5:17;
then A16: (E-min (L~ z)) .. z > (W-min (L~ z)) .. z by A2, XXREAL_0:1;
then (E-min (L~ z)) .. z > 1 by A15, XXREAL_0:2;
then A17: (S-min (L~ z)) .. z > 1 by A1, Lm9, XXREAL_0:2;
z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A7, PARTFUN1:def_6
.= S-min (L~ z) by A6, FINSEQ_4:19 ;
then (S-min (L~ z)) .. z < len z by A3, A8, A5, XXREAL_0:1;
then reconsider h = mid (z,((S-min (L~ z)) .. z),(len z)) as S-Sequence_in_R2 by A17, Th38;
A18: (E-min (L~ z)) .. z < (S-min (L~ z)) .. z by A1, Lm9;
A19: len z in dom z by FINSEQ_5:6;
then h /. (len h) = z /. (len z) by A7, Th9;
then A20: (h /. (len h)) `2 = N-bound (L~ z) by A3, EUCLID:52;
(E-min (L~ z)) .. z <= len z by A10, FINSEQ_3:25;
then (E-min (L~ z)) .. z < len z by A3, A11, A4, XXREAL_0:1;
then reconsider M = mid (z,((W-min (L~ z)) .. z),((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A15, A16, Th38;
M /. (len M) = z /. ((E-min (L~ z)) .. z) by A10, A13, Th9
.= E-min (L~ z) by A9, FINSEQ_5:38 ;
then A21: (M /. (len M)) `1 = E-bound (L~ z) by EUCLID:52;
z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A7, PARTFUN1:def_6
.= S-min (L~ z) by A6, FINSEQ_4:19 ;
then h /. 1 = S-min (L~ z) by A19, A7, Th8;
then A22: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
h is_in_the_area_of z by A19, A7, Th21, Th22;
then A23: h is_a_v.c._for z by A22, A20, Def3;
( W-max (L~ z) in L~ z & (N-min (L~ z)) `2 = N-bound (L~ z) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ z)) `2 <= (N-min (L~ z)) `2 by PSCOMP_1:24;
then N-min (L~ z) <> W-min (L~ z) by Th57;
then (W-min (L~ z)) .. z > 1 by A1, A15, A14, XXREAL_0:1;
then A24: L~ M misses L~ h by A2, A8, A18, Th47;
M /. 1 = W-min (L~ z) by A10, A13, A14, Th8;
then A25: (M /. 1) `1 = W-bound (L~ z) by EUCLID:52;
M is_in_the_area_of z by A10, A13, Th21, Th22;
then A26: M is_a_h.c._for z by A25, A21, Def2;
( len h >= 2 & len M >= 2 ) by TOPREAL1:def_8;
hence contradiction by A23, A26, A24, Th29; ::_thesis: verum
end;
theorem :: SPRECT_2:74
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) & S-min (L~ z) <> W-min (L~ z) holds
(S-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) & S-min (L~ z) <> W-min (L~ z) implies (S-min (L~ z)) .. z < (W-min (L~ z)) .. z )
set i1 = (E-min (L~ z)) .. z;
set i2 = (W-min (L~ z)) .. z;
set j = (S-min (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: ( S-min (L~ z) <> W-min (L~ z) & (S-min (L~ z)) .. z >= (W-min (L~ z)) .. z ) ; ::_thesis: contradiction
A3: z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
N-max (L~ z) in L~ z by SPRECT_1:11;
then (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
then (N-min (L~ z)) `1 < E-bound (L~ z) by Th51, XXREAL_0:2;
then A4: (N-min (L~ z)) `1 < (E-min (L~ z)) `1 by EUCLID:52;
A5: E-min (L~ z) in rng z by Th45;
then A6: (E-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A7: 1 <= (E-min (L~ z)) .. z by FINSEQ_3:25;
then A8: (S-min (L~ z)) .. z > 1 by A1, Lm9, XXREAL_0:2;
z /. ((E-min (L~ z)) .. z) = z . ((E-min (L~ z)) .. z) by A6, PARTFUN1:def_6
.= E-min (L~ z) by A5, FINSEQ_4:19 ;
then A9: (E-min (L~ z)) .. z > 1 by A1, A7, A4, XXREAL_0:1;
( (N-min (L~ z)) `2 = N-bound (L~ z) & (S-min (L~ z)) `2 = S-bound (L~ z) ) by EUCLID:52;
then A10: N-min (L~ z) <> S-min (L~ z) by TOPREAL5:16;
A11: S-min (L~ z) in rng z by Th41;
then A12: (S-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A13: (S-min (L~ z)) .. z <= len z by FINSEQ_3:25;
z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A12, PARTFUN1:def_6
.= S-min (L~ z) by A11, FINSEQ_4:19 ;
then (S-min (L~ z)) .. z < len z by A3, A13, A10, XXREAL_0:1;
then reconsider h = mid (z,((S-min (L~ z)) .. z),(len z)) as S-Sequence_in_R2 by A8, Th38;
A14: len z in dom z by FINSEQ_5:6;
then h /. (len h) = z /. (len z) by A12, Th9;
then A15: (h /. (len h)) `2 = N-bound (L~ z) by A3, EUCLID:52;
A16: z /. ((S-min (L~ z)) .. z) = z . ((S-min (L~ z)) .. z) by A12, PARTFUN1:def_6
.= S-min (L~ z) by A11, FINSEQ_4:19 ;
then h /. 1 = S-min (L~ z) by A12, A14, Th8;
then A17: (h /. 1) `2 = S-bound (L~ z) by EUCLID:52;
h is_in_the_area_of z by A12, A14, Th21, Th22;
then A18: h is_a_v.c._for z by A17, A15, Def3;
A19: (E-min (L~ z)) .. z < (W-min (L~ z)) .. z by A1, Lm11;
A20: W-min (L~ z) in rng z by Th43;
then A21: (W-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then (W-min (L~ z)) .. z <= len z by FINSEQ_3:25;
then reconsider M = mid (z,((W-min (L~ z)) .. z),((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A19, A9, Th37;
M /. (len M) = z /. ((E-min (L~ z)) .. z) by A6, A21, Th9
.= E-min (L~ z) by A5, FINSEQ_5:38 ;
then A22: (M /. (len M)) `1 = E-bound (L~ z) by EUCLID:52;
A23: z /. ((W-min (L~ z)) .. z) = z . ((W-min (L~ z)) .. z) by A21, PARTFUN1:def_6
.= W-min (L~ z) by A20, FINSEQ_4:19 ;
then M /. 1 = W-min (L~ z) by A6, A21, Th8;
then A24: (M /. 1) `1 = W-bound (L~ z) by EUCLID:52;
M is_in_the_area_of z by A6, A21, Th21, Th22;
then A25: M is_a_h.c._for z by A24, A22, Def2;
A26: ( len h >= 2 & len M >= 2 ) by TOPREAL1:def_8;
(S-min (L~ z)) .. z > (W-min (L~ z)) .. z by A2, A23, A16, XXREAL_0:1;
then L~ M misses L~ h by A19, A9, A13, Th50;
hence contradiction by A18, A26, A25, Th29; ::_thesis: verum
end;
theorem Th75: :: SPRECT_2:75
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) & N-min (L~ z) <> W-max (L~ z) holds
(W-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) & N-min (L~ z) <> W-max (L~ z) implies (W-min (L~ z)) .. z < (W-max (L~ z)) .. z )
set i1 = (W-min (L~ z)) .. z;
set i2 = (W-max (L~ z)) .. z;
set j = (E-min (L~ z)) .. z;
assume that
A1: z /. 1 = N-min (L~ z) and
A2: N-min (L~ z) <> W-max (L~ z) and
A3: (W-min (L~ z)) .. z >= (W-max (L~ z)) .. z ; ::_thesis: contradiction
A4: (W-max (L~ z)) .. z > (E-min (L~ z)) .. z by A1, A2, Lm10;
A5: E-min (L~ z) in rng z by Th45;
then A6: (E-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A7: z /. ((E-min (L~ z)) .. z) = z . ((E-min (L~ z)) .. z) by PARTFUN1:def_6
.= E-min (L~ z) by A5, FINSEQ_4:19 ;
then A8: (z /. ((E-min (L~ z)) .. z)) `1 = E-bound (L~ z) by EUCLID:52;
A9: (E-min (L~ z)) .. z <= len z by A6, FINSEQ_3:25;
A10: z /. (len z) = N-min (L~ z) by A1, FINSEQ_6:def_1;
A11: W-max (L~ z) in rng z by Th44;
then A12: (W-max (L~ z)) .. z in dom z by FINSEQ_4:20;
then A13: 1 <= (W-max (L~ z)) .. z by FINSEQ_3:25;
A14: W-min (L~ z) in rng z by Th43;
then A15: (W-min (L~ z)) .. z in dom z by FINSEQ_4:20;
then A16: z /. ((W-min (L~ z)) .. z) = z . ((W-min (L~ z)) .. z) by PARTFUN1:def_6
.= W-min (L~ z) by A14, FINSEQ_4:19 ;
A17: (W-min (L~ z)) .. z <= len z by A15, FINSEQ_3:25;
( W-max (L~ z) in L~ z & (N-min (L~ z)) `2 = N-bound (L~ z) ) by EUCLID:52, SPRECT_1:13;
then (W-max (L~ z)) `2 <= (N-min (L~ z)) `2 by PSCOMP_1:24;
then ( z /. 1 = z /. (len z) & N-min (L~ z) <> W-min (L~ z) ) by Th57, FINSEQ_6:def_1;
then A18: (W-min (L~ z)) .. z < len z by A1, A17, A16, XXREAL_0:1;
then ((W-min (L~ z)) .. z) + 1 <= len z by NAT_1:13;
then (len z) - ((W-min (L~ z)) .. z) >= 1 by XREAL_1:19;
then (len z) -' ((W-min (L~ z)) .. z) >= 1 by NAT_D:39;
then A19: ((len z) -' ((W-min (L~ z)) .. z)) + 1 >= 1 + 1 by XREAL_1:6;
A20: 1 <= (E-min (L~ z)) .. z by A6, FINSEQ_3:25;
then (W-min (L~ z)) .. z > 1 by A1, Lm11, XXREAL_0:2;
then reconsider M = mid (z,((W-min (L~ z)) .. z),(len z)) as S-Sequence_in_R2 by A18, Th38;
A21: len z in dom z by FINSEQ_5:6;
then A22: M /. 1 = z /. ((W-min (L~ z)) .. z) by A15, Th8;
1 <= (W-min (L~ z)) .. z by A15, FINSEQ_3:25;
then A23: len M = ((len z) -' ((W-min (L~ z)) .. z)) + 1 by A17, JORDAN4:8;
A24: M is_in_the_area_of z by A15, A21, Th21, Th22;
A25: M /. (len M) = z /. (len z) by A15, A21, Th9;
N-max (L~ z) in L~ z by SPRECT_1:11;
then (N-max (L~ z)) `1 <= E-bound (L~ z) by PSCOMP_1:24;
then (N-min (L~ z)) `1 < E-bound (L~ z) by Th51, XXREAL_0:2;
then (N-min (L~ z)) `1 < (E-min (L~ z)) `1 by EUCLID:52;
then A26: 1 < (E-min (L~ z)) .. z by A1, A20, A7, XXREAL_0:1;
A27: (W-max (L~ z)) .. z <= len z by A12, FINSEQ_3:25;
then reconsider h = mid (z,((W-max (L~ z)) .. z),((E-min (L~ z)) .. z)) as S-Sequence_in_R2 by A4, A26, Th37;
A28: z /. ((W-max (L~ z)) .. z) = z . ((W-max (L~ z)) .. z) by A12, PARTFUN1:def_6
.= W-max (L~ z) by A11, FINSEQ_4:19 ;
then h /. 1 = W-max (L~ z) by A12, A6, Th8;
then A29: (h /. 1) `1 = W-bound (L~ z) by EUCLID:52;
( h is_in_the_area_of z & h /. (len h) = z /. ((E-min (L~ z)) .. z) ) by A12, A6, Th9, Th21, Th22;
then A30: ( len h >= 2 & h is_a_h.c._for z ) by A8, A29, Def2, TOPREAL1:def_8;
W-max (L~ z) <> W-min (L~ z) by Th58;
then (W-min (L~ z)) .. z > (W-max (L~ z)) .. z by A3, A28, A16, XXREAL_0:1;
then A31: L~ M misses L~ h by A17, A4, A26, Th50;
percases ( SW-corner (L~ z) = W-min (L~ z) or SW-corner (L~ z) <> W-min (L~ z) ) ;
supposeA32: SW-corner (L~ z) = W-min (L~ z) ; ::_thesis: contradiction
A33: (M /. (len M)) `2 = N-bound (L~ z) by A10, A25, EUCLID:52;
(M /. 1) `2 = S-bound (L~ z) by A16, A22, A32, EUCLID:52;
then M is_a_v.c._for z by A24, A33, Def3;
hence contradiction by A30, A31, A23, A19, Th29; ::_thesis: verum
end;
suppose SW-corner (L~ z) <> W-min (L~ z) ; ::_thesis: contradiction
then reconsider g = <*(SW-corner (L~ z))*> ^ M as S-Sequence_in_R2 by A15, A16, A21, Th67;
g /. 1 = SW-corner (L~ z) by FINSEQ_5:15;
then A34: (g /. 1) `2 = S-bound (L~ z) by EUCLID:52;
( len M in dom M & len g = (len M) + (len <*(SW-corner (L~ z))*>) ) by FINSEQ_1:22, FINSEQ_5:6;
then g /. (len g) = M /. (len M) by FINSEQ_4:69;
then A35: (g /. (len g)) `2 = N-bound (L~ z) by A10, A25, EUCLID:52;
(LSeg ((M /. 1),(SW-corner (L~ z)))) /\ (L~ h) c= (LSeg ((M /. 1),(SW-corner (L~ z)))) /\ (L~ z) by A13, A27, A20, A9, JORDAN4:35, XBOOLE_1:26;
then A36: (LSeg ((M /. 1),(SW-corner (L~ z)))) /\ (L~ h) c= {(M /. 1)} by A16, A22, PSCOMP_1:35;
( L~ g = (L~ M) \/ (LSeg ((SW-corner (L~ z)),(M /. 1))) & M /. 1 in L~ M ) by A23, A19, JORDAN3:1, SPPOL_2:20;
then A37: L~ g misses L~ h by A31, A36, ZFMISC_1:125;
<*(SW-corner (L~ z))*> is_in_the_area_of z by Th28;
then g is_in_the_area_of z by A24, Th24;
then ( len g >= 2 & g is_a_v.c._for z ) by A34, A35, Def3, TOPREAL1:def_8;
hence contradiction by A30, A37, Th29; ::_thesis: verum
end;
end;
end;
theorem :: SPRECT_2:76
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(W-min (L~ z)) .. z < len z
proof
let z be non constant standard clockwise_oriented special_circular_sequence; ::_thesis: ( z /. 1 = N-min (L~ z) implies (W-min (L~ z)) .. z < len z )
assume A1: z /. 1 = N-min (L~ z) ; ::_thesis: (W-min (L~ z)) .. z < len z
A2: W-max (L~ z) in rng z by Th44;
A3: W-min (L~ z) in rng z by Th43;
percases ( N-min (L~ z) = W-max (L~ z) or N-min (L~ z) <> W-max (L~ z) ) ;
suppose N-min (L~ z) = W-max (L~ z) ; ::_thesis: (W-min (L~ z)) .. z < len z
then A4: z /. (len z) = W-max (L~ z) by A1, FINSEQ_6:def_1;
A5: (W-min (L~ z)) .. z in dom z by A3, FINSEQ_4:20;
then A6: (W-min (L~ z)) .. z <= len z by FINSEQ_3:25;
z /. ((W-min (L~ z)) .. z) = z . ((W-min (L~ z)) .. z) by A5, PARTFUN1:def_6
.= W-min (L~ z) by A3, FINSEQ_4:19 ;
then (W-min (L~ z)) .. z <> len z by A4, Th58;
hence (W-min (L~ z)) .. z < len z by A6, XXREAL_0:1; ::_thesis: verum
end;
supposeA7: N-min (L~ z) <> W-max (L~ z) ; ::_thesis: (W-min (L~ z)) .. z < len z
(W-max (L~ z)) .. z in dom z by A2, FINSEQ_4:20;
then (W-max (L~ z)) .. z <= len z by FINSEQ_3:25;
hence (W-min (L~ z)) .. z < len z by A1, A7, Th75, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
theorem :: SPRECT_2:77
for f being non constant standard special_circular_sequence st f /. 1 = N-min (L~ f) holds
(W-max (L~ f)) .. f < len f
proof
let f be non constant standard special_circular_sequence; ::_thesis: ( f /. 1 = N-min (L~ f) implies (W-max (L~ f)) .. f < len f )
assume A1: f /. 1 = N-min (L~ f) ; ::_thesis: (W-max (L~ f)) .. f < len f
then A2: f /. (len f) = N-min (L~ f) by FINSEQ_6:def_1;
A3: W-max (L~ f) in rng f by Th44;
then (W-max (L~ f)) .. f in dom f by FINSEQ_4:20;
then A4: f /. ((W-max (L~ f)) .. f) = f . ((W-max (L~ f)) .. f) by PARTFUN1:def_6
.= W-max (L~ f) by A3, FINSEQ_4:19 ;
percases ( N-min (L~ f) = W-max (L~ f) or N-min (L~ f) <> W-max (L~ f) ) ;
suppose N-min (L~ f) = W-max (L~ f) ; ::_thesis: (W-max (L~ f)) .. f < len f
then (W-max (L~ f)) .. f = 1 by A1, FINSEQ_6:43;
hence (W-max (L~ f)) .. f < len f by GOBOARD7:34, XXREAL_0:2; ::_thesis: verum
end;
supposeA5: N-min (L~ f) <> W-max (L~ f) ; ::_thesis: (W-max (L~ f)) .. f < len f
(W-max (L~ f)) .. f <= len f by A3, FINSEQ_4:21;
hence (W-max (L~ f)) .. f < len f by A2, A4, A5, XXREAL_0:1; ::_thesis: verum
end;
end;
end;