Article SPRECT_3, MML version 4.99.1005
:: SPRECT_3:th 2
theorem
for b1, b2, b3, b4 being set
st b1 c= b2 & b2 /\ b3 = {b4} & b4 in b1
holds b1 /\ b3 = {b4};
:: SPRECT_3:th 11
theorem
for b1 being non empty set
for b2 being non empty FinSequence of b1
for b3 being FinSequence of b1 holds
(b3 ^ b2) /. len (b3 ^ b2) = b2 /. len b2;
:: SPRECT_3:th 12
theorem
for b1, b2, b3, b4 being set holds
Indices ((b1,b2)][(b3,b4)) = {[1,1],[1,2],[2,1],[2,2]};
:: SPRECT_3:th 13
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of REAL
st {} < b4 &
b2 = ((1 - b4) * b2) + (b4 * b3)
holds b2 = b3;
:: SPRECT_3:th 14
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
for b4 being Element of REAL
st b4 < 1 &
b2 = ((1 - b4) * b3) + (b4 * b2)
holds b2 = b3;
:: SPRECT_3:th 15
theorem
for b1 being Element of NAT
for b2, b3 being Element of the carrier of TOP-REAL b1
st b2 = (1 / 2) * (b2 + b3)
holds b2 = b3;
:: SPRECT_3:th 16
theorem
for b1 being Element of NAT
for b2, b3, b4 being Element of the carrier of TOP-REAL b1
st b3 in LSeg(b2,b4) & b4 in LSeg(b2,b3)
holds b3 = b4;
:: SPRECT_3:th 17
theorem
for b1 being non empty Element of bool the carrier of TOP-REAL 2
for b2 being Element of the carrier of Euclid 2
for b3 being Element of REAL
st b1 = Ball(b2,b3)
holds b1 is connected(TOP-REAL 2);
:: SPRECT_3:th 18
theorem
for b1, b2 being Element of bool the carrier of TOP-REAL 2
st b1 is open(TOP-REAL 2) & b2 is_a_component_of b1
holds b2 is open(TOP-REAL 2);
:: SPRECT_3:th 21
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st LSeg(b1,b2) is horizontal & LSeg(b3,b4) is horizontal & LSeg(b1,b2) meets LSeg(b3,b4)
holds b1 `2 = b3 `2;
:: SPRECT_3:th 22
theorem
for b1, b2, b3 being Element of the carrier of TOP-REAL 2
st LSeg(b1,b2) is vertical & LSeg(b2,b3) is horizontal
holds (LSeg(b1,b2)) /\ LSeg(b2,b3) = {b2};
:: SPRECT_3:th 23
theorem
for b1, b2, b3, b4 being Element of the carrier of TOP-REAL 2
st LSeg(b1,b2) is horizontal & LSeg(b4,b3) is vertical & b3 in LSeg(b1,b2)
holds (LSeg(b1,b2)) /\ LSeg(b4,b3) = {b3};
:: SPRECT_3:th 24
theorem
for b1, b2, b3 being Element of NAT
for b4 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) *
st 1 <= b1 & b1 <= b2 & b2 <= width b4 & 1 <= b3 & b3 <= len b4
holds (b4 *(b3,b1)) `2 <= (b4 *(b3,b2)) `2;
:: SPRECT_3:th 25
theorem
for b1, b2, b3 being Element of NAT
for b4 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) *
st 1 <= b1 & b1 <= width b4 & 1 <= b2 & b2 <= b3 & b3 <= len b4
holds (b4 *(b2,b1)) `1 <= (b4 *(b3,b1)) `1;
:: SPRECT_3:th 26
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
LSeg(NW-corner b1,NE-corner b1) c= L~ SpStSeq b1;
:: SPRECT_3:th 28
theorem
for b1 being non empty compact Element of bool the carrier of TOP-REAL 2 holds
N-min b1 in LSeg(NW-corner b1,NE-corner b1);
:: SPRECT_3:th 29
theorem
for b1 being Element of bool the carrier of TOP-REAL 2 holds
LSeg(NW-corner b1,NE-corner b1) is horizontal;
:: SPRECT_3:th 31
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 /. 1 <> b2 &
((b1 /. 1) `1 = b2 `1 or (b1 /. 1) `2 = b2 `2) &
b1 is being_S-Seq &
(LSeg(b2,b1 /. 1)) /\ L~ b1 = {b1 /. 1}
holds <*b2*> ^ b1 is being_S-Seq;
:: SPRECT_3:th 33
theorem
for b1 being Element of NAT
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b3 being Element of the carrier of TOP-REAL 2
st 1 < b1 & b1 <= len b2 & b3 in L~ mid(b2,1,b1)
holds LE b3,b2 /. b1,L~ b2,b2 /. 1,b2 /. len b2;
:: SPRECT_3:th 34
theorem
for b1, b2 being Element of NAT
for b3 being FinSequence of the carrier of TOP-REAL 2
st b1 in dom b3 & b2 in dom b3
holds L~ mid(b3,b1,b2) c= L~ b3;
:: SPRECT_3:th 35
theorem
for b1, b2 being Element of NAT
st 1 <= b1 & b1 < b2
for b3 being FinSequence of the carrier of TOP-REAL 2
st b2 <= len b3
holds L~ mid(b3,b1,b2) = (LSeg(b3,b1)) \/ L~ mid(b3,b1 + 1,b2);
:: SPRECT_3:th 36
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) = (L~ mid(b3,b1,b2 -' 1)) \/ LSeg(b3,b2 -' 1);
:: SPRECT_3:th 38
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b1 is being_S-Seq &
b2 is being_S-Seq &
((b1 /. len b1) `1 = (b2 /. 1) `1 or (b1 /. len b1) `2 = (b2 /. 1) `2) &
L~ b1 misses L~ b2 &
(LSeg(b1 /. len b1,b2 /. 1)) /\ L~ b1 = {b1 /. len b1} &
(LSeg(b1 /. len b1,b2 /. 1)) /\ L~ b2 = {b2 /. 1}
holds b1 ^ b2 is being_S-Seq;
:: SPRECT_3:th 39
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)) /. 1 = b1 /. 1;
:: SPRECT_3:th 40
theorem
for b1 being Element of NAT
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
for b3, b4 being Element of the carrier of TOP-REAL 2
st 1 <= b1 & b1 < len b2 & b3 in LSeg(b2,b1) & b4 in LSeg(b2 /. b1,b3)
holds LE b4,b3,L~ b2,b2 /. 1,b2 /. len b2;
:: SPRECT_3: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 holds
LeftComp b1 is open(TOP-REAL 2) & RightComp b1 is open(TOP-REAL 2);
:: SPRECT_3:funcreg 1
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 -> non horizontal non vertical;
end;
:: SPRECT_3:funcreg 2
registration
let a1 be non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2;
cluster LeftComp a1 -> being_Region;
end;
:: SPRECT_3: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 RightComp a1 -> being_Region;
end;
:: SPRECT_3: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 holds
RightComp b1 misses L~ b1;
:: SPRECT_3:th 43
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
LeftComp b1 misses L~ b1;
:: SPRECT_3:th 44
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
i_w_n b1 < i_e_n b1;
:: SPRECT_3:th 45
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
ex b2 being Element of NAT st
1 <= b2 &
b2 < len GoB b1 &
N-min L~ b1 = (GoB b1) *(b2,width GoB b1);
:: SPRECT_3:th 46
theorem
for b1 being Element of NAT
for b2 being non constant non empty circular special unfolded s.c.c. standard clockwise_oriented FinSequence of the carrier of TOP-REAL 2
st b1 in dom GoB b2 &
b2 /. 1 = (GoB b2) *(b1,width GoB b2) &
b2 /. 1 = N-min L~ b2
holds b2 /. 2 = (GoB b2) *(b1 + 1,width GoB b2) &
b2 /. ((len b2) -' 1) = (GoB b2) *(b1,(width GoB b2) -' 1);
:: SPRECT_3:th 47
theorem
for b1, b2 being Element of NAT
for b3 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 <= len b3 & b3 /. 1 in L~ mid(b3,b1,b2) & b1 <> 1
holds b2 = len b3;
:: SPRECT_3:th 48
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 LSeg(b1 /. 1,b1 /. 2) c= L~ SpStSeq L~ b1;
:: SPRECT_3:th 49
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st b2 in L~ b1 & b2 `1 <> W-bound L~ b1 & b2 `1 <> E-bound L~ b1 & b2 `2 <> S-bound L~ b1
holds b2 `2 = N-bound L~ b1;
:: SPRECT_3:exreg 1
registration
cluster Relation-like Function-like non empty finite FinSequence-like circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2;
end;
:: SPRECT_3:th 50
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
st b2 /. 1 in LeftComp b1 & b2 /. len b2 in RightComp b1
holds L~ b1 meets L~ b2;
:: SPRECT_3:th 51
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2 holds
SpStSeq L~ b1 = b1;
:: SPRECT_3:th 52
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2 holds
L~ b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: ((b2 `1 = W-bound L~ b1 & b2 `2 <= N-bound L~ b1 implies b2 `2 < S-bound L~ b1) &
(b2 `1 <= E-bound L~ b1 & W-bound L~ b1 <= b2 `1 implies b2 `2 <> N-bound L~ b1) &
(b2 `1 <= E-bound L~ b1 & W-bound L~ b1 <= b2 `1 implies b2 `2 <> S-bound L~ b1) implies b2 `1 = E-bound L~ b1 & b2 `2 <= N-bound L~ b1 & S-bound L~ b1 <= b2 `2)};
:: SPRECT_3:th 53
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2 holds
GoB b1 = (b1 /. 4,b1 /. 1)][(b1 /. 3,b1 /. 2);
:: SPRECT_3:th 54
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2 holds
LeftComp b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: (W-bound L~ b1 <= b2 `1 & b2 `1 <= E-bound L~ b1 & S-bound L~ b1 <= b2 `2 implies N-bound L~ b1 < b2 `2)} &
RightComp b1 = {b2 where b2 is Element of the carrier of TOP-REAL 2: W-bound L~ b1 < b2 `1 & b2 `1 < E-bound L~ b1 & S-bound L~ b1 < b2 `2 & b2 `2 < N-bound L~ b1};
:: SPRECT_3:exreg 2
registration
cluster Relation-like Function-like non constant non empty finite FinSequence-like non trivial circular special unfolded s.c.c. standard rectangular clockwise_oriented FinSequence of the carrier of TOP-REAL 2;
end;
:: SPRECT_3:condreg 1
registration
cluster non empty circular special unfolded s.c.c. rectangular -> clockwise_oriented (FinSequence of the carrier of TOP-REAL 2);
end;
:: SPRECT_3:th 55
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
st b2 /. 1 in LeftComp b1 & b2 /. len b2 in RightComp b1
holds Last_Point(L~ b2,b2 /. 1,b2 /. len b2,L~ b1) <> NW-corner L~ b1;
:: SPRECT_3:th 56
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
st b2 /. 1 in LeftComp b1 & b2 /. len b2 in RightComp b1
holds Last_Point(L~ b2,b2 /. 1,b2 /. len b2,L~ b1) <> SE-corner L~ b1;
:: SPRECT_3:th 57
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st (W-bound L~ b1 <= b2 `1 & b2 `1 <= E-bound L~ b1 & S-bound L~ b1 <= b2 `2 implies N-bound L~ b1 < b2 `2)
holds b2 in LeftComp b1;
:: SPRECT_3:th 58
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 LeftComp SpStSeq L~ b1 c= LeftComp b1;
:: SPRECT_3:th 59
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,b3*> is_in_the_area_of b1
iff
<*b2*> is_in_the_area_of b1 & <*b3*> is_in_the_area_of b1;
:: SPRECT_3:th 60
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being Element of the carrier of TOP-REAL 2
st <*b2*> is_in_the_area_of b1 &
(b2 `1 <> W-bound L~ b1 & b2 `1 <> E-bound L~ b1 & b2 `2 <> S-bound L~ b1 implies b2 `2 = N-bound L~ b1)
holds b2 in L~ b1;
:: SPRECT_3:th 61
theorem
for b1 being FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
for b4 being Element of REAL
st {} <= b4 & b4 <= 1 & <*b2,b3*> is_in_the_area_of b1
holds <*((1 - b4) * b2) + (b4 * b3)*> is_in_the_area_of b1;
:: SPRECT_3:th 62
theorem
for b1 being Element of NAT
for b2, b3 being FinSequence of the carrier of TOP-REAL 2
st b3 is_in_the_area_of b2 & b1 in dom b3
holds <*b3 /. b1*> is_in_the_area_of b2;
:: SPRECT_3:th 63
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 b2 is_in_the_area_of b1 & b3 in L~ b2
holds <*b3*> is_in_the_area_of b1;
:: SPRECT_3:th 64
theorem
for b1 being rectangular FinSequence of the carrier of TOP-REAL 2
for b2, b3 being Element of the carrier of TOP-REAL 2
st not b3 in L~ b1 & <*b2,b3*> is_in_the_area_of b1
holds (LSeg(b2,b3)) /\ L~ b1 c= {b2};
:: SPRECT_3:th 65
theorem
for b1 being rectangular 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 & not b3 in L~ b1 & <*b3*> is_in_the_area_of b1
holds (LSeg(b2,b3)) /\ L~ b1 = {b2};
:: SPRECT_3:th 66
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 <= len GoB b3 & 1 <= b2 & b2 <= width GoB b3
holds <*(GoB b3) *(b1,b2)*> is_in_the_area_of b3;
:: SPRECT_3:th 67
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,b3*> is_in_the_area_of b1
holds <*(1 / 2) * (b2 + b3)*> is_in_the_area_of b1;
:: SPRECT_3:th 68
theorem
for b1, b2 being FinSequence of the carrier of TOP-REAL 2
st b2 is_in_the_area_of b1
holds Rev b2 is_in_the_area_of b1;
:: SPRECT_3:th 69
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 b2 is_in_the_area_of b1 & <*b3*> is_in_the_area_of b1 & b2 is being_S-Seq & b3 in L~ b2
holds R_Cut(b2,b3) is_in_the_area_of b1;
:: SPRECT_3: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 being FinSequence of the carrier of TOP-REAL 2 holds
b2 is_in_the_area_of b1
iff
b2 is_in_the_area_of SpStSeq L~ b1;
:: SPRECT_3:th 71
theorem
for b1 being non empty circular special unfolded s.c.c. rectangular FinSequence of the carrier of TOP-REAL 2
for b2 being being_S-Seq FinSequence of the carrier of TOP-REAL 2
st b2 /. 1 in LeftComp b1 & b2 /. len b2 in RightComp b1
holds L_Cut(b2,Last_Point(L~ b2,b2 /. 1,b2 /. len b2,L~ b1)) is_in_the_area_of b1;
:: SPRECT_3:th 72
theorem
for b1, b2 being Element of NAT
for b3 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2
st 1 <= b1 & b1 < len GoB b3 & 1 <= b2 & b2 < width GoB b3
holds Int cell(GoB b3,b1,b2) misses L~ SpStSeq L~ b3;
:: SPRECT_3:th 73
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 b2 is_in_the_area_of b1 & <*b3*> is_in_the_area_of b1 & b2 is being_S-Seq & b3 in L~ b2
holds L_Cut(b2,b3) is_in_the_area_of b1;