:: JORDAN3 semantic presentation
begin
theorem :: JORDAN3:1
for n being Element of NAT
for f being FinSequence of (TOP-REAL n) st 2 <= len f holds
( f . 1 in L~ f & f /. 1 in L~ f & f . (len f) in L~ f & f /. (len f) in L~ f )
proof
let n be Element of NAT ; ::_thesis: for f being FinSequence of (TOP-REAL n) st 2 <= len f holds
( f . 1 in L~ f & f /. 1 in L~ f & f . (len f) in L~ f & f /. (len f) in L~ f )
let f be FinSequence of (TOP-REAL n); ::_thesis: ( 2 <= len f implies ( f . 1 in L~ f & f /. 1 in L~ f & f . (len f) in L~ f & f /. (len f) in L~ f ) )
assume A1: 2 <= len f ; ::_thesis: ( f . 1 in L~ f & f /. 1 in L~ f & f . (len f) in L~ f & f /. (len f) in L~ f )
then A2: 1 + 1 <= len f ;
then A3: LSeg (f,1) in { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } ;
f /. 1 in LSeg ((f /. 1),(f /. (1 + 1))) by RLTOPSP1:68;
then f /. 1 in LSeg (f,1) by A1, TOPREAL1:def_3;
then f /. 1 in union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by A3, TARSKI:def_4;
then A4: f /. 1 in L~ f by TOPREAL1:def_4;
A5: ((len f) -' 1) + 1 = len f by A2, NAT_D:46, XREAL_1:235;
A6: 1 <= (len f) -' 1 by A2, NAT_D:49;
then A7: LSeg (f,((len f) -' 1)) in { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by A5;
f /. (len f) in LSeg ((f /. ((len f) -' 1)),(f /. (((len f) -' 1) + 1))) by A5, RLTOPSP1:68;
then f /. (len f) in LSeg (f,((len f) -' 1)) by A6, A5, TOPREAL1:def_3;
then f /. (len f) in union { (LSeg (f,i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len f ) } by A7, TARSKI:def_4;
then A8: f /. (len f) in L~ f by TOPREAL1:def_4;
1 <= len f by A2, NAT_D:46;
hence ( f . 1 in L~ f & f /. 1 in L~ f & f . (len f) in L~ f & f /. (len f) in L~ f ) by A4, A8, FINSEQ_4:15; ::_thesis: verum
end;
theorem Th2: :: JORDAN3:2
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & not q1 `1 = q2 `1 holds
q1 `2 = q2 `2
proof
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & not q1 `1 = q2 `1 implies q1 `2 = q2 `2 )
assume that
A1: ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) and
A2: q1 in LSeg (p1,p2) and
A3: q2 in LSeg (p1,p2) ; ::_thesis: ( q1 `1 = q2 `1 or q1 `2 = q2 `2 )
consider r2 being Real such that
A4: q2 = ((1 - r2) * p1) + (r2 * p2) and
0 <= r2 and
r2 <= 1 by A3;
consider r1 being Real such that
A5: q1 = ((1 - r1) * p1) + (r1 * p2) and
0 <= r1 and
r1 <= 1 by A2;
q1 `1 = (((1 - r1) * p1) `1) + ((r1 * p2) `1) by A5, TOPREAL3:2;
then q1 `1 = ((1 - r1) * (p1 `1)) + ((r1 * p2) `1) by TOPREAL3:4;
then A6: q1 `1 = ((1 - r1) * (p1 `1)) + (r1 * (p2 `1)) by TOPREAL3:4;
q2 `1 = (((1 - r2) * p1) `1) + ((r2 * p2) `1) by A4, TOPREAL3:2;
then q2 `1 = ((1 - r2) * (p1 `1)) + ((r2 * p2) `1) by TOPREAL3:4;
then A7: q2 `1 = ((1 - r2) * (p1 `1)) + (r2 * (p2 `1)) by TOPREAL3:4;
q1 `2 = (((1 - r1) * p1) `2) + ((r1 * p2) `2) by A5, TOPREAL3:2;
then q1 `2 = ((1 - r1) * (p1 `2)) + ((r1 * p2) `2) by TOPREAL3:4;
then A8: q1 `2 = ((1 - r1) * (p1 `2)) + (r1 * (p2 `2)) by TOPREAL3:4;
q2 `2 = (((1 - r2) * p1) `2) + ((r2 * p2) `2) by A4, TOPREAL3:2;
then q2 `2 = ((1 - r2) * (p1 `2)) + ((r2 * p2) `2) by TOPREAL3:4;
then A9: q2 `2 = ((1 - r2) * (p1 `2)) + (r2 * (p2 `2)) by TOPREAL3:4;
percases ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) by A1;
suppose p1 `1 = p2 `1 ; ::_thesis: ( q1 `1 = q2 `1 or q1 `2 = q2 `2 )
hence ( q1 `1 = q2 `1 or q1 `2 = q2 `2 ) by A6, A7; ::_thesis: verum
end;
suppose p1 `2 = p2 `2 ; ::_thesis: ( q1 `1 = q2 `1 or q1 `2 = q2 `2 )
hence ( q1 `1 = q2 `1 or q1 `2 = q2 `2 ) by A8, A9; ::_thesis: verum
end;
end;
end;
theorem Th3: :: JORDAN3:3
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & LSeg (q1,q2) c= LSeg (p1,p2) & not q1 `1 = q2 `1 holds
q1 `2 = q2 `2
proof
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & LSeg (q1,q2) c= LSeg (p1,p2) & not q1 `1 = q2 `1 implies q1 `2 = q2 `2 )
A1: q2 in LSeg (q1,q2) by RLTOPSP1:68;
q1 in LSeg (q1,q2) by RLTOPSP1:68;
hence ( ( p1 `1 = p2 `1 or p1 `2 = p2 `2 ) & LSeg (q1,q2) c= LSeg (p1,p2) & not q1 `1 = q2 `1 implies q1 `2 = q2 `2 ) by A1, Th2; ::_thesis: verum
end;
theorem Th4: :: JORDAN3:4
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st 2 <= n & f is being_S-Seq holds
f | n is being_S-Seq
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for n being Element of NAT st 2 <= n & f is being_S-Seq holds
f | n is being_S-Seq
let n be Element of NAT ; ::_thesis: ( 2 <= n & f is being_S-Seq implies f | n is being_S-Seq )
assume that
A1: 2 <= n and
A2: f is being_S-Seq ; ::_thesis: f | n is being_S-Seq
A3: len f >= 2 by A2, TOPREAL1:def_8;
A4: now__::_thesis:_(_(_n_<=_len_f_&_len_(f_|_n)_>=_2_)_or_(_n_>_len_f_&_len_(f_|_n)_>=_2_)_)
percases ( n <= len f or n > len f ) ;
case n <= len f ; ::_thesis: len (f | n) >= 2
hence len (f | n) >= 2 by A1, FINSEQ_1:59; ::_thesis: verum
end;
case n > len f ; ::_thesis: len (f | n) >= 2
hence len (f | n) >= 2 by A3, FINSEQ_1:58; ::_thesis: verum
end;
end;
end;
reconsider f9 = f as one-to-one special unfolded s.n.c. FinSequence of (TOP-REAL 2) by A2;
f9 | n is one-to-one ;
hence f | n is being_S-Seq by A4, TOPREAL1:def_8; ::_thesis: verum
end;
theorem Th5: :: JORDAN3:5
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT st n <= len f & 2 <= (len f) -' n & f is being_S-Seq holds
f /^ n is being_S-Seq
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for n being Element of NAT st n <= len f & 2 <= (len f) -' n & f is being_S-Seq holds
f /^ n is being_S-Seq
let n be Element of NAT ; ::_thesis: ( n <= len f & 2 <= (len f) -' n & f is being_S-Seq implies f /^ n is being_S-Seq )
assume that
A1: n <= len f and
A2: 2 <= (len f) -' n and
A3: f is being_S-Seq ; ::_thesis: f /^ n is being_S-Seq
reconsider f9 = f as one-to-one special unfolded s.n.c. FinSequence of (TOP-REAL 2) by A3;
len (f /^ n) = (len f) - n by A1, RFINSEQ:def_1;
then len (f9 /^ n) >= 2 by A1, A2, XREAL_1:233;
hence f /^ n is being_S-Seq by TOPREAL1:def_8; ::_thesis: verum
end;
theorem :: JORDAN3:6
for f being FinSequence of (TOP-REAL 2)
for k1, k2 being Element of NAT st f is being_S-Seq & 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & k1 <> k2 holds
mid (f,k1,k2) is being_S-Seq
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for k1, k2 being Element of NAT st f is being_S-Seq & 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & k1 <> k2 holds
mid (f,k1,k2) is being_S-Seq
let k1, k2 be Element of NAT ; ::_thesis: ( f is being_S-Seq & 1 <= k1 & k1 <= len f & 1 <= k2 & k2 <= len f & k1 <> k2 implies mid (f,k1,k2) is being_S-Seq )
assume that
A1: f is being_S-Seq and
A2: 1 <= k1 and
A3: k1 <= len f and
A4: 1 <= k2 and
A5: k2 <= len f and
A6: k1 <> k2 ; ::_thesis: mid (f,k1,k2) is being_S-Seq
percases ( k1 <= k2 or k1 > k2 ) ;
supposeA7: k1 <= k2 ; ::_thesis: mid (f,k1,k2) is being_S-Seq
then k1 < k2 by A6, XXREAL_0:1;
then A8: k1 + 1 <= k2 by NAT_1:13;
then (k1 + 1) - k1 <= k2 - k1 by XREAL_1:9;
then 1 <= k2 -' k1 by NAT_D:39;
then A9: 1 + 1 <= (k2 -' k1) + 1 by XREAL_1:6;
k1 + 1 <= len f by A5, A8, XXREAL_0:2;
then (k1 + 1) - k1 <= (len f) - k1 by XREAL_1:9;
then A10: 1 + 1 <= ((len f) - k1) + 1 by XREAL_1:6;
(len f) -' (k1 -' 1) = (len f) - (k1 -' 1) by A3, NAT_D:50, XREAL_1:233
.= (len f) - (k1 - 1) by A2, XREAL_1:233
.= ((len f) - k1) + 1 ;
then A11: f /^ (k1 -' 1) is being_S-Seq by A1, A3, A10, Th5, NAT_D:50;
mid (f,k1,k2) = (f /^ (k1 -' 1)) | ((k2 -' k1) + 1) by A7, FINSEQ_6:def_3;
hence mid (f,k1,k2) is being_S-Seq by A11, A9, Th4; ::_thesis: verum
end;
supposeA12: k1 > k2 ; ::_thesis: mid (f,k1,k2) is being_S-Seq
then A13: k2 + 1 <= k1 by NAT_1:13;
then (k2 + 1) - k2 <= k1 - k2 by XREAL_1:9;
then 1 <= k1 -' k2 by NAT_D:39;
then A14: 1 + 1 <= (k1 -' k2) + 1 by XREAL_1:6;
k2 + 1 <= len f by A3, A13, XXREAL_0:2;
then (k2 + 1) - k2 <= (len f) - k2 by XREAL_1:9;
then A15: 1 + 1 <= ((len f) - k2) + 1 by XREAL_1:6;
(len f) -' (k2 -' 1) = (len f) - (k2 -' 1) by A5, NAT_D:50, XREAL_1:233
.= (len f) - (k2 - 1) by A4, XREAL_1:233
.= ((len f) - k2) + 1 ;
then f /^ (k2 -' 1) is being_S-Seq by A1, A5, A15, Th5, NAT_D:50;
then A16: (f /^ (k2 -' 1)) | ((k1 -' k2) + 1) is S-Sequence_in_R2 by A14, Th4;
mid (f,k1,k2) = Rev ((f /^ (k2 -' 1)) | ((k1 -' k2) + 1)) by A12, FINSEQ_6:def_3;
hence mid (f,k1,k2) is being_S-Seq by A16; ::_thesis: verum
end;
end;
end;
begin
definition
let f be FinSequence of (TOP-REAL 2);
let p be Point of (TOP-REAL 2);
assume A1: p in L~ f ;
func Index (p,f) -> Element of NAT means :Def1: :: JORDAN3:def 1
ex S being non empty Subset of NAT st
( it = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } );
existence
ex b1 being Element of NAT ex S being non empty Subset of NAT st
( b1 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } )
proof
set S = { i where i is Element of NAT : p in LSeg (f,i) } ;
A2: { i where i is Element of NAT : p in LSeg (f,i) } c= NAT
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { i where i is Element of NAT : p in LSeg (f,i) } or x in NAT )
assume x in { i where i is Element of NAT : p in LSeg (f,i) } ; ::_thesis: x in NAT
then ex i being Element of NAT st
( x = i & p in LSeg (f,i) ) ;
hence x in NAT ; ::_thesis: verum
end;
consider i2 being Element of NAT such that
1 <= i2 and
i2 + 1 <= len f and
A3: p in LSeg (f,i2) by A1, SPPOL_2:13;
i2 in { i where i is Element of NAT : p in LSeg (f,i) } by A3;
then reconsider S = { i where i is Element of NAT : p in LSeg (f,i) } as non empty Subset of NAT by A2;
take min S ; ::_thesis: ex S being non empty Subset of NAT st
( min S = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } )
take S ; ::_thesis: ( min S = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } )
thus ( min S = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of NAT st ex S being non empty Subset of NAT st
( b1 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) & ex S being non empty Subset of NAT st
( b2 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) holds
b1 = b2 ;
end;
:: deftheorem Def1 defines Index JORDAN3:def_1_:_
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
for b3 being Element of NAT holds
( b3 = Index (p,f) iff ex S being non empty Subset of NAT st
( b3 = min S & S = { i where i is Element of NAT : p in LSeg (f,i) } ) );
theorem Th7: :: JORDAN3:7
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for i being Element of NAT st p in LSeg (f,i) holds
Index (p,f) <= i
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for i being Element of NAT st p in LSeg (f,i) holds
Index (p,f) <= i
let p be Point of (TOP-REAL 2); ::_thesis: for i being Element of NAT st p in LSeg (f,i) holds
Index (p,f) <= i
let i0 be Element of NAT ; ::_thesis: ( p in LSeg (f,i0) implies Index (p,f) <= i0 )
assume A1: p in LSeg (f,i0) ; ::_thesis: Index (p,f) <= i0
LSeg (f,i0) c= L~ f by TOPREAL3:19;
then consider S being non empty Subset of NAT such that
A2: Index (p,f) = min S and
A3: S = { i where i is Element of NAT : p in LSeg (f,i) } by A1, Def1;
i0 in S by A1, A3;
hence Index (p,f) <= i0 by A2, XXREAL_2:def_7; ::_thesis: verum
end;
theorem Th8: :: JORDAN3:8
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
( 1 <= Index (p,f) & Index (p,f) < len f )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
( 1 <= Index (p,f) & Index (p,f) < len f )
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies ( 1 <= Index (p,f) & Index (p,f) < len f ) )
assume p in L~ f ; ::_thesis: ( 1 <= Index (p,f) & Index (p,f) < len f )
then consider S being non empty Subset of NAT such that
A1: Index (p,f) = min S and
A2: S = { i where i is Element of NAT : p in LSeg (f,i) } by Def1;
Index (p,f) in S by A1, XXREAL_2:def_7;
then A3: ex i being Element of NAT st
( i = Index (p,f) & p in LSeg (f,i) ) by A2;
hence 1 <= Index (p,f) by TOPREAL1:def_3; ::_thesis: Index (p,f) < len f
(Index (p,f)) + 1 <= len f by A3, TOPREAL1:def_3;
hence Index (p,f) < len f by NAT_1:13; ::_thesis: verum
end;
theorem Th9: :: JORDAN3:9
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
p in LSeg (f,(Index (p,f)))
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
p in LSeg (f,(Index (p,f)))
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies p in LSeg (f,(Index (p,f))) )
assume p in L~ f ; ::_thesis: p in LSeg (f,(Index (p,f)))
then consider S being non empty Subset of NAT such that
A1: Index (p,f) = min S and
A2: S = { i where i is Element of NAT : p in LSeg (f,i) } by Def1;
Index (p,f) in S by A1, XXREAL_2:def_7;
then ex i being Element of NAT st
( i = Index (p,f) & p in LSeg (f,i) ) by A2;
hence p in LSeg (f,(Index (p,f))) ; ::_thesis: verum
end;
theorem Th10: :: JORDAN3:10
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in LSeg (f,1) holds
Index (p,f) = 1
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in LSeg (f,1) holds
Index (p,f) = 1
let p be Point of (TOP-REAL 2); ::_thesis: ( p in LSeg (f,1) implies Index (p,f) = 1 )
assume A1: p in LSeg (f,1) ; ::_thesis: Index (p,f) = 1
then A2: Index (p,f) <= 1 by Th7;
LSeg (f,1) c= L~ f by TOPREAL3:19;
then Index (p,f) >= 1 by A1, Th8;
hence Index (p,f) = 1 by A2, XXREAL_0:1; ::_thesis: verum
end;
theorem Th11: :: JORDAN3:11
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st len f >= 2 holds
Index ((f /. 1),f) = 1
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st len f >= 2 holds
Index ((f /. 1),f) = 1
let p be Point of (TOP-REAL 2); ::_thesis: ( len f >= 2 implies Index ((f /. 1),f) = 1 )
assume len f >= 2 ; ::_thesis: Index ((f /. 1),f) = 1
then len f >= 1 + 1 ;
then f /. 1 in LSeg (f,1) by TOPREAL1:21;
hence Index ((f /. 1),f) = 1 by Th10; ::_thesis: verum
end;
theorem Th12: :: JORDAN3:12
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for i1 being Nat st f is being_S-Seq & 1 < i1 & i1 <= len f & p = f . i1 holds
(Index (p,f)) + 1 = i1
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for i1 being Nat st f is being_S-Seq & 1 < i1 & i1 <= len f & p = f . i1 holds
(Index (p,f)) + 1 = i1
let p be Point of (TOP-REAL 2); ::_thesis: for i1 being Nat st f is being_S-Seq & 1 < i1 & i1 <= len f & p = f . i1 holds
(Index (p,f)) + 1 = i1
let i1 be Nat; ::_thesis: ( f is being_S-Seq & 1 < i1 & i1 <= len f & p = f . i1 implies (Index (p,f)) + 1 = i1 )
assume A1: f is being_S-Seq ; ::_thesis: ( not 1 < i1 or not i1 <= len f or not p = f . i1 or (Index (p,f)) + 1 = i1 )
assume that
A2: 1 < i1 and
A3: i1 <= len f ; ::_thesis: ( not p = f . i1 or (Index (p,f)) + 1 = i1 )
A4: i1 in dom f by A2, A3, FINSEQ_3:25;
assume p = f . i1 ; ::_thesis: (Index (p,f)) + 1 = i1
then A5: p = f /. i1 by A4, PARTFUN1:def_6;
assume A6: (Index (p,f)) + 1 <> i1 ; ::_thesis: contradiction
consider j being Nat such that
A7: i1 = j + 1 by A2, NAT_1:6;
reconsider j = j as Element of NAT by ORDINAL1:def_12;
A8: 1 + 0 <= j by A2, A7, NAT_1:13;
then A9: p in LSeg (f,j) by A3, A7, A5, TOPREAL1:21;
then Index (p,f) <= j by Th7;
then Index (p,f) < j by A7, A6, XXREAL_0:1;
then A10: (Index (p,f)) + 1 <= j by NAT_1:13;
A11: LSeg (f,j) c= L~ f by TOPREAL3:19;
then A12: p in LSeg (f,(Index (p,f))) by A9, Th9;
percases ( (Index (p,f)) + 1 = j or (Index (p,f)) + 1 < j ) by A10, XXREAL_0:1;
supposeA13: (Index (p,f)) + 1 = j ; ::_thesis: contradiction
then A14: (Index (p,f)) + (1 + 1) <= len f by A3, A7;
1 <= Index (p,f) by A9, A11, Th8;
then (LSeg (f,(Index (p,f)))) /\ (LSeg (f,j)) = {(f /. j)} by A1, A13, A14, TOPREAL1:def_6;
then p in {(f /. j)} by A9, A12, XBOOLE_0:def_4;
then A15: p = f /. j by TARSKI:def_1;
j < len f by A3, A7, NAT_1:13;
then A16: j in dom f by A8, FINSEQ_3:25;
j < i1 by A7, NAT_1:13;
hence contradiction by A1, A4, A5, A15, A16, PARTFUN2:10; ::_thesis: verum
end;
supposeA17: (Index (p,f)) + 1 < j ; ::_thesis: contradiction
p in (LSeg (f,(Index (p,f)))) /\ (LSeg (f,j)) by A9, A12, XBOOLE_0:def_4;
then LSeg (f,(Index (p,f))) meets LSeg (f,j) by XBOOLE_0:4;
hence contradiction by A1, A17, TOPREAL1:def_7; ::_thesis: verum
end;
end;
end;
theorem Th13: :: JORDAN3:13
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for i1 being Element of NAT st f is s.n.c. & p in LSeg (f,i1) & not i1 = Index (p,f) holds
i1 = (Index (p,f)) + 1
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for i1 being Element of NAT st f is s.n.c. & p in LSeg (f,i1) & not i1 = Index (p,f) holds
i1 = (Index (p,f)) + 1
let p be Point of (TOP-REAL 2); ::_thesis: for i1 being Element of NAT st f is s.n.c. & p in LSeg (f,i1) & not i1 = Index (p,f) holds
i1 = (Index (p,f)) + 1
let i1 be Element of NAT ; ::_thesis: ( f is s.n.c. & p in LSeg (f,i1) & not i1 = Index (p,f) implies i1 = (Index (p,f)) + 1 )
assume that
A1: f is s.n.c. and
A2: p in LSeg (f,i1) ; ::_thesis: ( i1 = Index (p,f) or i1 = (Index (p,f)) + 1 )
p in L~ f by A2, SPPOL_2:17;
then p in LSeg (f,(Index (p,f))) by Th9;
then p in (LSeg (f,(Index (p,f)))) /\ (LSeg (f,i1)) by A2, XBOOLE_0:def_4;
then A3: LSeg (f,(Index (p,f))) meets LSeg (f,i1) by XBOOLE_0:4;
assume A4: ( not i1 = Index (p,f) & not i1 = (Index (p,f)) + 1 ) ; ::_thesis: contradiction
Index (p,f) <= i1 by A2, Th7;
then Index (p,f) < i1 by A4, XXREAL_0:1;
then (Index (p,f)) + 1 <= i1 by NAT_1:13;
then (Index (p,f)) + 1 < i1 by A4, XXREAL_0:1;
hence contradiction by A1, A3, TOPREAL1:def_7; ::_thesis: verum
end;
theorem Th14: :: JORDAN3:14
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for i1 being Element of NAT st f is unfolded & f is s.n.c. & i1 + 1 <= len f & p in LSeg (f,i1) & p <> f . i1 holds
i1 = Index (p,f)
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for i1 being Element of NAT st f is unfolded & f is s.n.c. & i1 + 1 <= len f & p in LSeg (f,i1) & p <> f . i1 holds
i1 = Index (p,f)
let p be Point of (TOP-REAL 2); ::_thesis: for i1 being Element of NAT st f is unfolded & f is s.n.c. & i1 + 1 <= len f & p in LSeg (f,i1) & p <> f . i1 holds
i1 = Index (p,f)
let i1 be Element of NAT ; ::_thesis: ( f is unfolded & f is s.n.c. & i1 + 1 <= len f & p in LSeg (f,i1) & p <> f . i1 implies i1 = Index (p,f) )
assume that
A1: ( f is unfolded & f is s.n.c. ) and
A2: i1 + 1 <= len f and
A3: p in LSeg (f,i1) ; ::_thesis: ( not p <> f . i1 or i1 = Index (p,f) )
A4: i1 < len f by A2, NAT_1:13;
A5: 1 <= (Index (p,f)) + 1 by NAT_1:11;
Index (p,f) <= i1 by A3, Th7;
then Index (p,f) < len f by A4, XXREAL_0:2;
then (Index (p,f)) + 1 <= len f by NAT_1:13;
then A6: (Index (p,f)) + 1 in dom f by A5, FINSEQ_3:25;
assume A7: p <> f . i1 ; ::_thesis: i1 = Index (p,f)
A8: p in L~ f by A3, SPPOL_2:17;
then p in LSeg (f,(Index (p,f))) by Th9;
then A9: p in (LSeg (f,(Index (p,f)))) /\ (LSeg (f,i1)) by A3, XBOOLE_0:def_4;
A10: 1 <= Index (p,f) by A8, Th8;
now__::_thesis:_not_i1_=_(Index_(p,f))_+_1
assume A11: i1 = (Index (p,f)) + 1 ; ::_thesis: contradiction
then (Index (p,f)) + (1 + 1) <= len f by A2;
then p in {(f /. ((Index (p,f)) + 1))} by A1, A9, A10, A11, TOPREAL1:def_6;
then p = f /. ((Index (p,f)) + 1) by TARSKI:def_1;
hence contradiction by A7, A6, A11, PARTFUN1:def_6; ::_thesis: verum
end;
hence i1 = Index (p,f) by A1, A3, Th13; ::_thesis: verum
end;
definition
let g be FinSequence of (TOP-REAL 2);
let p1, p2 be Point of (TOP-REAL 2);
predg is_S-Seq_joining p1,p2 means :Def2: :: JORDAN3:def 2
( g is being_S-Seq & g . 1 = p1 & g . (len g) = p2 );
end;
:: deftheorem Def2 defines is_S-Seq_joining JORDAN3:def_2_:_
for g being FinSequence of (TOP-REAL 2)
for p1, p2 being Point of (TOP-REAL 2) holds
( g is_S-Seq_joining p1,p2 iff ( g is being_S-Seq & g . 1 = p1 & g . (len g) = p2 ) );
theorem Th15: :: JORDAN3:15
for g being FinSequence of (TOP-REAL 2)
for p1, p2 being Point of (TOP-REAL 2) st g is_S-Seq_joining p1,p2 holds
Rev g is_S-Seq_joining p2,p1
proof
let g be FinSequence of (TOP-REAL 2); ::_thesis: for p1, p2 being Point of (TOP-REAL 2) st g is_S-Seq_joining p1,p2 holds
Rev g is_S-Seq_joining p2,p1
let p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( g is_S-Seq_joining p1,p2 implies Rev g is_S-Seq_joining p2,p1 )
assume that
A1: g is being_S-Seq and
A2: g . 1 = p1 and
A3: g . (len g) = p2 ; :: according to JORDAN3:def_2 ::_thesis: Rev g is_S-Seq_joining p2,p1
thus Rev g is being_S-Seq by A1; :: according to JORDAN3:def_2 ::_thesis: ( (Rev g) . 1 = p2 & (Rev g) . (len (Rev g)) = p1 )
thus (Rev g) . 1 = p2 by A3, FINSEQ_5:62; ::_thesis: (Rev g) . (len (Rev g)) = p1
dom g = dom (Rev g) by FINSEQ_5:57;
hence (Rev g) . (len (Rev g)) = (Rev g) . (len g) by FINSEQ_3:29
.= p1 by A2, FINSEQ_5:62 ;
::_thesis: verum
end;
theorem Th16: :: JORDAN3:16
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for j being Nat st p in L~ f & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) & 1 <= j & j + 1 <= len g holds
LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1))
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for j being Nat st p in L~ f & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) & 1 <= j & j + 1 <= len g holds
LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1))
let p be Point of (TOP-REAL 2); ::_thesis: for j being Nat st p in L~ f & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) & 1 <= j & j + 1 <= len g holds
LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1))
let j be Nat; ::_thesis: ( p in L~ f & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) & 1 <= j & j + 1 <= len g implies LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) )
assume that
A1: p in L~ f and
A2: g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) and
A3: 1 <= j and
A4: j + 1 <= len g ; ::_thesis: LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1))
A5: j <= len g by A4, NAT_1:13;
len g = (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A2, FINSEQ_1:22;
then A6: len g = 1 + (len (mid (f,((Index (p,f)) + 1),(len f)))) by FINSEQ_1:39;
then A7: (j + 1) - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A4, XREAL_1:9;
j -' 1 <= j by NAT_D:35;
then A8: j -' 1 <= len (mid (f,((Index (p,f)) + 1),(len f))) by A7, XXREAL_0:2;
1 <= (Index (p,f)) + j by A3, NAT_1:12;
then A9: 1 - 1 <= ((Index (p,f)) + j) - 1 by XREAL_1:9;
A10: j -' 1 = j - 1 by A3, XREAL_1:233;
A11: j = 1 + (j - 1)
.= (len <*p*>) + (j -' 1) by A10, FINSEQ_1:39 ;
1 <= Index (p,f) by A1, Th8;
then 1 + 1 <= (Index (p,f)) + j by A3, XREAL_1:7;
then 1 <= ((Index (p,f)) + j) - 1 by XREAL_1:19;
then A12: 1 <= ((Index (p,f)) + j) -' 1 by NAT_D:39;
consider i being Element of NAT such that
1 <= i and
A13: i + 1 <= len f and
p in LSeg (f,i) by A1, SPPOL_2:13;
1 <= i + 1 by NAT_1:12;
then A14: 1 <= len f by A13, XXREAL_0:2;
A15: Index (p,f) < len f by A1, Th8;
then A16: (Index (p,f)) + 1 <= len f by NAT_1:13;
(Index (p,f)) + 1 <= len f by A15, NAT_1:13;
then ((Index (p,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then A17: 1 - 1 <= ((len f) - (Index (p,f))) - 1 by XREAL_1:9;
then A18: (len f) -' ((Index (p,f)) + 1) = (len f) - ((Index (p,f)) + 1) by XREAL_0:def_2
.= ((len f) - (Index (p,f))) - 1 ;
A19: 0 + 1 <= (Index (p,f)) + 1 by NAT_1:13;
then A20: 1 <= len f by A15, NAT_1:13;
(Index (p,f)) + 1 <= len f by A15, NAT_1:13;
then A21: len (mid (f,((Index (p,f)) + 1),(len f))) = ((len f) -' ((Index (p,f)) + 1)) + 1 by A14, A19, FINSEQ_6:118;
A22: len g = (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A2, FINSEQ_1:22
.= 1 + (len (mid (f,((Index (p,f)) + 1),(len f)))) by FINSEQ_1:39 ;
then len g = 1 + (((len f) - ((Index (p,f)) + 1)) + 1) by A17, A21, XREAL_0:def_2
.= 1 + ((len f) - (Index (p,f))) ;
then j <= (len f) - (Index (p,f)) by A4, XREAL_1:6;
then A23: j + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by XREAL_1:6;
then A24: (((Index (p,f)) + j) -' 1) + 1 <= len f by A3, NAT_1:12, XREAL_1:235;
A25: 1 <= j + 1 by A3, NAT_1:13;
then A26: g /. (j + 1) = g . (j + 1) by A4, FINSEQ_4:15;
A27: j + 1 = (len <*p*>) + ((j + 1) - 1) by FINSEQ_1:39
.= (len <*p*>) + ((j + 1) -' 1) by A25, XREAL_1:233 ;
A28: (j + 1) -' 1 = (j + 1) - 1 by A25, XREAL_1:233;
then (j + 1) -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A3, A7, FINSEQ_3:25;
then g . (j + 1) = (mid (f,((Index (p,f)) + 1),(len f))) . ((j + 1) -' 1) by A2, A27, FINSEQ_1:def_7
.= f . ((((j + 1) -' 1) + ((Index (p,f)) + 1)) -' 1) by A3, A19, A16, A20, A28, A7, FINSEQ_6:118
.= f . (((((j + 1) -' 1) + 1) + (Index (p,f))) -' 1)
.= f . (((j + 1) + (Index (p,f))) -' 1) by A25, XREAL_1:235
.= f . ((((Index (p,f)) + j) + 1) -' 1)
.= f . ((Index (p,f)) + j) by NAT_D:34
.= f . ((((Index (p,f)) + j) -' 1) + 1) by A3, NAT_1:12, XREAL_1:235 ;
then A29: f /. ((((Index (p,f)) + j) -' 1) + 1) = g /. (j + 1) by A24, A26, FINSEQ_4:15, NAT_1:11;
(j + 1) - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A4, A6, XREAL_1:9;
then j + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by A21, A18, XREAL_1:6;
then ((Index (p,f)) + (j - 1)) + 1 <= len f ;
then (((Index (p,f)) + j) -' 1) + 1 <= len f by A9, XREAL_0:def_2;
then A30: LSeg (f,(((Index (p,f)) + j) -' 1)) = LSeg ((f /. (((Index (p,f)) + j) -' 1)),(f /. ((((Index (p,f)) + j) -' 1) + 1))) by A12, TOPREAL1:def_3;
A31: 1 <= len g by A22, NAT_1:11;
now__::_thesis:_(_(_1_<_j_&_LSeg_(g,j)_c=_LSeg_(f,(((Index_(p,f))_+_j)_-'_1))_)_or_(_1_=_j_&_LSeg_(g,j)_c=_LSeg_(f,(((Index_(p,f))_+_j)_-'_1))_)_)
percases ( 1 < j or 1 = j ) by A3, XXREAL_0:1;
caseA32: 1 < j ; ::_thesis: LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1))
then A33: j -' 1 = j - 1 by XREAL_1:233;
then A34: 1 <= j -' 1 by A32, SPPOL_1:1;
j - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A6, A5, XREAL_1:9;
then j -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A33, A34, FINSEQ_3:25;
then A35: g . j = (mid (f,((Index (p,f)) + 1),(len f))) . (j -' 1) by A2, A11, FINSEQ_1:def_7
.= f . (((j -' 1) + ((Index (p,f)) + 1)) -' 1) by A19, A16, A20, A8, A34, FINSEQ_6:118
.= f . ((((j -' 1) + 1) + (Index (p,f))) -' 1)
.= f . (((Index (p,f)) + j) -' 1) by A3, XREAL_1:235 ;
g /. j = g . j by A3, A5, FINSEQ_4:15;
then LSeg (f,(((Index (p,f)) + j) -' 1)) = LSeg ((g /. j),(g /. (j + 1))) by A23, A29, A12, A30, A35, FINSEQ_4:15, NAT_D:50
.= LSeg (g,j) by A3, A4, TOPREAL1:def_3 ;
hence LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) ; ::_thesis: verum
end;
caseA36: 1 = j ; ::_thesis: LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1))
then j <= len <*p*> by FINSEQ_1:39;
then j in dom <*p*> by A36, FINSEQ_3:25;
then A37: g . j = <*p*> . j by A2, FINSEQ_1:def_7
.= p by A36, FINSEQ_1:40 ;
A38: f /. ((((Index (p,f)) + j) -' 1) + 1) in LSeg ((f /. (((Index (p,f)) + j) -' 1)),(f /. ((((Index (p,f)) + j) -' 1) + 1))) by RLTOPSP1:68;
A39: g /. j = g . j by A31, A36, FINSEQ_4:15;
A40: ((Index (p,f)) + j) -' 1 = Index (p,f) by A36, NAT_D:34;
p in LSeg (f,(Index (p,f))) by A1, Th9;
then LSeg (p,(g /. (j + 1))) c= LSeg (f,(((Index (p,f)) + j) -' 1)) by A29, A30, A38, A40, TOPREAL1:6;
hence LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) by A3, A4, A37, A39, TOPREAL1:def_3; ::_thesis: verum
end;
end;
end;
hence LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) ; ::_thesis: verum
end;
theorem :: JORDAN3:17
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . ((Index (p,f)) + 1) & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) holds
g is_S-Seq_joining p,f /. (len f)
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . ((Index (p,f)) + 1) & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) holds
g is_S-Seq_joining p,f /. (len f)
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p <> f . ((Index (p,f)) + 1) & g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) implies g is_S-Seq_joining p,f /. (len f) )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . ((Index (p,f)) + 1) and
A4: g = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) ; ::_thesis: g is_S-Seq_joining p,f /. (len f)
len g = (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A4, FINSEQ_1:22;
then A5: len g = 1 + (len (mid (f,((Index (p,f)) + 1),(len f)))) by FINSEQ_1:39;
consider i being Element of NAT such that
1 <= i and
A6: i + 1 <= len f and
p in LSeg (f,i) by A2, SPPOL_2:13;
1 <= 1 + i by NAT_1:12;
then A7: 1 <= len f by A6, XXREAL_0:2;
A8: for j1, j2 being Nat st j1 + 1 < j2 holds
LSeg (g,j1) misses LSeg (g,j2)
proof
let j1, j2 be Nat; ::_thesis: ( j1 + 1 < j2 implies LSeg (g,j1) misses LSeg (g,j2) )
assume A9: j1 + 1 < j2 ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
A10: ( j1 = 0 or j1 >= 0 + 1 ) by NAT_1:13;
now__::_thesis:_(_(_j1_=_0_&_LSeg_(g,j1)_misses_LSeg_(g,j2)_)_or_(_(_j1_=_1_or_j1_>_1_)_&_j2_+_1_<=_len_g_&_LSeg_(g,j1)_misses_LSeg_(g,j2)_)_or_(_j2_+_1_>_len_g_&_LSeg_(g,j1)_misses_LSeg_(g,j2)_)_)
percases ( j1 = 0 or ( ( j1 = 1 or j1 > 1 ) & j2 + 1 <= len g ) or j2 + 1 > len g ) by A10, XXREAL_0:1;
case j1 = 0 ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
then LSeg (g,j1) = {} by TOPREAL1:def_3;
then (LSeg (g,j1)) /\ (LSeg (g,j2)) = {} ;
hence LSeg (g,j1) misses LSeg (g,j2) by XBOOLE_0:def_7; ::_thesis: verum
end;
casethat A11: ( j1 = 1 or j1 > 1 ) and
A12: j2 + 1 <= len g ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
1 < j1 + 1 by A11, NAT_1:13;
then 1 <= j2 by A9, XXREAL_0:2;
then A13: LSeg (g,j2) c= LSeg (f,(((Index (p,f)) + j2) -' 1)) by A2, A4, A12, Th16;
1 <= (Index (p,f)) + j1 by A2, Th8, NAT_1:12;
then 1 - 1 <= ((Index (p,f)) + j1) - 1 by XREAL_1:9;
then A14: ((Index (p,f)) + j1) - 1 = ((Index (p,f)) + j1) -' 1 by XREAL_0:def_2;
(Index (p,f)) + (j1 + 1) < (Index (p,f)) + j2 by A9, XREAL_1:6;
then (((Index (p,f)) + j1) + 1) - 1 < ((Index (p,f)) + j2) - 1 by XREAL_1:9;
then (((Index (p,f)) + j1) -' 1) + 1 < ((Index (p,f)) + j2) -' 1 by A14, XREAL_0:def_2;
then LSeg (f,(((Index (p,f)) + j1) -' 1)) misses LSeg (f,(((Index (p,f)) + j2) -' 1)) by A1, TOPREAL1:def_7;
then A15: (LSeg (f,(((Index (p,f)) + j1) -' 1))) /\ (LSeg (f,(((Index (p,f)) + j2) -' 1))) = {} by XBOOLE_0:def_7;
j2 < len g by A12, NAT_1:13;
then j1 + 1 <= len g by A9, XXREAL_0:2;
then LSeg (g,j1) c= LSeg (f,(((Index (p,f)) + j1) -' 1)) by A2, A4, A11, Th16;
then (LSeg (g,j1)) /\ (LSeg (g,j2)) = {} by A13, A15, XBOOLE_1:3, XBOOLE_1:27;
hence LSeg (g,j1) misses LSeg (g,j2) by XBOOLE_0:def_7; ::_thesis: verum
end;
case j2 + 1 > len g ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
then LSeg (g,j2) = {} by TOPREAL1:def_3;
then (LSeg (g,j1)) /\ (LSeg (g,j2)) = {} ;
hence LSeg (g,j1) misses LSeg (g,j2) by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence LSeg (g,j1) misses LSeg (g,j2) ; ::_thesis: verum
end;
A16: Index (p,f) < len f by A2, Th8;
then A17: (Index (p,f)) + 1 <= len f by NAT_1:13;
(Index (p,f)) + 1 <= len f by A16, NAT_1:13;
then A18: ((Index (p,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then A19: 1 - 1 <= ((len f) - (Index (p,f))) - 1 by XREAL_1:9;
then A20: (len f) -' ((Index (p,f)) + 1) = (len f) - ((Index (p,f)) + 1) by XREAL_0:def_2
.= ((len f) - (Index (p,f))) - 1 ;
A21: 0 + 1 <= (Index (p,f)) + 1 by NAT_1:11;
then A22: len (mid (f,((Index (p,f)) + 1),(len f))) = ((len f) -' ((Index (p,f)) + 1)) + 1 by A7, A17, FINSEQ_6:118;
A23: for j being Nat st 1 <= j & j + 2 <= len g holds
(LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))}
proof
let j be Nat; ::_thesis: ( 1 <= j & j + 2 <= len g implies (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))} )
assume that
A24: 1 <= j and
A25: j + 2 <= len g ; ::_thesis: (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))}
A26: j + 2 = (j + 1) + 1 ;
then A27: j + 1 <= len g by A25, NAT_1:13;
then A28: LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) by A2, A4, A24, Th16;
1 <= j + 1 by A24, NAT_1:13;
then LSeg (g,(j + 1)) c= LSeg (f,(((Index (p,f)) + (j + 1)) -' 1)) by A2, A4, A25, A26, Th16;
then A29: (LSeg (g,j)) /\ (LSeg (g,(j + 1))) c= (LSeg (f,(((Index (p,f)) + j) -' 1))) /\ (LSeg (f,(((Index (p,f)) + (j + 1)) -' 1))) by A28, XBOOLE_1:27;
A30: 1 <= Index (p,f) by A2, Th8;
1 <= Index (p,f) by A2, Th8;
then 1 + 1 <= (Index (p,f)) + j by A24, XREAL_1:7;
then 1 <= ((Index (p,f)) + j) - 1 by XREAL_1:19;
then A31: 1 <= ((Index (p,f)) + j) -' 1 by NAT_D:39;
1 <= (Index (p,f)) + j by A2, Th8, NAT_1:12;
then A32: 1 - 1 <= ((Index (p,f)) + j) - 1 by XREAL_1:9;
((j + 1) + 1) - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A5, A25, XREAL_1:9;
then A33: (j + 1) + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by A22, A20, XREAL_1:6;
then ((Index (p,f)) + j) + 1 <= len f ;
then ((Index (p,f)) + (j - 1)) + 1 <= len f by NAT_D:46;
then A34: (((Index (p,f)) + j) -' 1) + 1 <= len f by A32, XREAL_0:def_2;
(((Index (p,f)) + j) - 1) + (1 + 1) <= len f by A33;
then (((Index (p,f)) + j) -' 1) + 2 <= len f by A32, XREAL_0:def_2;
then A35: {(f /. ((((Index (p,f)) + j) -' 1) + 1))} = (LSeg (f,(((Index (p,f)) + j) -' 1))) /\ (LSeg (f,((((Index (p,f)) + j) -' 1) + 1))) by A1, A31, TOPREAL1:def_6;
A36: 1 < j + 1 by A24, NAT_1:13;
then A37: g /. (j + 1) = g . (j + 1) by A27, FINSEQ_4:15;
A38: g /. (j + 1) in LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1))) by RLTOPSP1:68;
g /. (j + 1) in LSeg ((g /. j),(g /. (j + 1))) by RLTOPSP1:68;
then A39: g /. (j + 1) in (LSeg ((g /. j),(g /. (j + 1)))) /\ (LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1)))) by A38, XBOOLE_0:def_4;
A40: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A24, A27, TOPREAL1:def_3;
LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1))) = LSeg (g,(j + 1)) by A25, A36, TOPREAL1:def_3;
then A41: {(g /. (j + 1))} c= (LSeg (g,j)) /\ (LSeg (g,(j + 1))) by A40, A39, ZFMISC_1:31;
A42: j + 1 = ((j + 1) - 1) + 1
.= ((j + 1) -' 1) + 1 by A36, XREAL_1:233 ;
then A43: j + 1 = (len <*p*>) + ((j + 1) -' 1) by FINSEQ_1:39;
A44: (j + 1) -' 1 <= len (mid (f,((Index (p,f)) + 1),(len f))) by A5, A27, A42, XREAL_1:6;
then (j + 1) -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A24, A42, FINSEQ_3:25;
then g . (j + 1) = (mid (f,((Index (p,f)) + 1),(len f))) . ((j + 1) -' 1) by A4, A43, FINSEQ_1:def_7
.= f . ((((j + 1) -' 1) + ((Index (p,f)) + 1)) -' 1) by A7, A17, A21, A24, A42, A44, FINSEQ_6:118
.= f . (((((j + 1) -' 1) + 1) + (Index (p,f))) -' 1)
.= f . (((j + 1) + (Index (p,f))) -' 1) by A36, XREAL_1:235
.= f . ((((Index (p,f)) + j) + 1) -' 1)
.= f . ((Index (p,f)) + j) by NAT_D:34
.= f . ((((Index (p,f)) + j) -' 1) + 1) by A30, NAT_1:12, XREAL_1:235 ;
then A45: f /. ((((Index (p,f)) + j) -' 1) + 1) = g /. (j + 1) by A37, A34, FINSEQ_4:15, NAT_1:11;
((Index (p,f)) + (j + 1)) -' 1 = (((Index (p,f)) + j) + 1) - 1 by NAT_1:11, XREAL_1:233
.= (((Index (p,f)) + j) - 1) + 1
.= (((Index (p,f)) + j) -' 1) + 1 by A30, NAT_1:12, XREAL_1:233 ;
hence (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))} by A29, A35, A45, A41, XBOOLE_0:def_10; ::_thesis: verum
end;
A46: len g = (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A4, FINSEQ_1:22
.= 1 + (len (mid (f,((Index (p,f)) + 1),(len f)))) by FINSEQ_1:39 ;
then A47: len g = 1 + (((len f) - ((Index (p,f)) + 1)) + 1) by A19, A22, XREAL_0:def_2
.= 1 + ((len f) - (Index (p,f))) ;
then A48: (len g) -' 1 = (len g) - 1 by A18, XREAL_0:def_2;
then A49: (len g) -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A18, A22, A47, A20, FINSEQ_3:25;
A50: (len f) - (Index (p,f)) >= 0 by A2, Th8, XREAL_1:50;
then A51: (len f) - (Index (p,f)) = (len f) -' (Index (p,f)) by XREAL_0:def_2;
then A52: (mid (f,((Index (p,f)) + 1),(len f))) . ((len f) -' (Index (p,f))) = f . ((((len f) -' (Index (p,f))) + ((Index (p,f)) + 1)) -' 1) by A7, A18, A17, A21, A22, A20, FINSEQ_6:118;
A53: (len g) -' 1 = (len f) -' (Index (p,f)) by A47, A48, XREAL_0:def_2;
for x1, x2 being set st x1 in dom g & x2 in dom g & g . x1 = g . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom g & x2 in dom g & g . x1 = g . x2 implies x1 = x2 )
assume that
A54: x1 in dom g and
A55: x2 in dom g and
A56: g . x1 = g . x2 ; ::_thesis: x1 = x2
reconsider n1 = x1, n2 = x2 as Element of NAT by A54, A55;
A57: n1 <= len g by A54, FINSEQ_3:25;
A58: 1 <= n2 by A55, FINSEQ_3:25;
A59: n2 <= len g by A55, FINSEQ_3:25;
A60: 1 <= n1 by A54, FINSEQ_3:25;
now__::_thesis:_(_(_n1_=_1_&_n2_=_1_&_x1_=_x2_)_or_(_n1_=_1_&_n2_>_1_&_contradiction_)_or_(_n1_>_1_&_n2_=_1_&_contradiction_)_or_(_n1_>_1_&_n2_>_1_&_x1_=_x2_)_)
percases ( ( n1 = 1 & n2 = 1 ) or ( n1 = 1 & n2 > 1 ) or ( n1 > 1 & n2 = 1 ) or ( n1 > 1 & n2 > 1 ) ) by A60, A58, XXREAL_0:1;
case ( n1 = 1 & n2 = 1 ) ; ::_thesis: x1 = x2
hence x1 = x2 ; ::_thesis: verum
end;
casethat A61: n1 = 1 and
A62: n2 > 1 ; ::_thesis: contradiction
A63: n2 - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A5, A59, XREAL_1:9;
n1 <= len <*p*> by A61, FINSEQ_1:39;
then n1 in dom <*p*> by A61, FINSEQ_3:25;
then A64: g . n1 = <*p*> . n1 by A4, FINSEQ_1:def_7;
n2 - 1 > 0 by A62, XREAL_1:50;
then A65: n2 -' 1 = n2 - 1 by XREAL_0:def_2;
then A66: (len <*p*>) + (n2 -' 1) = 1 + (n2 - 1) by FINSEQ_1:39
.= n2 ;
A67: 1 <= n2 -' 1 by A62, A65, SPPOL_1:1;
then n2 -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A65, A63, FINSEQ_3:25;
then g . n2 = (mid (f,((Index (p,f)) + 1),(len f))) . (n2 -' 1) by A4, A66, FINSEQ_1:def_7
.= f . (((n2 -' 1) + ((Index (p,f)) + 1)) -' 1) by A7, A17, A21, A65, A63, A67, FINSEQ_6:118
.= f . ((n2 + (Index (p,f))) -' 1) by A65 ;
then A68: f . ((n2 + (Index (p,f))) -' 1) = p by A56, A61, A64, FINSEQ_1:40;
n2 -' 1 <= (len f) - (Index (p,f)) by A47, A48, A59, NAT_D:42;
then n2 - 1 <= (len f) - (Index (p,f)) by A62, XREAL_1:233;
then A69: (n2 - 1) + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by XREAL_1:6;
1 + 1 < n2 + (Index (p,f)) by A2, A62, Th8, XREAL_1:8;
then A70: 1 < (n2 + (Index (p,f))) - 1 by XREAL_1:20;
then (n2 + (Index (p,f))) -' 1 = (n2 + (Index (p,f))) - 1 by XREAL_0:def_2;
hence contradiction by A1, A3, A70, A68, A69, Th12; ::_thesis: verum
end;
casethat A71: n1 > 1 and
A72: n2 = 1 ; ::_thesis: contradiction
A73: n1 - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A5, A57, XREAL_1:9;
n2 <= len <*p*> by A72, FINSEQ_1:39;
then n2 in dom <*p*> by A72, FINSEQ_3:25;
then A74: g . n2 = <*p*> . n2 by A4, FINSEQ_1:def_7;
n1 - 1 > 0 by A71, XREAL_1:50;
then A75: n1 -' 1 = n1 - 1 by XREAL_0:def_2;
then A76: (len <*p*>) + (n1 -' 1) = 1 + (n1 - 1) by FINSEQ_1:39
.= n1 ;
A77: 1 <= n1 -' 1 by A71, A75, SPPOL_1:1;
then n1 -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A75, A73, FINSEQ_3:25;
then g . n1 = (mid (f,((Index (p,f)) + 1),(len f))) . (n1 -' 1) by A4, A76, FINSEQ_1:def_7
.= f . (((n1 -' 1) + ((Index (p,f)) + 1)) -' 1) by A7, A17, A21, A75, A73, A77, FINSEQ_6:118
.= f . ((n1 + (Index (p,f))) -' 1) by A75 ;
then A78: f . ((n1 + (Index (p,f))) -' 1) = p by A56, A72, A74, FINSEQ_1:40;
n1 -' 1 <= (len f) - (Index (p,f)) by A47, A48, A57, NAT_D:42;
then n1 - 1 <= (len f) - (Index (p,f)) by A71, XREAL_1:233;
then A79: (n1 - 1) + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by XREAL_1:6;
1 + 1 < n1 + (Index (p,f)) by A2, A71, Th8, XREAL_1:8;
then A80: 1 < (n1 + (Index (p,f))) - 1 by XREAL_1:20;
then (n1 + (Index (p,f))) -' 1 = (n1 + (Index (p,f))) - 1 by XREAL_0:def_2;
hence contradiction by A1, A3, A80, A78, A79, Th12; ::_thesis: verum
end;
casethat A81: n1 > 1 and
A82: n2 > 1 ; ::_thesis: x1 = x2
A83: n2 - 1 > 0 by A82, XREAL_1:50;
then A84: n2 -' 1 = n2 - 1 by XREAL_0:def_2;
then A85: (len <*p*>) + (n2 -' 1) = 1 + (n2 - 1) by FINSEQ_1:39
.= n2 ;
A86: n1 - 1 > 0 by A81, XREAL_1:50;
then A87: n1 -' 1 = n1 - 1 by XREAL_0:def_2;
then A88: 0 + 1 <= n1 -' 1 by A86, NAT_1:13;
then A89: 1 <= (n1 - 1) + (Index (p,f)) by A87, NAT_1:12;
then A90: (n1 + (Index (p,f))) -' 1 = (n1 + (Index (p,f))) - 1 by XREAL_0:def_2;
A91: (len <*p*>) + (n1 -' 1) = 1 + (n1 - 1) by A87, FINSEQ_1:39
.= n1 ;
A92: n1 - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A5, A57, XREAL_1:9;
then n1 -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A87, A88, FINSEQ_3:25;
then A93: g . n1 = (mid (f,((Index (p,f)) + 1),(len f))) . (n1 -' 1) by A4, A91, FINSEQ_1:def_7
.= f . (((n1 -' 1) + ((Index (p,f)) + 1)) -' 1) by A7, A17, A21, A87, A88, A92, FINSEQ_6:118
.= f . ((n1 + (Index (p,f))) -' 1) by A87 ;
n1 -' 1 <= (len f) -' (Index (p,f)) by A53, A57, NAT_D:42;
then (n1 -' 1) + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by A51, XREAL_1:6;
then A94: (n1 + (Index (p,f))) -' 1 in dom f by A87, A89, A90, FINSEQ_3:25;
A95: 0 + 1 <= n2 -' 1 by A83, A84, NAT_1:13;
then A96: 1 <= (n2 -' 1) + (Index (p,f)) by NAT_1:12;
then A97: (n2 + (Index (p,f))) -' 1 = (n2 + (Index (p,f))) - 1 by A84, XREAL_0:def_2;
A98: n2 - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A5, A59, XREAL_1:9;
then n2 -' 1 in dom (mid (f,((Index (p,f)) + 1),(len f))) by A84, A95, FINSEQ_3:25;
then A99: g . n2 = (mid (f,((Index (p,f)) + 1),(len f))) . (n2 -' 1) by A4, A85, FINSEQ_1:def_7
.= f . (((n2 -' 1) + ((Index (p,f)) + 1)) -' 1) by A7, A17, A21, A84, A95, A98, FINSEQ_6:118
.= f . ((n2 + (Index (p,f))) -' 1) by A84 ;
n2 -' 1 <= (len f) -' (Index (p,f)) by A53, A59, NAT_D:42;
then (n2 -' 1) + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by A51, XREAL_1:6;
then (n2 + (Index (p,f))) -' 1 in dom f by A84, A96, A97, FINSEQ_3:25;
then (n1 + (Index (p,f))) -' 1 = (n2 + (Index (p,f))) -' 1 by A1, A56, A99, A93, A94, FUNCT_1:def_4;
hence x1 = x2 by A97, A90; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
then A100: g is one-to-one by FUNCT_1:def_4;
A101: ((len g) - 1) + 1 >= 1 + 1 by A18, A47, XREAL_1:6;
A102: ((len f) -' ((Index (p,f)) + 1)) + 1 = (len f) - (Index (p,f)) by A20;
for j being Nat st 1 <= j & j + 1 <= len g & not (g /. j) `1 = (g /. (j + 1)) `1 holds
(g /. j) `2 = (g /. (j + 1)) `2
proof
1 <= Index (p,f) by A2, Th8;
then A103: 1 < (Index (p,f)) + 1 by NAT_1:13;
let j be Nat; ::_thesis: ( 1 <= j & j + 1 <= len g & not (g /. j) `1 = (g /. (j + 1)) `1 implies (g /. j) `2 = (g /. (j + 1)) `2 )
assume that
A104: 1 <= j and
A105: j + 1 <= len g ; ::_thesis: ( (g /. j) `1 = (g /. (j + 1)) `1 or (g /. j) `2 = (g /. (j + 1)) `2 )
A106: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A104, A105, TOPREAL1:def_3;
(j + 1) - 1 <= (1 + (len (mid (f,((Index (p,f)) + 1),(len f))))) - 1 by A5, A105, XREAL_1:9;
then (j + 1) - 1 <= (len f) - (Index (p,f)) by A7, A17, A21, A102, FINSEQ_6:118;
then A107: j + (Index (p,f)) <= ((len f) - (Index (p,f))) + (Index (p,f)) by XREAL_1:6;
(Index (p,f)) + 1 <= (Index (p,f)) + j by A104, XREAL_1:6;
then 1 < (Index (p,f)) + j by A103, XXREAL_0:2;
then A108: 1 <= ((Index (p,f)) + j) - 1 by SPPOL_1:1;
then A109: ((Index (p,f)) + j) - 1 = ((Index (p,f)) + j) -' 1 by XREAL_0:def_2;
then A110: LSeg (f,(((Index (p,f)) + j) -' 1)) = LSeg ((f /. (((Index (p,f)) + j) -' 1)),(f /. ((((Index (p,f)) + j) -' 1) + 1))) by A108, A107, TOPREAL1:def_3;
A111: ( (f /. (((Index (p,f)) + j) -' 1)) `1 = (f /. ((((Index (p,f)) + j) -' 1) + 1)) `1 or (f /. (((Index (p,f)) + j) -' 1)) `2 = (f /. ((((Index (p,f)) + j) -' 1) + 1)) `2 ) by A1, A108, A109, A107, TOPREAL1:def_5;
LSeg (g,j) c= LSeg (f,(((Index (p,f)) + j) -' 1)) by A2, A4, A104, A105, Th16;
hence ( (g /. j) `1 = (g /. (j + 1)) `1 or (g /. j) `2 = (g /. (j + 1)) `2 ) by A106, A110, A111, Th3; ::_thesis: verum
end;
then ( g is unfolded & g is s.n.c. & g is special ) by A23, A8, TOPREAL1:def_5, TOPREAL1:def_6, TOPREAL1:def_7;
then A112: g is being_S-Seq by A101, A100, TOPREAL1:def_8;
A113: ((len f) -' (Index (p,f))) + ((Index (p,f)) + 1) = ((len f) - (Index (p,f))) + ((Index (p,f)) + 1) by A50, XREAL_0:def_2
.= (len f) + 1 ;
1 + ((len g) -' 1) = 1 + ((len g) - 1) by A46, XREAL_0:def_2
.= len g ;
then g . (len g) = g . ((len <*p*>) + ((len g) -' 1)) by FINSEQ_1:39
.= (mid (f,((Index (p,f)) + 1),(len f))) . ((len g) -' 1) by A4, A49, FINSEQ_1:def_7 ;
then g . (len g) = f . (len f) by A47, A48, A52, A113, NAT_D:34;
then A114: g . (len g) = f /. (len f) by A7, FINSEQ_4:15;
g . 1 = p by A4, FINSEQ_1:41;
hence g is_S-Seq_joining p,f /. (len f) by A112, A114, Def2; ::_thesis: verum
end;
theorem Th18: :: JORDAN3:18
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2)
for j being Nat st p in L~ f & 1 <= j & j + 1 <= len g & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds
LSeg (g,j) c= LSeg (f,j)
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2)
for j being Nat st p in L~ f & 1 <= j & j + 1 <= len g & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds
LSeg (g,j) c= LSeg (f,j)
let p be Point of (TOP-REAL 2); ::_thesis: for j being Nat st p in L~ f & 1 <= j & j + 1 <= len g & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds
LSeg (g,j) c= LSeg (f,j)
let j be Nat; ::_thesis: ( p in L~ f & 1 <= j & j + 1 <= len g & g = (mid (f,1,(Index (p,f)))) ^ <*p*> implies LSeg (g,j) c= LSeg (f,j) )
assume that
A1: p in L~ f and
A2: 1 <= j and
A3: j + 1 <= len g and
A4: g = (mid (f,1,(Index (p,f)))) ^ <*p*> ; ::_thesis: LSeg (g,j) c= LSeg (f,j)
A5: Index (p,f) < len f by A1, Th8;
A6: j in NAT by ORDINAL1:def_12;
A7: 1 <= j + 1 by NAT_1:11;
A8: 1 <= Index (p,f) by A1, Th8;
1 <= Index (p,f) by A1, Th8;
then A9: 1 <= len f by A5, XXREAL_0:2;
j <= j + 1 by NAT_1:11;
then A10: j <= len g by A3, XXREAL_0:2;
now__::_thesis:_LSeg_(g,j)_c=_LSeg_(f,j)
len g = (len (mid (f,1,(Index (p,f))))) + (len <*p*>) by A4, FINSEQ_1:22
.= (len (mid (f,1,(Index (p,f))))) + 1 by FINSEQ_1:39 ;
then len g = (((Index (p,f)) -' 1) + 1) + 1 by A5, A9, A8, FINSEQ_6:118;
then A11: len g = (Index (p,f)) + 1 by A1, Th8, XREAL_1:235;
then A12: j <= Index (p,f) by A3, XREAL_1:6;
(Index (p,f)) + 1 <= (len f) + 1 by A5, XREAL_1:6;
then j + 1 <= (len f) + 1 by A3, A11, XXREAL_0:2;
then A13: (j + 1) - 1 <= ((len f) + 1) - 1 by XREAL_1:9;
A14: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A5, A9, A8, FINSEQ_6:118
.= Index (p,f) by A1, Th8, XREAL_1:235 ;
then A15: j in dom (mid (f,1,(Index (p,f)))) by A2, A12, FINSEQ_3:25;
A16: g /. j = g . j by A2, A10, FINSEQ_4:15
.= (mid (f,1,(Index (p,f)))) . j by A4, A15, FINSEQ_1:def_7
.= f . ((j + 1) -' 1) by A2, A6, A5, A9, A8, A12, A14, FINSEQ_6:118
.= f . j by NAT_D:34
.= f /. j by A2, A13, FINSEQ_4:15 ;
now__::_thesis:_(_(_j_+_1_<=_Index_(p,f)_&_LSeg_(g,j)_c=_LSeg_(f,j)_)_or_(_j_+_1_>_Index_(p,f)_&_LSeg_(g,j)_c=_LSeg_(f,j)_)_)
percases ( j + 1 <= Index (p,f) or j + 1 > Index (p,f) ) ;
caseA17: j + 1 <= Index (p,f) ; ::_thesis: LSeg (g,j) c= LSeg (f,j)
A18: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A5, A9, A8, FINSEQ_6:118
.= Index (p,f) by A1, Th8, XREAL_1:235 ;
then A19: j + 1 in dom (mid (f,1,(Index (p,f)))) by A7, A17, FINSEQ_3:25;
A20: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A2, A3, TOPREAL1:def_3;
A21: j + 1 <= len f by A5, A17, XXREAL_0:2;
g /. (j + 1) = g . (j + 1) by A3, FINSEQ_4:15, NAT_1:11
.= (mid (f,1,(Index (p,f)))) . (j + 1) by A4, A19, FINSEQ_1:def_7
.= f . (((j + 1) + 1) -' 1) by A5, A9, A8, A7, A17, A18, FINSEQ_6:118
.= f . (j + 1) by NAT_D:34
.= f /. (j + 1) by A21, FINSEQ_4:15, NAT_1:11 ;
hence LSeg (g,j) c= LSeg (f,j) by A2, A16, A21, A20, TOPREAL1:def_3; ::_thesis: verum
end;
case j + 1 > Index (p,f) ; ::_thesis: LSeg (g,j) c= LSeg (f,j)
then j >= Index (p,f) by NAT_1:13;
then A22: j = Index (p,f) by A12, XXREAL_0:1;
then A23: p in LSeg (f,j) by A1, Th9;
now__::_thesis:_not_j_+_1_>_len_f
assume j + 1 > len f ; ::_thesis: contradiction
then j >= len f by NAT_1:13;
hence contradiction by A1, A22, Th8; ::_thesis: verum
end;
then A24: LSeg (f,j) = LSeg ((f /. j),(f /. (j + 1))) by A2, TOPREAL1:def_3;
1 <= len <*p*> by FINSEQ_1:40;
then A25: 1 in dom <*p*> by FINSEQ_3:25;
A26: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A5, A9, A8, FINSEQ_6:118
.= Index (p,f) by A1, Th8, XREAL_1:235 ;
A27: f /. j in LSeg ((f /. j),(f /. (j + 1))) by RLTOPSP1:68;
g /. (j + 1) = g . (j + 1) by A3, FINSEQ_4:15, NAT_1:11
.= <*p*> . 1 by A4, A22, A25, A26, FINSEQ_1:def_7
.= p by FINSEQ_1:def_8 ;
then LSeg ((g /. j),(g /. (j + 1))) c= LSeg ((f /. j),(f /. (j + 1))) by A16, A27, A23, A24, TOPREAL1:6;
hence LSeg (g,j) c= LSeg (f,j) by A2, A3, A24, TOPREAL1:def_3; ::_thesis: verum
end;
end;
end;
hence LSeg (g,j) c= LSeg (f,j) ; ::_thesis: verum
end;
hence LSeg (g,j) c= LSeg (f,j) ; ::_thesis: verum
end;
theorem Th19: :: JORDAN3:19
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds
g is_S-Seq_joining f /. 1,p
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 & g = (mid (f,1,(Index (p,f)))) ^ <*p*> holds
g is_S-Seq_joining f /. 1,p
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p <> f . 1 & g = (mid (f,1,(Index (p,f)))) ^ <*p*> implies g is_S-Seq_joining f /. 1,p )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . 1 and
A4: g = (mid (f,1,(Index (p,f)))) ^ <*p*> ; ::_thesis: g is_S-Seq_joining f /. 1,p
A5: Index (p,f) <= len f by A2, Th8;
A6: for j1, j2 being Nat st j1 + 1 < j2 holds
LSeg (g,j1) misses LSeg (g,j2)
proof
let j1, j2 be Nat; ::_thesis: ( j1 + 1 < j2 implies LSeg (g,j1) misses LSeg (g,j2) )
assume A7: j1 + 1 < j2 ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
A8: ( j1 = 0 or j1 >= 0 + 1 ) by NAT_1:13;
now__::_thesis:_(_(_j1_=_0_&_LSeg_(g,j1)_misses_LSeg_(g,j2)_)_or_(_(_j1_=_1_or_j1_>_1_)_&_j2_+_1_<=_len_g_&_LSeg_(g,j1)_misses_LSeg_(g,j2)_)_or_(_j2_+_1_>_len_g_&_LSeg_(g,j1)_misses_LSeg_(g,j2)_)_)
percases ( j1 = 0 or ( ( j1 = 1 or j1 > 1 ) & j2 + 1 <= len g ) or j2 + 1 > len g ) by A8, XXREAL_0:1;
case j1 = 0 ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
then LSeg (g,j1) = {} by TOPREAL1:def_3;
then (LSeg (g,j1)) /\ (LSeg (g,j2)) = {} ;
hence LSeg (g,j1) misses LSeg (g,j2) by XBOOLE_0:def_7; ::_thesis: verum
end;
casethat A9: ( j1 = 1 or j1 > 1 ) and
A10: j2 + 1 <= len g ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
j2 < len g by A10, NAT_1:13;
then j1 + 1 < len g by A7, XXREAL_0:2;
then A11: LSeg (g,j1) c= LSeg (f,j1) by A2, A4, A9, Th18;
1 + 1 <= j1 + 1 by A9, XREAL_1:6;
then 2 <= j2 by A7, XXREAL_0:2;
then 1 <= j2 by XXREAL_0:2;
then A12: LSeg (g,j2) c= LSeg (f,j2) by A2, A4, A10, Th18;
LSeg (f,j1) misses LSeg (f,j2) by A1, A7, TOPREAL1:def_7;
then (LSeg (f,j1)) /\ (LSeg (f,j2)) = {} by XBOOLE_0:def_7;
then (LSeg (g,j1)) /\ (LSeg (g,j2)) = {} by A11, A12, XBOOLE_1:3, XBOOLE_1:27;
hence LSeg (g,j1) misses LSeg (g,j2) by XBOOLE_0:def_7; ::_thesis: verum
end;
case j2 + 1 > len g ; ::_thesis: LSeg (g,j1) misses LSeg (g,j2)
then LSeg (g,j2) = {} by TOPREAL1:def_3;
then (LSeg (g,j1)) /\ (LSeg (g,j2)) = {} ;
hence LSeg (g,j1) misses LSeg (g,j2) by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
end;
hence LSeg (g,j1) misses LSeg (g,j2) ; ::_thesis: verum
end;
A13: for n1, n2 being Element of NAT st 1 <= n1 & n1 <= len f & 1 <= n2 & n2 <= len f & f . n1 = f . n2 holds
n1 = n2
proof
let n1, n2 be Element of NAT ; ::_thesis: ( 1 <= n1 & n1 <= len f & 1 <= n2 & n2 <= len f & f . n1 = f . n2 implies n1 = n2 )
assume that
A14: 1 <= n1 and
A15: n1 <= len f and
A16: 1 <= n2 and
A17: n2 <= len f and
A18: f . n1 = f . n2 ; ::_thesis: n1 = n2
A19: n2 in dom f by A16, A17, FINSEQ_3:25;
n1 in dom f by A14, A15, FINSEQ_3:25;
hence n1 = n2 by A1, A18, A19, FUNCT_1:def_4; ::_thesis: verum
end;
A20: len g = (len (mid (f,1,(Index (p,f))))) + (len <*p*>) by A4, FINSEQ_1:22
.= (len (mid (f,1,(Index (p,f))))) + 1 by FINSEQ_1:39 ;
consider i being Element of NAT such that
1 <= i and
A21: i + 1 <= len f and
p in LSeg (f,i) by A2, SPPOL_2:13;
A22: 1 <= Index (p,f) by A2, Th8;
1 <= 1 + i by NAT_1:12;
then A23: 1 <= len f by A21, XXREAL_0:2;
then A24: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A22, A5, FINSEQ_6:118;
then A25: len (mid (f,1,(Index (p,f)))) = Index (p,f) by A2, Th8, XREAL_1:235;
then g . 1 = (mid (f,1,(Index (p,f)))) . 1 by A4, A22, FINSEQ_6:109;
then g . 1 = f . 1 by A22, A5, A23, FINSEQ_6:118;
then A26: g . 1 = f /. 1 by A23, FINSEQ_4:15;
A27: for j being Nat st 1 <= j & j + 2 <= len g holds
(LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))}
proof
let j be Nat; ::_thesis: ( 1 <= j & j + 2 <= len g implies (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))} )
assume that
A28: 1 <= j and
A29: j + 2 <= len g ; ::_thesis: (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))}
A30: j + 1 <= len g by A29, NAT_D:47;
then LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A28, TOPREAL1:def_3;
then A31: g /. (j + 1) in LSeg (g,j) by RLTOPSP1:68;
A32: 1 <= j + 1 by A28, NAT_D:48;
then LSeg (g,(j + 1)) = LSeg ((g /. (j + 1)),(g /. ((j + 1) + 1))) by A29, TOPREAL1:def_3;
then g /. (j + 1) in LSeg (g,(j + 1)) by RLTOPSP1:68;
then g /. (j + 1) in (LSeg (g,j)) /\ (LSeg (g,(j + 1))) by A31, XBOOLE_0:def_4;
then A33: {(g /. (j + 1))} c= (LSeg (g,j)) /\ (LSeg (g,(j + 1))) by ZFMISC_1:31;
j + 1 <= len g by A29, NAT_D:47;
then A34: LSeg (g,j) c= LSeg (f,j) by A2, A4, A28, Th18;
A35: Index (p,f) <= len f by A2, Th8;
A36: (j + 1) + 1 <= len g by A29;
then LSeg (g,(j + 1)) c= LSeg (f,(j + 1)) by A2, A4, A32, Th18;
then A37: (LSeg (g,j)) /\ (LSeg (g,(j + 1))) c= (LSeg (f,j)) /\ (LSeg (f,(j + 1))) by A34, XBOOLE_1:27;
A38: g /. (j + 1) = g . (j + 1) by A32, A30, FINSEQ_4:15;
now__::_thesis:_(LSeg_(g,j))_/\_(LSeg_(g,(j_+_1)))_=_{(g_/._(j_+_1))}
A39: len g = (len (mid (f,1,(Index (p,f))))) + 1 by A4, FINSEQ_2:16;
Index (p,f) <= len f by A2, Th8;
then A40: len g <= (len f) + 1 by A25, A39, XREAL_1:6;
now__::_thesis:_(_(_len_g_=_(len_f)_+_1_&_contradiction_)_or_(_len_g_<_(len_f)_+_1_&_(LSeg_(g,j))_/\_(LSeg_(g,(j_+_1)))_=_{(g_/._(j_+_1))}_)_)
percases ( len g = (len f) + 1 or len g < (len f) + 1 ) by A40, XXREAL_0:1;
case len g = (len f) + 1 ; ::_thesis: contradiction
hence contradiction by A2, A25, A39, Th8; ::_thesis: verum
end;
case len g < (len f) + 1 ; ::_thesis: (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))}
then len g <= len f by NAT_1:13;
then j + 2 <= len f by A29, XXREAL_0:2;
then A41: (LSeg (g,j)) /\ (LSeg (g,(j + 1))) c= {(f /. (j + 1))} by A1, A28, A37, TOPREAL1:def_6;
A42: j + 1 <= Index (p,f) by A25, A36, A39, XREAL_1:6;
then j + 1 <= len f by A35, XXREAL_0:2;
then A43: f . (j + 1) = f /. (j + 1) by A32, FINSEQ_4:15;
g . (j + 1) = (mid (f,1,(Index (p,f)))) . (j + 1) by A4, A25, A32, A42, FINSEQ_1:64
.= f . (j + 1) by A5, A32, A42, FINSEQ_6:123 ;
hence (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))} by A38, A33, A41, A43, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;
hence (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))} ; ::_thesis: verum
end;
hence (LSeg (g,j)) /\ (LSeg (g,(j + 1))) = {(g /. (j + 1))} ; ::_thesis: verum
end;
for j being Nat st 1 <= j & j + 1 <= len g & not (g /. j) `1 = (g /. (j + 1)) `1 holds
(g /. j) `2 = (g /. (j + 1)) `2
proof
A44: Index (p,f) < len f by A2, Th8;
let j be Nat; ::_thesis: ( 1 <= j & j + 1 <= len g & not (g /. j) `1 = (g /. (j + 1)) `1 implies (g /. j) `2 = (g /. (j + 1)) `2 )
assume that
A45: 1 <= j and
A46: j + 1 <= len g ; ::_thesis: ( (g /. j) `1 = (g /. (j + 1)) `1 or (g /. j) `2 = (g /. (j + 1)) `2 )
A47: LSeg (g,j) = LSeg ((g /. j),(g /. (j + 1))) by A45, A46, TOPREAL1:def_3;
j + 1 <= (Index (p,f)) + 1 by A4, A25, A46, FINSEQ_2:16;
then j <= Index (p,f) by XREAL_1:6;
then j < len f by A44, XXREAL_0:2;
then A48: j + 1 <= len f by NAT_1:13;
then A49: LSeg (f,j) = LSeg ((f /. j),(f /. (j + 1))) by A45, TOPREAL1:def_3;
A50: ( (f /. j) `1 = (f /. (j + 1)) `1 or (f /. j) `2 = (f /. (j + 1)) `2 ) by A1, A45, A48, TOPREAL1:def_5;
LSeg (g,j) c= LSeg (f,j) by A2, A4, A45, A46, Th18;
hence ( (g /. j) `1 = (g /. (j + 1)) `1 or (g /. j) `2 = (g /. (j + 1)) `2 ) by A47, A49, A50, Th3; ::_thesis: verum
end;
then A51: ( g is unfolded & g is s.n.c. & g is special ) by A27, A6, TOPREAL1:def_5, TOPREAL1:def_6, TOPREAL1:def_7;
1 <= len <*p*> by FINSEQ_1:39;
then A52: 1 in dom <*p*> by FINSEQ_3:25;
for x1, x2 being set st x1 in dom g & x2 in dom g & g . x1 = g . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom g & x2 in dom g & g . x1 = g . x2 implies x1 = x2 )
assume that
A53: x1 in dom g and
A54: x2 in dom g and
A55: g . x1 = g . x2 ; ::_thesis: x1 = x2
reconsider n1 = x1, n2 = x2 as Element of NAT by A53, A54;
A56: 1 <= n1 by A53, FINSEQ_3:25;
A57: n2 <= len g by A54, FINSEQ_3:25;
A58: 1 <= n2 by A54, FINSEQ_3:25;
A59: n1 <= len g by A53, FINSEQ_3:25;
now__::_thesis:_x1_=_x2
A60: g . (len g) = <*p*> . 1 by A4, A52, A20, FINSEQ_1:def_7
.= p by FINSEQ_1:def_8 ;
now__::_thesis:_(_(_n1_=_len_g_&_x1_=_x2_)_or_(_n2_=_len_g_&_x1_=_x2_)_or_(_n1_<>_len_g_&_n2_<>_len_g_&_x1_=_x2_)_)
percases ( n1 = len g or n2 = len g or ( n1 <> len g & n2 <> len g ) ) ;
caseA61: n1 = len g ; ::_thesis: x1 = x2
now__::_thesis:_not_n2_<>_len_g
assume A62: n2 <> len g ; ::_thesis: contradiction
then n2 < len g by A57, XXREAL_0:1;
then A63: n2 <= len (mid (f,1,(Index (p,f)))) by A20, NAT_1:13;
then A64: n2 <= len f by A5, A25, XXREAL_0:2;
g . n2 = (mid (f,1,(Index (p,f)))) . n2 by A4, A58, A63, FINSEQ_1:64;
then g . n2 = f . ((n2 + 1) -' 1) by A22, A5, A23, A58, A63, FINSEQ_6:118;
then A65: p = f . n2 by A55, A60, A61, NAT_D:34;
then 1 < n2 by A3, A58, XXREAL_0:1;
then (Index (p,f)) + 1 = n2 by A1, A65, A64, Th12;
hence contradiction by A2, A24, A20, A62, Th8, XREAL_1:235; ::_thesis: verum
end;
hence x1 = x2 by A61; ::_thesis: verum
end;
caseA66: n2 = len g ; ::_thesis: x1 = x2
now__::_thesis:_not_n1_<>_len_g
assume A67: n1 <> len g ; ::_thesis: contradiction
then n1 < len g by A59, XXREAL_0:1;
then A68: n1 <= len (mid (f,1,(Index (p,f)))) by A20, NAT_1:13;
then A69: n1 <= len f by A5, A25, XXREAL_0:2;
g . n1 = (mid (f,1,(Index (p,f)))) . n1 by A4, A56, A68, FINSEQ_1:64;
then g . n1 = f . ((n1 + 1) -' 1) by A22, A5, A23, A56, A68, FINSEQ_6:118;
then A70: p = f . n1 by A55, A60, A66, NAT_D:34;
then 1 < n1 by A3, A56, XXREAL_0:1;
then (Index (p,f)) + 1 = n1 by A1, A70, A69, Th12;
hence contradiction by A2, A24, A20, A67, Th8, XREAL_1:235; ::_thesis: verum
end;
hence x1 = x2 by A66; ::_thesis: verum
end;
casethat A71: n1 <> len g and
A72: n2 <> len g ; ::_thesis: x1 = x2
n1 < len g by A59, A71, XXREAL_0:1;
then A73: n1 <= len (mid (f,1,(Index (p,f)))) by A20, NAT_1:13;
then A74: n1 <= len f by A5, A25, XXREAL_0:2;
n2 < len g by A57, A72, XXREAL_0:1;
then A75: n2 <= len (mid (f,1,(Index (p,f)))) by A20, NAT_1:13;
then A76: g . n2 = (mid (f,1,(Index (p,f)))) . n2 by A4, A58, FINSEQ_1:64
.= f . n2 by A5, A25, A58, A75, FINSEQ_6:123 ;
A77: n2 <= len f by A5, A25, A75, XXREAL_0:2;
g . n1 = (mid (f,1,(Index (p,f)))) . n1 by A4, A56, A73, FINSEQ_1:64
.= f . n1 by A5, A25, A56, A73, FINSEQ_6:123 ;
hence x1 = x2 by A13, A55, A56, A58, A74, A77, A76; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
hence x1 = x2 ; ::_thesis: verum
end;
then A78: g is one-to-one by FUNCT_1:def_4;
1 + 1 <= len g by A22, A25, A20, XREAL_1:6;
then A79: g is being_S-Seq by A78, A51, TOPREAL1:def_8;
g . (len g) = p by A4, A20, FINSEQ_1:42;
hence g is_S-Seq_joining f /. 1,p by A26, A79, Def2; ::_thesis: verum
end;
begin
definition
let f be FinSequence of (TOP-REAL 2);
let p be Point of (TOP-REAL 2);
func L_Cut (f,p) -> FinSequence of (TOP-REAL 2) equals :Def3: :: JORDAN3:def 3
<*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) if p <> f . ((Index (p,f)) + 1)
otherwise mid (f,((Index (p,f)) + 1),(len f));
correctness
coherence
( ( p <> f . ((Index (p,f)) + 1) implies <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) is FinSequence of (TOP-REAL 2) ) & ( not p <> f . ((Index (p,f)) + 1) implies mid (f,((Index (p,f)) + 1),(len f)) is FinSequence of (TOP-REAL 2) ) );
consistency
for b1 being FinSequence of (TOP-REAL 2) holds verum;
;
func R_Cut (f,p) -> FinSequence of (TOP-REAL 2) equals :Def4: :: JORDAN3:def 4
(mid (f,1,(Index (p,f)))) ^ <*p*> if p <> f . 1
otherwise <*p*>;
correctness
coherence
( ( p <> f . 1 implies (mid (f,1,(Index (p,f)))) ^ <*p*> is FinSequence of (TOP-REAL 2) ) & ( not p <> f . 1 implies <*p*> is FinSequence of (TOP-REAL 2) ) );
consistency
for b1 being FinSequence of (TOP-REAL 2) holds verum;
;
end;
:: deftheorem Def3 defines L_Cut JORDAN3:def_3_:_
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) holds
( ( p <> f . ((Index (p,f)) + 1) implies L_Cut (f,p) = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) ) & ( not p <> f . ((Index (p,f)) + 1) implies L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) ) );
:: deftheorem Def4 defines R_Cut JORDAN3:def_4_:_
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) holds
( ( p <> f . 1 implies R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> ) & ( not p <> f . 1 implies R_Cut (f,p) = <*p*> ) );
theorem Th20: :: JORDAN3:20
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p = f . ((Index (p,f)) + 1) & p <> f . (len f) holds
((Index (p,(Rev f))) + (Index (p,f))) + 1 = len f
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 in L~ f & p = f . ((Index (p,f)) + 1) & p <> f . (len f) holds
((Index (p,(Rev f))) + (Index (p,f))) + 1 = len f
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p = f . ((Index (p,f)) + 1) & p <> f . (len f) implies ((Index (p,(Rev f))) + (Index (p,f))) + 1 = len f )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p = f . ((Index (p,f)) + 1) and
A4: p <> f . (len f) ; ::_thesis: ((Index (p,(Rev f))) + (Index (p,f))) + 1 = len f
A5: len f <= (len f) + (Index (p,f)) by NAT_1:11;
len f = len (Rev f) by FINSEQ_5:def_3;
then A6: (len f) - (Index (p,f)) <= len (Rev f) by A5, XREAL_1:20;
Index (p,f) <= len f by A2, Th8;
then A7: (len f) - (Index (p,f)) = (len f) -' (Index (p,f)) by XREAL_1:233;
Index (p,f) < len f by A2, Th8;
then A8: (Index (p,f)) + 1 <= len f by NAT_1:13;
then (Index (p,f)) + 1 < len f by A3, A4, XXREAL_0:1;
then A9: 1 < (len f) - (Index (p,f)) by XREAL_1:20;
1 <= (Index (p,f)) + 1 by NAT_1:11;
then (Index (p,f)) + 1 in dom f by A8, FINSEQ_3:25;
then A10: (Index (p,f)) + 1 in dom (Rev f) by FINSEQ_5:57;
p = (Rev (Rev f)) . ((Index (p,f)) + 1) by A3
.= (Rev f) . (((len (Rev f)) - ((Index (p,f)) + 1)) + 1) by A10, FINSEQ_5:58
.= (Rev f) . ((len (Rev f)) - (Index (p,f)))
.= (Rev f) . ((len f) - (Index (p,f))) by FINSEQ_5:def_3 ;
then (Index (p,(Rev f))) + 1 = (len f) -' (Index (p,f)) by A1, A6, A9, A7, Th12;
hence ((Index (p,(Rev f))) + (Index (p,f))) + 1 = len f by A7; ::_thesis: verum
end;
theorem Th21: :: JORDAN3:21
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is unfolded & f is s.n.c. & p in L~ f & p <> f . ((Index (p,f)) + 1) holds
(Index (p,(Rev f))) + (Index (p,f)) = len f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f is unfolded & f is s.n.c. & p in L~ f & p <> f . ((Index (p,f)) + 1) holds
(Index (p,(Rev f))) + (Index (p,f)) = len f
let p be Point of (TOP-REAL 2); ::_thesis: ( f is unfolded & f is s.n.c. & p in L~ f & p <> f . ((Index (p,f)) + 1) implies (Index (p,(Rev f))) + (Index (p,f)) = len f )
assume that
A1: ( f is unfolded & f is s.n.c. ) and
A2: p in L~ f and
A3: p <> f . ((Index (p,f)) + 1) ; ::_thesis: (Index (p,(Rev f))) + (Index (p,f)) = len f
A4: Index (p,f) < len f by A2, Th8;
then A5: ((len f) -' (Index (p,f))) + (Index (p,f)) = len f by XREAL_1:235;
0 + 1 <= Index (p,f) by A2, Th8;
then (len f) + 0 < (len f) + (Index (p,f)) by XREAL_1:6;
then (len f) - (Index (p,f)) < len f by XREAL_1:19;
then A6: (len f) -' (Index (p,f)) < len f by A4, XREAL_1:233;
A7: Index (p,f) < len f by A2, Th8;
then (Index (p,f)) + 1 <= len f by NAT_1:13;
then 1 <= (len f) - (Index (p,f)) by XREAL_1:19;
then 1 <= (len f) -' (Index (p,f)) by NAT_D:39;
then (len f) -' (Index (p,f)) in dom f by A6, FINSEQ_3:25;
then A8: (Rev f) . ((len f) -' (Index (p,f))) = f . (((len f) - ((len f) -' (Index (p,f)))) + 1) by FINSEQ_5:58
.= f . (((len f) - ((len f) - (Index (p,f)))) + 1) by A7, XREAL_1:233
.= f . ((0 + (Index (p,f))) + 1) ;
p in LSeg (f,(Index (p,f))) by A2, Th9;
then A9: p in LSeg ((Rev f),((len f) -' (Index (p,f)))) by A5, SPPOL_2:2;
len f = len (Rev f) by FINSEQ_5:def_3;
then A10: ((len f) -' (Index (p,f))) + 1 <= len (Rev f) by A6, NAT_1:13;
Rev f is s.n.c. by A1, SPPOL_2:35;
then (len f) -' (Index (p,f)) = Index (p,(Rev f)) by A1, A3, A9, A10, A8, Th14, SPPOL_2:28;
hence (Index (p,(Rev f))) + (Index (p,f)) = len f by A7, XREAL_1:235; ::_thesis: verum
end;
theorem Th22: :: JORDAN3:22
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f holds
L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
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 in L~ f holds
L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f implies L_Cut ((Rev f),p) = Rev (R_Cut (f,p)) )
assume that
A1: f is being_S-Seq and
A2: p in L~ f ; ::_thesis: L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
A3: len f = len (Rev f) by FINSEQ_5:def_3;
A4: p in L~ (Rev f) by A2, SPPOL_2:22;
A5: 1 <= Index (p,f) by A2, Th8;
A6: Rev f is being_S-Seq by A1;
A7: Rev (Rev f) = f ;
A8: Index (p,f) < len f by A2, Th8;
L~ f = L~ (Rev f) by SPPOL_2:22;
then Index (p,(Rev f)) < len (Rev f) by A2, Th8;
then A9: (Index (p,(Rev f))) + 1 <= len f by A3, NAT_1:13;
1 <= (Index (p,(Rev f))) + 1 by NAT_1:11;
then A10: (Index (p,(Rev f))) + 1 in dom f by A9, FINSEQ_3:25;
A11: 1 + 1 <= len f by A1, TOPREAL1:def_8;
then A12: 1 < len f by NAT_1:13;
then A13: 1 in dom f by FINSEQ_3:25;
A14: len f in dom f by A12, FINSEQ_3:25;
A15: 2 in dom f by A11, FINSEQ_3:25;
A16: dom (Rev f) = dom f by FINSEQ_5:57;
percases ( p = f . (len f) or p = f . 1 or ( p <> f . 1 & p <> f . (len f) & p = f . ((Index (p,f)) + 1) ) or ( p <> f . 1 & p <> f . ((Index (p,f)) + 1) ) ) ;
supposeA17: p = f . (len f) ; ::_thesis: L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
then A18: p <> f . 1 by A1, A12, A13, A14, FUNCT_1:def_4;
A19: p = (Rev f) . 1 by A17, FINSEQ_5:62;
then A20: p <> (Rev f) . (1 + 1) by A1, A16, A13, A15, FUNCT_1:def_4;
p = (Rev f) /. 1 by A16, A13, A19, PARTFUN1:def_6;
then A21: Index (p,(Rev f)) = 1 by A3, A11, Th11;
then (Index (p,(Rev f))) + (Index (p,f)) = len f by A6, A4, A7, A3, A20, Th21;
then A22: Index (p,(Rev f)) = (len f) - (Index (p,f)) ;
thus L_Cut ((Rev f),p) = <*p*> ^ (mid ((Rev f),((Index (p,(Rev f))) + 1),(len f))) by A3, A21, A20, Def3
.= <*p*> ^ (mid ((Rev f),(((len f) -' (Index (p,f))) + 1),(len f))) by A8, A22, XREAL_1:233
.= <*p*> ^ (mid ((Rev f),(((len f) -' (Index (p,f))) + 1),(((len f) -' 1) + 1))) by A12, XREAL_1:235
.= <*p*> ^ (Rev (mid (f,1,(Index (p,f))))) by A12, A5, A8, FINSEQ_6:113
.= Rev ((mid (f,1,(Index (p,f)))) ^ <*p*>) by FINSEQ_5:63
.= Rev (R_Cut (f,p)) by A18, Def4 ; ::_thesis: verum
end;
supposeA23: p = f . 1 ; ::_thesis: L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
A24: ((len (Rev f)) -' 1) + 1 = len (Rev f) by A3, A12, XREAL_1:235;
then A25: ((Rev f) /^ ((len (Rev f)) -' 1)) . 1 = (Rev f) . (len (Rev f)) by FINSEQ_6:114;
A26: len ((Rev f) /^ ((len (Rev f)) -' 1)) = (len (Rev f)) -' ((len (Rev f)) -' 1) by RFINSEQ:29;
1 <= (len (Rev f)) - ((len (Rev f)) -' 1) by A24;
then A27: 1 <= len ((Rev f) /^ ((len (Rev f)) -' 1)) by A26, NAT_D:39;
((len (Rev f)) -' (len (Rev f))) + 1 = ((len (Rev f)) - (len (Rev f))) + 1 by XREAL_1:233
.= 1 ;
then A28: mid ((Rev f),(len (Rev f)),(len (Rev f))) = ((Rev f) /^ ((len (Rev f)) -' 1)) | 1 by FINSEQ_6:def_3
.= <*(((Rev f) /^ ((len (Rev f)) -' 1)) /. 1)*> by A27, CARD_1:27, FINSEQ_5:20
.= <*((Rev f) . (len (Rev f)))*> by A25, A27, FINSEQ_4:15 ;
A29: p = (Rev f) . (len f) by A23, FINSEQ_5:62;
then (Index (p,(Rev f))) + 1 = len f by A1, A3, A12, Th12;
hence L_Cut ((Rev f),p) = <*p*> by A3, A29, A28, Def3
.= Rev <*p*> by FINSEQ_5:60
.= Rev (R_Cut (f,p)) by A23, Def4 ;
::_thesis: verum
end;
supposethat A30: p <> f . 1 and
A31: p <> f . (len f) and
A32: p = f . ((Index (p,f)) + 1) ; ::_thesis: L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
A33: len f = ((Index (p,(Rev f))) + (Index (p,f))) + 1 by A1, A2, A31, A32, Th20
.= (Index (p,f)) + ((Index (p,(Rev f))) + 1) ;
len f = ((Index (p,(Rev f))) + (Index (p,f))) + 1 by A1, A2, A31, A32, Th20
.= (Index (p,(Rev f))) + ((Index (p,f)) + 1) ;
then A34: p = f . (((len f) - ((Index (p,(Rev f))) + 1)) + 1) by A32
.= (Rev f) . ((Index (p,(Rev f))) + 1) by A10, FINSEQ_5:58 ;
A35: (len f) -' (Index (p,f)) = (len f) - (Index (p,f)) by A8, XREAL_1:233
.= (Index (p,(Rev f))) + 1 by A33 ;
p <> (Rev f) . (len f) by A30, FINSEQ_5:62;
then A36: (Index (p,(Rev f))) + 1 < len f by A9, A34, XXREAL_0:1;
thus L_Cut ((Rev f),p) = mid ((Rev f),((Index (p,(Rev f))) + 1),(len f)) by A3, A34, Def3
.= <*p*> ^ (mid ((Rev f),(((len f) -' (Index (p,f))) + 1),(len f))) by A16, A10, A34, A35, A36, FINSEQ_6:126
.= <*p*> ^ (mid ((Rev f),(((len f) -' (Index (p,f))) + 1),(((len f) -' 1) + 1))) by A12, XREAL_1:235
.= <*p*> ^ (Rev (mid (f,1,(Index (p,f))))) by A12, A5, A8, FINSEQ_6:113
.= Rev ((mid (f,1,(Index (p,f)))) ^ <*p*>) by FINSEQ_5:63
.= Rev (R_Cut (f,p)) by A30, Def4 ; ::_thesis: verum
end;
supposethat A37: p <> f . 1 and
A38: p <> f . ((Index (p,f)) + 1) ; ::_thesis: L_Cut ((Rev f),p) = Rev (R_Cut (f,p))
A39: p <> (Rev f) . (len f) by A37, FINSEQ_5:62;
A40: now__::_thesis:_not_p_=_(Rev_f)_._((Index_(p,(Rev_f)))_+_1)
assume A41: p = (Rev f) . ((Index (p,(Rev f))) + 1) ; ::_thesis: contradiction
then A42: len (Rev f) = ((Index (p,(Rev (Rev f)))) + (Index (p,(Rev f)))) + 1 by A1, A4, A3, A39, Th20
.= ((Index (p,f)) + 1) + (Index (p,(Rev f))) ;
p = f . (((len f) - ((Index (p,(Rev f))) + 1)) + 1) by A10, A41, FINSEQ_5:58
.= f . ((Index (p,f)) + 1) by A3, A42 ;
hence contradiction by A38; ::_thesis: verum
end;
A43: Index (p,f) < len f by A2, Th8;
len f = (Index (p,(Rev f))) + (Index (p,f)) by A1, A2, A38, Th21;
then Index (p,(Rev f)) = (len f) - (Index (p,f))
.= (len f) -' (Index (p,f)) by A43, XREAL_1:233 ;
hence L_Cut ((Rev f),p) = <*p*> ^ (mid ((Rev f),(((len f) -' (Index (p,f))) + 1),(len f))) by A3, A40, Def3
.= <*p*> ^ (mid ((Rev f),(((len f) -' (Index (p,f))) + 1),(((len f) -' 1) + 1))) by A12, XREAL_1:235
.= <*p*> ^ (Rev (mid (f,1,(Index (p,f))))) by A12, A5, A8, FINSEQ_6:113
.= Rev ((mid (f,1,(Index (p,f)))) ^ <*p*>) by FINSEQ_5:63
.= Rev (R_Cut (f,p)) by A37, Def4 ;
::_thesis: verum
end;
end;
end;
theorem Th23: :: JORDAN3:23
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
( (L_Cut (f,p)) . 1 = p & ( for i being Element of NAT st 1 < i & i <= len (L_Cut (f,p)) holds
( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) ) ) )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
( (L_Cut (f,p)) . 1 = p & ( for i being Element of NAT st 1 < i & i <= len (L_Cut (f,p)) holds
( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) ) ) )
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies ( (L_Cut (f,p)) . 1 = p & ( for i being Element of NAT st 1 < i & i <= len (L_Cut (f,p)) holds
( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) ) ) ) )
assume A1: p in L~ f ; ::_thesis: ( (L_Cut (f,p)) . 1 = p & ( for i being Element of NAT st 1 < i & i <= len (L_Cut (f,p)) holds
( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) ) ) )
then Index (p,f) < len f by Th8;
then A2: (Index (p,f)) + 1 <= len f by NAT_1:13;
A3: not f is empty by A1, CARD_1:27, TOPREAL1:22;
now__::_thesis:_(L_Cut_(f,p))_._1_=_p
percases ( p = f . ((Index (p,f)) + 1) or p <> f . ((Index (p,f)) + 1) ) ;
supposeA4: p = f . ((Index (p,f)) + 1) ; ::_thesis: (L_Cut (f,p)) . 1 = p
1 in dom f by A3, FINSEQ_5:6;
then A5: 1 <= len f by FINSEQ_3:25;
Index (p,f) < len f by A1, Th8;
then A6: (Index (p,f)) + 1 <= len f by NAT_1:13;
A7: 1 <= (Index (p,f)) + 1 by NAT_1:11;
L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) by A4, Def3;
hence (L_Cut (f,p)) . 1 = p by A4, A7, A6, A5, FINSEQ_6:118; ::_thesis: verum
end;
suppose p <> f . ((Index (p,f)) + 1) ; ::_thesis: (L_Cut (f,p)) . 1 = p
then L_Cut (f,p) = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) by Def3;
hence (L_Cut (f,p)) . 1 = p by FINSEQ_1:41; ::_thesis: verum
end;
end;
end;
hence (L_Cut (f,p)) . 1 = p ; ::_thesis: for i being Element of NAT st 1 < i & i <= len (L_Cut (f,p)) holds
( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) )
let i be Element of NAT ; ::_thesis: ( 1 < i & i <= len (L_Cut (f,p)) implies ( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) ) )
assume that
A8: 1 < i and
A9: i <= len (L_Cut (f,p)) ; ::_thesis: ( ( p = f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . ((Index (p,f)) + i) ) & ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) ) )
A10: len <*p*> <= i by A8, FINSEQ_1:40;
A11: 1 <= (Index (p,f)) + 1 by NAT_1:11;
then A12: 1 <= len f by A2, XXREAL_0:2;
then len (mid (f,((Index (p,f)) + 1),(len f))) = ((len f) -' ((Index (p,f)) + 1)) + 1 by A11, A2, FINSEQ_6:118;
then A13: (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) = 1 + (((len f) -' ((Index (p,f)) + 1)) + 1) by FINSEQ_1:40
.= 1 + (((len f) - ((Index (p,f)) + 1)) + 1) by A2, XREAL_1:233
.= ((len f) - (Index (p,f))) + 1 ;
A14: (i -' 1) + 1 = (i - 1) + 1 by A8, XREAL_1:233
.= i ;
A15: 1 <= i - 1 by A8, SPPOL_1:1;
then A16: 1 <= i -' 1 by NAT_D:39;
hereby ::_thesis: ( p <> f . ((Index (p,f)) + 1) implies (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) )
assume p = f . ((Index (p,f)) + 1) ; ::_thesis: (L_Cut (f,p)) . i = f . ((Index (p,f)) + i)
then L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) by Def3;
hence (L_Cut (f,p)) . i = f . ((i + ((Index (p,f)) + 1)) -' 1) by A8, A9, A11, A2, A12, FINSEQ_6:118
.= f . (((i + (Index (p,f))) + 1) -' 1)
.= f . ((Index (p,f)) + i) by NAT_D:34 ;
::_thesis: verum
end;
A17: i <= i + (Index (p,f)) by NAT_1:11;
assume p <> f . ((Index (p,f)) + 1) ; ::_thesis: (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1)
then A18: L_Cut (f,p) = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) by Def3;
then i <= (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A9, FINSEQ_1:22;
then i - 1 <= (((len f) - (Index (p,f))) + 1) - 1 by A13, XREAL_1:9;
then A19: i -' 1 <= ((len f) - ((Index (p,f)) + 1)) + 1 by A15, NAT_D:39;
len <*p*> < i by A8, FINSEQ_1:39;
then (L_Cut (f,p)) . i = (mid (f,((Index (p,f)) + 1),(len f))) . (i - (len <*p*>)) by A9, A18, FINSEQ_6:108
.= (mid (f,((Index (p,f)) + 1),(len f))) . (i -' (len <*p*>)) by A10, XREAL_1:233
.= (mid (f,((Index (p,f)) + 1),(len f))) . (i -' 1) by FINSEQ_1:39
.= f . (((i -' 1) + ((Index (p,f)) + 1)) -' 1) by A11, A2, A16, A19, FINSEQ_6:122
.= f . (((Index (p,f)) + i) -' 1) by A14 ;
hence (L_Cut (f,p)) . i = f . (((Index (p,f)) + i) - 1) by A8, A17, XREAL_1:233, XXREAL_0:2; ::_thesis: verum
end;
theorem Th24: :: JORDAN3:24
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
( (R_Cut (f,p)) . (len (R_Cut (f,p))) = p & ( for i being Element of NAT st 1 <= i & i <= Index (p,f) holds
(R_Cut (f,p)) . i = f . i ) )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
( (R_Cut (f,p)) . (len (R_Cut (f,p))) = p & ( for i being Element of NAT st 1 <= i & i <= Index (p,f) holds
(R_Cut (f,p)) . i = f . i ) )
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies ( (R_Cut (f,p)) . (len (R_Cut (f,p))) = p & ( for i being Element of NAT st 1 <= i & i <= Index (p,f) holds
(R_Cut (f,p)) . i = f . i ) ) )
assume A1: p in L~ f ; ::_thesis: ( (R_Cut (f,p)) . (len (R_Cut (f,p))) = p & ( for i being Element of NAT st 1 <= i & i <= Index (p,f) holds
(R_Cut (f,p)) . i = f . i ) )
then A2: Index (p,f) < len f by Th8;
now__::_thesis:_(R_Cut_(f,p))_._(len_(R_Cut_(f,p)))_=_p
percases ( p <> f . 1 or p = f . 1 ) ;
supposeA3: p <> f . 1 ; ::_thesis: (R_Cut (f,p)) . (len (R_Cut (f,p))) = p
A4: len ((mid (f,1,(Index (p,f)))) ^ <*p*>) = (len (mid (f,1,(Index (p,f))))) + (len <*p*>) by FINSEQ_1:22
.= (len (mid (f,1,(Index (p,f))))) + 1 by FINSEQ_1:39 ;
R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by A3, Def4;
hence (R_Cut (f,p)) . (len (R_Cut (f,p))) = p by A4, FINSEQ_1:42; ::_thesis: verum
end;
suppose p = f . 1 ; ::_thesis: (R_Cut (f,p)) . (len (R_Cut (f,p))) = p
then A5: R_Cut (f,p) = <*p*> by Def4;
then len (R_Cut (f,p)) = 1 by FINSEQ_1:40;
hence (R_Cut (f,p)) . (len (R_Cut (f,p))) = p by A5, FINSEQ_1:40; ::_thesis: verum
end;
end;
end;
hence (R_Cut (f,p)) . (len (R_Cut (f,p))) = p ; ::_thesis: for i being Element of NAT st 1 <= i & i <= Index (p,f) holds
(R_Cut (f,p)) . i = f . i
A6: 1 <= Index (p,f) by A1, Th8;
then len f > 1 by A2, XXREAL_0:2;
then A7: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A6, A2, FINSEQ_6:118
.= Index (p,f) by A1, Th8, XREAL_1:235 ;
thus for i being Element of NAT st 1 <= i & i <= Index (p,f) holds
(R_Cut (f,p)) . i = f . i ::_thesis: verum
proof
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= Index (p,f) implies (R_Cut (f,p)) . i = f . i )
assume that
A8: 1 <= i and
A9: i <= Index (p,f) ; ::_thesis: (R_Cut (f,p)) . i = f . i
now__::_thesis:_(_(_p_<>_f_._1_&_(R_Cut_(f,p))_._i_=_f_._i_)_or_(_p_=_f_._1_&_(R_Cut_(f,p))_._i_=_f_._i_)_)
percases ( p <> f . 1 or p = f . 1 ) ;
case p <> f . 1 ; ::_thesis: (R_Cut (f,p)) . i = f . i
then (R_Cut (f,p)) . i = ((mid (f,1,(Index (p,f)))) ^ <*p*>) . i by Def4
.= (mid (f,1,(Index (p,f)))) . i by A7, A8, A9, FINSEQ_1:64
.= f . i by A2, A8, A9, FINSEQ_6:123 ;
hence (R_Cut (f,p)) . i = f . i ; ::_thesis: verum
end;
caseA10: p = f . 1 ; ::_thesis: (R_Cut (f,p)) . i = f . i
A11: len f > 1 by A6, A2, XXREAL_0:2;
then 1 in dom f by FINSEQ_3:25;
then A12: p = f /. 1 by A10, PARTFUN1:def_6;
len f >= 1 + 1 by A11, NAT_1:13;
then Index (p,f) = 1 by A12, Th11;
then A13: i = 1 by A8, A9, XXREAL_0:1;
R_Cut (f,p) = <*p*> by A10, Def4;
hence (R_Cut (f,p)) . i = f . i by A10, A13, FINSEQ_1:40; ::_thesis: verum
end;
end;
end;
hence (R_Cut (f,p)) . i = f . i ; ::_thesis: verum
end;
end;
theorem :: JORDAN3:25
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
( ( p <> f . 1 implies len (R_Cut (f,p)) = (Index (p,f)) + 1 ) & ( p = f . 1 implies len (R_Cut (f,p)) = Index (p,f) ) )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
( ( p <> f . 1 implies len (R_Cut (f,p)) = (Index (p,f)) + 1 ) & ( p = f . 1 implies len (R_Cut (f,p)) = Index (p,f) ) )
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies ( ( p <> f . 1 implies len (R_Cut (f,p)) = (Index (p,f)) + 1 ) & ( p = f . 1 implies len (R_Cut (f,p)) = Index (p,f) ) ) )
assume A1: p in L~ f ; ::_thesis: ( ( p <> f . 1 implies len (R_Cut (f,p)) = (Index (p,f)) + 1 ) & ( p = f . 1 implies len (R_Cut (f,p)) = Index (p,f) ) )
then consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
p in LSeg (f,i) by SPPOL_2:13;
A4: 1 <= Index (p,f) by A1, Th8;
A5: Index (p,f) <= len f by A1, Th8;
i <= len f by A3, NAT_D:46;
then A6: 1 <= len f by A2, XXREAL_0:2;
now__::_thesis:_(_(_p_<>_f_._1_&_len_(R_Cut_(f,p))_=_(Index_(p,f))_+_1_)_or_(_p_=_f_._1_&_len_(R_Cut_(f,p))_=_Index_(p,f)_)_)
percases ( p <> f . 1 or p = f . 1 ) ;
case p <> f . 1 ; ::_thesis: len (R_Cut (f,p)) = (Index (p,f)) + 1
then R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by Def4;
hence len (R_Cut (f,p)) = (len (mid (f,1,(Index (p,f))))) + (len <*p*>) by FINSEQ_1:22
.= (len (mid (f,1,(Index (p,f))))) + 1 by FINSEQ_1:39
.= (((Index (p,f)) -' 1) + 1) + 1 by A6, A4, A5, FINSEQ_6:118
.= (Index (p,f)) + 1 by A1, Th8, XREAL_1:235 ;
::_thesis: verum
end;
caseA7: p = f . 1 ; ::_thesis: len (R_Cut (f,p)) = Index (p,f)
len f > i by A3, NAT_1:13;
then len f > 1 by A2, XXREAL_0:2;
then A8: len f >= 1 + 1 by NAT_1:13;
1 in dom f by A3, CARD_1:27, FINSEQ_5:6;
then A9: p = f /. 1 by A7, PARTFUN1:def_6;
R_Cut (f,p) = <*p*> by A7, Def4;
hence len (R_Cut (f,p)) = 1 by FINSEQ_1:39
.= Index (p,f) by A8, A9, Th11 ;
::_thesis: verum
end;
end;
end;
hence ( ( p <> f . 1 implies len (R_Cut (f,p)) = (Index (p,f)) + 1 ) & ( p = f . 1 implies len (R_Cut (f,p)) = Index (p,f) ) ) ; ::_thesis: verum
end;
theorem Th26: :: JORDAN3:26
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
( ( p = f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = (len f) - (Index (p,f)) ) & ( p <> f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 ) )
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
( ( p = f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = (len f) - (Index (p,f)) ) & ( p <> f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 ) )
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies ( ( p = f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = (len f) - (Index (p,f)) ) & ( p <> f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 ) ) )
assume A1: p in L~ f ; ::_thesis: ( ( p = f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = (len f) - (Index (p,f)) ) & ( p <> f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 ) )
then consider i being Element of NAT such that
A2: 1 <= i and
A3: i + 1 <= len f and
p in LSeg (f,i) by SPPOL_2:13;
i <= len f by A3, NAT_D:46;
then A4: 1 <= len f by A2, XXREAL_0:2;
1 <= Index (p,f) by A1, Th8;
then A5: 1 < (Index (p,f)) + 1 by NAT_1:13;
Index (p,f) < len f by A1, Th8;
then A6: ((Index (p,f)) + 1) + 0 <= len f by NAT_1:13;
then A7: (len f) - ((Index (p,f)) + 1) >= 0 by XREAL_1:19;
now__::_thesis:_(_(_p_<>_f_._((Index_(p,f))_+_1)_&_len_(L_Cut_(f,p))_=_((len_f)_-_(Index_(p,f)))_+_1_)_or_(_p_=_f_._((Index_(p,f))_+_1)_&_len_(L_Cut_(f,p))_=_(len_f)_-_(Index_(p,f))_)_)
percases ( p <> f . ((Index (p,f)) + 1) or p = f . ((Index (p,f)) + 1) ) ;
case p <> f . ((Index (p,f)) + 1) ; ::_thesis: len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1
then L_Cut (f,p) = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) by Def3;
hence len (L_Cut (f,p)) = 1 + (len (mid (f,((Index (p,f)) + 1),(len f)))) by FINSEQ_5:8
.= (((len f) -' ((Index (p,f)) + 1)) + 1) + 1 by A4, A5, A6, FINSEQ_6:118
.= (((len f) - ((Index (p,f)) + 1)) + 1) + 1 by A7, XREAL_0:def_2
.= ((len f) - (Index (p,f))) + 1 ;
::_thesis: verum
end;
case p = f . ((Index (p,f)) + 1) ; ::_thesis: len (L_Cut (f,p)) = (len f) - (Index (p,f))
then L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) by Def3;
hence len (L_Cut (f,p)) = ((len f) -' ((Index (p,f)) + 1)) + 1 by A4, A5, A6, FINSEQ_6:118
.= ((len f) - ((Index (p,f)) + 1)) + 1 by A7, XREAL_0:def_2
.= (len f) - (Index (p,f)) ;
::_thesis: verum
end;
end;
end;
hence ( ( p = f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = (len f) - (Index (p,f)) ) & ( p <> f . ((Index (p,f)) + 1) implies len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 ) ) ; ::_thesis: verum
end;
definition
let p1, p2, q1, q2 be Point of (TOP-REAL 2);
pred LE q1,q2,p1,p2 means :Def5: :: JORDAN3:def 5
( q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & ( for r1, r2 being Real st 0 <= r1 & r1 <= 1 & q1 = ((1 - r1) * p1) + (r1 * p2) & 0 <= r2 & r2 <= 1 & q2 = ((1 - r2) * p1) + (r2 * p2) holds
r1 <= r2 ) );
end;
:: deftheorem Def5 defines LE JORDAN3:def_5_:_
for p1, p2, q1, q2 being Point of (TOP-REAL 2) holds
( LE q1,q2,p1,p2 iff ( q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & ( for r1, r2 being Real st 0 <= r1 & r1 <= 1 & q1 = ((1 - r1) * p1) + (r1 * p2) & 0 <= r2 & r2 <= 1 & q2 = ((1 - r2) * p1) + (r2 * p2) holds
r1 <= r2 ) ) );
definition
let p1, p2, q1, q2 be Point of (TOP-REAL 2);
pred LT q1,q2,p1,p2 means :Def6: :: JORDAN3:def 6
( LE q1,q2,p1,p2 & q1 <> q2 );
end;
:: deftheorem Def6 defines LT JORDAN3:def_6_:_
for p1, p2, q1, q2 being Point of (TOP-REAL 2) holds
( LT q1,q2,p1,p2 iff ( LE q1,q2,p1,p2 & q1 <> q2 ) );
theorem :: JORDAN3:27
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st LE q1,q2,p1,p2 & LE q2,q1,p1,p2 holds
q1 = q2
proof
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( LE q1,q2,p1,p2 & LE q2,q1,p1,p2 implies q1 = q2 )
assume that
A1: LE q1,q2,p1,p2 and
A2: LE q2,q1,p1,p2 ; ::_thesis: q1 = q2
q1 in LSeg (p1,p2) by A1, Def5;
then consider r1 being Real such that
A3: q1 = ((1 - r1) * p1) + (r1 * p2) and
A4: 0 <= r1 and
A5: r1 <= 1 ;
q2 in LSeg (p1,p2) by A1, Def5;
then consider r2 being Real such that
A6: q2 = ((1 - r2) * p1) + (r2 * p2) and
A7: 0 <= r2 and
A8: r2 <= 1 ;
A9: r2 <= r1 by A2, A3, A4, A5, A6, A8, Def5;
r1 <= r2 by A1, A3, A5, A6, A7, A8, Def5;
then r1 = r2 by A9, XXREAL_0:1;
hence q1 = q2 by A3, A6; ::_thesis: verum
end;
theorem Th28: :: JORDAN3:28
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & p1 <> p2 holds
( ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) & ( not LE q1,q2,p1,p2 or not LT q2,q1,p1,p2 ) )
proof
let p1, p2, q1, q2 be Point of (TOP-REAL 2); ::_thesis: ( q1 in LSeg (p1,p2) & q2 in LSeg (p1,p2) & p1 <> p2 implies ( ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) & ( not LE q1,q2,p1,p2 or not LT q2,q1,p1,p2 ) ) )
assume that
A1: q1 in LSeg (p1,p2) and
A2: q2 in LSeg (p1,p2) and
A3: p1 <> p2 ; ::_thesis: ( ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) & ( not LE q1,q2,p1,p2 or not LT q2,q1,p1,p2 ) )
consider r1 being Real such that
A4: q1 = ((1 - r1) * p1) + (r1 * p2) and
A5: 0 <= r1 and
A6: r1 <= 1 by A1;
consider r2 being Real such that
A7: q2 = ((1 - r2) * p1) + (r2 * p2) and
A8: 0 <= r2 and
A9: r2 <= 1 by A2;
A10: now__::_thesis:_(_(_r1_<=_r2_&_(_LE_q1,q2,p1,p2_or_LT_q2,q1,p1,p2_)_)_or_(_r1_>_r2_&_(_LE_q1,q2,p1,p2_or_LT_q2,q1,p1,p2_)_)_)
percases ( r1 <= r2 or r1 > r2 ) ;
caseA11: r1 <= r2 ; ::_thesis: ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 )
for s1, s2 being Real st 0 <= s1 & s1 <= 1 & q1 = ((1 - s1) * p1) + (s1 * p2) & 0 <= s2 & s2 <= 1 & q2 = ((1 - s2) * p1) + (s2 * p2) holds
s1 <= s2
proof
let s1, s2 be Real; ::_thesis: ( 0 <= s1 & s1 <= 1 & q1 = ((1 - s1) * p1) + (s1 * p2) & 0 <= s2 & s2 <= 1 & q2 = ((1 - s2) * p1) + (s2 * p2) implies s1 <= s2 )
assume that
0 <= s1 and
s1 <= 1 and
A12: q1 = ((1 - s1) * p1) + (s1 * p2) and
0 <= s2 and
s2 <= 1 and
A13: q2 = ((1 - s2) * p1) + (s2 * p2) ; ::_thesis: s1 <= s2
((1 - s2) * p1) + ((s2 * p2) - (s2 * p2)) = (((1 - r2) * p1) + (r2 * p2)) - (s2 * p2) by A7, A13, EUCLID:45;
then ((1 - s2) * p1) + ((s2 * p2) - (s2 * p2)) = ((1 - r2) * p1) + ((r2 * p2) - (s2 * p2)) by EUCLID:45;
then ((1 - s2) * p1) + (0. (TOP-REAL 2)) = ((1 - r2) * p1) + ((r2 * p2) - (s2 * p2)) by EUCLID:42;
then (1 - s2) * p1 = ((1 - r2) * p1) + ((r2 * p2) - (s2 * p2)) by EUCLID:27;
then (1 - s2) * p1 = ((1 - r2) * p1) + ((r2 - s2) * p2) by EUCLID:50;
then ((1 - s2) * p1) - ((1 - r2) * p1) = ((r2 - s2) * p2) + (((1 - r2) * p1) - ((1 - r2) * p1)) by EUCLID:45;
then ((1 - s2) * p1) - ((1 - r2) * p1) = ((r2 - s2) * p2) + (0. (TOP-REAL 2)) by EUCLID:42;
then ((1 - s2) * p1) - ((1 - r2) * p1) = (r2 - s2) * p2 by EUCLID:27;
then ((1 - s2) - (1 - r2)) * p1 = (r2 - s2) * p2 by EUCLID:50;
then A14: ( r2 - s2 = 0 or p1 = p2 ) by EUCLID:34;
((1 - s1) * p1) + ((s1 * p2) - (s1 * p2)) = (((1 - r1) * p1) + (r1 * p2)) - (s1 * p2) by A4, A12, EUCLID:45;
then ((1 - s1) * p1) + ((s1 * p2) - (s1 * p2)) = ((1 - r1) * p1) + ((r1 * p2) - (s1 * p2)) by EUCLID:45;
then ((1 - s1) * p1) + (0. (TOP-REAL 2)) = ((1 - r1) * p1) + ((r1 * p2) - (s1 * p2)) by EUCLID:42;
then (1 - s1) * p1 = ((1 - r1) * p1) + ((r1 * p2) - (s1 * p2)) by EUCLID:27;
then (1 - s1) * p1 = ((1 - r1) * p1) + ((r1 - s1) * p2) by EUCLID:50;
then ((1 - s1) * p1) - ((1 - r1) * p1) = ((r1 - s1) * p2) + (((1 - r1) * p1) - ((1 - r1) * p1)) by EUCLID:45;
then ((1 - s1) * p1) - ((1 - r1) * p1) = ((r1 - s1) * p2) + (0. (TOP-REAL 2)) by EUCLID:42;
then ((1 - s1) * p1) - ((1 - r1) * p1) = (r1 - s1) * p2 by EUCLID:27;
then ((1 - s1) - (1 - r1)) * p1 = (r1 - s1) * p2 by EUCLID:50;
then ( r1 - s1 = 0 or p1 = p2 ) by EUCLID:34;
hence s1 <= s2 by A3, A11, A14; ::_thesis: verum
end;
hence ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) by A1, A2, Def5; ::_thesis: verum
end;
caseA15: r1 > r2 ; ::_thesis: ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 )
A16: for s2, s1 being Real st 0 <= s2 & s2 <= 1 & q2 = ((1 - s2) * p1) + (s2 * p2) & 0 <= s1 & s1 <= 1 & q1 = ((1 - s1) * p1) + (s1 * p2) holds
s1 >= s2
proof
let s2, s1 be Real; ::_thesis: ( 0 <= s2 & s2 <= 1 & q2 = ((1 - s2) * p1) + (s2 * p2) & 0 <= s1 & s1 <= 1 & q1 = ((1 - s1) * p1) + (s1 * p2) implies s1 >= s2 )
assume that
0 <= s2 and
s2 <= 1 and
A17: q2 = ((1 - s2) * p1) + (s2 * p2) and
0 <= s1 and
s1 <= 1 and
A18: q1 = ((1 - s1) * p1) + (s1 * p2) ; ::_thesis: s1 >= s2
((1 - s1) * p1) + ((s1 * p2) - (s1 * p2)) = (((1 - r1) * p1) + (r1 * p2)) - (s1 * p2) by A4, A18, EUCLID:45;
then ((1 - s1) * p1) + ((s1 * p2) - (s1 * p2)) = ((1 - r1) * p1) + ((r1 * p2) - (s1 * p2)) by EUCLID:45;
then ((1 - s1) * p1) + (0. (TOP-REAL 2)) = ((1 - r1) * p1) + ((r1 * p2) - (s1 * p2)) by EUCLID:42;
then (1 - s1) * p1 = ((1 - r1) * p1) + ((r1 * p2) - (s1 * p2)) by EUCLID:27;
then (1 - s1) * p1 = ((1 - r1) * p1) + ((r1 - s1) * p2) by EUCLID:50;
then ((1 - s1) * p1) - ((1 - r1) * p1) = ((r1 - s1) * p2) + (((1 - r1) * p1) - ((1 - r1) * p1)) by EUCLID:45;
then ((1 - s1) * p1) - ((1 - r1) * p1) = ((r1 - s1) * p2) + (0. (TOP-REAL 2)) by EUCLID:42;
then ((1 - s1) * p1) - ((1 - r1) * p1) = (r1 - s1) * p2 by EUCLID:27;
then ((1 - s1) - (1 - r1)) * p1 = (r1 - s1) * p2 by EUCLID:50;
then A19: ( r1 - s1 = 0 or p1 = p2 ) by EUCLID:34;
((1 - s2) * p1) + ((s2 * p2) - (s2 * p2)) = (((1 - r2) * p1) + (r2 * p2)) - (s2 * p2) by A7, A17, EUCLID:45;
then ((1 - s2) * p1) + ((s2 * p2) - (s2 * p2)) = ((1 - r2) * p1) + ((r2 * p2) - (s2 * p2)) by EUCLID:45;
then ((1 - s2) * p1) + (0. (TOP-REAL 2)) = ((1 - r2) * p1) + ((r2 * p2) - (s2 * p2)) by EUCLID:42;
then (1 - s2) * p1 = ((1 - r2) * p1) + ((r2 * p2) - (s2 * p2)) by EUCLID:27
.= ((r2 - s2) * p2) + ((1 - r2) * p1) by EUCLID:50 ;
then ((1 - s2) * p1) - ((1 - r2) * p1) = ((r2 - s2) * p2) + (((1 - r2) * p1) - ((1 - r2) * p1)) by EUCLID:45;
then ((1 - s2) * p1) - ((1 - r2) * p1) = ((r2 - s2) * p2) + (0. (TOP-REAL 2)) by EUCLID:42;
then ((1 - s2) * p1) - ((1 - r2) * p1) = (r2 - s2) * p2 by EUCLID:27;
then ((1 - s2) - (1 - r2)) * p1 = (r2 - s2) * p2 by EUCLID:50;
then ( r2 - s2 = 0 or p1 = p2 ) by EUCLID:34;
hence s1 >= s2 by A3, A15, A19; ::_thesis: verum
end;
then A20: LE q2,q1,p1,p2 by A1, A2, Def5;
q1 <> q2 by A4, A6, A7, A8, A9, A15, A16;
hence ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) by A20, Def6; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_LE_q1,q2,p1,p2_implies_not_LT_q2,q1,p1,p2_)
assume that
A21: LE q1,q2,p1,p2 and
A22: LT q2,q1,p1,p2 ; ::_thesis: contradiction
LE q2,q1,p1,p2 by A22, Def6;
then A23: r2 <= r1 by A4, A5, A6, A7, A9, Def5;
r1 <= r2 by A4, A6, A7, A8, A9, A21, Def5;
then r1 = r2 by A23, XXREAL_0:1;
hence contradiction by A4, A7, A22, Def6; ::_thesis: verum
end;
hence ( ( LE q1,q2,p1,p2 or LT q2,q1,p1,p2 ) & ( not LE q1,q2,p1,p2 or not LT q2,q1,p1,p2 ) ) by A10; ::_thesis: verum
end;
theorem Th29: :: JORDAN3:29
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & Index (p,f) < Index (q,f) holds
q in L~ (L_Cut (f,p))
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & Index (p,f) < Index (q,f) holds
q in L~ (L_Cut (f,p))
let p, q be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) implies q in L~ (L_Cut (f,p)) )
assume that
A1: p in L~ f and
A2: q in L~ f and
A3: Index (p,f) < Index (q,f) ; ::_thesis: q in L~ (L_Cut (f,p))
A4: Index (q,f) < len f by A2, Th8;
then A5: (Index (q,f)) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then A6: ((Index (q,f)) - (Index (p,f))) + 1 <= ((len f) - (Index (p,f))) + 1 by XREAL_1:6;
(Index (q,f)) - (Index (p,f)) <= (((len f) - (Index (p,f))) - 1) + 1 by A4, XREAL_1:9;
then A7: (Index (q,f)) -' (Index (p,f)) <= ((len f) - ((Index (p,f)) + 1)) + 1 by A3, XREAL_1:233;
set i1 = ((Index (q,f)) -' (Index (p,f))) + 1;
A8: 1 <= (Index (p,f)) + 1 by NAT_1:11;
A9: (Index (p,f)) + 1 <= Index (q,f) by A3, NAT_1:13;
then A10: ((Index (p,f)) + 1) - (Index (p,f)) <= (Index (q,f)) - (Index (p,f)) by XREAL_1:9;
then A11: 1 <= (Index (q,f)) -' (Index (p,f)) by XREAL_0:def_2;
then A12: 1 <= ((Index (q,f)) -' (Index (p,f))) + 1 by NAT_D:48;
1 + 1 <= ((Index (q,f)) -' (Index (p,f))) + 1 by A11, XREAL_1:6;
then A13: 1 < ((Index (q,f)) -' (Index (p,f))) + 1 by XXREAL_0:2;
then A14: len <*p*> < ((Index (q,f)) -' (Index (p,f))) + 1 by FINSEQ_1:40;
then A15: len <*p*> < (((Index (q,f)) -' (Index (p,f))) + 1) + 1 by NAT_1:13;
A16: (Index (p,f)) + 1 <= len f by A4, A9, XXREAL_0:2;
A17: 1 <= Index (q,f) by A2, Th8;
then 1 < len f by A4, XXREAL_0:2;
then len (mid (f,((Index (p,f)) + 1),(len f))) = ((len f) -' ((Index (p,f)) + 1)) + 1 by A8, A16, FINSEQ_6:118;
then A18: (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) = 1 + (((len f) -' ((Index (p,f)) + 1)) + 1) by FINSEQ_1:40
.= 1 + (((len f) - ((Index (p,f)) + 1)) + 1) by A4, A9, XREAL_1:233, XXREAL_0:2
.= ((len f) - (Index (p,f))) + 1 ;
then A19: ((Index (q,f)) -' (Index (p,f))) + 1 <= (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A3, A6, XREAL_1:233;
percases ( p = f . ((Index (p,f)) + 1) or p <> f . ((Index (p,f)) + 1) ) ;
supposeA20: p = f . ((Index (p,f)) + 1) ; ::_thesis: q in L~ (L_Cut (f,p))
then A21: len (L_Cut (f,p)) = (len f) - (Index (p,f)) by A1, Th26;
then len (L_Cut (f,p)) >= (Index (q,f)) -' (Index (p,f)) by A3, A5, XREAL_1:233;
then (L_Cut (f,p)) /. ((Index (q,f)) -' (Index (p,f))) = (L_Cut (f,p)) . ((Index (q,f)) -' (Index (p,f))) by A11, FINSEQ_4:15
.= (mid (f,((Index (p,f)) + 1),(len f))) . ((Index (q,f)) -' (Index (p,f))) by A20, Def3
.= f . ((((Index (p,f)) + 1) + ((Index (q,f)) -' (Index (p,f)))) - 1) by A11, A8, A16, A7, FINSEQ_6:122
.= f . ((((Index (p,f)) + 1) + ((Index (q,f)) - (Index (p,f)))) - 1) by A3, XREAL_1:233
.= f . (Index (q,f)) ;
then A22: (L_Cut (f,p)) /. ((Index (q,f)) -' (Index (p,f))) = f /. (Index (q,f)) by A2, A4, Th8, FINSEQ_4:15;
1 <= Index (q,f) by A2, Th8;
then A23: 1 <= (Index (q,f)) + 1 by NAT_D:48;
A24: q in LSeg (f,(Index (q,f))) by A2, Th9;
A25: Index (q,f) < len f by A2, Th8;
then A26: (Index (q,f)) + 1 <= len f by NAT_1:13;
then A27: ((Index (q,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then ((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) ;
then A28: ((Index (q,f)) -' (Index (p,f))) + 1 <= ((len f) - ((Index (p,f)) + 1)) + 1 by A3, XREAL_1:233;
((Index (q,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by A26, XREAL_1:9;
then ((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) ;
then A29: ((Index (q,f)) -' (Index (p,f))) + 1 <= len (L_Cut (f,p)) by A10, A21, XREAL_0:def_2;
A30: (Index (q,f)) + 1 <= len f by A25, NAT_1:13;
((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) by A27;
then len (L_Cut (f,p)) >= ((Index (q,f)) -' (Index (p,f))) + 1 by A3, A21, XREAL_1:233;
then (L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1) = (L_Cut (f,p)) . (((Index (q,f)) -' (Index (p,f))) + 1) by A13, FINSEQ_4:15
.= (mid (f,((Index (p,f)) + 1),(len f))) . (((Index (q,f)) -' (Index (p,f))) + 1) by A20, Def3
.= f . ((((Index (p,f)) + 1) + (((Index (q,f)) -' (Index (p,f))) + 1)) - 1) by A12, A8, A16, A28, FINSEQ_6:122
.= f . ((((Index (p,f)) + 1) + (((Index (q,f)) - (Index (p,f))) + 1)) - 1) by A3, XREAL_1:233
.= f /. ((Index (q,f)) + 1) by A23, A30, FINSEQ_4:15 ;
then q in LSeg (((L_Cut (f,p)) /. ((Index (q,f)) -' (Index (p,f)))),((L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1))) by A17, A22, A26, A24, TOPREAL1:def_3;
hence q in L~ (L_Cut (f,p)) by A11, A29, SPPOL_2:15; ::_thesis: verum
end;
supposeA31: p <> f . ((Index (p,f)) + 1) ; ::_thesis: q in L~ (L_Cut (f,p))
A32: len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 by A1, A31, Th26;
then len (L_Cut (f,p)) >= ((Index (q,f)) -' (Index (p,f))) + 1 by A3, A6, XREAL_1:233;
then (L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1) = (L_Cut (f,p)) . (((Index (q,f)) -' (Index (p,f))) + 1) by FINSEQ_4:15, NAT_1:11
.= (<*p*> ^ (mid (f,((Index (p,f)) + 1),(len f)))) . (((Index (q,f)) -' (Index (p,f))) + 1) by A31, Def3
.= (mid (f,((Index (p,f)) + 1),(len f))) . ((((Index (q,f)) -' (Index (p,f))) + 1) - (len <*p*>)) by A14, A19, FINSEQ_6:108
.= (mid (f,((Index (p,f)) + 1),(len f))) . ((((Index (q,f)) -' (Index (p,f))) + 1) - 1) by FINSEQ_1:40
.= f . ((((Index (p,f)) + 1) + ((Index (q,f)) -' (Index (p,f)))) - 1) by A11, A8, A16, A7, FINSEQ_6:122
.= f . ((((Index (p,f)) + 1) + ((Index (q,f)) - (Index (p,f)))) - 1) by A3, XREAL_1:233
.= f . (Index (q,f)) ;
then A33: (L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1) = f /. (Index (q,f)) by A2, A4, Th8, FINSEQ_4:15;
A34: Index (q,f) < len f by A2, Th8;
then A35: (Index (q,f)) + 1 <= len f by NAT_1:13;
then A36: ((Index (q,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then ((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) ;
then A37: ((Index (q,f)) -' (Index (p,f))) + 1 <= ((len f) - ((Index (p,f)) + 1)) + 1 by A3, XREAL_1:233;
((Index (q,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by A35, XREAL_1:9;
then ((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) ;
then ((Index (q,f)) -' (Index (p,f))) + 1 <= (len f) - (Index (p,f)) by A10, XREAL_0:def_2;
then A38: (((Index (q,f)) -' (Index (p,f))) + 1) + 1 <= len (L_Cut (f,p)) by A32, XREAL_1:6;
1 <= Index (q,f) by A2, Th8;
then A39: 1 <= (Index (q,f)) + 1 by NAT_D:48;
A40: q in LSeg (f,(Index (q,f))) by A2, Th9;
A41: (Index (q,f)) + 1 <= len f by A34, NAT_1:13;
A42: (((Index (q,f)) - (Index (p,f))) + 1) + 1 <= ((len f) - (Index (p,f))) + 1 by A36, XREAL_1:6;
then A43: (((Index (q,f)) -' (Index (p,f))) + 1) + 1 <= (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A3, A18, XREAL_1:233;
len (L_Cut (f,p)) >= (((Index (q,f)) -' (Index (p,f))) + 1) + 1 by A3, A32, A42, XREAL_1:233;
then (L_Cut (f,p)) /. ((((Index (q,f)) -' (Index (p,f))) + 1) + 1) = (L_Cut (f,p)) . ((((Index (q,f)) -' (Index (p,f))) + 1) + 1) by FINSEQ_4:15, NAT_1:11
.= (<*p*> ^ (mid (f,((Index (p,f)) + 1),(len f)))) . ((((Index (q,f)) -' (Index (p,f))) + 1) + 1) by A31, Def3
.= (mid (f,((Index (p,f)) + 1),(len f))) . (((((Index (q,f)) -' (Index (p,f))) + 1) + 1) - (len <*p*>)) by A15, A43, FINSEQ_6:108
.= (mid (f,((Index (p,f)) + 1),(len f))) . (((((Index (q,f)) -' (Index (p,f))) + 1) + 1) - 1) by FINSEQ_1:40
.= f . ((((Index (p,f)) + 1) + (((Index (q,f)) -' (Index (p,f))) + 1)) - 1) by A12, A8, A16, A37, FINSEQ_6:122
.= f . ((((Index (p,f)) + 1) + (((Index (q,f)) - (Index (p,f))) + 1)) - 1) by A3, XREAL_1:233
.= f /. ((Index (q,f)) + 1) by A39, A41, FINSEQ_4:15 ;
then q in LSeg (((L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1)),((L_Cut (f,p)) /. ((((Index (q,f)) -' (Index (p,f))) + 1) + 1))) by A17, A33, A35, A40, TOPREAL1:def_3;
hence q in L~ (L_Cut (f,p)) by A38, NAT_1:11, SPPOL_2:15; ::_thesis: verum
end;
end;
end;
theorem Th30: :: JORDAN3:30
for p, q, p1, p2 being Point of (TOP-REAL 2) st LE p,q,p1,p2 holds
( q in LSeg (p,p2) & p in LSeg (p1,q) )
proof
let p, q, p1, p2 be Point of (TOP-REAL 2); ::_thesis: ( LE p,q,p1,p2 implies ( q in LSeg (p,p2) & p in LSeg (p1,q) ) )
assume A1: LE p,q,p1,p2 ; ::_thesis: ( q in LSeg (p,p2) & p in LSeg (p1,q) )
then p in LSeg (p1,p2) by Def5;
then consider s1 being Real such that
A2: p = ((1 - s1) * p1) + (s1 * p2) and
A3: 0 <= s1 and
A4: s1 <= 1 ;
q in LSeg (p1,p2) by A1, Def5;
then consider s2 being Real such that
A5: q = ((1 - s2) * p1) + (s2 * p2) and
A6: 0 <= s2 and
A7: s2 <= 1 ;
A8: s1 <= s2 by A1, A2, A4, A5, A6, A7, Def5;
A9: 1 - s1 >= 0 by A4, XREAL_1:48;
A10: now__::_thesis:_(_(_1_-_s1_<>_0_&_q_in_LSeg_(p,p2)_)_or_(_1_-_s1_=_0_&_q_in_LSeg_(p,p2)_)_)
percases ( 1 - s1 <> 0 or 1 - s1 = 0 ) ;
caseA11: 1 - s1 <> 0 ; ::_thesis: q in LSeg (p,p2)
set s = (s2 - s1) / (1 - s1);
A12: (1 - s1) * ((1 - s2) / (1 - s1)) = 1 - s2 by A11, XCMPLX_1:87;
A13: (1 - s1) * ((s2 - s1) / (1 - s1)) = s2 - s1 by A11, XCMPLX_1:87;
1 - ((s2 - s1) / (1 - s1)) = ((1 * (1 - s1)) - (s2 - s1)) / (1 - s1) by A11, XCMPLX_1:127
.= (((1 - s1) + s1) - s2) / (1 - s1) ;
then (1 - s1) * (((1 - ((s2 - s1) / (1 - s1))) * p) + (((s2 - s1) / (1 - s1)) * p2)) = ((1 - s1) * (((1 - s2) / (1 - s1)) * (((1 - s1) * p1) + (s1 * p2)))) + ((1 - s1) * (((s2 - s1) / (1 - s1)) * p2)) by A2, EUCLID:32
.= (((1 - s1) * ((1 - s2) / (1 - s1))) * (((1 - s1) * p1) + (s1 * p2))) + ((1 - s1) * (((s2 - s1) / (1 - s1)) * p2)) by EUCLID:30
.= ((1 - s2) * (((1 - s1) * p1) + (s1 * p2))) + (((1 - s1) * ((s2 - s1) / (1 - s1))) * p2) by A12, EUCLID:30
.= (((1 - s2) * ((1 - s1) * p1)) + ((1 - s2) * (s1 * p2))) + ((s2 - s1) * p2) by A13, EUCLID:32
.= ((((1 - s2) * (1 - s1)) * p1) + ((1 - s2) * (s1 * p2))) + ((s2 - s1) * p2) by EUCLID:30
.= ((((1 - s2) * (1 - s1)) * p1) + (((1 - s2) * s1) * p2)) + ((s2 - s1) * p2) by EUCLID:30
.= (((1 - s2) * (1 - s1)) * p1) + ((((1 - s2) * s1) * p2) + ((s2 - s1) * p2)) by EUCLID:26
.= (((1 - s2) * (1 - s1)) * p1) + ((((1 * s1) - (s2 * s1)) + (s2 - s1)) * p2) by EUCLID:33
.= ((1 - s1) * ((1 - s2) * p1)) + (((1 - s1) * s2) * p2) by EUCLID:30
.= ((1 - s1) * ((1 - s2) * p1)) + ((1 - s1) * (s2 * p2)) by EUCLID:30
.= (1 - s1) * q by A5, EUCLID:32 ;
then A14: q = ((1 - ((s2 - s1) / (1 - s1))) * p) + (((s2 - s1) / (1 - s1)) * p2) by A11, EUCLID:34;
1 - s1 >= s2 - s1 by A7, XREAL_1:9;
then (1 - s1) / (1 - s1) >= (s2 - s1) / (1 - s1) by A9, XREAL_1:72;
then A15: 1 >= (s2 - s1) / (1 - s1) by A11, XCMPLX_1:60;
s2 - s1 >= 0 by A8, XREAL_1:48;
hence q in LSeg (p,p2) by A9, A15, A14; ::_thesis: verum
end;
case 1 - s1 = 0 ; ::_thesis: q in LSeg (p,p2)
then s2 = 1 by A7, A8, XXREAL_0:1;
then q = (0. (TOP-REAL 2)) + (1 * p2) by A5, EUCLID:29
.= (0. (TOP-REAL 2)) + p2 by EUCLID:29
.= p2 by EUCLID:27 ;
hence q in LSeg (p,p2) by RLTOPSP1:68; ::_thesis: verum
end;
end;
end;
now__::_thesis:_(_(_s2_<>_0_&_p_in_LSeg_(p1,q)_)_or_(_s2_=_0_&_p_in_LSeg_(p1,q)_)_)
percases ( s2 <> 0 or s2 = 0 ) ;
caseA16: s2 <> 0 ; ::_thesis: p in LSeg (p1,q)
set s = s1 / s2;
s2 / s2 >= s1 / s2 by A6, A8, XREAL_1:72;
then A17: 1 >= s1 / s2 by A16, XCMPLX_1:60;
A18: (s2 - s1) + (s1 * (1 - s2)) = s2 * (1 - s1) ;
A19: s2 * (s1 / s2) = s1 by A16, XCMPLX_1:87;
A20: s2 * ((s2 - s1) / s2) = s2 - s1 by A16, XCMPLX_1:87;
s2 * (((1 - (s1 / s2)) * p1) + ((s1 / s2) * q)) = s2 * (((((1 * s2) - s1) / s2) * p1) + ((s1 / s2) * (((1 - s2) * p1) + (s2 * p2)))) by A5, A16, XCMPLX_1:127
.= (s2 * (((s2 - s1) / s2) * p1)) + (s2 * ((s1 / s2) * (((1 - s2) * p1) + (s2 * p2)))) by EUCLID:32
.= ((s2 * ((s2 - s1) / s2)) * p1) + (s2 * ((s1 / s2) * (((1 - s2) * p1) + (s2 * p2)))) by EUCLID:30
.= ((s2 - s1) * p1) + ((s2 * (s1 / s2)) * (((1 - s2) * p1) + (s2 * p2))) by A20, EUCLID:30
.= ((s2 - s1) * p1) + ((s1 * ((1 - s2) * p1)) + (s1 * (s2 * p2))) by A19, EUCLID:32
.= ((s2 - s1) * p1) + (((s1 * (1 - s2)) * p1) + (s1 * (s2 * p2))) by EUCLID:30
.= ((s2 - s1) * p1) + (((s1 * (1 - s2)) * p1) + ((s1 * s2) * p2)) by EUCLID:30
.= (((s2 - s1) * p1) + ((s1 * (1 - s2)) * p1)) + ((s1 * s2) * p2) by EUCLID:26
.= (((s2 - s1) + (s1 * (1 - s2))) * p1) + ((s1 * s2) * p2) by EUCLID:33
.= (s2 * ((1 - s1) * p1)) + ((s2 * s1) * p2) by A18, EUCLID:30
.= (s2 * ((1 - s1) * p1)) + (s2 * (s1 * p2)) by EUCLID:30
.= s2 * p by A2, EUCLID:32 ;
then p = ((1 - (s1 / s2)) * p1) + ((s1 / s2) * q) by A16, EUCLID:34;
hence p in LSeg (p1,q) by A3, A6, A17; ::_thesis: verum
end;
case s2 = 0 ; ::_thesis: p in LSeg (p1,q)
then s1 = 0 by A1, A2, A3, A4, A5, Def5;
then p = (1 * p1) + (0. (TOP-REAL 2)) by A2, EUCLID:29
.= p1 + (0. (TOP-REAL 2)) by EUCLID:29
.= p1 by EUCLID:27 ;
hence p in LSeg (p1,q) by RLTOPSP1:68; ::_thesis: verum
end;
end;
end;
hence ( q in LSeg (p,p2) & p in LSeg (p1,q) ) by A10; ::_thesis: verum
end;
theorem Th31: :: JORDAN3:31
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & p <> q & Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) holds
q in L~ (L_Cut (f,p))
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p, q being Point of (TOP-REAL 2) st p in L~ f & q in L~ f & p <> q & Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) holds
q in L~ (L_Cut (f,p))
let p, q be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f & q in L~ f & p <> q & Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) implies q in L~ (L_Cut (f,p)) )
assume that
A1: p in L~ f and
A2: q in L~ f and
A3: p <> q and
A4: Index (p,f) = Index (q,f) and
A5: LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ; ::_thesis: q in L~ (L_Cut (f,p))
A6: (Index (q,f)) -' (Index (p,f)) = (Index (q,f)) - (Index (p,f)) by A4, XREAL_1:233
.= 0 by A4 ;
Index (q,f) < len f by A2, Th8;
then A7: (Index (q,f)) + 1 <= len f by NAT_1:13;
A8: now__::_thesis:_not_p_=_f_._((Index_(p,f))_+_1)
q in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A5, Def5;
then consider r being Real such that
A9: q = ((1 - r) * (f /. (Index (p,f)))) + (r * (f /. ((Index (p,f)) + 1))) and
A10: 0 <= r and
A11: r <= 1 ;
A12: p = 1 * p by EUCLID:29
.= (0. (TOP-REAL 2)) + (1 * p) by EUCLID:27
.= ((1 - 1) * (f /. (Index (p,f)))) + (1 * p) by EUCLID:29 ;
assume A13: p = f . ((Index (p,f)) + 1) ; ::_thesis: contradiction
then p = f /. ((Index (p,f)) + 1) by A4, A7, FINSEQ_4:15, NAT_1:11;
then 1 <= r by A5, A9, A10, A12, Def5;
then r = 1 by A11, XXREAL_0:1;
hence contradiction by A3, A4, A7, A13, A9, A12, FINSEQ_4:15, NAT_1:11; ::_thesis: verum
end;
then A14: len (L_Cut (f,p)) = ((len f) - (Index (p,f))) + 1 by A1, Th26;
1 <= Index (q,f) by A2, Th8;
then A15: 1 <= (Index (q,f)) + 1 by NAT_D:48;
1 < (0 + 1) + 1 ;
then A16: len <*p*> < (((Index (q,f)) -' (Index (p,f))) + 1) + 1 by A6, FINSEQ_1:40;
A17: Index (q,f) < len f by A2, Th8;
then A18: (Index (q,f)) + 1 <= len f by NAT_1:13;
then A19: ((Index (q,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then A20: (((Index (q,f)) - (Index (p,f))) + 1) + 1 <= ((len f) - (Index (p,f))) + 1 by XREAL_1:6;
((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) by A19;
then A21: ((Index (q,f)) -' (Index (p,f))) + 1 <= ((len f) - ((Index (p,f)) + 1)) + 1 by A4, XREAL_1:233;
A22: 1 <= (Index (p,f)) + 1 by NAT_1:11;
A23: Index (q,f) < len f by A2, Th8;
then (Index (q,f)) - (Index (p,f)) <= (len f) - (Index (p,f)) by XREAL_1:9;
then ((Index (q,f)) - (Index (p,f))) + 1 <= ((len f) - (Index (p,f))) + 1 by XREAL_1:6;
then A24: (L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1) = (L_Cut (f,p)) . (((Index (q,f)) -' (Index (p,f))) + 1) by A4, A6, A14, FINSEQ_4:15
.= (<*p*> ^ (mid (f,((Index (p,f)) + 1),(len f)))) . (((Index (q,f)) -' (Index (p,f))) + 1) by A8, Def3
.= p by A6, FINSEQ_1:41 ;
set i1 = ((Index (q,f)) -' (Index (p,f))) + 1;
A25: (Index (q,f)) + 1 <= len f by A17, NAT_1:13;
((Index (q,f)) + 1) - (Index (p,f)) <= (len f) - (Index (p,f)) by A18, XREAL_1:9;
then ((Index (q,f)) - (Index (p,f))) + 1 <= (len f) - (Index (p,f)) ;
then ((Index (q,f)) -' (Index (p,f))) + 1 <= (len f) - (Index (p,f)) by A4, XREAL_0:def_2;
then A26: (((Index (q,f)) -' (Index (p,f))) + 1) + 1 <= len (L_Cut (f,p)) by A14, XREAL_1:6;
1 <= Index (q,f) by A2, Th8;
then 1 < len f by A23, XXREAL_0:2;
then len (mid (f,((Index (p,f)) + 1),(len f))) = ((len f) -' ((Index (p,f)) + 1)) + 1 by A4, A7, A22, FINSEQ_6:118;
then (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) = 1 + (((len f) -' ((Index (p,f)) + 1)) + 1) by FINSEQ_1:40
.= 1 + (((len f) - ((Index (p,f)) + 1)) + 1) by A4, A7, XREAL_1:233
.= ((len f) - (Index (p,f))) + 1 ;
then A27: (((Index (q,f)) -' (Index (p,f))) + 1) + 1 <= (len <*p*>) + (len (mid (f,((Index (p,f)) + 1),(len f)))) by A4, A20, XREAL_1:233;
(L_Cut (f,p)) /. ((((Index (q,f)) -' (Index (p,f))) + 1) + 1) = (L_Cut (f,p)) . ((((Index (q,f)) -' (Index (p,f))) + 1) + 1) by A4, A6, A14, A20, FINSEQ_4:15
.= (<*p*> ^ (mid (f,((Index (p,f)) + 1),(len f)))) . ((((Index (q,f)) -' (Index (p,f))) + 1) + 1) by A8, Def3
.= (mid (f,((Index (p,f)) + 1),(len f))) . (((((Index (q,f)) -' (Index (p,f))) + 1) + 1) - (len <*p*>)) by A16, A27, FINSEQ_6:108
.= (mid (f,((Index (p,f)) + 1),(len f))) . (((((Index (q,f)) -' (Index (p,f))) + 1) + 1) - 1) by FINSEQ_1:40
.= f . ((((Index (p,f)) + 1) + (((Index (q,f)) - (Index (p,f))) + 1)) - 1) by A4, A7, A6, A22, A21, FINSEQ_6:122
.= f /. ((Index (q,f)) + 1) by A15, A25, FINSEQ_4:15 ;
then q in LSeg (((L_Cut (f,p)) /. (((Index (q,f)) -' (Index (p,f))) + 1)),((L_Cut (f,p)) /. ((((Index (q,f)) -' (Index (p,f))) + 1) + 1))) by A4, A5, A24, Th30;
hence q in L~ (L_Cut (f,p)) by A6, A26, SPPOL_2:15; ::_thesis: verum
end;
begin
definition
let f be FinSequence of (TOP-REAL 2);
let p, q be Point of (TOP-REAL 2);
func B_Cut (f,p,q) -> FinSequence of (TOP-REAL 2) equals :Def7: :: JORDAN3:def 7
R_Cut ((L_Cut (f,p)),q) if ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) )
otherwise Rev (R_Cut ((L_Cut (f,q)),p));
correctness
coherence
( ( ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) implies R_Cut ((L_Cut (f,p)),q) is FinSequence of (TOP-REAL 2) ) & ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) or Rev (R_Cut ((L_Cut (f,q)),p)) is FinSequence of (TOP-REAL 2) ) );
consistency
for b1 being FinSequence of (TOP-REAL 2) holds verum;
;
end;
:: deftheorem Def7 defines B_Cut JORDAN3:def_7_:_
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) holds
( ( ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) implies B_Cut (f,p,q) = R_Cut ((L_Cut (f,p)),q) ) & ( ( p in L~ f & q in L~ f & Index (p,f) < Index (q,f) ) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) or B_Cut (f,p,q) = Rev (R_Cut ((L_Cut (f,q)),p)) ) );
theorem Th32: :: JORDAN3:32
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 holds
R_Cut (f,p) is_S-Seq_joining f /. 1,p
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 in L~ f & p <> f . 1 holds
R_Cut (f,p) is_S-Seq_joining f /. 1,p
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p <> f . 1 implies R_Cut (f,p) is_S-Seq_joining f /. 1,p )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . 1 ; ::_thesis: R_Cut (f,p) is_S-Seq_joining f /. 1,p
R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by A3, Def4;
hence R_Cut (f,p) is_S-Seq_joining f /. 1,p by A1, A2, A3, Th19; ::_thesis: verum
end;
theorem Th33: :: JORDAN3:33
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . (len f) holds
L_Cut (f,p) is_S-Seq_joining p,f /. (len f)
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 in L~ f & p <> f . (len f) holds
L_Cut (f,p) is_S-Seq_joining p,f /. (len f)
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p <> f . (len f) implies L_Cut (f,p) is_S-Seq_joining p,f /. (len f) )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . (len f) ; ::_thesis: L_Cut (f,p) is_S-Seq_joining p,f /. (len f)
A4: f <> {} by A2, CARD_1:27, TOPREAL1:22;
A5: Rev f is being_S-Seq by A1;
A6: p in L~ (Rev f) by A2, SPPOL_2:22;
A7: p <> (Rev f) . 1 by A3, FINSEQ_5:62;
L_Cut (f,p) = L_Cut ((Rev (Rev f)),p)
.= Rev (R_Cut ((Rev f),p)) by A1, A6, Th22 ;
then L_Cut (f,p) is_S-Seq_joining p,(Rev f) /. 1 by A5, A6, A7, Th15, Th32;
hence L_Cut (f,p) is_S-Seq_joining p,f /. (len f) by A4, FINSEQ_5:65; ::_thesis: verum
end;
theorem Th34: :: JORDAN3:34
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . (len f) holds
L_Cut (f,p) is being_S-Seq
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 in L~ f & p <> f . (len f) holds
L_Cut (f,p) is being_S-Seq
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p <> f . (len f) implies L_Cut (f,p) is being_S-Seq )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . (len f) ; ::_thesis: L_Cut (f,p) is being_S-Seq
L_Cut (f,p) is_S-Seq_joining p,f /. (len f) by A1, A2, A3, Th33;
hence L_Cut (f,p) is being_S-Seq by Def2; ::_thesis: verum
end;
theorem Th35: :: JORDAN3:35
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & p <> f . 1 holds
R_Cut (f,p) is being_S-Seq
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 in L~ f & p <> f . 1 holds
R_Cut (f,p) is being_S-Seq
let p be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & p <> f . 1 implies R_Cut (f,p) is being_S-Seq )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: p <> f . 1 ; ::_thesis: R_Cut (f,p) is being_S-Seq
R_Cut (f,p) is_S-Seq_joining f /. 1,p by A1, A2, A3, Th32;
hence R_Cut (f,p) is being_S-Seq by Def2; ::_thesis: verum
end;
Lm1: for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) holds
B_Cut (f,p,q) is_S-Seq_joining p,q
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) holds
B_Cut (f,p,q) is_S-Seq_joining p,q
let p, q be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & q in L~ f & p <> q & ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) implies B_Cut (f,p,q) is_S-Seq_joining p,q )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: q in L~ f and
A4: p <> q ; ::_thesis: ( ( not Index (p,f) < Index (q,f) & not ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) or B_Cut (f,p,q) is_S-Seq_joining p,q )
assume A5: ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) ; ::_thesis: B_Cut (f,p,q) is_S-Seq_joining p,q
then A6: B_Cut (f,p,q) = R_Cut ((L_Cut (f,p)),q) by A2, A3, Def7;
Index (p,f) < len f by A2, Th8;
then A7: (Index (p,f)) + 1 <= len f by NAT_1:13;
A8: Index (q,f) < len f by A3, Th8;
1 <= Index (q,f) by A3, Th8;
then A9: 1 < len f by A8, XXREAL_0:2;
A10: now__::_thesis:_(_(_Index_(p,f)_<_Index_(q,f)_&_not_p_=_f_._(len_f)_)_or_(_Index_(p,f)_=_Index_(q,f)_&_LE_p,q,f_/._(Index_(p,f)),f_/._((Index_(p,f))_+_1)_&_not_p_=_f_._(len_f)_)_)
percases ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) by A5;
caseA11: Index (p,f) < Index (q,f) ; ::_thesis: not p = f . (len f)
assume A12: p = f . (len f) ; ::_thesis: contradiction
(Index (p,f)) + 1 <= Index (q,f) by A11, NAT_1:13;
then len f <= Index (q,f) by A1, A9, A12, Th12;
hence contradiction by A3, Th8; ::_thesis: verum
end;
caseA13: ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ; ::_thesis: not p = f . (len f)
A14: now__::_thesis:_not_p_=_f_._((Index_(p,f))_+_1)
q in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A13, Def5;
then consider r being Real such that
A15: q = ((1 - r) * (f /. (Index (p,f)))) + (r * (f /. ((Index (p,f)) + 1))) and
A16: 0 <= r and
A17: r <= 1 ;
A18: p = 1 * p by EUCLID:29
.= (0. (TOP-REAL 2)) + (1 * p) by EUCLID:27
.= ((1 - 1) * (f /. (Index (p,f)))) + (1 * p) by EUCLID:29 ;
assume A19: p = f . ((Index (p,f)) + 1) ; ::_thesis: contradiction
then p = f /. ((Index (p,f)) + 1) by A7, FINSEQ_4:15, NAT_1:11;
then 1 <= r by A13, A15, A16, A18, Def5;
then r = 1 by A17, XXREAL_0:1;
hence contradiction by A4, A7, A19, A15, A18, FINSEQ_4:15, NAT_1:11; ::_thesis: verum
end;
assume p = f . (len f) ; ::_thesis: contradiction
hence contradiction by A1, A9, A14, Th12; ::_thesis: verum
end;
end;
end;
then L_Cut (f,p) is_S-Seq_joining p,f /. (len f) by A1, A2, Th33;
then A20: (L_Cut (f,p)) . 1 = p by Def2;
now__::_thesis:_(_(_Index_(p,f)_<_Index_(q,f)_&_ex_i1_being_Element_of_NAT_st_
(_1_<=_i1_&_i1_+_1_<=_len_(L_Cut_(f,p))_&_q_in_LSeg_((L_Cut_(f,p)),i1)_)_)_or_(_Index_(p,f)_=_Index_(q,f)_&_LE_p,q,f_/._(Index_(p,f)),f_/._((Index_(p,f))_+_1)_&_ex_i1_being_Element_of_NAT_st_
(_1_<=_i1_&_i1_+_1_<=_len_(L_Cut_(f,p))_&_q_in_LSeg_((L_Cut_(f,p)),i1)_)_)_)
percases ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) by A5;
case Index (p,f) < Index (q,f) ; ::_thesis: ex i1 being Element of NAT st
( 1 <= i1 & i1 + 1 <= len (L_Cut (f,p)) & q in LSeg ((L_Cut (f,p)),i1) )
then q in L~ (L_Cut (f,p)) by A2, A3, Th29;
hence ex i1 being Element of NAT st
( 1 <= i1 & i1 + 1 <= len (L_Cut (f,p)) & q in LSeg ((L_Cut (f,p)),i1) ) by SPPOL_2:13; ::_thesis: verum
end;
case ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ; ::_thesis: ex i1 being Element of NAT st
( 1 <= i1 & i1 + 1 <= len (L_Cut (f,p)) & q in LSeg ((L_Cut (f,p)),i1) )
then q in L~ (L_Cut (f,p)) by A2, A3, A4, Th31;
hence ex i1 being Element of NAT st
( 1 <= i1 & i1 + 1 <= len (L_Cut (f,p)) & q in LSeg ((L_Cut (f,p)),i1) ) by SPPOL_2:13; ::_thesis: verum
end;
end;
end;
then A21: q in L~ (L_Cut (f,p)) by SPPOL_2:17;
then A22: Index (q,(L_Cut (f,p))) < len (L_Cut (f,p)) by Th8;
1 <= Index (q,(L_Cut (f,p))) by A21, Th8;
then 1 <= len (L_Cut (f,p)) by A22, XXREAL_0:2;
then p = (L_Cut (f,p)) /. 1 by A20, FINSEQ_4:15;
hence B_Cut (f,p,q) is_S-Seq_joining p,q by A1, A2, A4, A6, A10, A21, A20, Th32, Th34; ::_thesis: verum
end;
theorem Th36: :: JORDAN3:36
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds
B_Cut (f,p,q) is_S-Seq_joining p,q
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds
B_Cut (f,p,q) is_S-Seq_joining p,q
let p, q be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & q in L~ f & p <> q implies B_Cut (f,p,q) is_S-Seq_joining p,q )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: q in L~ f and
A4: p <> q ; ::_thesis: B_Cut (f,p,q) is_S-Seq_joining p,q
percases ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) or ( not Index (p,f) < Index (q,f) & not ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) ) ;
suppose ( Index (p,f) < Index (q,f) or ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) ; ::_thesis: B_Cut (f,p,q) is_S-Seq_joining p,q
hence B_Cut (f,p,q) is_S-Seq_joining p,q by A1, A2, A3, A4, Lm1; ::_thesis: verum
end;
supposeA5: ( not Index (p,f) < Index (q,f) & not ( Index (p,f) = Index (q,f) & LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) ; ::_thesis: B_Cut (f,p,q) is_S-Seq_joining p,q
A6: now__::_thesis:_(_Index_(p,f)_=_Index_(q,f)_&_not_LE_p,q,f_/._(Index_(p,f)),f_/._((Index_(p,f))_+_1)_implies_LE_q,p,f_/._(Index_(q,f)),f_/._((Index_(q,f))_+_1)_)
A7: Index (p,f) < len f by A2, Th8;
then A8: (Index (p,f)) + 1 <= len f by NAT_1:13;
1 <= (Index (p,f)) + 1 by NAT_1:11;
then A9: (Index (p,f)) + 1 in dom f by A8, FINSEQ_3:25;
A10: (Index (p,f)) + 0 <> (Index (p,f)) + 1 ;
A11: 1 <= Index (p,f) by A2, Th8;
then A12: LSeg (f,(Index (p,f))) = LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A8, TOPREAL1:def_3;
then A13: p in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A2, Th9;
Index (p,f) in dom f by A11, A7, FINSEQ_3:25;
then A14: f /. (Index (p,f)) <> f /. ((Index (p,f)) + 1) by A1, A9, A10, PARTFUN2:10;
assume that
A15: Index (p,f) = Index (q,f) and
A16: not LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ; ::_thesis: LE q,p,f /. (Index (q,f)),f /. ((Index (q,f)) + 1)
q in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A3, A15, A12, Th9;
then LT q,p,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) by A16, A13, A14, Th28;
hence LE q,p,f /. (Index (q,f)),f /. ((Index (q,f)) + 1) by A15, Def6; ::_thesis: verum
end;
A17: ( Index (q,f) < Index (p,f) or ( Index (p,f) = Index (q,f) & not LE p,q,f /. (Index (p,f)),f /. ((Index (p,f)) + 1) ) ) by A5, XXREAL_0:1;
B_Cut (f,p,q) = Rev (R_Cut ((L_Cut (f,q)),p)) by A5, Def7;
then A18: Rev (B_Cut (f,q,p)) = B_Cut (f,p,q) by A2, A3, A17, A6, Def7;
B_Cut (f,q,p) is_S-Seq_joining q,p by A1, A2, A3, A4, A17, A6, Lm1;
hence B_Cut (f,p,q) is_S-Seq_joining p,q by A18, Th15; ::_thesis: verum
end;
end;
end;
theorem :: JORDAN3:37
for f being FinSequence of (TOP-REAL 2)
for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds
B_Cut (f,p,q) is being_S-Seq
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p, q being Point of (TOP-REAL 2) st f is being_S-Seq & p in L~ f & q in L~ f & p <> q holds
B_Cut (f,p,q) is being_S-Seq
let p, q be Point of (TOP-REAL 2); ::_thesis: ( f is being_S-Seq & p in L~ f & q in L~ f & p <> q implies B_Cut (f,p,q) is being_S-Seq )
assume that
A1: f is being_S-Seq and
A2: p in L~ f and
A3: q in L~ f and
A4: p <> q ; ::_thesis: B_Cut (f,p,q) is being_S-Seq
B_Cut (f,p,q) is_S-Seq_joining p,q by A1, A2, A3, A4, Th36;
hence B_Cut (f,p,q) is being_S-Seq by Def2; ::_thesis: verum
end;
theorem Th38: :: JORDAN3:38
for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds
f ^ (mid (g,2,(len g))) is being_S-Seq
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} implies f ^ (mid (g,2,(len g))) is being_S-Seq )
assume that
A1: f . (len f) = g . 1 and
A2: f is being_S-Seq and
A3: g is being_S-Seq and
A4: (L~ f) /\ (L~ g) = {(g . 1)} ; ::_thesis: f ^ (mid (g,2,(len g))) is being_S-Seq
A5: len f >= 2 by A2, TOPREAL1:def_8;
A6: len (f ^ (mid (g,2,(len g)))) = (len f) + (len (mid (g,2,(len g)))) by FINSEQ_1:22;
then len f <= len (f ^ (mid (g,2,(len g)))) by NAT_1:11;
then A7: len (f ^ (mid (g,2,(len g)))) >= 2 by A5, XXREAL_0:2;
A8: len g >= 2 by A3, TOPREAL1:def_8;
then A9: 1 <= len g by XXREAL_0:2;
then A10: len (mid (g,2,(len g))) = ((len g) -' 2) + 1 by A8, FINSEQ_6:118
.= ((len g) - 2) + 1 by A8, XREAL_1:233
.= (len g) - 1 ;
for x1, x2 being set st x1 in dom (f ^ (mid (g,2,(len g)))) & x2 in dom (f ^ (mid (g,2,(len g)))) & (f ^ (mid (g,2,(len g)))) . x1 = (f ^ (mid (g,2,(len g)))) . x2 holds
x1 = x2
proof
A11: rng g c= L~ g by A8, SPPOL_2:18;
A12: rng (f ^ (mid (g,2,(len g)))) c= the carrier of (TOP-REAL 2) by FINSEQ_1:def_4;
let x1, x2 be set ; ::_thesis: ( x1 in dom (f ^ (mid (g,2,(len g)))) & x2 in dom (f ^ (mid (g,2,(len g)))) & (f ^ (mid (g,2,(len g)))) . x1 = (f ^ (mid (g,2,(len g)))) . x2 implies x1 = x2 )
assume that
A13: x1 in dom (f ^ (mid (g,2,(len g)))) and
A14: x2 in dom (f ^ (mid (g,2,(len g)))) and
A15: (f ^ (mid (g,2,(len g)))) . x1 = (f ^ (mid (g,2,(len g)))) . x2 ; ::_thesis: x1 = x2
reconsider n1 = x1, n2 = x2 as Element of NAT by A13, A14;
A16: x2 in Seg (len (f ^ (mid (g,2,(len g))))) by A14, FINSEQ_1:def_3;
then A17: 1 <= n2 by FINSEQ_1:1;
(f ^ (mid (g,2,(len g)))) . x1 in rng (f ^ (mid (g,2,(len g)))) by A13, FUNCT_1:def_3;
then reconsider q = (f ^ (mid (g,2,(len g)))) . x1 as Point of (TOP-REAL 2) by A12;
A18: rng (mid (g,2,(len g))) c= rng g by FINSEQ_6:119;
A19: rng f c= L~ f by A5, SPPOL_2:18;
A20: now__::_thesis:_(_q_in_rng_f_implies_not_q_in_rng_(mid_(g,2,(len_g)))_)
A21: now__::_thesis:_not_g_._1_in_rng_(mid_(g,2,(len_g)))
g | 1 = g | (Seg 1) by FINSEQ_1:def_15;
then A22: (g | 1) . 1 = g . 1 by FINSEQ_1:3, FUNCT_1:49;
len (g | 1) = 1 by A8, FINSEQ_1:59, XXREAL_0:2;
then 1 in dom (g | 1) by FINSEQ_3:25;
then A23: g . 1 in rng (g | 1) by A22, FUNCT_1:def_3;
A24: 2 -' 1 = 2 - 1 by XREAL_1:233;
assume g . 1 in rng (mid (g,2,(len g))) ; ::_thesis: contradiction
then A25: g . 1 in rng (g /^ 1) by A8, A24, FINSEQ_6:117;
rng (g | 1) misses rng (g /^ 1) by A3, FINSEQ_5:34;
hence contradiction by A25, A23, XBOOLE_0:3; ::_thesis: verum
end;
assume that
A26: q in rng f and
A27: q in rng (mid (g,2,(len g))) ; ::_thesis: contradiction
q in rng g by A18, A27;
then q in (L~ f) /\ (L~ g) by A19, A11, A26, XBOOLE_0:def_4;
hence contradiction by A4, A27, A21, TARSKI:def_1; ::_thesis: verum
end;
n2 <= len (f ^ (mid (g,2,(len g)))) by A16, FINSEQ_1:1;
then A28: n2 - (len f) <= ((len f) + (len (mid (g,2,(len g))))) - (len f) by A6, XREAL_1:9;
A29: x1 in Seg (len (f ^ (mid (g,2,(len g))))) by A13, FINSEQ_1:def_3;
then n1 <= len (f ^ (mid (g,2,(len g)))) by FINSEQ_1:1;
then A30: n1 - (len f) <= ((len f) + (len (mid (g,2,(len g))))) - (len f) by A6, XREAL_1:9;
A31: 1 <= n1 by A29, FINSEQ_1:1;
now__::_thesis:_(_(_n1_<=_len_f_&_x1_=_x2_)_or_(_n1_>_len_f_&_x1_=_x2_)_)
percases ( n1 <= len f or n1 > len f ) ;
case n1 <= len f ; ::_thesis: x1 = x2
then A32: n1 in dom f by A31, FINSEQ_3:25;
then A33: (f ^ (mid (g,2,(len g)))) . x1 = f . n1 by FINSEQ_1:def_7;
now__::_thesis:_(_(_n2_<=_len_f_&_x1_=_x2_)_or_(_n2_>_len_f_&_contradiction_)_)
percases ( n2 <= len f or n2 > len f ) ;
case n2 <= len f ; ::_thesis: x1 = x2
then A34: n2 in dom f by A17, FINSEQ_3:25;
then (f ^ (mid (g,2,(len g)))) . x2 = f . n2 by FINSEQ_1:def_7;
hence x1 = x2 by A2, A15, A32, A33, A34, FUNCT_1:def_4; ::_thesis: verum
end;
caseA35: n2 > len f ; ::_thesis: contradiction
then (len f) + 1 <= n2 by NAT_1:13;
then A36: ((len f) + 1) - (len f) <= n2 - (len f) by XREAL_1:9;
then A37: 1 <= n2 -' (len f) by NAT_D:39;
A38: (len f) + (n2 -' (len f)) = (len f) + (n2 - (len f)) by A35, XREAL_1:233
.= n2 ;
n2 -' (len f) <= len (mid (g,2,(len g))) by A28, A36, NAT_D:39;
then A39: n2 -' (len f) in dom (mid (g,2,(len g))) by A37, FINSEQ_3:25;
then (f ^ (mid (g,2,(len g)))) . ((len f) + (n2 -' (len f))) = (mid (g,2,(len g))) . (n2 -' (len f)) by FINSEQ_1:def_7;
hence contradiction by A15, A20, A32, A33, A39, A38, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
caseA40: n1 > len f ; ::_thesis: x1 = x2
then (len f) + 1 <= n1 by NAT_1:13;
then A41: ((len f) + 1) - (len f) <= n1 - (len f) by XREAL_1:9;
then A42: 1 <= n1 -' (len f) by NAT_D:39;
then A43: 1 <= (n1 -' (len f)) + 1 by NAT_D:48;
n1 -' (len f) <= (n1 -' (len f)) + 2 by NAT_1:11;
then A44: ((n1 -' (len f)) + 2) -' 1 = ((n1 -' (len f)) + 2) - 1 by A42, XREAL_1:233, XXREAL_0:2
.= (n1 -' (len f)) + ((1 + 1) - 1) ;
A45: (len f) + (n1 -' (len f)) = (len f) + (n1 - (len f)) by A40, XREAL_1:233
.= n1 ;
A46: n1 -' (len f) <= len (mid (g,2,(len g))) by A30, A41, NAT_D:39;
then A47: n1 -' (len f) in dom (mid (g,2,(len g))) by A42, FINSEQ_3:25;
then A48: (f ^ (mid (g,2,(len g)))) . ((len f) + (n1 -' (len f))) = (mid (g,2,(len g))) . (n1 -' (len f)) by FINSEQ_1:def_7;
(n1 -' (len f)) + 1 <= ((len g) - 1) + 1 by A10, A46, XREAL_1:6;
then A49: (n1 -' (len f)) + 1 in dom g by A43, FINSEQ_3:25;
(len f) + (n1 -' (len f)) = (len f) + (n1 - (len f)) by A40, XREAL_1:233
.= n1 ;
then A50: (f ^ (mid (g,2,(len g)))) . n1 = g . ((n1 -' (len f)) + 1) by A8, A9, A30, A41, A48, A44, FINSEQ_6:118;
now__::_thesis:_(_(_n2_<=_len_f_&_contradiction_)_or_(_n2_>_len_f_&_x1_=_x2_)_)
percases ( n2 <= len f or n2 > len f ) ;
case n2 <= len f ; ::_thesis: contradiction
then A51: n2 in dom f by A17, FINSEQ_3:25;
then (f ^ (mid (g,2,(len g)))) . x2 = f . n2 by FINSEQ_1:def_7;
hence contradiction by A15, A20, A47, A48, A45, A51, FUNCT_1:def_3; ::_thesis: verum
end;
caseA52: n2 > len f ; ::_thesis: x1 = x2
then (len f) + 1 <= n2 by NAT_1:13;
then A53: ((len f) + 1) - (len f) <= n2 - (len f) by XREAL_1:9;
then A54: 1 <= n2 -' (len f) by NAT_D:39;
then A55: 1 <= (n2 -' (len f)) + 1 by NAT_D:48;
A56: n2 -' (len f) <= len (mid (g,2,(len g))) by A28, A53, NAT_D:39;
then (n2 -' (len f)) + 1 <= ((len g) - 1) + 1 by A10, XREAL_1:6;
then A57: (n2 -' (len f)) + 1 in dom g by A55, FINSEQ_3:25;
n2 -' (len f) <= (n2 -' (len f)) + 2 by NAT_1:11;
then A58: ((n2 -' (len f)) + 2) -' 1 = ((n2 -' (len f)) + 2) - 1 by A54, XREAL_1:233, XXREAL_0:2
.= (n2 -' (len f)) + 1 ;
1 <= n2 -' (len f) by A53, NAT_D:39;
then n2 -' (len f) in dom (mid (g,2,(len g))) by A56, FINSEQ_3:25;
then A59: (f ^ (mid (g,2,(len g)))) . ((len f) + (n2 -' (len f))) = (mid (g,2,(len g))) . (n2 -' (len f)) by FINSEQ_1:def_7;
(len f) + (n2 -' (len f)) = (len f) + (n2 - (len f)) by A52, XREAL_1:233
.= n2 ;
then (f ^ (mid (g,2,(len g)))) . n2 = g . ((n2 -' (len f)) + 1) by A8, A9, A28, A53, A59, A58, FINSEQ_6:118;
then (n1 -' (len f)) + 1 = (n2 -' (len f)) + 1 by A3, A15, A49, A50, A57, FUNCT_1:def_4;
then n1 - (len f) = n2 -' (len f) by A40, XREAL_1:233;
then n1 - (len f) = n2 - (len f) by A52, XREAL_1:233;
hence x1 = x2 ; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
then A60: f ^ (mid (g,2,(len g))) is one-to-one by FUNCT_1:def_4;
A61: 1 <= len f by A5, XXREAL_0:2;
A62: for i, j being Nat st i + 1 < j holds
LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j)
proof
let i, j be Nat; ::_thesis: ( i + 1 < j implies LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) )
assume A63: i + 1 < j ; ::_thesis: LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j)
now__::_thesis:_(_(_j_<_len_f_&_j_+_1_<=_len_(f_^_(mid_(g,2,(len_g))))_&_LSeg_((f_^_(mid_(g,2,(len_g)))),i)_misses_LSeg_((f_^_(mid_(g,2,(len_g)))),j)_)_or_(_i_+_1_<=_len_f_&_len_f_<=_j_&_j_+_1_<=_len_(f_^_(mid_(g,2,(len_g))))_&_LSeg_((f_^_(mid_(g,2,(len_g)))),i)_misses_LSeg_((f_^_(mid_(g,2,(len_g)))),j)_)_or_(_len_f_<_i_+_1_&_j_+_1_<=_len_(f_^_(mid_(g,2,(len_g))))_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_or_(_j_+_1_>_len_(f_^_(mid_(g,2,(len_g))))_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( ( j < len f & j + 1 <= len (f ^ (mid (g,2,(len g)))) ) or ( i + 1 <= len f & len f <= j & j + 1 <= len (f ^ (mid (g,2,(len g)))) ) or ( len f < i + 1 & j + 1 <= len (f ^ (mid (g,2,(len g)))) ) or j + 1 > len (f ^ (mid (g,2,(len g)))) ) ;
caseA64: ( j < len f & j + 1 <= len (f ^ (mid (g,2,(len g)))) ) ; ::_thesis: LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j)
then A65: i + 1 < len f by A63, XXREAL_0:2;
then A66: i < len f by NAT_1:13;
A67: j <= len (f ^ (mid (g,2,(len g)))) by A64, NAT_D:46;
then A68: i + 1 < len (f ^ (mid (g,2,(len g)))) by A63, XXREAL_0:2;
then A69: i <= len (f ^ (mid (g,2,(len g)))) by NAT_D:46;
now__::_thesis:_(_(_1_<=_i_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_or_(_i_<_1_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( 1 <= i or i < 1 ) ;
caseA70: 1 <= i ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then A71: f /. i = f . i by A66, FINSEQ_4:15;
(f ^ (mid (g,2,(len g)))) /. i = (f ^ (mid (g,2,(len g)))) . i by A69, A70, FINSEQ_4:15;
then A72: (f ^ (mid (g,2,(len g)))) /. i = f /. i by A66, A70, A71, FINSEQ_1:64;
A73: LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),i) by A68, A70, TOPREAL1:def_3;
A74: 1 <= i + 1 by A70, NAT_D:48;
then A75: f /. (i + 1) = f . (i + 1) by A65, FINSEQ_4:15;
(f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A68, A74, FINSEQ_4:15;
then LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f /. i),(f /. (i + 1))) by A65, A74, A72, A75, FINSEQ_1:64;
then A76: LSeg ((f ^ (mid (g,2,(len g)))),i) = LSeg (f,i) by A65, A70, A73, TOPREAL1:def_3;
A77: 1 < j by A63, A74, XXREAL_0:2;
then A78: f /. j = f . j by A64, FINSEQ_4:15;
(f ^ (mid (g,2,(len g)))) /. j = (f ^ (mid (g,2,(len g)))) . j by A67, A77, FINSEQ_4:15;
then A79: (f ^ (mid (g,2,(len g)))) /. j = f /. j by A64, A77, A78, FINSEQ_1:64;
A80: 1 <= j + 1 by A77, NAT_D:48;
then A81: (f ^ (mid (g,2,(len g)))) /. (j + 1) = (f ^ (mid (g,2,(len g)))) . (j + 1) by A64, FINSEQ_4:15;
A82: j + 1 <= len f by A64, NAT_1:13;
then A83: LSeg (f,j) = LSeg ((f /. j),(f /. (j + 1))) by A77, TOPREAL1:def_3;
f /. (j + 1) = f . (j + 1) by A80, A82, FINSEQ_4:15;
then A84: LSeg (((f ^ (mid (g,2,(len g)))) /. j),((f ^ (mid (g,2,(len g)))) /. (j + 1))) = LSeg ((f /. j),(f /. (j + 1))) by A80, A82, A79, A81, FINSEQ_1:64;
LSeg (((f ^ (mid (g,2,(len g)))) /. j),((f ^ (mid (g,2,(len g)))) /. (j + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),j) by A64, A77, TOPREAL1:def_3;
then LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) by A2, A63, A76, A84, A83, TOPREAL1:def_7;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} by XBOOLE_0:def_7; ::_thesis: verum
end;
case i < 1 ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then LSeg ((f ^ (mid (g,2,(len g)))),i) = {} by TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) by XBOOLE_0:def_7; ::_thesis: verum
end;
caseA85: ( i + 1 <= len f & len f <= j & j + 1 <= len (f ^ (mid (g,2,(len g)))) ) ; ::_thesis: LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j)
now__::_thesis:_(_(_i_+_1_<_len_f_&_len_f_<=_j_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_or_(_i_+_1_<=_len_f_&_len_f_<_j_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( ( i + 1 < len f & len f <= j ) or ( i + 1 <= len f & len f < j ) ) by A63, A85, XXREAL_0:1;
caseA86: ( i + 1 < len f & len f <= j ) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
len f <= (len f) + (len (mid (g,2,(len g)))) by NAT_1:11;
then A87: i + 1 < len (f ^ (mid (g,2,(len g)))) by A6, A86, XXREAL_0:2;
A88: len f <= j + 1 by A86, NAT_D:48;
A89: 1 + 1 <= j by A5, A86, XXREAL_0:2;
then A90: 1 <= j by NAT_D:46;
now__::_thesis:_(_(_1_<=_i_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_or_(_i_<_1_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( 1 <= i or i < 1 ) ;
caseA91: 1 <= i ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
i <= len f by A85, NAT_D:46;
then A92: f /. i = f . i by A91, FINSEQ_4:15;
i <= len (f ^ (mid (g,2,(len g)))) by A87, NAT_D:46;
then A93: (f ^ (mid (g,2,(len g)))) /. i = (f ^ (mid (g,2,(len g)))) . i by A91, FINSEQ_4:15;
i <= len f by A85, NAT_D:46;
then A94: (f ^ (mid (g,2,(len g)))) /. i = f /. i by A91, A93, A92, FINSEQ_1:64;
A95: j <= len (f ^ (mid (g,2,(len g)))) by A85, NAT_D:46;
A96: now__::_thesis:_(_1_>_j_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._j_=_g_._((j_-'_(len_f))_+_1)_)
assume 1 > j -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1)
then (j -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then A97: j -' (len f) = 0 by XREAL_1:6;
then j - (len f) = 0 by A85, XREAL_1:233;
hence (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1) by A1, A61, A97, FINSEQ_1:64; ::_thesis: verum
end;
1 <= j + 1 by A90, NAT_D:48;
then A98: (f ^ (mid (g,2,(len g)))) /. (j + 1) = (f ^ (mid (g,2,(len g)))) . (j + 1) by A85, FINSEQ_4:15;
A99: LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),i) by A87, A91, TOPREAL1:def_3;
A100: 1 <= i + 1 by A91, NAT_D:48;
then A101: f /. (i + 1) = f . (i + 1) by A85, FINSEQ_4:15;
A102: now__::_thesis:_(_1_>_(j_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._(j_+_1)_=_g_._(((j_+_1)_-'_(len_f))_+_1)_)
assume 1 > (j + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1)
then ((j + 1) -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then A103: (j + 1) -' (len f) = 0 by XREAL_1:6;
then (j + 1) - (len f) = 0 by A88, XREAL_1:233;
hence (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1) by A1, A61, A103, FINSEQ_1:64; ::_thesis: verum
end;
(j + 1) + 1 <= ((len f) + ((len g) - 1)) + 1 by A6, A10, A85, XREAL_1:6;
then ((j + 1) + 1) - (len f) <= ((len f) + (len g)) - (len f) by XREAL_1:9;
then ((j - (len f)) + 1) + 1 <= len g ;
then A104: ((j -' (len f)) + 1) + 1 <= len g by A86, XREAL_1:233;
then (j -' (len f)) + 1 <= len g by NAT_D:46;
then A105: g /. ((j -' (len f)) + 1) = g . ((j -' (len f)) + 1) by FINSEQ_4:15, NAT_1:11;
(((j -' (len f)) + 1) + 1) - 1 <= (len g) - 1 by A104, XREAL_1:9;
then A106: (j -' (len f)) + 1 <= ((len g) - 2) + 1 ;
then (j -' (len f)) + 1 <= ((len g) -' 2) + 1 by A8, XREAL_1:233;
then A107: j -' (len f) <= ((len g) -' 2) + 1 by NAT_D:46;
A108: now__::_thesis:_(_1_<=_j_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._j_=_g_._((j_-'_(len_f))_+_1)_)
assume A109: 1 <= j -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1)
then 1 <= j - (len f) by NAT_D:39;
then 1 + (len f) <= (j - (len f)) + (len f) by XREAL_1:6;
then A110: len f < j by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . j = (mid (g,2,(len g))) . (j - (len f)) by A95, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . j = (mid (g,2,(len g))) . (j -' (len f)) by A110, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . j = g . (((j -' (len f)) + 2) - 1) by A8, A107, A109, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1) ; ::_thesis: verum
end;
A111: (j -' (len f)) + 1 = (j - (len f)) + 1 by A85, XREAL_1:233
.= (j + 1) - (len f)
.= (j + 1) -' (len f) by A88, XREAL_1:233 ;
A112: now__::_thesis:_(_1_<=_(j_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._(j_+_1)_=_g_._(((j_+_1)_-'_(len_f))_+_1)_)
assume A113: 1 <= (j + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1)
then 1 <= (j + 1) - (len f) by NAT_D:39;
then 1 + (len f) <= ((j + 1) - (len f)) + (len f) by XREAL_1:6;
then A114: len f < j + 1 by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . (j + 1) = (mid (g,2,(len g))) . ((j + 1) - (len f)) by A85, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . (j + 1) = (mid (g,2,(len g))) . ((j + 1) -' (len f)) by A114, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . (j + 1) = g . ((((j + 1) -' (len f)) + 2) - 1) by A8, A106, A111, A113, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1) ; ::_thesis: verum
end;
1 <= 1 + (j -' (len f)) by NAT_1:11;
then A115: LSeg (g,((j -' (len f)) + 1)) = LSeg ((g /. ((j -' (len f)) + 1)),(g /. (((j -' (len f)) + 1) + 1))) by A104, TOPREAL1:def_3;
1 <= j by A89, NAT_D:46;
then A116: (f ^ (mid (g,2,(len g)))) /. j = g /. ((j -' (len f)) + 1) by A95, A105, A108, A96, FINSEQ_4:15;
g /. (((j -' (len f)) + 1) + 1) = g . (((j -' (len f)) + 1) + 1) by A104, FINSEQ_4:15, NAT_1:11;
then LSeg ((f ^ (mid (g,2,(len g)))),j) = LSeg (g,((j -' (len f)) + 1)) by A85, A90, A111, A115, A116, A98, A112, A102, TOPREAL1:def_3;
then A117: LSeg ((f ^ (mid (g,2,(len g)))),j) c= L~ g by TOPREAL3:19;
A118: (i + 1) + 1 <= len f by A86, NAT_1:13;
(f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A87, A100, FINSEQ_4:15;
then LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f /. i),(f /. (i + 1))) by A85, A100, A94, A101, FINSEQ_1:64;
then A119: LSeg ((f ^ (mid (g,2,(len g)))),i) = LSeg (f,i) by A85, A91, A99, TOPREAL1:def_3;
then LSeg ((f ^ (mid (g,2,(len g)))),i) c= L~ f by TOPREAL3:19;
then A120: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) c= {(g . 1)} by A4, A117, XBOOLE_1:27;
now__::_thesis:_(_(_i_+_1_<_(len_f)_-'_1_&_LSeg_((f_^_(mid_(g,2,(len_g)))),i)_misses_LSeg_((f_^_(mid_(g,2,(len_g)))),j)_)_or_(_i_+_1_>=_(len_f)_-'_1_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( i + 1 < (len f) -' 1 or i + 1 >= (len f) -' 1 ) ;
caseA121: i + 1 < (len f) -' 1 ; ::_thesis: LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j)
A122: 1 <= len f by A5, XXREAL_0:2;
A123: ((len f) -' 1) + 1 = ((len f) - 1) + 1 by A5, XREAL_1:233, XXREAL_0:2
.= len f ;
A124: (1 + 1) - 1 <= (len f) - 1 by A5, XREAL_1:9;
now__::_thesis:_for_x_being_set_holds_not_x_in_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))
f /. (len f) in LSeg (f,((len f) -' 1)) by A124, A123, TOPREAL1:21;
then A125: g . 1 in LSeg (f,((len f) -' 1)) by A1, A122, FINSEQ_4:15;
given x being set such that A126: x in (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) ; ::_thesis: contradiction
A127: x in LSeg (f,i) by A119, A126, XBOOLE_0:def_4;
x = g . 1 by A120, A126, TARSKI:def_1;
then x in (LSeg (f,i)) /\ (LSeg (f,((len f) -' 1))) by A127, A125, XBOOLE_0:def_4;
then LSeg (f,i) meets LSeg (f,((len f) -' 1)) by XBOOLE_0:4;
hence contradiction by A2, A121, TOPREAL1:def_7; ::_thesis: verum
end;
hence LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) by XBOOLE_0:4; ::_thesis: verum
end;
case i + 1 >= (len f) -' 1 ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then i + 1 >= (len f) - 1 by A5, XREAL_1:233, XXREAL_0:2;
then A128: (i + 1) + 1 >= ((len f) - 1) + 1 by XREAL_1:6;
then A129: (i + 1) + 1 = len f by A118, XXREAL_0:1;
then A130: i + 1 <= len f by NAT_1:11;
i + 1 = (len f) - 1 by A129;
then A131: i + 1 = (len f) -' 1 by A5, XREAL_1:233, XXREAL_0:2;
A132: ((len f) -' 1) + 1 = ((len f) - 1) + 1 by A5, XREAL_1:233, XXREAL_0:2
.= len f ;
then 1 + 1 <= ((len f) -' 1) + 1 by A2, TOPREAL1:def_8;
then A133: 1 <= (len f) -' 1 by XREAL_1:6;
A134: i + (1 + 1) = len f by A118, A128, XXREAL_0:1;
now__::_thesis:_for_x_being_set_holds_not_x_in_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))
(1 + 1) - 1 <= (len f) - 1 by A5, XREAL_1:9;
then A135: 1 <= (len f) -' 1 by NAT_D:39;
(len f) -' 1 <= len f by NAT_D:35;
then A136: (len f) -' 1 in dom f by A135, FINSEQ_3:25;
A137: (LSeg (f,i)) /\ (LSeg (f,((len f) -' 1))) = {(f /. (i + 1))} by A2, A91, A131, A134, TOPREAL1:def_6;
f /. (len f) in LSeg (f,((len f) -' 1)) by A132, A133, TOPREAL1:21;
then A138: g . 1 in LSeg (f,((len f) -' 1)) by A1, A61, FINSEQ_4:15;
given x being set such that A139: x in (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) ; ::_thesis: contradiction
A140: x = g . 1 by A120, A139, TARSKI:def_1;
x in LSeg (f,i) by A119, A139, XBOOLE_0:def_4;
then x in (LSeg (f,i)) /\ (LSeg (f,((len f) -' 1))) by A140, A138, XBOOLE_0:def_4;
then f . (len f) = f /. (i + 1) by A1, A140, A137, TARSKI:def_1;
then A141: f . (len f) = f . ((len f) -' 1) by A131, A130, FINSEQ_4:15, NAT_1:11;
len f in dom f by A61, FINSEQ_3:25;
then len f = (len f) -' 1 by A2, A141, A136, FUNCT_1:def_4;
then len f = (len f) - 1 by A5, XREAL_1:233, XXREAL_0:2;
hence contradiction ; ::_thesis: verum
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
end;
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} by XBOOLE_0:def_7; ::_thesis: verum
end;
case i < 1 ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then LSeg ((f ^ (mid (g,2,(len g)))),i) = {} by TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
end;
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
caseA142: ( i + 1 <= len f & len f < j ) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
(1 + 1) - 1 <= (len g) - 1 by A8, XREAL_1:9;
then (len f) + 1 <= (len f) + (len (mid (g,2,(len g)))) by A10, XREAL_1:7;
then len f < (len f) + (len (mid (g,2,(len g)))) by NAT_1:13;
then A143: i + 1 < len (f ^ (mid (g,2,(len g)))) by A6, A142, XXREAL_0:2;
A144: len f <= j + 1 by A142, NAT_D:48;
A145: 1 + 1 <= j by A5, A142, XXREAL_0:2;
then A146: 1 <= j by NAT_D:46;
now__::_thesis:_(_(_1_<=_i_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_or_(_i_<_1_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( 1 <= i or i < 1 ) ;
caseA147: 1 <= i ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
i <= len f by A85, NAT_D:46;
then A148: f /. i = f . i by A147, FINSEQ_4:15;
i <= len (f ^ (mid (g,2,(len g)))) by A143, NAT_D:46;
then A149: (f ^ (mid (g,2,(len g)))) /. i = (f ^ (mid (g,2,(len g)))) . i by A147, FINSEQ_4:15;
i <= len f by A85, NAT_D:46;
then A150: (f ^ (mid (g,2,(len g)))) /. i = f /. i by A147, A149, A148, FINSEQ_1:64;
A151: LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),i) by A143, A147, TOPREAL1:def_3;
A152: 1 <= i + 1 by A147, NAT_D:48;
then A153: f /. (i + 1) = f . (i + 1) by A85, FINSEQ_4:15;
(f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A143, A152, FINSEQ_4:15;
then LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f /. i),(f /. (i + 1))) by A85, A152, A150, A153, FINSEQ_1:64;
then LSeg ((f ^ (mid (g,2,(len g)))),i) = LSeg (f,i) by A85, A147, A151, TOPREAL1:def_3;
then A154: LSeg ((f ^ (mid (g,2,(len g)))),i) c= L~ f by TOPREAL3:19;
A155: j <= len (f ^ (mid (g,2,(len g)))) by A85, NAT_D:46;
A156: now__::_thesis:_(_1_>_j_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._j_=_g_._((j_-'_(len_f))_+_1)_)
assume 1 > j -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1)
then (j -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then A157: j -' (len f) = 0 by XREAL_1:6;
then j - (len f) = 0 by A85, XREAL_1:233;
hence (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1) by A1, A61, A157, FINSEQ_1:64; ::_thesis: verum
end;
1 <= j + 1 by A146, NAT_D:48;
then A158: (f ^ (mid (g,2,(len g)))) /. (j + 1) = (f ^ (mid (g,2,(len g)))) . (j + 1) by A85, FINSEQ_4:15;
A159: now__::_thesis:_(_1_>_(j_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._(j_+_1)_=_g_._(((j_+_1)_-'_(len_f))_+_1)_)
assume 1 > (j + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1)
then ((j + 1) -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then A160: (j + 1) -' (len f) = 0 by XREAL_1:6;
then (j + 1) - (len f) = 0 by A144, XREAL_1:233;
hence (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1) by A1, A61, A160, FINSEQ_1:64; ::_thesis: verum
end;
(j + 1) + 1 <= ((len f) + ((len g) - 1)) + 1 by A6, A10, A85, XREAL_1:6;
then ((j + 1) + 1) - (len f) <= ((len f) + (len g)) - (len f) by XREAL_1:9;
then ((j - (len f)) + 1) + 1 <= len g ;
then A161: ((j -' (len f)) + 1) + 1 <= len g by A142, XREAL_1:233;
then (j -' (len f)) + 1 <= len g by NAT_D:46;
then A162: g /. ((j -' (len f)) + 1) = g . ((j -' (len f)) + 1) by FINSEQ_4:15, NAT_1:11;
(((j -' (len f)) + 1) + 1) - 1 <= (len g) - 1 by A161, XREAL_1:9;
then A163: (j -' (len f)) + 1 <= ((len g) - 2) + 1 ;
then (j -' (len f)) + 1 <= ((len g) -' 2) + 1 by A8, XREAL_1:233;
then A164: j -' (len f) <= ((len g) -' 2) + 1 by NAT_D:46;
A165: now__::_thesis:_(_1_<=_j_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._j_=_g_._((j_-'_(len_f))_+_1)_)
assume A166: 1 <= j -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1)
then 1 <= j - (len f) by NAT_D:39;
then 1 + (len f) <= (j - (len f)) + (len f) by XREAL_1:6;
then A167: len f < j by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . j = (mid (g,2,(len g))) . (j - (len f)) by A155, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . j = (mid (g,2,(len g))) . (j -' (len f)) by A167, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . j = g . (((j -' (len f)) + 2) - 1) by A8, A164, A166, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . j = g . ((j -' (len f)) + 1) ; ::_thesis: verum
end;
A168: (j -' (len f)) + 1 = (j - (len f)) + 1 by A85, XREAL_1:233
.= (j + 1) - (len f)
.= (j + 1) -' (len f) by A144, XREAL_1:233 ;
A169: now__::_thesis:_(_1_<=_(j_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._(j_+_1)_=_g_._(((j_+_1)_-'_(len_f))_+_1)_)
assume A170: 1 <= (j + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1)
then 1 <= (j + 1) - (len f) by NAT_D:39;
then 1 + (len f) <= ((j + 1) - (len f)) + (len f) by XREAL_1:6;
then A171: len f < j + 1 by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . (j + 1) = (mid (g,2,(len g))) . ((j + 1) - (len f)) by A85, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . (j + 1) = (mid (g,2,(len g))) . ((j + 1) -' (len f)) by A171, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . (j + 1) = g . ((((j + 1) -' (len f)) + 2) - 1) by A8, A163, A168, A170, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . (j + 1) = g . (((j + 1) -' (len f)) + 1) ; ::_thesis: verum
end;
1 <= 1 + (j -' (len f)) by NAT_1:11;
then A172: LSeg (g,((j -' (len f)) + 1)) = LSeg ((g /. ((j -' (len f)) + 1)),(g /. (((j -' (len f)) + 1) + 1))) by A161, TOPREAL1:def_3;
1 <= j by A145, NAT_D:46;
then A173: (f ^ (mid (g,2,(len g)))) /. j = g /. ((j -' (len f)) + 1) by A155, A162, A165, A156, FINSEQ_4:15;
g /. (((j -' (len f)) + 1) + 1) = g . (((j -' (len f)) + 1) + 1) by A161, FINSEQ_4:15, NAT_1:11;
then A174: LSeg ((f ^ (mid (g,2,(len g)))),j) = LSeg (g,((j -' (len f)) + 1)) by A85, A146, A168, A172, A173, A158, A169, A159, TOPREAL1:def_3;
then LSeg ((f ^ (mid (g,2,(len g)))),j) c= L~ g by TOPREAL3:19;
then A175: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) c= {(g . 1)} by A4, A154, XBOOLE_1:27;
now__::_thesis:_for_x_being_set_holds_not_x_in_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))
A176: 1 + 1 in dom g by A8, FINSEQ_3:25;
A177: (j -' (len f)) + 1 = (j - (len f)) + 1 by A142, XREAL_1:233;
A178: 1 + 1 <= len g by A3, TOPREAL1:def_8;
given x being set such that A179: x in (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) ; ::_thesis: contradiction
A180: x in LSeg (g,((j -' (len f)) + 1)) by A174, A179, XBOOLE_0:def_4;
A181: x = g . 1 by A175, A179, TARSKI:def_1;
then g /. 1 = x by A9, FINSEQ_4:15;
then x in LSeg (g,1) by A178, TOPREAL1:21;
then A182: x in (LSeg (g,1)) /\ (LSeg (g,((j -' (len f)) + 1))) by A180, XBOOLE_0:def_4;
then LSeg (g,1) meets LSeg (g,((j -' (len f)) + 1)) by XBOOLE_0:4;
then 1 + 1 >= (j -' (len f)) + 1 by A3, TOPREAL1:def_7;
then 1 >= j -' (len f) by XREAL_1:6;
then 1 >= j - (len f) by NAT_D:39;
then A183: 1 + (len f) >= (j - (len f)) + (len f) by XREAL_1:6;
j >= (len f) + 1 by A142, NAT_1:13;
then A184: j = (len f) + 1 by A183, XXREAL_0:1;
LSeg (g,((j -' (len f)) + 1)) <> {} by A174, A179;
then 1 + 2 <= len g by A184, A177, TOPREAL1:def_3;
then (LSeg (g,1)) /\ (LSeg (g,((j -' (len f)) + 1))) = {(g /. (1 + 1))} by A3, A184, A177, TOPREAL1:def_6;
then A185: x = g /. (1 + 1) by A182, TARSKI:def_1
.= g . (1 + 1) by A8, FINSEQ_4:15 ;
1 in dom g by A9, FINSEQ_3:25;
hence contradiction by A3, A181, A185, A176, FUNCT_1:def_4; ::_thesis: verum
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
case i < 1 ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then LSeg ((f ^ (mid (g,2,(len g)))),i) = {} by TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
end;
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) by XBOOLE_0:def_7; ::_thesis: verum
end;
caseA186: ( len f < i + 1 & j + 1 <= len (f ^ (mid (g,2,(len g)))) ) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then A187: len f <= i by NAT_1:13;
then A188: i -' (len f) = i - (len f) by XREAL_1:233;
A189: 1 + 1 <= i by A5, A187, XXREAL_0:2;
then A190: 1 <= i by NAT_D:46;
then A191: 1 <= i + 1 by NAT_D:48;
then A192: 1 <= j by A63, XXREAL_0:2;
A193: 1 <= (j -' (len f)) + 1 by NAT_1:11;
A194: len f < j by A63, A186, XXREAL_0:2;
j <= j + 1 by NAT_1:11;
then A195: len f < j + 1 by A194, XXREAL_0:2;
A196: 1 <= (i -' (len f)) + 1 by NAT_1:11;
A197: j -' (len f) = j - (len f) by A63, A186, XREAL_1:233, XXREAL_0:2;
(i + 1) - (len f) < j - (len f) by A63, XREAL_1:9;
then A198: ((i -' (len f)) + 1) + 1 < (j -' (len f)) + 1 by A188, A197, XREAL_1:6;
now__::_thesis:_(_(_j_+_1_<=_len_(f_^_(mid_(g,2,(len_g))))_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_or_(_j_+_1_>_len_(f_^_(mid_(g,2,(len_g))))_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),j))_=_{}_)_)
percases ( j + 1 <= len (f ^ (mid (g,2,(len g)))) or j + 1 > len (f ^ (mid (g,2,(len g)))) ) ;
caseA199: j + 1 <= len (f ^ (mid (g,2,(len g)))) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
A200: 1 <= j by A63, A191, XXREAL_0:2;
then 1 <= j + 1 by NAT_D:48;
then A201: (f ^ (mid (g,2,(len g)))) /. (j + 1) = (f ^ (mid (g,2,(len g)))) . (j + 1) by A199, FINSEQ_4:15;
(len f) + 1 <= j by A194, NAT_1:13;
then A202: ((len f) + 1) - (len f) <= j - (len f) by XREAL_1:9;
A203: 1 <= i by A189, NAT_D:46;
then A204: 1 <= i + 1 by NAT_D:48;
A205: j <= len (f ^ (mid (g,2,(len g)))) by A199, NAT_D:46;
then A206: i + 1 < len (f ^ (mid (g,2,(len g)))) by A63, XXREAL_0:2;
then A207: i <= len (f ^ (mid (g,2,(len g)))) by NAT_D:46;
i + 1 < (len f) + ((len g) - 1) by A10, A206, FINSEQ_1:22;
then A208: (i + 1) - (len f) < ((len f) + ((len g) - 1)) - (len f) by XREAL_1:9;
then A209: ((i - (len f)) + 1) + 1 < ((len g) - 1) + 1 by XREAL_1:6;
then ((i -' (len f)) + 1) + 1 <= len g by A187, XREAL_1:233;
then A210: g /. (((i -' (len f)) + 1) + 1) = g . (((i -' (len f)) + 1) + 1) by FINSEQ_4:15, NAT_1:11;
i + 1 <= len (f ^ (mid (g,2,(len g)))) by A63, A205, XXREAL_0:2;
then A211: (f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A204, FINSEQ_4:15;
A212: LSeg (((f ^ (mid (g,2,(len g)))) /. j),((f ^ (mid (g,2,(len g)))) /. (j + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),j) by A192, A199, TOPREAL1:def_3;
A213: LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),i) by A190, A206, TOPREAL1:def_3;
A214: (i -' (len f)) + 1 <= len g by A188, A209, NAT_D:46;
then A215: g /. ((i -' (len f)) + 1) = g . ((i -' (len f)) + 1) by FINSEQ_4:15, NAT_1:11;
A216: now__::_thesis:_(_(_i_<=_len_f_&_(f_^_(mid_(g,2,(len_g))))_/._i_=_g_/._((i_-'_(len_f))_+_1)_)_or_(_i_>_len_f_&_(f_^_(mid_(g,2,(len_g))))_/._i_=_g_/._((i_-'_(len_f))_+_1)_)_)
percases ( i <= len f or i > len f ) ;
case i <= len f ; ::_thesis: (f ^ (mid (g,2,(len g)))) /. i = g /. ((i -' (len f)) + 1)
then A217: i = len f by A187, XXREAL_0:1;
then (f ^ (mid (g,2,(len g)))) . i = g . (0 + 1) by A1, A190, FINSEQ_1:64
.= g . ((i -' (len f)) + 1) by A217, XREAL_1:232 ;
hence (f ^ (mid (g,2,(len g)))) /. i = g /. ((i -' (len f)) + 1) by A203, A207, A215, FINSEQ_4:15; ::_thesis: verum
end;
caseA218: i > len f ; ::_thesis: (f ^ (mid (g,2,(len g)))) /. i = g /. ((i -' (len f)) + 1)
then (len f) + 1 <= i by NAT_1:13;
then A219: ((len f) + 1) - (len f) <= i - (len f) by XREAL_1:9;
((i -' (len f)) + 1) - 1 <= (len g) - 1 by A214, XREAL_1:9;
then A220: i -' (len f) <= ((len g) - 2) + 1 ;
(f ^ (mid (g,2,(len g)))) . i = (mid (g,2,(len g))) . (i -' (len f)) by A188, A207, A218, FINSEQ_6:108
.= g . (((i -' (len f)) + 2) - 1) by A8, A188, A219, A220, FINSEQ_6:122
.= g . ((i -' (len f)) + 1) ;
hence (f ^ (mid (g,2,(len g)))) /. i = g /. ((i -' (len f)) + 1) by A203, A207, A215, FINSEQ_4:15; ::_thesis: verum
end;
end;
end;
j + 1 <= (len f) + ((len g) - 1) by A10, A199, FINSEQ_1:22;
then A221: (j + 1) - (len f) <= ((len f) + ((len g) - 1)) - (len f) by XREAL_1:9;
then A222: (j -' (len f)) + 1 <= ((len g) - 2) + 1 by A197;
A223: (((j -' (len f)) + 1) + 2) - 1 = ((j -' (len f)) + 1) + 1 ;
A224: (f ^ (mid (g,2,(len g)))) . (j + 1) = (mid (g,2,(len g))) . ((j + 1) - (len f)) by A195, A199, FINSEQ_6:108
.= g . (((j -' (len f)) + 1) + 1) by A8, A197, A193, A222, A223, FINSEQ_6:122 ;
A225: (((i -' (len f)) + 1) + 2) - 1 = ((i -' (len f)) + 1) + 1 ;
A226: (i -' (len f)) + 1 <= ((len g) - 2) + 1 by A188, A208;
(f ^ (mid (g,2,(len g)))) . (i + 1) = (mid (g,2,(len g))) . ((i + 1) - (len f)) by A186, A206, FINSEQ_6:108
.= g . (((i -' (len f)) + 1) + 1) by A8, A188, A196, A226, A225, FINSEQ_6:122 ;
then A227: LSeg ((f ^ (mid (g,2,(len g)))),i) = LSeg (g,((i -' (len f)) + 1)) by A188, A196, A209, A216, A211, A210, A213, TOPREAL1:def_3;
A228: ((j - (len f)) + 1) + 1 <= ((len g) - 1) + 1 by A221, XREAL_1:6;
then A229: (j -' (len f)) + 1 <= len g by A197, NAT_D:46;
then A230: g /. ((j -' (len f)) + 1) = g . ((j -' (len f)) + 1) by FINSEQ_4:15, NAT_1:11;
A231: j <= len (f ^ (mid (g,2,(len g)))) by A199, NAT_D:46;
then A232: (f ^ (mid (g,2,(len g)))) /. j = (f ^ (mid (g,2,(len g)))) . j by A200, FINSEQ_4:15;
((j -' (len f)) + 1) + 1 <= len g by A63, A186, A228, XREAL_1:233, XXREAL_0:2;
then A233: g /. (((j -' (len f)) + 1) + 1) = g . (((j -' (len f)) + 1) + 1) by FINSEQ_4:15, NAT_1:11;
((j -' (len f)) + 1) - 1 <= (len g) - 1 by A229, XREAL_1:9;
then A234: j -' (len f) <= ((len g) - 2) + 1 ;
(f ^ (mid (g,2,(len g)))) . j = (mid (g,2,(len g))) . (j - (len f)) by A194, A231, FINSEQ_6:108
.= g . (((j -' (len f)) + 2) - 1) by A8, A197, A202, A234, FINSEQ_6:122
.= g . ((j -' (len f)) + 1) ;
then LSeg ((f ^ (mid (g,2,(len g)))),j) = LSeg (g,((j -' (len f)) + 1)) by A197, A193, A228, A232, A230, A201, A233, A224, A212, TOPREAL1:def_3;
then LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) by A3, A198, A227, TOPREAL1:def_7;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} by XBOOLE_0:def_7; ::_thesis: verum
end;
case j + 1 > len (f ^ (mid (g,2,(len g)))) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then LSeg ((f ^ (mid (g,2,(len g)))),j) = {} by TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
end;
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
case j + 1 > len (f ^ (mid (g,2,(len g)))) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {}
then LSeg ((f ^ (mid (g,2,(len g)))),j) = {} by TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),j)) = {} ; ::_thesis: verum
end;
end;
end;
hence LSeg ((f ^ (mid (g,2,(len g)))),i) misses LSeg ((f ^ (mid (g,2,(len g)))),j) by XBOOLE_0:def_7; ::_thesis: verum
end;
A235: for i being Nat st 1 <= i & i + 2 <= len (f ^ (mid (g,2,(len g)))) holds
(LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))}
proof
let i be Nat; ::_thesis: ( 1 <= i & i + 2 <= len (f ^ (mid (g,2,(len g)))) implies (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))} )
assume that
A236: 1 <= i and
A237: i + 2 <= len (f ^ (mid (g,2,(len g)))) ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))}
A238: 1 <= i + 1 by A236, NAT_D:48;
A239: i <= len (f ^ (mid (g,2,(len g)))) by A237, NAT_D:47;
A240: 1 <= (i + 1) + 1 by A236, NAT_D:48;
A241: i + 1 <= len (f ^ (mid (g,2,(len g)))) by A237, NAT_D:47;
(i + 1) + 1 <= (len f) + (len (mid (g,2,(len g)))) by A237, FINSEQ_1:22;
then (i + 1) + 1 <= (len f) + (((len g) -' 2) + 1) by A8, A9, FINSEQ_6:118;
then (i + 1) + 1 <= (len f) + (((len g) - (1 + 1)) + 1) by A8, XREAL_1:233;
then A242: ((i + 1) + 1) - (len f) <= ((len f) + (((len g) - (1 + 1)) + 1)) - (len f) by XREAL_1:9;
then A243: (((i + 1) - (len f)) + 1) + 1 <= ((len g) - 1) + 1 by XREAL_1:6;
then A244: (((i - (len f)) + 1) + 1) + 1 <= len g ;
now__::_thesis:_(_(_i_+_2_<=_len_f_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),(i_+_1)))_=_{((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))}_)_or_(_i_+_2_>_len_f_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),(i_+_1)))_=_{((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))}_)_)
percases ( i + 2 <= len f or i + 2 > len f ) ;
caseA245: i + 2 <= len f ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))}
A246: (f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A238, A241, FINSEQ_4:15;
(i + 1) + 1 <= len f by A245;
then A247: i + 1 <= len f by NAT_D:46;
then f /. (i + 1) = f . (i + 1) by A238, FINSEQ_4:15;
then A248: (f ^ (mid (g,2,(len g)))) /. (i + 1) = f /. (i + 1) by A238, A247, A246, FINSEQ_1:64;
A249: f /. ((i + 1) + 1) = f . ((i + 1) + 1) by A240, A245, FINSEQ_4:15;
A250: LSeg (((f ^ (mid (g,2,(len g)))) /. (i + 1)),((f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),(i + 1)) by A237, A238, TOPREAL1:def_3;
A251: (f ^ (mid (g,2,(len g)))) /. i = (f ^ (mid (g,2,(len g)))) . i by A236, A239, FINSEQ_4:15;
A252: i <= len f by A247, NAT_D:46;
then f /. i = f . i by A236, FINSEQ_4:15;
then A253: (f ^ (mid (g,2,(len g)))) /. i = f /. i by A236, A252, A251, FINSEQ_1:64;
(f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1) = (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) by A237, A240, FINSEQ_4:15;
then LSeg (((f ^ (mid (g,2,(len g)))) /. (i + 1)),((f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1))) = LSeg ((f /. (i + 1)),(f /. ((i + 1) + 1))) by A240, A245, A248, A249, FINSEQ_1:64;
then A254: LSeg ((f ^ (mid (g,2,(len g)))),(i + 1)) = LSeg (f,(i + 1)) by A238, A245, A250, TOPREAL1:def_3;
LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),i) by A236, A241, TOPREAL1:def_3;
then LSeg ((f ^ (mid (g,2,(len g)))),i) = LSeg (f,i) by A236, A247, A253, A248, TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))} by A2, A236, A245, A248, A254, TOPREAL1:def_6; ::_thesis: verum
end;
case i + 2 > len f ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))}
then A255: i + 2 >= (len f) + 1 by NAT_1:13;
now__::_thesis:_(_(_i_+_2_>_(len_f)_+_1_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),(i_+_1)))_=_{((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))}_)_or_(_i_+_2_=_(len_f)_+_1_&_(LSeg_((f_^_(mid_(g,2,(len_g)))),i))_/\_(LSeg_((f_^_(mid_(g,2,(len_g)))),(i_+_1)))_=_{((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))}_)_)
percases ( i + 2 > (len f) + 1 or i + 2 = (len f) + 1 ) by A255, XXREAL_0:1;
caseA256: i + 2 > (len f) + 1 ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))}
then (i + 1) + 1 > (len f) + 1 ;
then A257: i + 1 >= (len f) + 1 by NAT_1:13;
then A258: i >= len f by XREAL_1:6;
A259: now__::_thesis:_(_1_>_i_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._i_=_g_._((i_-'_(len_f))_+_1)_)
assume 1 > i -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1)
then (i -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then A260: i -' (len f) = 0 by XREAL_1:6;
then i - (len f) = 0 by A258, XREAL_1:233;
hence (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1) by A1, A61, A260, FINSEQ_1:64; ::_thesis: verum
end;
A261: i + 1 >= len f by A257, NAT_D:48;
A262: now__::_thesis:_(_1_>_(i_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._(i_+_1)_=_g_._(((i_+_1)_-'_(len_f))_+_1)_)
assume 1 > (i + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . (i + 1) = g . (((i + 1) -' (len f)) + 1)
then ((i + 1) -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then A263: (i + 1) -' (len f) = 0 by XREAL_1:6;
then (i + 1) - (len f) = 0 by A261, XREAL_1:233;
hence (f ^ (mid (g,2,(len g)))) . (i + 1) = g . (((i + 1) -' (len f)) + 1) by A1, A61, A263, FINSEQ_1:64; ::_thesis: verum
end;
(i + 1) + 1 >= ((len f) + 1) + 1 by A256, NAT_1:13;
then ((i + 1) + 1) - ((len f) + 1) >= (((len f) + 1) + 1) - ((len f) + 1) by XREAL_1:9;
then (i - (len f)) + 1 >= 1 ;
then A264: (i -' (len f)) + 1 >= 1 by A258, XREAL_1:233;
then A265: ((i -' (len f)) + 1) + 1 >= 1 by NAT_D:48;
then A266: ((i - (len f)) + 1) + 1 >= 1 by A258, XREAL_1:233;
then ((i + 1) - (len f)) + 1 >= 1 ;
then A267: ((i + 1) -' (len f)) + 1 >= 1 by A261, XREAL_1:233;
then A268: (((i + 1) -' (len f)) + 1) + 1 >= 1 by NAT_D:48;
((i + 1) - (len f)) + 1 >= 0 + 1 by A266;
then A269: (i + 1) - (len f) >= 0 by XREAL_1:6;
then (((i + 1) -' (len f)) + 1) + 1 <= len g by A243, XREAL_0:def_2;
then A270: g /. ((((i + 1) -' (len f)) + 1) + 1) = g . ((((i + 1) -' (len f)) + 1) + 1) by A268, FINSEQ_4:15;
(((i + 1) -' (len f)) + 1) + 1 <= len g by A243, A261, XREAL_1:233;
then A271: LSeg (g,(((i + 1) -' (len f)) + 1)) = LSeg ((g /. (((i + 1) -' (len f)) + 1)),(g /. ((((i + 1) -' (len f)) + 1) + 1))) by A267, TOPREAL1:def_3;
(((i + 1) + 1) - (len f)) + 1 = (((i + 1) - (len f)) + 1) + 1 ;
then A272: (((i + 1) + 1) - (len f)) + 1 = (((i + 1) -' (len f)) + 1) + 1 by A269, XREAL_0:def_2;
A273: (((i -' (len f)) + 1) + 1) + 1 <= len g by A244, A258, XREAL_1:233;
then A274: ((i -' (len f)) + 1) + 1 <= len g by NAT_D:46;
then A275: LSeg (g,((i -' (len f)) + 1)) = LSeg ((g /. ((i -' (len f)) + 1)),(g /. (((i -' (len f)) + 1) + 1))) by A264, TOPREAL1:def_3;
(((i -' (len f)) + 1) + 1) - 1 <= (len g) - 1 by A274, XREAL_1:9;
then (i -' (len f)) + 1 <= ((len g) - 2) + 1 ;
then A276: (i -' (len f)) + 1 <= ((len g) -' 2) + 1 by A8, XREAL_1:233;
then A277: i -' (len f) <= ((len g) -' 2) + 1 by NAT_D:46;
A278: now__::_thesis:_(_1_<=_i_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._i_=_g_._((i_-'_(len_f))_+_1)_)
assume A279: 1 <= i -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1)
then 1 <= i - (len f) by NAT_D:39;
then 1 + (len f) <= (i - (len f)) + (len f) by XREAL_1:6;
then A280: len f < i by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . i = (mid (g,2,(len g))) . (i - (len f)) by A239, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . i = (mid (g,2,(len g))) . (i -' (len f)) by A280, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . i = g . (((i -' (len f)) + 2) - 1) by A8, A277, A279, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1) ; ::_thesis: verum
end;
(i -' (len f)) + 1 <= len g by A274, NAT_D:46;
then g /. ((i -' (len f)) + 1) = g . ((i -' (len f)) + 1) by A264, FINSEQ_4:15;
then A281: (f ^ (mid (g,2,(len g)))) /. i = g /. ((i -' (len f)) + 1) by A236, A239, A278, A259, FINSEQ_4:15;
A282: ((i -' (len f)) + 1) + (1 + 1) <= len g by A273;
A283: g /. (((i -' (len f)) + 1) + 1) = g . (((i -' (len f)) + 1) + 1) by A274, A265, FINSEQ_4:15;
(i - (len f)) + 1 <= ((len g) -' 2) + 1 by A258, A276, XREAL_1:233;
then (i + 1) - (len f) <= ((len g) -' 2) + 1 ;
then A284: (i + 1) -' (len f) <= ((len g) -' 2) + 1 by A261, XREAL_1:233;
A285: now__::_thesis:_(_1_<=_(i_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._(i_+_1)_=_g_._(((i_+_1)_-'_(len_f))_+_1)_)
assume A286: 1 <= (i + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . (i + 1) = g . (((i + 1) -' (len f)) + 1)
then 1 <= (i + 1) - (len f) by NAT_D:39;
then 1 + (len f) <= ((i + 1) - (len f)) + (len f) by XREAL_1:6;
then A287: len f < i + 1 by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . (i + 1) = (mid (g,2,(len g))) . ((i + 1) - (len f)) by A241, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . (i + 1) = (mid (g,2,(len g))) . ((i + 1) -' (len f)) by A287, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . (i + 1) = g . ((((i + 1) -' (len f)) + 2) - 1) by A8, A284, A286, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . (i + 1) = g . (((i + 1) -' (len f)) + 1) ; ::_thesis: verum
end;
A288: now__::_thesis:_(_1_>_((i_+_1)_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._((i_+_1)_+_1)_=_g_._((((i_+_1)_+_1)_-'_(len_f))_+_1)_)
assume 1 > ((i + 1) + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = g . ((((i + 1) + 1) -' (len f)) + 1)
then A289: (((i + 1) + 1) -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then ((i + 1) + 1) -' (len f) <= 0 by XREAL_1:6;
then A290: ((i + 1) + 1) - (len f) = 0 by A266, XREAL_0:def_2;
((i + 1) + 1) -' (len f) = 0 by A289, XREAL_1:6;
hence (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = g . ((((i + 1) + 1) -' (len f)) + 1) by A1, A61, A290, FINSEQ_1:64; ::_thesis: verum
end;
((i + 1) - (len f)) + 1 = ((i - (len f)) + 1) + 1 ;
then A291: ((i + 1) - (len f)) + 1 = ((i -' (len f)) + 1) + 1 by A258, XREAL_1:233;
then A292: ((i + 1) -' (len f)) + 1 = ((i -' (len f)) + 1) + 1 by A261, XREAL_1:233;
A293: ((i + 1) + 1) -' (len f) <= ((len g) - 2) + 1 by A242, A266, XREAL_0:def_2;
A294: now__::_thesis:_(_1_<=_((i_+_1)_+_1)_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._((i_+_1)_+_1)_=_g_._((((i_+_1)_+_1)_-'_(len_f))_+_1)_)
assume A295: 1 <= ((i + 1) + 1) -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = g . ((((i + 1) + 1) -' (len f)) + 1)
then 1 <= ((i + 1) + 1) - (len f) by NAT_D:39;
then 1 + (len f) <= (((i + 1) + 1) - (len f)) + (len f) by XREAL_1:6;
then A296: len f < (i + 1) + 1 by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = (mid (g,2,(len g))) . (((i + 1) + 1) - (len f)) by A237, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = (mid (g,2,(len g))) . (((i + 1) + 1) -' (len f)) by A296, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = g . (((((i + 1) + 1) -' (len f)) + 2) - 1) by A8, A293, A295, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = g . ((((i + 1) + 1) -' (len f)) + 1) ; ::_thesis: verum
end;
A297: (f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1) = (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) by A237, A240, FINSEQ_4:15;
A298: LSeg ((f ^ (mid (g,2,(len g)))),i) = LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) by A236, A241, TOPREAL1:def_3;
A299: (f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A238, A241, FINSEQ_4:15;
LSeg ((f ^ (mid (g,2,(len g)))),(i + 1)) = LSeg (((f ^ (mid (g,2,(len g)))) /. (i + 1)),((f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1))) by A237, A238, TOPREAL1:def_3;
then LSeg ((f ^ (mid (g,2,(len g)))),(i + 1)) = LSeg (g,(((i + 1) -' (len f)) + 1)) by A291, A272, A299, A283, A285, A262, A271, A297, A270, A294, A288, XREAL_0:def_2;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))} by A3, A264, A292, A298, A275, A281, A299, A283, A285, A282, TOPREAL1:def_6; ::_thesis: verum
end;
caseA300: i + 2 = (len f) + 1 ; ::_thesis: (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))}
then A301: f /. (i + 1) = f . (i + 1) by A238, FINSEQ_4:15;
then A302: f /. (i + 1) = g /. 1 by A1, A9, A300, FINSEQ_4:15;
(f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A238, A241, FINSEQ_4:15;
then A303: (f ^ (mid (g,2,(len g)))) /. (i + 1) = f /. (i + 1) by A238, A300, A301, FINSEQ_1:64;
A304: LSeg (f,i) = LSeg ((f /. i),(f /. (i + 1))) by A236, A300, TOPREAL1:def_3;
A305: (f ^ (mid (g,2,(len g)))) /. i = (f ^ (mid (g,2,(len g)))) . i by A236, A239, FINSEQ_4:15;
(i + 1) + 1 = (len f) + 1 by A300;
then A306: i <= len f by NAT_D:46;
then f /. i = f . i by A236, FINSEQ_4:15;
then A307: (f ^ (mid (g,2,(len g)))) /. i = f /. i by A236, A306, A305, FINSEQ_1:64;
A308: LSeg (((f ^ (mid (g,2,(len g)))) /. i),((f ^ (mid (g,2,(len g)))) /. (i + 1))) = LSeg ((f ^ (mid (g,2,(len g)))),i) by A236, A241, TOPREAL1:def_3;
i = (len f) - 1 by A300;
then A309: i = (len f) -' 1 by A5, XREAL_1:233, XXREAL_0:2;
A310: g /. 1 in LSeg ((g /. 1),(g /. (1 + 1))) by RLTOPSP1:68;
A311: g /. 1 = g . 1 by A9, FINSEQ_4:15;
then g /. 1 = f /. (len f) by A1, A61, FINSEQ_4:15;
then A312: g /. 1 in LSeg ((f /. ((len f) -' 1)),(f /. (len f))) by RLTOPSP1:68;
(len g) - 2 >= 0 by A8, XREAL_1:48;
then A313: 0 + 1 <= ((len g) - 2) + 1 by XREAL_1:6;
len f < (i + 1) + 1 by A300, NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = (mid (g,2,(len g))) . (((i + 1) + 1) - (len f)) by A237, FINSEQ_6:108;
then A314: (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) = g . ((2 + 1) -' 1) by A8, A300, A313, FINSEQ_6:122
.= g . 2 by NAT_D:34 ;
A315: LSeg (g,1) c= L~ g by TOPREAL3:19;
LSeg (f,i) c= L~ f by TOPREAL3:19;
then A316: (LSeg (f,i)) /\ (LSeg (g,1)) c= {(g /. 1)} by A4, A311, A315, XBOOLE_1:27;
A317: ((i + 1) -' (len f)) + 1 = 0 + 1 by A300, XREAL_1:232
.= 1 ;
then A318: g /. ((((i + 1) -' (len f)) + 1) + 1) = g . ((((i + 1) -' (len f)) + 1) + 1) by A8, FINSEQ_4:15;
LSeg (g,1) = LSeg ((g /. 1),(g /. (1 + 1))) by A8, TOPREAL1:def_3;
then g /. 1 in (LSeg (f,i)) /\ (LSeg (g,1)) by A300, A309, A304, A312, A310, XBOOLE_0:def_4;
then A319: {(g /. 1)} c= (LSeg (f,i)) /\ (LSeg (g,1)) by ZFMISC_1:31;
A320: (f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1) = (f ^ (mid (g,2,(len g)))) . ((i + 1) + 1) by A237, A240, FINSEQ_4:15;
A321: LSeg ((f ^ (mid (g,2,(len g)))),(i + 1)) = LSeg (((f ^ (mid (g,2,(len g)))) /. (i + 1)),((f ^ (mid (g,2,(len g)))) /. ((i + 1) + 1))) by A237, A238, TOPREAL1:def_3;
LSeg (g,(((i + 1) -' (len f)) + 1)) = LSeg ((g /. (((i + 1) -' (len f)) + 1)),(g /. ((((i + 1) -' (len f)) + 1) + 1))) by A8, A317, TOPREAL1:def_3;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))} by A307, A302, A303, A308, A304, A321, A317, A320, A318, A314, A319, A316, XBOOLE_0:def_10; ::_thesis: verum
end;
end;
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))} ; ::_thesis: verum
end;
end;
end;
hence (LSeg ((f ^ (mid (g,2,(len g)))),i)) /\ (LSeg ((f ^ (mid (g,2,(len g)))),(i + 1))) = {((f ^ (mid (g,2,(len g)))) /. (i + 1))} ; ::_thesis: verum
end;
for i being Nat st 1 <= i & i + 1 <= len (f ^ (mid (g,2,(len g)))) & not ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 holds
((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2
proof
let i be Nat; ::_thesis: ( 1 <= i & i + 1 <= len (f ^ (mid (g,2,(len g)))) & not ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 implies ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 )
assume that
A322: 1 <= i and
A323: i + 1 <= len (f ^ (mid (g,2,(len g)))) ; ::_thesis: ( ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 or ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 )
now__::_thesis:_(_(_i_<_len_f_&_(_((f_^_(mid_(g,2,(len_g))))_/._i)_`1_=_((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))_`1_or_((f_^_(mid_(g,2,(len_g))))_/._i)_`2_=_((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))_`2_)_)_or_(_i_>=_len_f_&_(_((f_^_(mid_(g,2,(len_g))))_/._i)_`1_=_((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))_`1_or_((f_^_(mid_(g,2,(len_g))))_/._i)_`2_=_((f_^_(mid_(g,2,(len_g))))_/._(i_+_1))_`2_)_)_)
percases ( i < len f or i >= len f ) ;
caseA324: i < len f ; ::_thesis: ( ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 or ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 )
i <= len (f ^ (mid (g,2,(len g)))) by A323, NAT_D:46;
then A325: (f ^ (mid (g,2,(len g)))) /. i = (f ^ (mid (g,2,(len g)))) . i by A322, FINSEQ_4:15;
f /. i = f . i by A322, A324, FINSEQ_4:15;
then A326: (f ^ (mid (g,2,(len g)))) /. i = f /. i by A322, A324, A325, FINSEQ_1:64;
A327: 1 <= i + 1 by A322, NAT_D:48;
then A328: (f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A323, FINSEQ_4:15;
A329: i + 1 <= len f by A324, NAT_1:13;
then A330: (f ^ (mid (g,2,(len g)))) . (i + 1) = f . (i + 1) by A327, FINSEQ_1:64;
f /. (i + 1) = f . (i + 1) by A327, A329, FINSEQ_4:15;
hence ( ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 or ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 ) by A2, A322, A329, A326, A328, A330, TOPREAL1:def_5; ::_thesis: verum
end;
caseA331: i >= len f ; ::_thesis: ( ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 or ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 )
1 <= 1 + (i -' (len f)) by NAT_1:11;
then 1 <= 1 + (i - (len f)) by A331, XREAL_1:233;
then 1 <= (1 + i) - (len f) ;
then A332: 1 <= (i + 1) -' (len f) by NAT_D:39;
A333: i <= len (f ^ (mid (g,2,(len g)))) by A323, NAT_D:46;
A334: i - (len f) >= 0 by A331, XREAL_1:48;
then A335: i -' (len f) = i - (len f) by XREAL_0:def_2;
A336: now__::_thesis:_(_1_>_i_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._i_=_g_._((i_-'_(len_f))_+_1)_)
assume 1 > i -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1)
then (i -' (len f)) + 1 <= 0 + 1 by NAT_1:13;
then i -' (len f) = 0 by XREAL_1:6;
hence (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1) by A1, A61, A335, FINSEQ_1:64; ::_thesis: verum
end;
A337: i + 1 >= len f by A331, NAT_D:48;
then A338: ((i + 1) -' (len f)) + 1 = ((i + 1) - (len f)) + 1 by XREAL_1:233
.= ((i - (len f)) + 1) + 1
.= ((i -' (len f)) + 1) + 1 by A331, XREAL_1:233 ;
A339: (i + 1) - (len f) <= ((len f) + ((len g) - 1)) - (len f) by A6, A10, A323, XREAL_1:9;
then A340: ((i - (len f)) + 1) + 1 <= ((len g) - 1) + 1 by XREAL_1:6;
then A341: ((i -' (len f)) + 1) + 1 <= len g by A334, XREAL_0:def_2;
i -' (len f) <= (i -' (len f)) + 1 by NAT_1:11;
then A342: i -' (len f) <= ((len g) - 2) + 1 by A335, A339, XXREAL_0:2;
A343: now__::_thesis:_(_1_<=_i_-'_(len_f)_implies_(f_^_(mid_(g,2,(len_g))))_._i_=_g_._((i_-'_(len_f))_+_1)_)
assume A344: 1 <= i -' (len f) ; ::_thesis: (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1)
then 1 <= i - (len f) by NAT_D:39;
then 1 + (len f) <= (i - (len f)) + (len f) by XREAL_1:6;
then A345: len f < i by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . i = (mid (g,2,(len g))) . (i - (len f)) by A333, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . i = (mid (g,2,(len g))) . (i -' (len f)) by A345, XREAL_1:233;
then (f ^ (mid (g,2,(len g)))) . i = g . (((i -' (len f)) + 2) - 1) by A8, A342, A344, FINSEQ_6:122;
hence (f ^ (mid (g,2,(len g)))) . i = g . ((i -' (len f)) + 1) ; ::_thesis: verum
end;
1 <= i + 1 by A322, NAT_D:48;
then A346: (f ^ (mid (g,2,(len g)))) /. (i + 1) = (f ^ (mid (g,2,(len g)))) . (i + 1) by A323, FINSEQ_4:15;
A347: 1 <= (i -' (len f)) + 1 by NAT_1:11;
(i + 1) - (len f) <= ((len g) - 2) + 1 by A339;
then (i + 1) - (len f) <= ((len g) -' 2) + 1 by A8, XREAL_1:233;
then A348: (i + 1) -' (len f) <= ((len g) -' 2) + 1 by A337, XREAL_1:233;
len f < i + 1 by A331, NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . (i + 1) = (mid (g,2,(len g))) . ((i + 1) - (len f)) by A323, FINSEQ_6:108;
then (f ^ (mid (g,2,(len g)))) . (i + 1) = (mid (g,2,(len g))) . ((i + 1) -' (len f)) by A337, XREAL_1:233;
then A349: (f ^ (mid (g,2,(len g)))) . (i + 1) = g . ((((i + 1) -' (len f)) + 2) - 1) by A8, A348, A332, FINSEQ_6:122;
((i -' (len f)) + 1) + 1 <= len g by A334, A340, XREAL_0:def_2;
then A350: g /. (((i -' (len f)) + 1) + 1) = g . (((i -' (len f)) + 1) + 1) by FINSEQ_4:15, NAT_1:11;
(i -' (len f)) + 1 <= len g by A335, A340, NAT_D:46;
then g /. ((i -' (len f)) + 1) = g . ((i -' (len f)) + 1) by FINSEQ_4:15, NAT_1:11;
then (f ^ (mid (g,2,(len g)))) /. i = g /. ((i -' (len f)) + 1) by A322, A333, A343, A336, FINSEQ_4:15;
hence ( ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 or ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 ) by A3, A347, A341, A338, A346, A350, A349, TOPREAL1:def_5; ::_thesis: verum
end;
end;
end;
hence ( ((f ^ (mid (g,2,(len g)))) /. i) `1 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `1 or ((f ^ (mid (g,2,(len g)))) /. i) `2 = ((f ^ (mid (g,2,(len g)))) /. (i + 1)) `2 ) ; ::_thesis: verum
end;
then ( f ^ (mid (g,2,(len g))) is unfolded & f ^ (mid (g,2,(len g))) is s.n.c. & f ^ (mid (g,2,(len g))) is special ) by A235, A62, TOPREAL1:def_5, TOPREAL1:def_6, TOPREAL1:def_7;
hence f ^ (mid (g,2,(len g))) is being_S-Seq by A60, A7, TOPREAL1:def_8; ::_thesis: verum
end;
theorem Th39: :: JORDAN3:39
for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds
f ^ (mid (g,2,(len g))) is_S-Seq_joining f /. 1,g /. (len g)
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} implies f ^ (mid (g,2,(len g))) is_S-Seq_joining f /. 1,g /. (len g) )
assume that
A1: f . (len f) = g . 1 and
A2: f is being_S-Seq and
A3: g is being_S-Seq and
A4: (L~ f) /\ (L~ g) = {(g . 1)} ; ::_thesis: f ^ (mid (g,2,(len g))) is_S-Seq_joining f /. 1,g /. (len g)
A5: f ^ (mid (g,2,(len g))) is being_S-Seq by A1, A2, A3, A4, Th38;
A6: len g >= 2 by A3, TOPREAL1:def_8;
then A7: (1 + 1) - 1 <= (len g) - 1 by XREAL_1:9;
len f >= 2 by A2, TOPREAL1:def_8;
then A8: 1 <= len f by XXREAL_0:2;
then A9: (f ^ (mid (g,2,(len g)))) . 1 = f . 1 by FINSEQ_1:64
.= f /. 1 by A8, FINSEQ_4:15 ;
A10: len (f ^ (mid (g,2,(len g)))) = (len f) + (len (mid (g,2,(len g)))) by FINSEQ_1:22;
A11: 1 <= len g by A6, XXREAL_0:2;
then A12: len (mid (g,2,(len g))) = ((len g) -' 2) + 1 by A6, FINSEQ_6:118;
then A13: len (mid (g,2,(len g))) = ((len g) - 2) + 1 by A6, XREAL_1:233
.= (len g) - 1 ;
then A14: ((len (mid (g,2,(len g)))) + 2) - 1 = len g ;
(len g) - 1 >= (1 + 1) - 1 by A6, XREAL_1:9;
then (len f) + 1 <= len (f ^ (mid (g,2,(len g)))) by A10, A13, XREAL_1:6;
then len f < len (f ^ (mid (g,2,(len g)))) by NAT_1:13;
then (f ^ (mid (g,2,(len g)))) . (len (f ^ (mid (g,2,(len g))))) = (mid (g,2,(len g))) . ((len (f ^ (mid (g,2,(len g))))) - (len f)) by FINSEQ_6:108
.= g . (len g) by A6, A10, A12, A7, A14, FINSEQ_6:122
.= g /. (len g) by A11, FINSEQ_4:15 ;
hence f ^ (mid (g,2,(len g))) is_S-Seq_joining f /. 1,g /. (len g) by A5, A9, Def2; ::_thesis: verum
end;
theorem :: JORDAN3:40
for f being FinSequence of (TOP-REAL 2)
for n being Element of NAT holds L~ (f /^ n) c= L~ f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for n being Element of NAT holds L~ (f /^ n) c= L~ f
let n be Element of NAT ; ::_thesis: L~ (f /^ n) c= L~ f
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ (f /^ n) or x in L~ f )
assume x in L~ (f /^ n) ; ::_thesis: x in L~ f
then x in union { (LSeg ((f /^ n),i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len (f /^ n) ) } by TOPREAL1:def_4;
then consider Y being set such that
A1: ( x in Y & Y in { (LSeg ((f /^ n),i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len (f /^ n) ) } ) by TARSKI:def_4;
consider i being Element of NAT such that
A2: Y = LSeg ((f /^ n),i) and
A3: 1 <= i and
A4: i + 1 <= len (f /^ n) by A1;
now__::_thesis:_(_(_n_<=_len_f_&_x_in_L~_f_)_or_(_n_>_len_f_&_contradiction_)_)
percases ( n <= len f or n > len f ) ;
case n <= len f ; ::_thesis: x in L~ f
then LSeg ((f /^ n),i) = LSeg (f,(n + i)) by A3, SPPOL_2:4;
then Y c= L~ f by A2, TOPREAL3:19;
hence x in L~ f by A1; ::_thesis: verum
end;
case n > len f ; ::_thesis: contradiction
then f /^ n = <*> the carrier of (TOP-REAL 2) by RFINSEQ:def_1;
hence contradiction by A4; ::_thesis: verum
end;
end;
end;
hence x in L~ f ; ::_thesis: verum
end;
theorem Th41: :: JORDAN3:41
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
L~ (R_Cut (f,p)) c= L~ f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
L~ (R_Cut (f,p)) c= L~ f
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies L~ (R_Cut (f,p)) c= L~ f )
assume A1: p in L~ f ; ::_thesis: L~ (R_Cut (f,p)) c= L~ f
A2: 1 <= Index (p,f) by A1, Th8;
len f <> 0 by A1, TOPREAL1:22;
then A3: len f >= 0 + 1 by NAT_1:13;
A4: Index (p,f) <= len f by A1, Th8;
percases ( p = f . 1 or p <> f . 1 ) ;
suppose p = f . 1 ; ::_thesis: L~ (R_Cut (f,p)) c= L~ f
then R_Cut (f,p) = <*p*> by Def4;
then len (R_Cut (f,p)) = 1 by FINSEQ_1:39;
then L~ (R_Cut (f,p)) = {} by TOPREAL1:22;
hence L~ (R_Cut (f,p)) c= L~ f by XBOOLE_1:2; ::_thesis: verum
end;
supposeA5: p <> f . 1 ; ::_thesis: L~ (R_Cut (f,p)) c= L~ f
A6: f /. (Index (p,f)) in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by RLTOPSP1:68;
A7: len (mid (f,1,(Index (p,f)))) = ((Index (p,f)) -' 1) + 1 by A3, A2, A4, FINSEQ_6:118
.= Index (p,f) by A1, Th8, XREAL_1:235 ;
then mid (f,1,(Index (p,f))) <> <*> the carrier of (TOP-REAL 2) by A2;
then A8: L~ ((mid (f,1,(Index (p,f)))) ^ <*p*>) = (L~ (mid (f,1,(Index (p,f))))) \/ (LSeg (((mid (f,1,(Index (p,f)))) /. (Index (p,f))),p)) by A7, SPPOL_2:19;
mid (f,1,(Index (p,f))) = (f /^ (1 -' 1)) | (((Index (p,f)) -' 1) + 1) by A2, FINSEQ_6:def_3
.= (f /^ 0) | (((Index (p,f)) -' 1) + 1) by XREAL_1:232
.= f | (((Index (p,f)) -' 1) + 1) by FINSEQ_5:28
.= f | (Index (p,f)) by A1, Th8, XREAL_1:235 ;
then A9: L~ (mid (f,1,(Index (p,f)))) c= L~ f by TOPREAL3:20;
Index (p,f) < len f by A1, Th8;
then A10: (Index (p,f)) + 1 <= len f by NAT_1:13;
then A11: LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) c= L~ f by A1, Th8, SPPOL_2:16;
p in LSeg (f,(Index (p,f))) by A1, Th9;
then A12: p in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A2, A10, TOPREAL1:def_3;
(mid (f,1,(Index (p,f)))) /. (Index (p,f)) = (mid (f,1,(Index (p,f)))) . (Index (p,f)) by A2, A7, FINSEQ_4:15
.= f . (Index (p,f)) by A2, A4, FINSEQ_6:123
.= f /. (Index (p,f)) by A1, A4, Th8, FINSEQ_4:15 ;
then LSeg (((mid (f,1,(Index (p,f)))) /. (Index (p,f))),p) c= LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A12, A6, TOPREAL1:6;
then A13: LSeg (((mid (f,1,(Index (p,f)))) /. (Index (p,f))),p) c= L~ f by A11, XBOOLE_1:1;
R_Cut (f,p) = (mid (f,1,(Index (p,f)))) ^ <*p*> by A5, Def4;
hence L~ (R_Cut (f,p)) c= L~ f by A8, A13, A9, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
Lm2: for i being Element of NAT
for D being non empty set holds (<*> D) | i = <*> D
proof
let i be Element of NAT ; ::_thesis: for D being non empty set holds (<*> D) | i = <*> D
let D be non empty set ; ::_thesis: (<*> D) | i = <*> D
len (<*> D) = 0 ;
hence (<*> D) | i = <*> D by FINSEQ_1:58; ::_thesis: verum
end;
Lm3: for D being non empty set
for f1 being FinSequence of D
for k being Element of NAT st 1 <= k & k <= len f1 holds
( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 )
proof
let D be non empty set ; ::_thesis: for f1 being FinSequence of D
for k being Element of NAT st 1 <= k & k <= len f1 holds
( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 )
let f1 be FinSequence of D; ::_thesis: for k being Element of NAT st 1 <= k & k <= len f1 holds
( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 )
let k be Element of NAT ; ::_thesis: ( 1 <= k & k <= len f1 implies ( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 ) )
assume that
A1: 1 <= k and
A2: k <= len f1 ; ::_thesis: ( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 )
A3: f1 /. k = f1 . k by A1, A2, FINSEQ_4:15;
(k -' 1) + 1 <= len f1 by A1, A2, XREAL_1:235;
then A4: ((k -' 1) + 1) - (k -' 1) <= (len f1) - (k -' 1) by XREAL_1:9;
len (f1 /^ (k -' 1)) = (len f1) -' (k -' 1) by RFINSEQ:29;
then A5: 1 <= len (f1 /^ (k -' 1)) by A4, NAT_D:39;
(k -' 1) + 1 = k by A1, XREAL_1:235;
then A6: (f1 /^ (k -' 1)) . 1 = f1 . k by A2, FINSEQ_6:114;
(k -' k) + 1 = (k - k) + 1 by XREAL_1:233
.= 1 ;
then mid (f1,k,k) = (f1 /^ (k -' 1)) | 1 by FINSEQ_6:def_3
.= <*((f1 /^ (k -' 1)) /. 1)*> by A5, CARD_1:27, FINSEQ_5:20 ;
hence ( mid (f1,k,k) = <*(f1 /. k)*> & len (mid (f1,k,k)) = 1 ) by A6, A3, A5, FINSEQ_1:39, FINSEQ_4:15; ::_thesis: verum
end;
Lm4: for D being non empty set
for f1 being FinSequence of D holds mid (f1,0,0) = f1 | 1
proof
let D be non empty set ; ::_thesis: for f1 being FinSequence of D holds mid (f1,0,0) = f1 | 1
let f1 be FinSequence of D; ::_thesis: mid (f1,0,0) = f1 | 1
0 - 1 < 0 ;
then A1: 0 -' 1 = 0 by XREAL_0:def_2;
(0 -' 0) + 1 = (0 - 0) + 1 by XREAL_1:233
.= 1 ;
then mid (f1,0,0) = (f1 /^ (0 -' 1)) | 1 by FINSEQ_6:def_3;
hence mid (f1,0,0) = f1 | 1 by A1, FINSEQ_5:28; ::_thesis: verum
end;
Lm5: for D being non empty set
for f1 being FinSequence of D
for k being Element of NAT st len f1 < k holds
mid (f1,k,k) = <*> D
proof
let D be non empty set ; ::_thesis: for f1 being FinSequence of D
for k being Element of NAT st len f1 < k holds
mid (f1,k,k) = <*> D
let f1 be FinSequence of D; ::_thesis: for k being Element of NAT st len f1 < k holds
mid (f1,k,k) = <*> D
let k be Element of NAT ; ::_thesis: ( len f1 < k implies mid (f1,k,k) = <*> D )
assume A1: len f1 < k ; ::_thesis: mid (f1,k,k) = <*> D
then (len f1) + 1 <= k by NAT_1:13;
then A2: ((len f1) + 1) - 1 <= k - 1 by XREAL_1:9;
0 + 1 <= k by A1, NAT_1:13;
then len f1 <= k -' 1 by A2, XREAL_1:233;
then A3: f1 /^ (k -' 1) = <*> D by FINSEQ_5:32;
(k -' k) + 1 = (k - k) + 1 by XREAL_1:233
.= 1 ;
then mid (f1,k,k) = (f1 /^ (k -' 1)) | 1 by FINSEQ_6:def_3;
hence mid (f1,k,k) = <*> D by A3, Lm2; ::_thesis: verum
end;
Lm6: for D being non empty set
for f1 being FinSequence of D
for i1, i2 being Element of NAT holds mid (f1,i1,i2) = Rev (mid (f1,i2,i1))
proof
let D be non empty set ; ::_thesis: for f1 being FinSequence of D
for i1, i2 being Element of NAT holds mid (f1,i1,i2) = Rev (mid (f1,i2,i1))
let f1 be FinSequence of D; ::_thesis: for i1, i2 being Element of NAT holds mid (f1,i1,i2) = Rev (mid (f1,i2,i1))
let k1, k2 be Element of NAT ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
now__::_thesis:_(_(_k1_<=_k2_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_or_(_k1_>_k2_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_)
percases ( k1 <= k2 or k1 > k2 ) ;
caseA1: k1 <= k2 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then A2: mid (f1,k1,k2) = (f1 /^ (k1 -' 1)) | ((k2 -' k1) + 1) by FINSEQ_6:def_3;
now__::_thesis:_(_(_k1_<_k2_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_or_(_k1_=_k2_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_)
percases ( k1 < k2 or k1 = k2 ) by A1, XXREAL_0:1;
case k1 < k2 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then mid (f1,k2,k1) = Rev (mid (f1,k1,k2)) by A2, FINSEQ_6:def_3;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) ; ::_thesis: verum
end;
caseA3: k1 = k2 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
A4: ( k1 = 0 or 0 + 1 <= k1 ) by NAT_1:13;
now__::_thesis:_(_(_k1_=_0_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_or_(_1_<=_k1_&_k1_<=_len_f1_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_or_(_len_f1_<_k1_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_)
percases ( k1 = 0 or ( 1 <= k1 & k1 <= len f1 ) or len f1 < k1 ) by A4;
case k1 = 0 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then A5: mid (f1,k1,k2) = f1 | 1 by A3, Lm4;
now__::_thesis:_(_(_len_f1_=_0_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_or_(_len_f1_<>_0_&_mid_(f1,k1,k2)_=_Rev_(mid_(f1,k2,k1))_)_)
percases ( len f1 = 0 or len f1 <> 0 ) ;
case len f1 = 0 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then f1 = <*> D ;
then f1 | 1 = <*> D by Lm2;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) by A3, A5; ::_thesis: verum
end;
case len f1 <> 0 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then f1 <> <*> D ;
then f1 | 1 = <*(f1 /. 1)*> by FINSEQ_5:20;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) by A3, A5, FINSEQ_5:60; ::_thesis: verum
end;
end;
end;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) ; ::_thesis: verum
end;
case ( 1 <= k1 & k1 <= len f1 ) ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then mid (f1,k1,k1) = <*(f1 /. k1)*> by Lm3;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) by A3, FINSEQ_5:60; ::_thesis: verum
end;
case len f1 < k1 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then mid (f1,k1,k1) = <*> D by Lm5;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) by A3; ::_thesis: verum
end;
end;
end;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) ; ::_thesis: verum
end;
end;
end;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) ; ::_thesis: verum
end;
caseA6: k1 > k2 ; ::_thesis: mid (f1,k1,k2) = Rev (mid (f1,k2,k1))
then mid (f1,k1,k2) = Rev ((f1 /^ (k2 -' 1)) | ((k1 -' k2) + 1)) by FINSEQ_6:def_3;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) by A6, FINSEQ_6:def_3; ::_thesis: verum
end;
end;
end;
hence mid (f1,k1,k2) = Rev (mid (f1,k2,k1)) ; ::_thesis: verum
end;
Lm7: for h being FinSequence of (TOP-REAL 2)
for i1, i2 being Element of NAT st 1 <= i1 & i1 <= len h & 1 <= i2 & i2 <= len h holds
L~ (mid (h,i1,i2)) c= L~ h
proof
let h be FinSequence of (TOP-REAL 2); ::_thesis: for i1, i2 being Element of NAT st 1 <= i1 & i1 <= len h & 1 <= i2 & i2 <= len h holds
L~ (mid (h,i1,i2)) c= L~ h
let i1, i2 be Element of NAT ; ::_thesis: ( 1 <= i1 & i1 <= len h & 1 <= i2 & i2 <= len h implies L~ (mid (h,i1,i2)) c= L~ h )
assume that
A1: 1 <= i1 and
A2: i1 <= len h and
A3: 1 <= i2 and
A4: i2 <= len h ; ::_thesis: L~ (mid (h,i1,i2)) c= L~ h
thus L~ (mid (h,i1,i2)) c= L~ h ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L~ (mid (h,i1,i2)) or x in L~ h )
assume A5: x in L~ (mid (h,i1,i2)) ; ::_thesis: x in L~ h
now__::_thesis:_(_(_i1_<=_i2_&_x_in_L~_h_)_or_(_i1_>_i2_&_x_in_L~_h_)_)
percases ( i1 <= i2 or i1 > i2 ) ;
caseA6: i1 <= i2 ; ::_thesis: x in L~ h
x in union { (LSeg ((mid (h,i1,i2)),i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len (mid (h,i1,i2)) ) } by A5, TOPREAL1:def_4;
then consider Y being set such that
A7: ( x in Y & Y in { (LSeg ((mid (h,i1,i2)),i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len (mid (h,i1,i2)) ) } ) by TARSKI:def_4;
consider i being Element of NAT such that
A8: Y = LSeg ((mid (h,i1,i2)),i) and
A9: 1 <= i and
A10: i + 1 <= len (mid (h,i1,i2)) by A7;
A11: LSeg ((mid (h,i1,i2)),i) = LSeg (((mid (h,i1,i2)) /. i),((mid (h,i1,i2)) /. (i + 1))) by A9, A10, TOPREAL1:def_3;
len (mid (h,i1,i2)) = (i2 -' i1) + 1 by A1, A2, A3, A4, A6, FINSEQ_6:118;
then (i + 1) - 1 <= ((i2 -' i1) + 1) - 1 by A10, XREAL_1:9;
then i <= i2 - i1 by A6, XREAL_1:233;
then A12: i + i1 <= (i2 - i1) + i1 by XREAL_1:6;
then A13: i + i1 <= len h by A4, XXREAL_0:2;
1 + 1 <= i + i1 by A1, A9, XREAL_1:7;
then A14: (1 + 1) - 1 <= (i + i1) - 1 by XREAL_1:9;
then 1 <= (i + i1) -' 1 by A1, NAT_1:12, XREAL_1:233;
then A15: h /. ((i + i1) -' 1) = h . ((i + i1) -' 1) by A13, FINSEQ_4:15, NAT_D:44;
1 <= i + 1 by NAT_1:11;
then A16: (mid (h,i1,i2)) . (i + 1) = h . (((i + 1) + i1) -' 1) by A1, A2, A3, A4, A6, A10, FINSEQ_6:118;
A17: ((i + 1) + i1) -' 1 = ((i + 1) + i1) - 1 by A1, NAT_1:12, XREAL_1:233
.= i + i1 ;
then A18: ((i + 1) + i1) -' 1 = ((i + i1) - 1) + 1
.= ((i + i1) -' 1) + 1 by A1, NAT_1:12, XREAL_1:233 ;
i <= i + 1 by NAT_1:11;
then A19: i <= len (mid (h,i1,i2)) by A10, XXREAL_0:2;
then A20: (mid (h,i1,i2)) /. i = (mid (h,i1,i2)) . i by A9, FINSEQ_4:15;
A21: (mid (h,i1,i2)) /. (i + 1) = (mid (h,i1,i2)) . (i + 1) by A10, FINSEQ_4:15, NAT_1:11;
A22: i + i1 <= len h by A4, A12, XXREAL_0:2;
(mid (h,i1,i2)) . i = h . ((i + i1) -' 1) by A1, A2, A3, A4, A6, A9, A19, FINSEQ_6:118;
then LSeg ((mid (h,i1,i2)),i) = LSeg ((h /. ((i + i1) -' 1)),(h /. (((i + 1) + i1) -' 1))) by A1, A11, A16, A20, A21, A15, A17, A22, FINSEQ_4:15, NAT_1:12
.= LSeg (h,((i + i1) -' 1)) by A14, A13, A17, A18, TOPREAL1:def_3 ;
then LSeg ((mid (h,i1,i2)),i) in { (LSeg (h,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len h ) } by A14, A17, A18, A22;
then x in union { (LSeg (h,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len h ) } by A7, A8, TARSKI:def_4;
hence x in L~ h by TOPREAL1:def_4; ::_thesis: verum
end;
caseA23: i1 > i2 ; ::_thesis: x in L~ h
mid (h,i1,i2) = Rev (mid (h,i2,i1)) by Lm6;
then x in L~ (mid (h,i2,i1)) by A5, SPPOL_2:22;
then x in union { (LSeg ((mid (h,i2,i1)),i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len (mid (h,i2,i1)) ) } by TOPREAL1:def_4;
then consider Y being set such that
A24: ( x in Y & Y in { (LSeg ((mid (h,i2,i1)),i)) where i is Element of NAT : ( 1 <= i & i + 1 <= len (mid (h,i2,i1)) ) } ) by TARSKI:def_4;
consider i being Element of NAT such that
A25: Y = LSeg ((mid (h,i2,i1)),i) and
A26: 1 <= i and
A27: i + 1 <= len (mid (h,i2,i1)) by A24;
A28: LSeg ((mid (h,i2,i1)),i) = LSeg (((mid (h,i2,i1)) /. i),((mid (h,i2,i1)) /. (i + 1))) by A26, A27, TOPREAL1:def_3;
len (mid (h,i2,i1)) = (i1 -' i2) + 1 by A1, A2, A3, A4, A23, FINSEQ_6:118;
then (i + 1) - 1 <= ((i1 -' i2) + 1) - 1 by A27, XREAL_1:9;
then i <= i1 - i2 by A23, XREAL_1:233;
then A29: i + i2 <= (i1 - i2) + i2 by XREAL_1:6;
then A30: i + i2 <= len h by A2, XXREAL_0:2;
1 + 1 <= i + i2 by A3, A26, XREAL_1:7;
then A31: (1 + 1) - 1 <= (i + i2) - 1 by XREAL_1:9;
then 1 <= (i + i2) -' 1 by A3, NAT_1:12, XREAL_1:233;
then A32: h /. ((i + i2) -' 1) = h . ((i + i2) -' 1) by A30, FINSEQ_4:15, NAT_D:44;
1 <= i + 1 by NAT_1:11;
then A33: (mid (h,i2,i1)) . (i + 1) = h . (((i + 1) + i2) -' 1) by A1, A2, A3, A4, A23, A27, FINSEQ_6:118;
A34: ((i + 1) + i2) -' 1 = ((i + 1) + i2) - 1 by A3, NAT_1:12, XREAL_1:233
.= i + i2 ;
then A35: ((i + 1) + i2) -' 1 = ((i + i2) - 1) + 1
.= ((i + i2) -' 1) + 1 by A3, NAT_1:12, XREAL_1:233 ;
i <= i + 1 by NAT_1:11;
then A36: i <= len (mid (h,i2,i1)) by A27, XXREAL_0:2;
then A37: (mid (h,i2,i1)) /. i = (mid (h,i2,i1)) . i by A26, FINSEQ_4:15;
A38: (mid (h,i2,i1)) /. (i + 1) = (mid (h,i2,i1)) . (i + 1) by A27, FINSEQ_4:15, NAT_1:11;
A39: i + i2 <= len h by A2, A29, XXREAL_0:2;
(mid (h,i2,i1)) . i = h . ((i + i2) -' 1) by A1, A2, A3, A4, A23, A26, A36, FINSEQ_6:118;
then LSeg ((mid (h,i2,i1)),i) = LSeg ((h /. ((i + i2) -' 1)),(h /. (((i + 1) + i2) -' 1))) by A3, A28, A33, A37, A38, A32, A34, A39, FINSEQ_4:15, NAT_1:12
.= LSeg (h,((i + i2) -' 1)) by A31, A30, A34, A35, TOPREAL1:def_3 ;
then LSeg ((mid (h,i2,i1)),i) in { (LSeg (h,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len h ) } by A31, A34, A35, A39;
then x in union { (LSeg (h,j)) where j is Element of NAT : ( 1 <= j & j + 1 <= len h ) } by A24, A25, TARSKI:def_4;
hence x in L~ h by TOPREAL1:def_4; ::_thesis: verum
end;
end;
end;
hence x in L~ h ; ::_thesis: verum
end;
end;
Lm8: 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 A2: i in dom f ; ::_thesis: ( not j in dom f or len (mid (f,i,j)) >= 1 )
then A3: i <= len f by FINSEQ_3:25;
assume A4: j in dom f ; ::_thesis: len (mid (f,i,j)) >= 1
then A5: 1 <= j by FINSEQ_3:25;
A6: j <= len f by A4, FINSEQ_3:25;
1 <= i by A2, FINSEQ_3:25;
then ( len (mid (f,i,j)) = (i -' j) + 1 or len (mid (f,i,j)) = (j -' i) + 1 ) by A3, A5, A6, A1, FINSEQ_6:118;
hence len (mid (f,i,j)) >= 1 by NAT_1:11; ::_thesis: verum
end;
Lm9: 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 that
A1: i in dom f and
A2: j in dom f ; ::_thesis: not mid (f,i,j) is empty
len (mid (f,i,j)) >= 1 by A1, A2, Lm8;
hence not mid (f,i,j) is empty ; ::_thesis: verum
end;
Lm10: 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 by FINSEQ_3:25;
A3: i <= len f by A1, FINSEQ_3:25;
assume A4: j in dom f ; ::_thesis: (mid (f,i,j)) /. 1 = f /. i
then A5: 1 <= j by FINSEQ_3:25;
A6: j <= len f by A4, FINSEQ_3:25;
not mid (f,i,j) is empty by A1, A4, Lm9;
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, A3, A5, A6, FINSEQ_6:118
.= f /. i by A1, PARTFUN1:def_6 ;
::_thesis: verum
end;
theorem Th42: :: JORDAN3:42
for f being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st p in L~ f holds
L~ (L_Cut (f,p)) c= L~ f
proof
let f be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st p in L~ f holds
L~ (L_Cut (f,p)) c= L~ f
let p be Point of (TOP-REAL 2); ::_thesis: ( p in L~ f implies L~ (L_Cut (f,p)) c= L~ f )
assume A1: p in L~ f ; ::_thesis: L~ (L_Cut (f,p)) c= L~ f
Index (p,f) < len f by A1, Th8;
then A2: (Index (p,f)) + 1 <= len f by NAT_1:13;
A3: 1 <= Index (p,f) by A1, Th8;
then A4: 1 < (Index (p,f)) + 1 by NAT_1:13;
then A5: (Index (p,f)) + 1 in dom f by A2, FINSEQ_3:25;
len f <> 0 by A1, TOPREAL1:22;
then A6: len f >= 0 + 1 by NAT_1:13;
then A7: len f in dom f by FINSEQ_3:25;
percases ( p = f . ((Index (p,f)) + 1) or p <> f . ((Index (p,f)) + 1) ) ;
suppose p = f . ((Index (p,f)) + 1) ; ::_thesis: L~ (L_Cut (f,p)) c= L~ f
then L_Cut (f,p) = mid (f,((Index (p,f)) + 1),(len f)) by Def3;
hence L~ (L_Cut (f,p)) c= L~ f by A6, A4, A2, Lm7; ::_thesis: verum
end;
suppose p <> f . ((Index (p,f)) + 1) ; ::_thesis: L~ (L_Cut (f,p)) c= L~ f
then A8: L_Cut (f,p) = <*p*> ^ (mid (f,((Index (p,f)) + 1),(len f))) by Def3;
A9: f /. ((Index (p,f)) + 1) in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by RLTOPSP1:68;
p in LSeg (f,(Index (p,f))) by A1, Th9;
then A10: p in LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A3, A2, TOPREAL1:def_3;
A11: LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) c= L~ f by A1, A2, Th8, SPPOL_2:16;
(mid (f,((Index (p,f)) + 1),(len f))) /. 1 = f /. ((Index (p,f)) + 1) by A7, A5, Lm10;
then LSeg (p,((mid (f,((Index (p,f)) + 1),(len f))) /. 1)) c= LSeg ((f /. (Index (p,f))),(f /. ((Index (p,f)) + 1))) by A10, A9, TOPREAL1:6;
then A12: LSeg (p,((mid (f,((Index (p,f)) + 1),(len f))) /. 1)) c= L~ f by A11, XBOOLE_1:1;
mid (f,((Index (p,f)) + 1),(len f)) <> {} by A7, A5, Lm8, CARD_1:27;
then A13: L~ (<*p*> ^ (mid (f,((Index (p,f)) + 1),(len f)))) = (LSeg (p,((mid (f,((Index (p,f)) + 1),(len f))) /. 1))) \/ (L~ (mid (f,((Index (p,f)) + 1),(len f)))) by SPPOL_2:20;
L~ (mid (f,((Index (p,f)) + 1),(len f))) c= L~ f by A6, A4, A2, Lm7;
hence L~ (L_Cut (f,p)) c= L~ f by A8, A13, A12, XBOOLE_1:8; ::_thesis: verum
end;
end;
end;
theorem Th43: :: JORDAN3:43
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g)
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g)
let p be Point of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) implies (L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g) )
assume that
A1: f . (len f) = g . 1 and
A2: p in L~ f and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: (L~ f) /\ (L~ g) = {(g . 1)} and
A6: p <> f . (len f) ; ::_thesis: (L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g)
L_Cut (f,p) is_S-Seq_joining p,f /. (len f) by A2, A3, A6, Th33;
then A7: (L_Cut (f,p)) . (len (L_Cut (f,p))) = f /. (len f) by Def2;
A8: len g >= 2 by A4, TOPREAL1:def_8;
then A9: 1 <= len g by XXREAL_0:2;
g /. 1 in LSeg ((g /. 1),(g /. (1 + 1))) by RLTOPSP1:68;
then g /. 1 in LSeg (g,1) by A8, TOPREAL1:def_3;
then g . 1 in LSeg (g,1) by A9, FINSEQ_4:15;
then A10: g . 1 in L~ g by SPPOL_2:17;
L~ (L_Cut (f,p)) c= L~ f by A2, Th42;
then A11: (L~ (L_Cut (f,p))) /\ (L~ g) c= (L~ f) /\ (L~ g) by XBOOLE_1:27;
len f >= 2 by A3, TOPREAL1:def_8;
then A12: 1 <= len f by XXREAL_0:2;
A13: L_Cut (f,p) is being_S-Seq by A2, A3, A6, Th34;
then A14: 1 + 1 <= len (L_Cut (f,p)) by TOPREAL1:def_8;
then A15: (1 + 1) - 1 <= (len (L_Cut (f,p))) - 1 by XREAL_1:9;
A16: 1 <= len (L_Cut (f,p)) by A14, XXREAL_0:2;
then (L_Cut (f,p)) . 1 = (L_Cut (f,p)) /. 1 by FINSEQ_4:15;
then A17: (L_Cut (f,p)) /. 1 = p by A2, Th23;
A18: ((len (L_Cut (f,p))) -' 1) + 1 = len (L_Cut (f,p)) by A14, XREAL_1:235, XXREAL_0:2;
then (L_Cut (f,p)) /. (len (L_Cut (f,p))) in LSeg (((L_Cut (f,p)) /. ((len (L_Cut (f,p))) -' 1)),((L_Cut (f,p)) /. (((len (L_Cut (f,p))) -' 1) + 1))) by RLTOPSP1:68;
then (L_Cut (f,p)) . (len (L_Cut (f,p))) in LSeg (((L_Cut (f,p)) /. ((len (L_Cut (f,p))) -' 1)),((L_Cut (f,p)) /. (((len (L_Cut (f,p))) -' 1) + 1))) by A16, FINSEQ_4:15;
then (L_Cut (f,p)) . (len (L_Cut (f,p))) in LSeg ((L_Cut (f,p)),((len (L_Cut (f,p))) -' 1)) by A15, A18, TOPREAL1:def_3;
then f /. (len f) in L~ (L_Cut (f,p)) by A7, SPPOL_2:17;
then f . (len f) in L~ (L_Cut (f,p)) by A12, FINSEQ_4:15;
then g . 1 in (L~ (L_Cut (f,p))) /\ (L~ g) by A1, A10, XBOOLE_0:def_4;
then {(g . 1)} c= (L~ (L_Cut (f,p))) /\ (L~ g) by ZFMISC_1:31;
then (L~ (L_Cut (f,p))) /\ (L~ g) = {(g . 1)} by A5, A11, XBOOLE_0:def_10;
hence (L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g) by A1, A4, A12, A13, A7, A17, Th39, FINSEQ_4:15; ::_thesis: verum
end;
theorem :: JORDAN3:44
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) holds
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq
let p be Point of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & p in L~ f & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> f . (len f) implies (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq )
assume that
A1: f . (len f) = g . 1 and
A2: p in L~ f and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: (L~ f) /\ (L~ g) = {(g . 1)} and
A6: p <> f . (len f) ; ::_thesis: (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq
(L_Cut (f,p)) ^ (mid (g,2,(len g))) is_S-Seq_joining p,g /. (len g) by A1, A2, A3, A4, A5, A6, Th43;
hence (L_Cut (f,p)) ^ (mid (g,2,(len g))) is being_S-Seq by Def2; ::_thesis: verum
end;
theorem Th45: :: JORDAN3:45
for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds
(mid (f,1,((len f) -' 1))) ^ g is being_S-Seq
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} implies (mid (f,1,((len f) -' 1))) ^ g is being_S-Seq )
assume that
A1: f . (len f) = g . 1 and
A2: f is being_S-Seq and
A3: g is being_S-Seq and
A4: (L~ f) /\ (L~ g) = {(g . 1)} ; ::_thesis: (mid (f,1,((len f) -' 1))) ^ g is being_S-Seq
A5: Rev f is being_S-Seq by A2;
L~ (Rev f) = L~ f by SPPOL_2:22;
then A6: (L~ (Rev g)) /\ (L~ (Rev f)) = {(g . 1)} by A4, SPPOL_2:22;
A7: (Rev f) . 1 = f . (len f) by FINSEQ_5:62;
A8: Rev g is being_S-Seq by A3;
(Rev g) . (len (Rev g)) = (Rev (Rev g)) . 1 by FINSEQ_5:62
.= (Rev f) . 1 by A1, A7 ;
then A9: (Rev g) ^ (mid ((Rev f),2,(len (Rev f)))) is being_S-Seq by A1, A5, A8, A6, A7, Th38;
A10: (len f) -' 1 <= len f by NAT_D:50;
A11: len (Rev f) = len f by FINSEQ_5:def_3;
A12: len f >= 2 by A2, TOPREAL1:def_8;
then A13: (len f) - 1 >= (1 + 1) - 1 by XREAL_1:9;
A14: ((len f) -' 1) + 1 = ((len f) - 1) + 1 by A12, XREAL_1:233, XXREAL_0:2
.= len f ;
A15: ((len f) -' ((len f) -' 1)) + 1 = ((len f) - ((len f) -' 1)) + 1 by NAT_D:50, XREAL_1:233
.= ((len f) - ((len f) - 1)) + 1 by A12, XREAL_1:233, XXREAL_0:2
.= 2 ;
1 <= len f by A12, XXREAL_0:2;
then (Rev g) ^ (Rev (mid (f,1,((len f) -' 1)))) is being_S-Seq by A13, A10, A15, A11, A14, A9, FINSEQ_6:113;
then Rev ((mid (f,1,((len f) -' 1))) ^ g) is being_S-Seq by FINSEQ_5:64;
then Rev (Rev ((mid (f,1,((len f) -' 1))) ^ g)) is being_S-Seq ;
hence (mid (f,1,((len f) -' 1))) ^ g is being_S-Seq ; ::_thesis: verum
end;
theorem Th46: :: JORDAN3:46
for f, g being FinSequence of (TOP-REAL 2) st f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} holds
(mid (f,1,((len f) -' 1))) ^ g is_S-Seq_joining f /. 1,g /. (len g)
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} implies (mid (f,1,((len f) -' 1))) ^ g is_S-Seq_joining f /. 1,g /. (len g) )
assume that
A1: f . (len f) = g . 1 and
A2: f is being_S-Seq and
A3: g is being_S-Seq and
A4: (L~ f) /\ (L~ g) = {(g . 1)} ; ::_thesis: (mid (f,1,((len f) -' 1))) ^ g is_S-Seq_joining f /. 1,g /. (len g)
A5: (len f) -' 1 <= len f by NAT_D:50;
A6: len f >= 2 by A2, TOPREAL1:def_8;
then (1 + 1) - 1 <= (len f) - 1 by XREAL_1:9;
then A7: 1 <= (len f) -' 1 by NAT_D:39;
A8: 1 <= len f by A6, XXREAL_0:2;
then len (mid (f,1,((len f) -' 1))) = (((len f) -' 1) -' 1) + 1 by A5, A7, FINSEQ_6:118
.= (((len f) -' 1) - 1) + 1 by A7, XREAL_1:233
.= (len f) -' 1 ;
then A9: ((mid (f,1,((len f) -' 1))) ^ g) . 1 = (mid (f,1,((len f) -' 1))) . 1 by A7, FINSEQ_1:64
.= f . 1 by A5, A7, FINSEQ_6:123
.= f /. 1 by A8, FINSEQ_4:15 ;
A10: len ((mid (f,1,((len f) -' 1))) ^ g) = (len (mid (f,1,((len f) -' 1)))) + (len g) by FINSEQ_1:22;
A11: len g >= 2 by A3, TOPREAL1:def_8;
then A12: 1 <= len g by XXREAL_0:2;
0 + (len (mid (f,1,((len f) -' 1)))) < (len g) + (len (mid (f,1,((len f) -' 1)))) by A11, XREAL_1:6;
then A13: ((mid (f,1,((len f) -' 1))) ^ g) . (len ((mid (f,1,((len f) -' 1))) ^ g)) = g . ((len ((mid (f,1,((len f) -' 1))) ^ g)) - (len (mid (f,1,((len f) -' 1))))) by A10, FINSEQ_6:108
.= g /. (len g) by A12, A10, FINSEQ_4:15 ;
(mid (f,1,((len f) -' 1))) ^ g is being_S-Seq by A1, A2, A3, A4, Th45;
hence (mid (f,1,((len f) -' 1))) ^ g is_S-Seq_joining f /. 1,g /. (len g) by A9, A13, Def2; ::_thesis: verum
end;
theorem Th47: :: JORDAN3:47
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 holds
(mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 holds
(mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p
let p be Point of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 implies (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p )
assume that
A1: f . (len f) = g . 1 and
A2: p in L~ g and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: (L~ f) /\ (L~ g) = {(g . 1)} and
A6: p <> g . 1 ; ::_thesis: (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p
len g >= 2 by A4, TOPREAL1:def_8;
then A7: 1 <= len g by XXREAL_0:2;
R_Cut (g,p) is_S-Seq_joining g /. 1,p by A2, A4, A6, Th32;
then A8: (R_Cut (g,p)) . 1 = g /. 1 by Def2;
then A9: (R_Cut (g,p)) . 1 = f . (len f) by A1, A7, FINSEQ_4:15;
A10: len f >= 2 by A3, TOPREAL1:def_8;
then A11: 1 <= len f by XXREAL_0:2;
A12: (1 + 1) - 1 <= (len f) - 1 by A10, XREAL_1:9;
A13: ((len f) -' 1) + 1 = len f by A10, XREAL_1:235, XXREAL_0:2;
then f /. (len f) in LSeg ((f /. ((len f) -' 1)),(f /. (((len f) -' 1) + 1))) by RLTOPSP1:68;
then f /. (len f) in LSeg (f,((len f) -' 1)) by A12, A13, TOPREAL1:def_3;
then f . (len f) in LSeg (f,((len f) -' 1)) by A11, FINSEQ_4:15;
then A14: f . (len f) in L~ f by SPPOL_2:17;
A15: R_Cut (g,p) is being_S-Seq by A2, A4, A6, Th35;
then A16: 1 + 1 <= len (R_Cut (g,p)) by TOPREAL1:def_8;
then A17: 1 <= len (R_Cut (g,p)) by XXREAL_0:2;
then (R_Cut (g,p)) . (len (R_Cut (g,p))) = (R_Cut (g,p)) /. (len (R_Cut (g,p))) by FINSEQ_4:15;
then A18: (R_Cut (g,p)) /. (len (R_Cut (g,p))) = p by A2, Th24;
(R_Cut (g,p)) /. 1 in LSeg (((R_Cut (g,p)) /. 1),((R_Cut (g,p)) /. (1 + 1))) by RLTOPSP1:68;
then (R_Cut (g,p)) . 1 in LSeg (((R_Cut (g,p)) /. 1),((R_Cut (g,p)) /. (1 + 1))) by A17, FINSEQ_4:15;
then (R_Cut (g,p)) . 1 in LSeg ((R_Cut (g,p)),1) by A16, TOPREAL1:def_3;
then g /. 1 in L~ (R_Cut (g,p)) by A8, SPPOL_2:17;
then g . 1 in L~ (R_Cut (g,p)) by A7, FINSEQ_4:15;
then f . (len f) in (L~ f) /\ (L~ (R_Cut (g,p))) by A1, A14, XBOOLE_0:def_4;
then A19: {(f . (len f))} c= (L~ f) /\ (L~ (R_Cut (g,p))) by ZFMISC_1:31;
L~ (R_Cut (g,p)) c= L~ g by A2, Th41;
then (L~ f) /\ (L~ (R_Cut (g,p))) c= (L~ f) /\ (L~ g) by XBOOLE_1:27;
then (L~ f) /\ (L~ (R_Cut (g,p))) = {((R_Cut (g,p)) . 1)} by A1, A5, A9, A19, XBOOLE_0:def_10;
hence (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p by A3, A15, A9, A18, Th46; ::_thesis: verum
end;
theorem :: JORDAN3:48
for f, g being FinSequence of (TOP-REAL 2)
for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 holds
(mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is being_S-Seq
proof
let f, g be FinSequence of (TOP-REAL 2); ::_thesis: for p being Point of (TOP-REAL 2) st f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 holds
(mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is being_S-Seq
let p be Point of (TOP-REAL 2); ::_thesis: ( f . (len f) = g . 1 & p in L~ g & f is being_S-Seq & g is being_S-Seq & (L~ f) /\ (L~ g) = {(g . 1)} & p <> g . 1 implies (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is being_S-Seq )
assume that
A1: f . (len f) = g . 1 and
A2: p in L~ g and
A3: f is being_S-Seq and
A4: g is being_S-Seq and
A5: (L~ f) /\ (L~ g) = {(g . 1)} and
A6: p <> g . 1 ; ::_thesis: (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is being_S-Seq
(mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is_S-Seq_joining f /. 1,p by A1, A2, A3, A4, A5, A6, Th47;
hence (mid (f,1,((len f) -' 1))) ^ (R_Cut (g,p)) is being_S-Seq by Def2; ::_thesis: verum
end;