Article SPRECT_5, MML version 4.99.1005

:: SPRECT_5:th 1
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of b1
      st b3 in proj2 (b2 | (b4 .. b2))
   holds b3 .. b2 <= b4 .. b2;

:: SPRECT_5:th 2
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of b1
      st b3 in proj2 b2 & b4 in proj2 b2 & b3 .. b2 <= b4 .. b2
   holds b4 .. (b2 :- b3) = ((b4 .. b2) - (b3 .. b2)) + 1;

:: SPRECT_5:th 3
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of b1
      st b3 in proj2 b2 & b4 in proj2 b2 & b3 .. b2 <= b4 .. b2
   holds b3 .. (b2 -: b4) = b3 .. b2;

:: SPRECT_5:th 4
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of b1
      st b3 in proj2 b2 & b4 in proj2 b2 & b3 .. b2 <= b4 .. b2
   holds b4 .. Rotate(b2,b3) = ((b4 .. b2) - (b3 .. b2)) + 1;

:: SPRECT_5:th 5
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of b1
      st b3 in proj2 b2 & b4 in proj2 b2 & b5 in proj2 b2 & b3 .. b2 <= b4 .. b2 & b4 .. b2 < b5 .. b2
   holds b4 .. Rotate(b2,b3) < b5 .. Rotate(b2,b3);

:: SPRECT_5:th 6
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4, b5 being Element of b1
      st b3 in proj2 b2 & b4 in proj2 b2 & b5 in proj2 b2 & b3 .. b2 < b4 .. b2 & b4 .. b2 <= b5 .. b2
   holds b4 .. Rotate(b2,b3) <= b5 .. Rotate(b2,b3);

:: SPRECT_5:th 7
theorem
for b1 being non empty set
for b2 being circular FinSequence of b1
for b3 being Element of b1
      st b3 in proj2 b2 & 1 < len b2
   holds b3 .. b2 < len b2;

:: SPRECT_5:th 8
theorem
for b1 being non constant non empty circular special unfolded s.c.c. standard FinSequence of the carrier of TOP-REAL 2 holds
   b1 /^ 1 is one-to-one;

:: SPRECT_5:th 9
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 Element of the carrier of TOP-REAL 2
      st 1 < b2 .. b1 & b2 in proj2 b1
   holds (b1 /. 1) .. Rotate(b1,b2) = ((len b1) + 1) - (b2 .. b1);

:: SPRECT_5:th 10
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 the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b3 in proj2 b1 & b2 .. b1 < b3 .. b1
   holds b2 .. Rotate(b1,b3) = ((len b1) + (b2 .. b1)) - (b3 .. b1);

:: SPRECT_5:th 11
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, b4 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b3 in proj2 b1 & b4 in proj2 b1 & b2 .. b1 < b3 .. b1 & b3 .. b1 < b4 .. b1
   holds b4 .. Rotate(b1,b3) < b2 .. Rotate(b1,b3);

:: SPRECT_5:th 12
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, b4 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b3 in proj2 b1 & b4 in proj2 b1 & b2 .. b1 < b3 .. b1 & b3 .. b1 < b4 .. b1
   holds b2 .. Rotate(b1,b4) < b3 .. Rotate(b1,b4);

:: SPRECT_5:th 13
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, b4 being Element of the carrier of TOP-REAL 2
      st b2 in proj2 b1 & b3 in proj2 b1 & b4 in proj2 b1 & b2 .. b1 <= b3 .. b1 & b3 .. b1 < b4 .. b1
   holds b2 .. Rotate(b1,b4) <= b3 .. Rotate(b1,b4);

:: SPRECT_5:th 14
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) .. b1 < len b1;

:: SPRECT_5:th 15
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-max L~ b1) .. b1 < len b1;

:: SPRECT_5:th 16
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) .. b1 < len b1;

:: SPRECT_5:th 17
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-max L~ b1) .. b1 < len b1;

:: SPRECT_5:th 18
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) .. b1 < len b1;

:: SPRECT_5:th 19
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-max L~ b1) .. b1 < len b1;

:: SPRECT_5:th 20
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-max L~ b1) .. b1 < len b1;

:: SPRECT_5:th 21
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) .. b1 < len b1;

:: SPRECT_5:th 22
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 = W-min L~ b1
   holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 23
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 = W-min L~ b1
   holds 1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 24
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 = W-min L~ b1 & W-max L~ b1 <> N-min L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5:th 25
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 = W-min L~ b1
   holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;

:: SPRECT_5:th 26
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 = W-min L~ b1 & N-max L~ b1 <> E-max L~ b1
   holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;

:: SPRECT_5:th 27
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 = W-min L~ b1
   holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;

:: SPRECT_5:th 28
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 = W-min L~ b1 & E-min L~ b1 <> S-max L~ b1
   holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;

:: SPRECT_5:th 29
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 = W-min L~ b1 & S-min L~ b1 <> W-min L~ b1
   holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 30
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 = S-max L~ b1
   holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 31
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 = S-max L~ b1
   holds 1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 32
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 = S-max L~ b1 & S-min L~ b1 <> W-min L~ b1
   holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;

:: SPRECT_5:th 33
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 = S-max L~ b1
   holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 34
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 = S-max L~ b1 & W-max L~ b1 <> N-min L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5:th 35
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 = S-max L~ b1
   holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;

:: SPRECT_5:th 36
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 = S-max L~ b1 & N-max L~ b1 <> E-max L~ b1
   holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;

:: SPRECT_5:th 37
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 = S-max L~ b1 & E-min L~ b1 <> S-max L~ b1
   holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;

:: SPRECT_5:th 38
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 = E-max L~ b1
   holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;

:: SPRECT_5:th 39
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 = E-max L~ b1
   holds 1 < (E-min L~ b1) .. b1;

:: SPRECT_5:th 40
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 = E-max L~ b1 & S-max L~ b1 <> E-min L~ b1
   holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;

:: SPRECT_5:th 41
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 = E-max L~ b1
   holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 42
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 = E-max L~ b1 & S-min L~ b1 <> W-min L~ b1
   holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;

:: SPRECT_5:th 43
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 = E-max L~ b1
   holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 44
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 = E-max L~ b1 & W-max L~ b1 <> N-min L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5:th 45
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 = E-max L~ b1 & N-max L~ b1 <> E-max L~ b1
   holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;

:: SPRECT_5:th 46
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-max L~ b1 & N-max L~ b1 <> E-max L~ b1
   holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;

:: SPRECT_5:th 47
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-max L~ b1
   holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;

:: SPRECT_5: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-max L~ b1 & E-min L~ b1 <> S-max L~ b1
   holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;

:: SPRECT_5:th 49
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-max L~ b1
   holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 50
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-max L~ b1 & S-min L~ b1 <> W-min L~ b1
   holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;

:: SPRECT_5:th 51
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-max L~ b1
   holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 52
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-max L~ b1 & N-min L~ b1 <> W-max L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5:th 53
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 = E-min L~ b1 & E-min L~ b1 <> S-max L~ b1
   holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;

:: SPRECT_5:th 54
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 = E-min L~ b1
   holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 55
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 = E-min L~ b1 & S-min L~ b1 <> W-min L~ b1
   holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;

:: SPRECT_5:th 56
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 = E-min L~ b1
   holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 57
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 = E-min L~ b1 & W-max L~ b1 <> N-min L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5: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 = E-min L~ b1
   holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;

:: SPRECT_5:th 59
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 = E-min L~ b1 & E-max L~ b1 <> N-max L~ b1
   holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;

:: SPRECT_5: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
      st b1 /. 1 = S-min L~ b1 & S-min L~ b1 <> W-min L~ b1
   holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;

:: SPRECT_5:th 61
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 = S-min L~ b1
   holds (W-min L~ b1) .. b1 < (W-max L~ b1) .. b1;

:: SPRECT_5:th 62
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 = S-min L~ b1 & W-max L~ b1 <> N-min L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5:th 63
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 = S-min L~ b1
   holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;

:: SPRECT_5:th 64
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 = S-min L~ b1 & N-max L~ b1 <> E-max L~ b1
   holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;

:: SPRECT_5:th 65
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 = S-min L~ b1
   holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;

:: SPRECT_5:th 66
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 = S-min L~ b1 & S-max L~ b1 <> E-min L~ b1
   holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;

:: SPRECT_5: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
      st b1 /. 1 = W-max L~ b1 & W-max L~ b1 <> N-min L~ b1
   holds (W-max L~ b1) .. b1 < (N-min L~ b1) .. b1;

:: SPRECT_5:th 68
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 = W-max L~ b1
   holds (N-min L~ b1) .. b1 < (N-max L~ b1) .. b1;

:: SPRECT_5:th 69
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 = W-max L~ b1 & N-max L~ b1 <> E-max L~ b1
   holds (N-max L~ b1) .. b1 < (E-max L~ b1) .. b1;

:: SPRECT_5:th 70
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 = W-max L~ b1
   holds (E-max L~ b1) .. b1 < (E-min L~ b1) .. b1;

:: SPRECT_5:th 71
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 = W-max L~ b1 & E-min L~ b1 <> S-max L~ b1
   holds (E-min L~ b1) .. b1 < (S-max L~ b1) .. b1;

:: SPRECT_5:th 72
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 = W-max L~ b1
   holds (S-max L~ b1) .. b1 < (S-min L~ b1) .. b1;

:: SPRECT_5:th 73
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 = W-max L~ b1 & W-min L~ b1 <> S-min L~ b1
   holds (S-min L~ b1) .. b1 < (W-min L~ b1) .. b1;