Article SPRECT_2, MML version 4.99.1005
:: SPRECT_2:th 1
theorem
for b1, b2, b3, b4 being set
st b1 /\ b2 c= {b4} & b4 in b3 & b3 misses b2
holds b1 \/ b3 misses b2;
:: SPRECT_2:th 2
theorem
for b1, b2, b3, b4 being set
st b1 /\ b3 = {b4} & b4 in b2 & b2 c= b3
holds b1 /\ b2 = {b4};
:: SPRECT_2:th 4
theorem
for b1, b2 being set
st for b3, b4 being set
st b3 in b1 & b4 in b2
holds b3 misses b4
holds union b1 misses union b2;
:: SPRECT_2:th 5
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty set
for b5 being FinSequence of b4
st b1 <= b2 & b1 in dom b5 & b2 in dom b5 & b3 in dom mid(b5,b1,b2)
holds (b3 + b1) -' 1 in dom b5;
:: SPRECT_2:th 6
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty set
for b5 being FinSequence of b4
st b2 < b1 & b1 in dom b5 & b2 in dom b5 & b3 in dom mid(b5,b1,b2)
holds (b1 -' b3) + 1 in dom b5;
:: SPRECT_2:th 7
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty set
for b5 being FinSequence of b4
st b1 <= b2 & b1 in dom b5 & b2 in dom b5 & b3 in dom mid(b5,b1,b2)
holds (mid(b5,b1,b2)) /. b3 = b5 /. ((b3 + b1) -' 1);
:: SPRECT_2:th 8
theorem
for b1, b2, b3 being Element of NAT
for b4 being non empty set
for b5 being FinSequence of b4
st b2 < b1 & b1 in dom b5 & b2 in dom b5 & b3 in dom mid(b5,b1,b2)
holds (mid(b5,b1,b2)) /. b3 = b5 /. ((b1 -' b3) + 1);
:: SPRECT_2:th 9
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being FinSequence of b3
st b1 in dom b4 & b2 in dom b4
holds 1 <= len mid(b4,b1,b2);
:: SPRECT_2:th 10
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being FinSequence of b3
st b1 in dom b4 & b2 in dom b4 & len mid(b4,b1,b2) = 1
holds b1 = b2;
:: SPRECT_2:th 11
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being FinSequence of b3
st b1 in dom b4 & b2 in dom b4
holds mid(b4,b1,b2) is not empty;
:: SPRECT_2:th 12
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being FinSequence of b3
st b1 in dom b4 & b2 in dom b4
holds (mid(b4,b1,b2)) /. 1 = b4 /. b1;
:: SPRECT_2:th 13
theorem
for b1, b2 being Element of NAT
for b3 being non empty set
for b4 being FinSequence of b3
st b1 in dom b4 & b2 in dom b4
holds (mid(b4,b1,b2)) /. len mid(b4,b1,b2) = b4 /. b2;
:: SPRECT_2:th 14
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in b1 & b2 `2 = N-bound b1
holds b2 in N-most b1;
:: SPRECT_2:th 15
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in b1 & b2 `2 = S-bound b1
holds b2 in S-most b1;
:: SPRECT_2:th 16
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in b1 & b2 `1 = W-bound b1
holds b2 in W-most b1;
:: SPRECT_2:th 17
theorem
for b1 being compact Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in b1 & b2 `1 = E-bound b1
holds b2 in E-most b1;
:: SPRECT_2:th 18
theorem
for b1, b2 being Element of NAT
for b3 being FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 <= b2 & b2 <= len b3
holds L~ mid(b3,b1,b2) = union {LSeg(b3,b4) where b4 is Element of NAT: b1 <= b4 & b4 < b2};
:: SPRECT_2:th 19
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
dom X_axis b1 = dom b1;
:: SPRECT_2:th 20
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
dom Y_axis b1 = dom b1;
:: SPRECT_2:th 21
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in LSeg(b1,b3) & b1 `1 <= b2 `1 & b3 `1 <= b2 `1 & b1 <> b2 & b2 <> b3
holds b1 `1 = b2 `1 & b3 `1 = b2 `1;
:: SPRECT_2:th 22
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in LSeg(b1,b3) & b1 `2 <= b2 `2 & b3 `2 <= b2 `2 & b1 <> b2 & b2 <> b3
holds b1 `2 = b2 `2 & b3 `2 = b2 `2;
:: SPRECT_2:th 23
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in LSeg(b1,b3) & b2 `1 <= b1 `1 & b2 `1 <= b3 `1 & b1 <> b2 & b2 <> b3
holds b1 `1 = b2 `1 & b3 `1 = b2 `1;
:: SPRECT_2:th 24
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st b2 in LSeg(b1,b3) & b2 `2 <= b1 `2 & b2 `2 <= b3 `2 & b1 <> b2 & b2 <> b3
holds b1 `2 = b2 `2 & b3 `2 = b2 `2;
:: SPRECT_2:prednot 1 => SPRECT_2:pred 1
definition
let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
pred A2 is_in_the_area_of A1 means
for b1 being Element of NAT
st b1 in dom a2
holds W-bound L~ a1 <= (a2 /. b1) `1 & (a2 /. b1) `1 <= E-bound L~ a1 & S-bound L~ a1 <= (a2 /. b1) `2 & (a2 /. b1) `2 <= N-bound L~ a1;
end;
:: SPRECT_2:dfs 1
definiens
let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
To prove
a2 is_in_the_area_of a1
it is sufficient to prove
thus for b1 being Element of NAT
st b1 in dom a2
holds W-bound L~ a1 <= (a2 /. b1) `1 & (a2 /. b1) `1 <= E-bound L~ a1 & S-bound L~ a1 <= (a2 /. b1) `2 & (a2 /. b1) `2 <= N-bound L~ a1;
:: SPRECT_2:def 1
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2 holds
b2 is_in_the_area_of b1
iff
for b3 being Element of NAT
st b3 in dom b2
holds W-bound L~ b1 <= (b2 /. b3) `1 & (b2 /. b3) `1 <= E-bound L~ b1 & S-bound L~ b1 <= (b2 /. b3) `2 & (b2 /. b3) `2 <= N-bound L~ b1;
:: SPRECT_2:th 25
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
b1 is_in_the_area_of b1;
:: SPRECT_2:th 26
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b2 is_in_the_area_of b1
for b3, b4 being Element of NAT
st b3 in dom b2 & b4 in dom b2
holds mid(b2,b3,b4) is_in_the_area_of b1;
:: SPRECT_2:th 27
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 & b3 in dom b1
holds mid(b1,b2,b3) is_in_the_area_of b1;
:: SPRECT_2:th 28
theorem
for b1, b2, b3 being FinSequence of the carrier of TOP-REAL 2
st b2 is_in_the_area_of b1 & b3 is_in_the_area_of b1
holds b2 ^ b3 is_in_the_area_of b1;
:: SPRECT_2:th 29
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
<*NE-corner L~ b1*> is_in_the_area_of b1;
:: SPRECT_2:th 30
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
<*NW-corner L~ b1*> is_in_the_area_of b1;
:: SPRECT_2:th 31
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
<*SE-corner L~ b1*> is_in_the_area_of b1;
:: SPRECT_2:th 32
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
<*SW-corner L~ b1*> is_in_the_area_of b1;
:: SPRECT_2:prednot 2 => SPRECT_2:pred 2
definition
let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
pred A2 is_a_h.c._for A1 means
a2 is_in_the_area_of a1 &
(a2 /. 1) `1 = W-bound L~ a1 &
(a2 /. len a2) `1 = E-bound L~ a1;
end;
:: SPRECT_2:dfs 2
definiens
let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
To prove
a2 is_a_h.c._for a1
it is sufficient to prove
thus a2 is_in_the_area_of a1 &
(a2 /. 1) `1 = W-bound L~ a1 &
(a2 /. len a2) `1 = E-bound L~ a1;
:: SPRECT_2:def 2
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2 holds
b2 is_a_h.c._for b1
iff
b2 is_in_the_area_of b1 &
(b2 /. 1) `1 = W-bound L~ b1 &
(b2 /. len b2) `1 = E-bound L~ b1;
:: SPRECT_2:prednot 3 => SPRECT_2:pred 3
definition
let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
pred A2 is_a_v.c._for A1 means
a2 is_in_the_area_of a1 &
(a2 /. 1) `2 = S-bound L~ a1 &
(a2 /. len a2) `2 = N-bound L~ a1;
end;
:: SPRECT_2:dfs 3
definiens
let a1, a2 be FinSequence of the carrier of TOP-REAL 2;
To prove
a2 is_a_v.c._for a1
it is sufficient to prove
thus a2 is_in_the_area_of a1 &
(a2 /. 1) `2 = S-bound L~ a1 &
(a2 /. len a2) `2 = N-bound L~ a1;
:: SPRECT_2:def 3
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2 holds
b2 is_a_v.c._for b1
iff
b2 is_in_the_area_of b1 &
(b2 /. 1) `2 = S-bound L~ b1 &
(b2 /. len b2) `2 = N-bound L~ b1;
:: SPRECT_2:th 33
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being one-to-one special FinSequence of the carrier of TOP-REAL 2
st 2 <= len b2 & 2 <= len b3 & b2 is_a_h.c._for b1 & b3 is_a_v.c._for b1
holds L~ b2 meets L~ b3;
:: SPRECT_2:attrnot 1 => SPRECT_2:attr 1
definition
let a1 be FinSequence of the carrier of TOP-REAL 2;
attr a1 is clockwise_oriented means
(Rotate(a1,N-min L~ a1)) /. 2 in N-most L~ a1;
end;
:: SPRECT_2:dfs 4
definiens
let a1 be FinSequence of the carrier of TOP-REAL 2;
To prove
a1 is clockwise_oriented
it is sufficient to prove
thus (Rotate(a1,N-min L~ a1)) /. 2 in N-most L~ a1;
:: SPRECT_2:def 4
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2 holds
b1 is clockwise_oriented
iff
(Rotate(b1,N-min L~ b1)) /. 2 in N-most L~ b1;
:: SPRECT_2:th 34
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds b1 is clockwise_oriented
iff
b1 /. 2 in N-most L~ b1;
:: SPRECT_2:funcreg 1
registration
cluster R^2-unit_square -> compact;
end;
:: SPRECT_2:th 35
theorem
N-bound R^2-unit_square = 1;
:: SPRECT_2:th 36
theorem
W-bound R^2-unit_square = 0;
:: SPRECT_2:th 37
theorem
E-bound R^2-unit_square = 1;
:: SPRECT_2:th 38
theorem
S-bound R^2-unit_square = 0;
:: SPRECT_2:th 39
theorem
N-most R^2-unit_square = LSeg(|[0,1]|,|[1,1]|);
:: SPRECT_2:th 40
theorem
N-min R^2-unit_square = |[0,1]|;
:: SPRECT_2:funcreg 2
registration
let a1 be non empty compact non horizontal non vertical Element of bool the carrier of TOP-REAL 2;
cluster SpStSeq a1 -> clockwise_oriented;
end;
:: SPRECT_2:exreg 1
registration
cluster Relation-like Function-like non constant non empty finite FinSequence-like non trivial circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2;
end;
:: SPRECT_2:th 41
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b3 < b2 &
(1 < b3 & b2 <= len b1 or 1 <= b3 & b2 < len b1)
holds mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 42
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 < b3 &
(1 < b2 & b3 <= len b1 or 1 <= b2 & b3 < len b1)
holds mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 43
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
N-min L~ b1 in rng b1;
:: SPRECT_2:th 44
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
N-max L~ b1 in rng b1;
:: SPRECT_2:th 45
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
S-min L~ b1 in rng b1;
:: SPRECT_2:th 46
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
S-max L~ b1 in rng b1;
:: SPRECT_2:th 47
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
W-min L~ b1 in rng b1;
:: SPRECT_2:th 48
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
W-max L~ b1 in rng b1;
:: SPRECT_2:th 49
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
E-min L~ b1 in rng b1;
:: SPRECT_2:th 50
theorem
for b1 being non trivial FinSequence of the carrier of TOP-REAL 2 holds
E-max L~ b1 in rng b1;
:: SPRECT_2:th 51
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 <= b2 & b2 < b3 & b3 <= b4 & b4 <= len b5 & (b1 <= 1 implies b4 < len b5)
holds L~ mid(b5,b1,b2) misses L~ mid(b5,b3,b4);
:: SPRECT_2:th 52
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 <= b2 & b2 < b3 & b3 <= b4 & b4 <= len b5 & (b1 <= 1 implies b4 < len b5)
holds L~ mid(b5,b1,b2) misses L~ mid(b5,b4,b3);
:: SPRECT_2:th 53
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 <= b2 & b2 < b3 & b3 <= b4 & b4 <= len b5 & (b1 <= 1 implies b4 < len b5)
holds L~ mid(b5,b2,b1) misses L~ mid(b5,b4,b3);
:: SPRECT_2:th 54
theorem
for b1, b2, b3, b4 being Element of NAT
for b5 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 <= b2 & b2 < b3 & b3 <= b4 & b4 <= len b5 & (b1 <= 1 implies b4 < len b5)
holds L~ mid(b5,b2,b1) misses L~ mid(b5,b3,b4);
:: SPRECT_2:th 55
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
(N-min L~ b1) `1 < (N-max L~ b1) `1;
:: SPRECT_2:th 56
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
N-min L~ b1 <> N-max L~ b1;
:: SPRECT_2:th 57
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
(E-min L~ b1) `2 < (E-max L~ b1) `2;
:: SPRECT_2:th 58
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
E-min L~ b1 <> E-max L~ b1;
:: SPRECT_2:th 59
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
(S-min L~ b1) `1 < (S-max L~ b1) `1;
:: SPRECT_2:th 60
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
S-min L~ b1 <> S-max L~ b1;
:: SPRECT_2:th 61
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
(W-min L~ b1) `2 < (W-max L~ b1) `2;
:: SPRECT_2:th 62
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
W-min L~ b1 <> W-max L~ b1;
:: SPRECT_2:th 63
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
LSeg(NW-corner L~ b1,N-min L~ b1) misses LSeg(N-max L~ b1,NE-corner L~ b1);
:: SPRECT_2:th 64
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 <> b1 /. 1 &
(b2 `1 = (b1 /. 1) `1 or b2 `2 = (b1 /. 1) `2) &
(LSeg(b2,b1 /. 1)) /\ L~ b1 = {b1 /. 1}
holds <*b2*> ^ b1 is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 65
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 <> b1 /. len b1 &
(b2 `1 = (b1 /. len b1) `1 or b2 `2 = (b1 /. len b1) `2) &
(LSeg(b2,b1 /. len b1)) /\ L~ b1 = {b1 /. len b1}
holds b1 ^ <*b2*> is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 66
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 &
b3 in dom b1 &
mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2 &
b1 /. b3 = N-max L~ b1 &
N-max L~ b1 <> NE-corner L~ b1
holds (mid(b1,b2,b3)) ^ <*NE-corner L~ b1*> is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 67
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 &
b3 in dom b1 &
mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2 &
b1 /. b3 = E-max L~ b1 &
E-max L~ b1 <> NE-corner L~ b1
holds (mid(b1,b2,b3)) ^ <*NE-corner L~ b1*> is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 68
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 &
b3 in dom b1 &
mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2 &
b1 /. b3 = S-max L~ b1 &
S-max L~ b1 <> SE-corner L~ b1
holds (mid(b1,b2,b3)) ^ <*SE-corner L~ b1*> is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 69
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 &
b3 in dom b1 &
mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2 &
b1 /. b3 = E-max L~ b1 &
E-max L~ b1 <> NE-corner L~ b1
holds (mid(b1,b2,b3)) ^ <*NE-corner L~ b1*> is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 70
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 &
b3 in dom b1 &
mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2 &
b1 /. b2 = N-min L~ b1 &
N-min L~ b1 <> NW-corner L~ b1
holds <*NW-corner L~ b1*> ^ mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:th 71
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of NAT
st b2 in dom b1 &
b3 in dom b1 &
mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2 &
b1 /. b2 = W-min L~ b1 &
W-min L~ b1 <> SW-corner L~ b1
holds <*SW-corner L~ b1*> ^ mid(b1,b2,b3) is being_S-Seq FinSequence of the carrier of TOP-REAL 2;
:: SPRECT_2:funcreg 3
registration
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
cluster L~ a1 -> being_simple_closed_curve;
end;
:: SPRECT_2:th 72
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;
:: SPRECT_2:th 73
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds 1 < (N-max L~ b1) .. b1;
:: SPRECT_2:th 74
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1 & N-max L~ b1 <> E-max L~ b1
holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;
:: SPRECT_2:th 75
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;
:: SPRECT_2:th 76
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1 & E-min L~ b1 <> S-max L~ b1
holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;
:: SPRECT_2:th 77
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;
:: SPRECT_2:th 78
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1 & S-min L~ b1 <> W-min L~ b1
holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;
:: SPRECT_2:th 79
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1 & N-min L~ b1 <> W-max L~ b1
holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;
:: SPRECT_2:th 80
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds (W-min L~ b1) .. b1 < len b1;
:: SPRECT_2:th 81
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st b1 /. 1 = N-min L~ b1
holds (W-max L~ b1) .. b1 < len b1;