:: SPRECT_5 semantic presentation

theorem Th1: :: SPRECT_5:1
for D being non empty set
for f being FinSequence of D
for q, p being Element of D st q in rng (f | (p .. f)) holds
q .. f <= p .. f
proof end;

theorem Th2: :: SPRECT_5:2
for D being non empty set
for f being FinSequence of D
for p, q being Element of D st p in rng f & q in rng f & p .. f <= q .. f holds
q .. (f :- p) = ((q .. f) - (p .. f)) + 1
proof end;

theorem Th3: :: SPRECT_5:3
for D being non empty set
for f being FinSequence of D
for p, q being Element of D st p in rng f & q in rng f & p .. f <= q .. f holds
p .. (f -: q) = p .. f
proof end;

theorem Th4: :: SPRECT_5:4
for D being non empty set
for f being FinSequence of D
for p, q being Element of D st p in rng f & q in rng f & p .. f <= q .. f holds
q .. (Rotate f,p) = ((q .. f) - (p .. f)) + 1
proof end;

theorem Th5: :: SPRECT_5:5
for D being non empty set
for f being FinSequence of D
for p1, p2, p3 being Element of D st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f <= p2 .. f & p2 .. f < p3 .. f holds
p2 .. (Rotate f,p1) < p3 .. (Rotate f,p1)
proof end;

theorem Th6: :: SPRECT_5:6
for D being non empty set
for f being FinSequence of D
for p1, p2, p3 being Element of D st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f < p2 .. f & p2 .. f <= p3 .. f holds
p2 .. (Rotate f,p1) <= p3 .. (Rotate f,p1)
proof end;

theorem Th7: :: SPRECT_5:7
for D being non empty set
for g being circular FinSequence of D
for p being Element of D st p in rng g & len g > 1 holds
p .. g < len g
proof end;

theorem Th8: :: SPRECT_5:8
for f being non constant standard special_circular_sequence holds f /^ 1 is one-to-one
proof end;

theorem Th9: :: SPRECT_5:9
for f being non constant standard special_circular_sequence
for q being Point of (TOP-REAL 2) st 1 < q .. f & q in rng f holds
(f /. 1) .. (Rotate f,q) = ((len f) + 1) - (q .. f)
proof end;

theorem Th10: :: SPRECT_5:10
for f being non constant standard special_circular_sequence
for p, q being Point of (TOP-REAL 2) st p in rng f & q in rng f & p .. f < q .. f holds
p .. (Rotate f,q) = ((len f) + (p .. f)) - (q .. f)
proof end;

theorem Th11: :: SPRECT_5:11
for f being non constant standard special_circular_sequence
for p1, p2, p3 being Point of (TOP-REAL 2) st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f < p2 .. f & p2 .. f < p3 .. f holds
p3 .. (Rotate f,p2) < p1 .. (Rotate f,p2)
proof end;

theorem Th12: :: SPRECT_5:12
for f being non constant standard special_circular_sequence
for p1, p2, p3 being Point of (TOP-REAL 2) st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f < p2 .. f & p2 .. f < p3 .. f holds
p1 .. (Rotate f,p3) < p2 .. (Rotate f,p3)
proof end;

theorem Th13: :: SPRECT_5:13
for f being non constant standard special_circular_sequence
for p1, p2, p3 being Point of (TOP-REAL 2) st p1 in rng f & p2 in rng f & p3 in rng f & p1 .. f <= p2 .. f & p2 .. f < p3 .. f holds
p1 .. (Rotate f,p3) <= p2 .. (Rotate f,p3)
proof end;

theorem Th14: :: SPRECT_5:14
for f being non constant standard special_circular_sequence holds (S-min (L~ f)) .. f < len f
proof end;

theorem Th15: :: SPRECT_5:15
for f being non constant standard special_circular_sequence holds (S-max (L~ f)) .. f < len f
proof end;

theorem Th16: :: SPRECT_5:16
for f being non constant standard special_circular_sequence holds (E-min (L~ f)) .. f < len f
proof end;

theorem Th17: :: SPRECT_5:17
for f being non constant standard special_circular_sequence holds (E-max (L~ f)) .. f < len f
proof end;

theorem Th18: :: SPRECT_5:18
for f being non constant standard special_circular_sequence holds (N-min (L~ f)) .. f < len f
proof end;

theorem Th19: :: SPRECT_5:19
for f being non constant standard special_circular_sequence holds (N-max (L~ f)) .. f < len f
proof end;

theorem Th20: :: SPRECT_5:20
for f being non constant standard special_circular_sequence holds (W-max (L~ f)) .. f < len f
proof end;

theorem Th21: :: SPRECT_5:21
for f being non constant standard special_circular_sequence holds (W-min (L~ f)) .. f < len f
proof end;

Lemma64: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

Lemma65: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

Lemma66: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-max (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma67: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-min (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

Lemma68: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(E-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma69: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(S-max (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma70: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(N-max (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma71: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(N-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma72: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(N-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

Lemma73: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-min (L~ z) holds
(N-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

theorem Th22: :: SPRECT_5:22
for f being non constant standard special_circular_sequence st f /. 1 = W-min (L~ f) holds
(W-min (L~ f)) .. f < (W-max (L~ f)) .. f
proof end;

theorem Th23: :: SPRECT_5:23
for f being non constant standard special_circular_sequence st f /. 1 = W-min (L~ f) holds
(W-max (L~ f)) .. f > 1
proof end;

theorem Th24: :: SPRECT_5:24
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) & W-max (L~ z) <> N-min (L~ z) holds
(W-max (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

theorem Th25: :: SPRECT_5:25
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th26: :: SPRECT_5:26
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) & N-max (L~ z) <> E-max (L~ z) holds
(N-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

theorem Th27: :: SPRECT_5:27
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(E-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

theorem Th28: :: SPRECT_5:28
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) & E-min (L~ z) <> S-max (L~ z) holds
(E-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

theorem Th29: :: SPRECT_5:29
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) & S-min (L~ z) <> W-min (L~ z) holds
(S-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

theorem Th30: :: SPRECT_5:30
for f being non constant standard special_circular_sequence st f /. 1 = S-max (L~ f) holds
(S-max (L~ f)) .. f < (S-min (L~ f)) .. f
proof end;

theorem Th31: :: SPRECT_5:31
for f being non constant standard special_circular_sequence st f /. 1 = S-max (L~ f) holds
(S-min (L~ f)) .. f > 1
proof end;

Lemma81: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(E-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

Lemma82: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(N-min (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

Lemma83: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(N-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

Lemma84: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(N-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

Lemma85: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(W-max (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

Lemma86: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(N-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

Lemma87: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(N-min (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

Lemma88: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(W-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

Lemma89: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-min (L~ z) holds
(W-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

theorem Th32: :: SPRECT_5:32
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) & S-min (L~ z) <> W-min (L~ z) holds
(S-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

theorem Th33: :: SPRECT_5:33
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(W-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

theorem Th34: :: SPRECT_5:34
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) & W-max (L~ z) <> N-min (L~ z) holds
(W-max (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

theorem Th35: :: SPRECT_5:35
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th36: :: SPRECT_5:36
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) & N-max (L~ z) <> E-max (L~ z) holds
(N-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

theorem Th37: :: SPRECT_5:37
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) & E-min (L~ z) <> S-max (L~ z) holds
(E-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

theorem Th38: :: SPRECT_5:38
for f being non constant standard special_circular_sequence st f /. 1 = E-max (L~ f) holds
(E-max (L~ f)) .. f < (E-min (L~ f)) .. f
proof end;

theorem Th39: :: SPRECT_5:39
for f being non constant standard special_circular_sequence st f /. 1 = E-max (L~ f) holds
(E-min (L~ f)) .. f > 1
proof end;

theorem Th40: :: SPRECT_5:40
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) & S-max (L~ z) <> E-min (L~ z) holds
(E-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

theorem Th41: :: SPRECT_5:41
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) holds
(S-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

Lemma98: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(N-min (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

Lemma99: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(W-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

Lemma100: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(W-min (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

Lemma101: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(W-max (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

Lemma102: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(W-min (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

Lemma103: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(S-min (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

Lemma104: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-max (L~ z) holds
(S-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th42: :: SPRECT_5:42
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) & S-min (L~ z) <> W-min (L~ z) holds
(S-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

theorem Th43: :: SPRECT_5:43
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) holds
(W-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

theorem Th44: :: SPRECT_5:44
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) & W-max (L~ z) <> N-min (L~ z) holds
(W-max (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

theorem Th45: :: SPRECT_5:45
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) & N-max (L~ z) <> E-max (L~ z) holds
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th46: :: SPRECT_5:46
for f being non constant standard special_circular_sequence st f /. 1 = N-max (L~ f) & N-max (L~ f) <> E-max (L~ f) holds
(N-max (L~ f)) .. f < (E-max (L~ f)) .. f
proof end;

theorem Th47: :: SPRECT_5:47
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-max (L~ z) holds
(E-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

theorem Th48: :: SPRECT_5:48
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-max (L~ z) & E-min (L~ z) <> S-max (L~ z) holds
(E-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

theorem Th49: :: SPRECT_5:49
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-max (L~ z) holds
(S-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

theorem Th50: :: SPRECT_5:50
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-max (L~ z) & S-min (L~ z) <> W-min (L~ z) holds
(S-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

theorem Th51: :: SPRECT_5:51
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-max (L~ z) holds
(W-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

theorem Th52: :: SPRECT_5:52
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = N-max (L~ z) & N-min (L~ z) <> W-max (L~ z) holds
(W-max (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

theorem Th53: :: SPRECT_5:53
for f being non constant standard special_circular_sequence st f /. 1 = E-min (L~ f) & E-min (L~ f) <> S-max (L~ f) holds
(E-min (L~ f)) .. f < (S-max (L~ f)) .. f
proof end;

theorem Th54: :: SPRECT_5:54
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-min (L~ z) holds
(S-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

theorem Th55: :: SPRECT_5:55
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-min (L~ z) & S-min (L~ z) <> W-min (L~ z) holds
(S-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma108: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) holds
(S-max (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma109: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) holds
(E-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;

Lemma110: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) holds
(E-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

Lemma111: for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-max (L~ z) holds
(S-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

theorem Th56: :: SPRECT_5:56
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-min (L~ z) holds
(W-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

theorem Th57: :: SPRECT_5:57
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-min (L~ z) & W-max (L~ z) <> N-min (L~ z) holds
(W-max (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

theorem Th58: :: SPRECT_5:58
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-min (L~ z) holds
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th59: :: SPRECT_5:59
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = E-min (L~ z) & E-max (L~ z) <> N-max (L~ z) holds
(N-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

theorem Th60: :: SPRECT_5:60
for f being non constant standard special_circular_sequence st f /. 1 = S-min (L~ f) & S-min (L~ f) <> W-min (L~ f) holds
(S-min (L~ f)) .. f < (W-min (L~ f)) .. f
proof end;

theorem Th61: :: SPRECT_5:61
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-min (L~ z) holds
(W-min (L~ z)) .. z < (W-max (L~ z)) .. z
proof end;

theorem Th62: :: SPRECT_5:62
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-min (L~ z) & W-max (L~ z) <> N-min (L~ z) holds
(W-max (L~ z)) .. z < (N-min (L~ z)) .. z
proof end;

theorem Th63: :: SPRECT_5:63
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-min (L~ z) holds
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th64: :: SPRECT_5:64
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-min (L~ z) & N-max (L~ z) <> E-max (L~ z) holds
(N-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

theorem Th65: :: SPRECT_5:65
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-min (L~ z) holds
(E-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

theorem Th66: :: SPRECT_5:66
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = S-min (L~ z) & S-max (L~ z) <> E-min (L~ z) holds
(E-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

theorem Th67: :: SPRECT_5:67
for f being non constant standard special_circular_sequence st f /. 1 = W-max (L~ f) & W-max (L~ f) <> N-min (L~ f) holds
(W-max (L~ f)) .. f < (N-min (L~ f)) .. f
proof end;

theorem Th68: :: SPRECT_5:68
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-max (L~ z) holds
(N-min (L~ z)) .. z < (N-max (L~ z)) .. z
proof end;

theorem Th69: :: SPRECT_5:69
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-max (L~ z) & N-max (L~ z) <> E-max (L~ z) holds
(N-max (L~ z)) .. z < (E-max (L~ z)) .. z
proof end;

theorem Th70: :: SPRECT_5:70
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-max (L~ z) holds
(E-max (L~ z)) .. z < (E-min (L~ z)) .. z
proof end;

theorem Th71: :: SPRECT_5:71
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-max (L~ z) & E-min (L~ z) <> S-max (L~ z) holds
(E-min (L~ z)) .. z < (S-max (L~ z)) .. z
proof end;

theorem Th72: :: SPRECT_5:72
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-max (L~ z) holds
(S-max (L~ z)) .. z < (S-min (L~ z)) .. z
proof end;

theorem Th73: :: SPRECT_5:73
for z being non constant standard clockwise_oriented special_circular_sequence st z /. 1 = W-max (L~ z) & W-min (L~ z) <> S-min (L~ z) holds
(S-min (L~ z)) .. z < (W-min (L~ z)) .. z
proof end;