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;