Article JORDAN3, MML version 4.99.1005
:: JORDAN3:th 15
theorem
for b1 being natural set
for b2, b3 being Relation-like Function-like FinSequence-like set
st len b2 < b1 &
(b1 <= (len b2) + len b3 or b1 <= len (b2 ^ b3))
holds (b2 ^ b3) . b1 = b3 . (b1 - len b2);
:: JORDAN3:th 17
theorem
for b1 being non empty set
for b2 being set
for b3 being FinSequence of b1
st 1 <= len b3
holds (b3 ^ <*b2*>) . 1 = b3 . 1 &
(b3 ^ <*b2*>) . 1 = b3 /. 1 &
(<*b2*> ^ b3) . ((len b3) + 1) = b3 . len b3 &
(<*b2*> ^ b3) . ((len b3) + 1) = b3 /. len b3;
:: JORDAN3:th 18
theorem
for b1 being Relation-like Function-like FinSequence-like set
st len b1 = 1
holds Rev b1 = b1;
:: JORDAN3:th 19
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being natural set holds
len (b2 /^ b3) = (len b2) -' b3;
:: JORDAN3:th 21
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being natural set holds
(b2 /^ b3) | (b4 -' b3) = (b2 | b4) /^ b3;
:: JORDAN3:funcnot 1 => JORDAN3:func 1
definition
let a1 be non empty set;
let a2 be FinSequence of a1;
let a3, a4 be natural set;
func mid(A2,A3,A4) -> FinSequence of a1 equals
(a2 /^ (a3 -' 1)) | ((a4 -' a3) + 1)
if a3 <= a4
otherwise Rev ((a2 /^ (a4 -' 1)) | ((a3 -' a4) + 1));
end;
:: JORDAN3:def 1
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being natural set holds
(b3 <= b4 implies mid(b2,b3,b4) = (b2 /^ (b3 -' 1)) | ((b4 -' b3) + 1)) &
(b3 <= b4 or mid(b2,b3,b4) = Rev ((b2 /^ (b4 -' 1)) | ((b3 -' b4) + 1)));
:: JORDAN3:th 22
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st 1 <= b3 & b3 <= len b2 & 1 <= b4 & b4 <= len b2
holds Rev mid(b2,b3,b4) = mid(Rev b2,((len b2) -' b4) + 1,((len b2) -' b3) + 1);
:: JORDAN3:th 23
theorem
for b1 being non empty set
for b2, b3 being Element of NAT
for b4 being FinSequence of b1
st 1 <= b3 & b3 + b2 <= len b4
holds (b4 /^ b2) . b3 = b4 . (b3 + b2);
:: JORDAN3:th 24
theorem
for b1 being non empty set
for b2 being Element of NAT
for b3 being FinSequence of b1
st 1 <= b2 & b2 <= len b3
holds (Rev b3) . b2 = b3 . (((len b3) - b2) + 1);
:: JORDAN3:th 25
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
st 1 <= b3
holds mid(b2,1,b3) = b2 | b3;
:: JORDAN3:th 26
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
st b3 <= len b2
holds mid(b2,b3,len b2) = b2 /^ (b3 -' 1);
:: JORDAN3:th 27
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st 1 <= b3 & b3 <= len b2 & 1 <= b4 & b4 <= len b2
holds (mid(b2,b3,b4)) . 1 = b2 . b3 &
(b3 <= b4 implies len mid(b2,b3,b4) = (b4 -' b3) + 1 &
(for b5 being Element of NAT
st 1 <= b5 & b5 <= len mid(b2,b3,b4)
holds (mid(b2,b3,b4)) . b5 = b2 . ((b5 + b3) -' 1))) &
(b3 <= b4 or len mid(b2,b3,b4) = (b3 -' b4) + 1 &
(for b5 being Element of NAT
st 1 <= b5 & b5 <= len mid(b2,b3,b4)
holds (mid(b2,b3,b4)) . b5 = b2 . ((b3 -' b5) + 1)));
:: JORDAN3:th 28
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT holds
proj2 mid(b2,b3,b4) c= proj2 b2;
:: JORDAN3:th 29
theorem
for b1 being non empty set
for b2 being FinSequence of b1
st 1 <= len b2
holds mid(b2,1,len b2) = b2;
:: JORDAN3:th 30
theorem
for b1 being non empty set
for b2 being FinSequence of b1
st 1 <= len b2
holds mid(b2,len b2,1) = Rev b2;
:: JORDAN3:th 31
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of NAT
st 1 <= b3 &
b3 <= b4 &
b4 <= len b2 &
1 <= b5 &
((b4 -' b3) + 1 < b5 & (b4 - b3) + 1 < b5 implies b5 <= (b4 + 1) - b3)
holds (mid(b2,b3,b4)) . b5 = b2 . ((b5 + b3) -' 1) &
(mid(b2,b3,b4)) . b5 = b2 . ((b5 -' 1) + b3) &
(mid(b2,b3,b4)) . b5 = b2 . ((b5 + b3) - 1) &
(mid(b2,b3,b4)) . b5 = b2 . ((b5 - 1) + b3);
:: JORDAN3:th 32
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st 1 <= b4 & b4 <= b3 & b3 <= len b2
holds (mid(b2,1,b3)) . b4 = b2 . b4;
:: JORDAN3:th 33
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st 1 <= b3 & b3 <= b4 & b4 <= len b2
holds len mid(b2,b3,b4) <= len b2;
:: JORDAN3:th 34
theorem
for b1 being Element of NAT
for b2 being FinSequence of the carrier of TOP-REAL b1
st 2 <= len b2
holds b2 . 1 in L~ b2 & b2 /. 1 in L~ b2 & b2 . len b2 in L~ b2 & b2 /. len b2 in L~ b2;
:: JORDAN3:th 35
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st (b1 `1 = b2 `1 or b1 `2 = b2 `2) &
b3 in LSeg(b1,b2) &
b4 in LSeg(b1,b2) &
b3 `1 <> b4 `1
holds b3 `2 = b4 `2;
:: JORDAN3:th 36
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st (b1 `1 = b2 `1 or b1 `2 = b2 `2) &
LSeg(b3,b4) c= LSeg(b1,b2) &
b3 `1 <> b4 `1
holds b3 `2 = b4 `2;
:: JORDAN3:th 37
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 2 <= b2 & b1 is being_S-Seq
holds b1 | b2 is being_S-Seq;
:: JORDAN3:th 38
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st b2 <= len b1 & 2 <= (len b1) -' b2 & b1 is being_S-Seq
holds b1 /^ b2 is being_S-Seq;
:: JORDAN3:th 39
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b1 is being_S-Seq & 1 <= b2 & b2 <= len b1 & 1 <= b3 & b3 <= len b1 & b2 <> b3
holds mid(b1,b2,b3) is being_S-Seq;
:: JORDAN3:funcnot 2 => JORDAN3:func 2
definition
let a1 be FinSequence of the carrier of TOP-REAL 2;
let a2 be Element of the carrier of TOP-REAL 2;
assume a2 in L~ a1;
func Index(A2,A1) -> Element of NAT means
ex b1 being non empty Element of bool NAT st
it = min b1 &
b1 = {b2 where b2 is Element of NAT: a2 in LSeg(a1,b2)};
end;
:: JORDAN3:def 2
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
for b3 being Element of NAT holds
b3 = Index(b2,b1)
iff
ex b4 being non empty Element of bool NAT st
b3 = min b4 &
b4 = {b5 where b5 is Element of NAT: b2 in LSeg(b1,b5)};
:: JORDAN3:th 40
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b2 in LSeg(b1,b3)
holds Index(b2,b1) <= b3;
:: JORDAN3:th 41
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds 1 <= Index(b2,b1) & Index(b2,b1) < len b1;
:: JORDAN3:th 42
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds b2 in LSeg(b1,Index(b2,b1));
:: JORDAN3:th 43
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in LSeg(b1,1)
holds Index(b2,b1) = 1;
:: JORDAN3:th 44
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st 2 <= len b1
holds Index(b1 /. 1,b1) = 1;
:: JORDAN3:th 45
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being natural set
st b1 is being_S-Seq & 1 < b3 & b3 <= len b1 & b2 = b1 . b3
holds (Index(b2,b1)) + 1 = b3;
:: JORDAN3:th 46
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 is s.n.c. & b2 in LSeg(b1,b3) & b3 <> Index(b2,b1)
holds b3 = (Index(b2,b1)) + 1;
:: JORDAN3:th 47
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
for b3 being Element of NAT
st b1 is unfolded & b1 is s.n.c. & b3 + 1 <= len b1 & b2 in LSeg(b1,b3) & b2 <> b1 . b3
holds b3 = Index(b2,b1);
:: JORDAN3:prednot 1 => JORDAN3:pred 1
definition
let a1 be FinSequence of the carrier of TOP-REAL 2;
let a2, a3 be Element of the carrier of TOP-REAL 2;
pred A1 is_S-Seq_joining A2,A3 means
a1 is being_S-Seq & a1 . 1 = a2 & a1 . len a1 = a3;
end;
:: JORDAN3:dfs 3
definiens
let a1 be FinSequence of the carrier of TOP-REAL 2;
let a2, a3 be Element of the carrier of TOP-REAL 2;
To prove
a1 is_S-Seq_joining a2,a3
it is sufficient to prove
thus a1 is being_S-Seq & a1 . 1 = a2 & a1 . len a1 = a3;
:: JORDAN3:def 3
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
b1 is_S-Seq_joining b2,b3
iff
b1 is being_S-Seq & b1 . 1 = b2 & b1 . len b1 = b3;
:: JORDAN3:th 48
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b1 is_S-Seq_joining b2,b3
holds Rev b1 is_S-Seq_joining b3,b2;
:: JORDAN3:th 49
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4 being natural set
st b3 in L~ b1 &
b2 = <*b3*> ^ mid(b1,(Index(b3,b1)) + 1,len b1) &
1 <= b4 &
b4 + 1 <= len b2
holds LSeg(b2,b4) c= LSeg(b1,((Index(b3,b1)) + b4) -' 1);
:: JORDAN3:th 50
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq &
b3 in L~ b1 &
b3 <> b1 . ((Index(b3,b1)) + 1) &
b2 = <*b3*> ^ mid(b1,(Index(b3,b1)) + 1,len b1)
holds b2 is_S-Seq_joining b3,b1 /. len b1;
:: JORDAN3:th 51
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
for b4 being natural set
st b3 in L~ b1 &
1 <= b4 &
b4 + 1 <= len b2 &
b2 = (mid(b1,1,Index(b3,b1))) ^ <*b3*>
holds LSeg(b2,b4) c= LSeg(b1,b4);
:: JORDAN3:th 52
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq &
b3 in L~ b1 &
b3 <> b1 . 1 &
b2 = (mid(b1,1,Index(b3,b1))) ^ <*b3*>
holds b2 is_S-Seq_joining b1 /. 1,b3;
:: JORDAN3:funcnot 3 => JORDAN3:func 3
definition
let a1 be FinSequence of the carrier of TOP-REAL 2;
let a2 be Element of the carrier of TOP-REAL 2;
func L_Cut(A1,A2) -> FinSequence of the carrier of TOP-REAL 2 equals
<*a2*> ^ mid(a1,(Index(a2,a1)) + 1,len a1)
if a2 <> a1 . ((Index(a2,a1)) + 1)
otherwise mid(a1,(Index(a2,a1)) + 1,len a1);
end;
:: JORDAN3:def 4
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
(b2 = b1 . ((Index(b2,b1)) + 1) or L_Cut(b1,b2) = <*b2*> ^ mid(b1,(Index(b2,b1)) + 1,len b1)) &
(b2 = b1 . ((Index(b2,b1)) + 1) implies L_Cut(b1,b2) = mid(b1,(Index(b2,b1)) + 1,len b1));
:: JORDAN3:funcnot 4 => JORDAN3:func 4
definition
let a1 be FinSequence of the carrier of TOP-REAL 2;
let a2 be Element of the carrier of TOP-REAL 2;
func R_Cut(A1,A2) -> FinSequence of the carrier of TOP-REAL 2 equals
(mid(a1,1,Index(a2,a1))) ^ <*a2*>
if a2 <> a1 . 1
otherwise <*a2*>;
end;
:: JORDAN3:def 5
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2 holds
(b2 = b1 . 1 or R_Cut(b1,b2) = (mid(b1,1,Index(b2,b1))) ^ <*b2*>) &
(b2 = b1 . 1 implies R_Cut(b1,b2) = <*b2*>);
:: JORDAN3:th 53
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b2 = b1 . ((Index(b2,b1)) + 1) & b2 <> b1 . len b1
holds ((Index(b2,Rev b1)) + Index(b2,b1)) + 1 = len b1;
:: JORDAN3:th 54
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is unfolded & b1 is s.n.c. & b2 in L~ b1 & b2 <> b1 . ((Index(b2,b1)) + 1)
holds (Index(b2,Rev b1)) + Index(b2,b1) = len b1;
:: JORDAN3:th 55
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3 being Element of NAT
for b4 being Element of b1 holds
(<*b4*> ^ b2) | (b3 + 1) = <*b4*> ^ (b2 | b3);
:: JORDAN3:th 56
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
st b3 < b4 & b3 in dom b2
holds mid(b2,b3,b4) = <*b2 . b3*> ^ mid(b2,b3 + 1,b4);
:: JORDAN3:th 57
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1
holds L_Cut(Rev b1,b2) = Rev R_Cut(b1,b2);
:: JORDAN3:th 58
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds (L_Cut(b1,b2)) . 1 = b2 &
(for b3 being Element of NAT
st 1 < b3 & b3 <= len L_Cut(b1,b2)
holds (b2 = b1 . ((Index(b2,b1)) + 1) implies (L_Cut(b1,b2)) . b3 = b1 . ((Index(b2,b1)) + b3)) &
(b2 = b1 . ((Index(b2,b1)) + 1) or (L_Cut(b1,b2)) . b3 = b1 . (((Index(b2,b1)) + b3) - 1)));
:: JORDAN3:th 59
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds (R_Cut(b1,b2)) . len R_Cut(b1,b2) = b2 &
(for b3 being Element of NAT
st 1 <= b3 & b3 <= Index(b2,b1)
holds (R_Cut(b1,b2)) . b3 = b1 . b3);
:: JORDAN3:th 60
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds (b2 = b1 . 1 or len R_Cut(b1,b2) = (Index(b2,b1)) + 1) &
(b2 = b1 . 1 implies len R_Cut(b1,b2) = Index(b2,b1));
:: JORDAN3:th 61
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds (b2 = b1 . ((Index(b2,b1)) + 1) implies len L_Cut(b1,b2) = (len b1) - Index(b2,b1)) &
(b2 = b1 . ((Index(b2,b1)) + 1) or len L_Cut(b1,b2) = ((len b1) - Index(b2,b1)) + 1);
:: JORDAN3:prednot 2 => JORDAN3:pred 2
definition
let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
pred LE A3,A4,A1,A2 means
a3 in LSeg(a1,a2) &
a4 in LSeg(a1,a2) &
(for b1, b2 being Element of REAL
st 0 <= b1 &
b1 <= 1 &
a3 = ((1 - b1) * a1) + (b1 * a2) &
0 <= b2 &
b2 <= 1 &
a4 = ((1 - b2) * a1) + (b2 * a2)
holds b1 <= b2);
end;
:: JORDAN3:dfs 6
definiens
let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
To prove
LE a3,a4,a1,a2
it is sufficient to prove
thus a3 in LSeg(a1,a2) &
a4 in LSeg(a1,a2) &
(for b1, b2 being Element of REAL
st 0 <= b1 &
b1 <= 1 &
a3 = ((1 - b1) * a1) + (b1 * a2) &
0 <= b2 &
b2 <= 1 &
a4 = ((1 - b2) * a1) + (b2 * a2)
holds b1 <= b2);
:: JORDAN3:def 6
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
LE b3,b4,b1,b2
iff
b3 in LSeg(b1,b2) &
b4 in LSeg(b1,b2) &
(for b5, b6 being Element of REAL
st 0 <= b5 &
b5 <= 1 &
b3 = ((1 - b5) * b1) + (b5 * b2) &
0 <= b6 &
b6 <= 1 &
b4 = ((1 - b6) * b1) + (b6 * b2)
holds b5 <= b6);
:: JORDAN3:prednot 3 => JORDAN3:pred 3
definition
let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
pred LT A3,A4,A1,A2 means
LE a3,a4,a1,a2 & a3 <> a4;
end;
:: JORDAN3:dfs 7
definiens
let a1, a2, a3, a4 be Element of the carrier of TOP-REAL 2;
To prove
LT a3,a4,a1,a2
it is sufficient to prove
thus LE a3,a4,a1,a2 & a3 <> a4;
:: JORDAN3:def 7
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2 holds
LT b3,b4,b1,b2
iff
LE b3,b4,b1,b2 & b3 <> b4;
:: JORDAN3:th 62
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st LE b3,b4,b1,b2 & LE b4,b3,b1,b2
holds b3 = b4;
:: JORDAN3:th 63
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st b3 in LSeg(b1,b2) & b4 in LSeg(b1,b2) & b1 <> b2
holds (LE b3,b4,b1,b2 or LT b4,b3,b1,b2) & (LE b3,b4,b1,b2 implies not LT b4,b3,b1,b2);
:: JORDAN3:th 64
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1 & b3 in L~ b1 & Index(b2,b1) < Index(b3,b1)
holds b3 in L~ L_Cut(b1,b2);
:: JORDAN3:th 65
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st LE b1,b2,b3,b4
holds b2 in LSeg(b1,b4) & b1 in LSeg(b3,b2);
:: JORDAN3:th 66
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1 &
b3 in L~ b1 &
b2 <> b3 &
Index(b2,b1) = Index(b3,b1) &
LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)
holds b3 in L~ L_Cut(b1,b2);
:: JORDAN3:funcnot 5 => JORDAN3:func 5
definition
let a1 be FinSequence of the carrier of TOP-REAL 2;
let a2, a3 be Element of the carrier of TOP-REAL 2;
func B_Cut(A1,A2,A3) -> FinSequence of the carrier of TOP-REAL 2 equals
R_Cut(L_Cut(a1,a2),a3)
if (a2 in L~ a1 & a3 in L~ a1 & Index(a2,a1) < Index(a3,a1) or Index(a2,a1) = Index(a3,a1) &
LE a2,a3,a1 /. Index(a2,a1),a1 /. ((Index(a2,a1)) + 1))
otherwise Rev R_Cut(L_Cut(a1,a3),a2);
end;
:: JORDAN3:def 8
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2 holds
((b2 in L~ b1 & b3 in L~ b1 implies Index(b3,b1) <= Index(b2,b1)) &
(Index(b2,b1) = Index(b3,b1) implies not LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)) or B_Cut(b1,b2,b3) = R_Cut(L_Cut(b1,b2),b3)) &
((b2 in L~ b1 & b3 in L~ b1 implies Index(b3,b1) <= Index(b2,b1)) &
(Index(b2,b1) = Index(b3,b1) implies not LE b2,b3,b1 /. Index(b2,b1),b1 /. ((Index(b2,b1)) + 1)) implies B_Cut(b1,b2,b3) = Rev R_Cut(L_Cut(b1,b3),b2));
:: JORDAN3:th 67
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . 1
holds R_Cut(b1,b2) is_S-Seq_joining b1 /. 1,b2;
:: JORDAN3:th 68
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . len b1
holds L_Cut(b1,b2) is_S-Seq_joining b2,b1 /. len b1;
:: JORDAN3:th 69
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . len b1
holds L_Cut(b1,b2) is being_S-Seq;
:: JORDAN3:th 70
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b2 <> b1 . 1
holds R_Cut(b1,b2) is being_S-Seq;
:: JORDAN3:th 71
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b3 in L~ b1 & b2 <> b3
holds B_Cut(b1,b2,b3) is_S-Seq_joining b2,b3;
:: JORDAN3:th 72
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st b1 is being_S-Seq & b2 in L~ b1 & b3 in L~ b1 & b2 <> b3
holds B_Cut(b1,b2,b3) is being_S-Seq;
:: JORDAN3:th 73
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1}
holds b1 ^ mid(b2,2,len b2) is being_S-Seq;
:: JORDAN3:th 74
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1}
holds b1 ^ mid(b2,2,len b2) is_S-Seq_joining b1 /. 1,b2 /. len b2;
:: JORDAN3:th 75
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT holds
L~ (b1 /^ b2) c= L~ b1;
:: JORDAN3:th 76
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds L~ R_Cut(b1,b2) c= L~ b1;
:: JORDAN3:th 77
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1
holds L~ L_Cut(b1,b2) c= L~ b1;
:: JORDAN3:th 78
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b3 in L~ b1 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1} &
b3 <> b1 . len b1
holds (L_Cut(b1,b3)) ^ mid(b2,2,len b2) is_S-Seq_joining b3,b2 /. len b2;
:: JORDAN3:th 79
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b3 in L~ b1 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1} &
b3 <> b1 . len b1
holds (L_Cut(b1,b3)) ^ mid(b2,2,len b2) is being_S-Seq;
:: JORDAN3:th 80
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1}
holds (mid(b1,1,(len b1) -' 1)) ^ b2 is being_S-Seq;
:: JORDAN3:th 81
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1}
holds (mid(b1,1,(len b1) -' 1)) ^ b2 is_S-Seq_joining b1 /. 1,b2 /. len b2;
:: JORDAN3:th 82
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b3 in L~ b2 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1} &
b3 <> b2 . 1
holds (mid(b1,1,(len b1) -' 1)) ^ R_Cut(b2,b3) is_S-Seq_joining b1 /. 1,b3;
:: JORDAN3:th 83
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st b1 . len b1 = b2 . 1 &
b3 in L~ b2 &
b1 is being_S-Seq &
b2 is being_S-Seq &
(L~ b1) /\ L~ b2 = {b2 . 1} &
b3 <> b2 . 1
holds (mid(b1,1,(len b1) -' 1)) ^ R_Cut(b2,b3) is being_S-Seq;