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;