Article TOPREAL8, MML version 4.99.1005
:: TOPREAL8:th 1
theorem
for b1, b2, b3 being set
st b1 c= {b2,b3} & b2 in b1 & not b3 in b1
holds b1 = {b2};
:: TOPREAL8:exreg 1
registration
cluster Relation-like Function-like trivial set;
end;
:: TOPREAL8:exreg 2
registration
cluster Relation-like Function-like non constant finite FinSequence-like set;
end;
:: TOPREAL8:th 2
theorem
for b1 being Relation-like Function-like FinSequence-like non trivial set holds
1 < len b1;
:: TOPREAL8:th 3
theorem
for b1 being non trivial set
for b2 being non constant circular FinSequence of b1 holds
2 < len b2;
:: TOPREAL8:th 4
theorem
for b1 being Relation-like Function-like FinSequence-like set
for b2 being set
st not b2 in proj2 b1
holds b2 .. b1 = 0;
:: TOPREAL8:th 5
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty FinSequence of b2
for b4 being FinSequence of b2
st b1 .. b3 = len b3
holds (b3 ^ b4) |-- b1 = b4;
:: TOPREAL8:th 6
theorem
for b1 being non empty set
for b2 being one-to-one non empty FinSequence of b1 holds
(b2 /. len b2) .. b2 = len b2;
:: TOPREAL8:th 7
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
len b1 <= len (b1 ^' b2);
:: TOPREAL8:th 8
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set
for b3 being set
st b3 in proj2 b1
holds b3 .. b1 = b3 .. (b1 ^' b2);
:: TOPREAL8:th 9
theorem
for b1 being Relation-like Function-like non empty FinSequence-like set
for b2 being Relation-like Function-like FinSequence-like set holds
len b2 <= len (b1 ^' b2);
:: TOPREAL8:th 10
theorem
for b1, b2 being Relation-like Function-like FinSequence-like set holds
proj2 b1 c= proj2 (b1 ^' b2);
:: TOPREAL8:th 11
theorem
for b1 being non empty set
for b2 being non empty FinSequence of b1
for b3 being non trivial FinSequence of b1
st b3 /. len b3 = b2 /. 1
holds b2 ^' b3 is circular(b1);
:: TOPREAL8:th 12
theorem
for b1 being non empty set
for b2 being tabular FinSequence of b1 *
for b3 being FinSequence of b1
for b4 being non empty FinSequence of b1
st b3 /. len b3 = b4 /. 1 & b3 is_sequence_on b2 & b4 is_sequence_on b2
holds b3 ^' b4 is_sequence_on b2;
:: TOPREAL8:th 13
theorem
for b1 being Element of NAT
for b2 being set
for b3 being FinSequence of b2
st 1 <= b1
holds (b1 + 1,len b3)-cut b3 = b3 /^ b1;
:: TOPREAL8:th 14
theorem
for b1 being Element of NAT
for b2 being set
for b3 being FinSequence of b2
st b1 <= len b3
holds (1,b1)-cut b3 = b3 | b1;
:: TOPREAL8:th 15
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty FinSequence of b2
for b4 being FinSequence of b2
st b1 .. b3 = len b3
holds (b3 ^ b4) -| b1 = (1,(len b3) -' 1)-cut b3;
:: TOPREAL8:th 16
theorem
for b1 being non empty set
for b2, b3 being non empty FinSequence of b1
st (b3 /. 1) .. b2 = len b2
holds (b2 ^' b3) :- (b3 /. 1) = b3;
:: TOPREAL8:th 17
theorem
for b1 being non empty set
for b2, b3 being non empty FinSequence of b1
st (b3 /. 1) .. b2 = len b2
holds (b2 ^' b3) -: (b3 /. 1) = b2;
:: TOPREAL8:th 18
theorem
for b1 being non trivial set
for b2 being non empty FinSequence of b1
for b3 being non trivial FinSequence of b1
st b3 /. 1 = b2 /. len b2 &
(for b4 being Element of NAT
st 1 <= b4 & b4 < len b2
holds b2 /. b4 <> b3 /. 1)
holds Rotate(b2 ^' b3,b3 /. 1) = b3 ^' b2;
:: TOPREAL8:th 19
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
LSeg(b1,1) = L~ (b1 | 2);
:: TOPREAL8:th 20
theorem
for b1 being s.c.c. FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st b2 < len b1
holds b1 | b2 is s.n.c.;
:: TOPREAL8:th 21
theorem
for b1 being s.c.c. FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2
holds b1 /^ b2 is s.n.c.;
:: TOPREAL8:th 22
theorem
for b1 being circular s.c.c. FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st b2 < len b1 & 4 < len b1
holds b1 | b2 is one-to-one;
:: TOPREAL8:th 23
theorem
for b1 being circular s.c.c. FinSequence of the carrier of TOP-REAL 2
st 4 < len b1
for b2, b3 being Element of NAT
st 1 < b2 & b2 < b3 & b3 <= len b1
holds b1 /. b2 <> b1 /. b3;
:: TOPREAL8:th 24
theorem
for b1 being circular s.c.c. FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 & 4 < len b1
holds b1 /^ b2 is one-to-one;
:: TOPREAL8:th 25
theorem
for b1, b2 being Element of NAT
for b3 being non empty special FinSequence of the carrier of TOP-REAL 2 holds
(b1,b2)-cut b3 is special;
:: TOPREAL8:th 26
theorem
for b1 being non empty special FinSequence of the carrier of TOP-REAL 2
for b2 being non trivial special FinSequence of the carrier of TOP-REAL 2
st b1 /. len b1 = b2 /. 1
holds b1 ^' b2 is special;
:: TOPREAL8:th 27
theorem
for b1 being circular unfolded s.c.c. FinSequence of the carrier of TOP-REAL 2
st 4 < len b1
holds (LSeg(b1,1)) /\ L~ (b1 /^ 1) = {b1 /. 1,b1 /. 2};
:: TOPREAL8:th 28
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
st b1 < len b2
holds LSeg(b2 ^' b3,b1) = LSeg(b2,b1);
:: TOPREAL8:th 29
theorem
for b1 being Element of NAT
for b2, b3 being non empty FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 + 1 < len b3
holds LSeg(b2 ^' b3,(len b2) + b1) = LSeg(b3,b1 + 1);
:: TOPREAL8:th 30
theorem
for b1 being non empty FinSequence of the carrier of TOP-REAL 2
for b2 being non trivial FinSequence of the carrier of TOP-REAL 2
st b1 /. len b1 = b2 /. 1
holds LSeg(b1 ^' b2,len b1) = LSeg(b2,1);
:: TOPREAL8:th 31
theorem
for b1 being Element of NAT
for b2 being non empty FinSequence of the carrier of TOP-REAL 2
for b3 being non trivial FinSequence of the carrier of TOP-REAL 2
st b1 + 1 < len b3 & b2 /. len b2 = b3 /. 1
holds LSeg(b2 ^' b3,(len b2) + b1) = LSeg(b3,b1 + 1);
:: TOPREAL8:th 32
theorem
for b1 being non empty unfolded s.n.c. FinSequence of the carrier of TOP-REAL 2
for b2 being Element of NAT
st 1 <= b2 & b2 < len b1
holds (LSeg(b1,b2)) /\ rng b1 = {b1 /. b2,b1 /. (b2 + 1)};
:: TOPREAL8:th 33
theorem
for b1, b2 being one-to-one non trivial unfolded s.n.c. FinSequence of the carrier of TOP-REAL 2
st (L~ b1) /\ L~ b2 = {b1 /. 1,b2 /. 1} &
b1 /. 1 = b2 /. len b2 &
b2 /. 1 = b1 /. len b1
holds b1 ^' b2 is s.c.c.;
:: TOPREAL8:th 34
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 is unfolded &
b2 is unfolded &
b1 /. len b1 = b2 /. 1 &
(LSeg(b1,(len b1) -' 1)) /\ LSeg(b2,1) = {b1 /. len b1}
holds b1 ^' b2 is unfolded;
:: TOPREAL8:th 35
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 is not empty & b2 is not trivial & b1 /. len b1 = b2 /. 1
holds L~ (b1 ^' b2) = (L~ b1) \/ L~ b2;
:: TOPREAL8:th 36
theorem
for b1 being non empty-yielding tabular X_equal-in-line Y_equal-in-column Y_increasing-in-line X_increasing-in-column FinSequence of (the carrier of TOP-REAL 2) *
for b2 being FinSequence of the carrier of TOP-REAL 2
st (for b3 being natural set
st b3 in dom b2
holds ex b4, b5 being natural set st
[b4,b5] in Indices b1 & b2 /. b3 = b1 *(b4,b5)) &
b2 is not constant &
b2 is circular(the carrier of TOP-REAL 2) &
b2 is unfolded &
b2 is s.c.c. &
b2 is special &
4 < len b2
holds ex b3 being FinSequence of the carrier of TOP-REAL 2 st
b3 is_sequence_on b1 & b3 is unfolded & b3 is s.c.c. & b3 is special & L~ b2 = L~ b3 & b2 /. 1 = b3 /. 1 & b2 /. len b2 = b3 /. len b3 & len b2 <= len b3;