:: SPRECT_3 semantic presentation
theorem Th1: :: SPRECT_3:1
canceled;
theorem Th2: :: SPRECT_3:2
for b
1, b
2, b
3, b
4 being
set holds
( b
1 c= b
2 & b
2 /\ b
3 = {b4} & b
4 in b
1 implies b
1 /\ b
3 = {b4} )
Lemma1:
for b1, b2, b3, b4 being Real holds
( b4 >= 0 & b4 <= 1 & b3 = ((1 - b4) * b1) + (b4 * b2) & b1 <= b3 & b2 < b3 implies b4 = 0 )
by XREAL_1:181;
Lemma2:
for b1, b2, b3, b4 being Real holds
( b4 >= 0 & b4 <= 1 & b3 = ((1 - b4) * b1) + (b4 * b2) & b1 >= b3 & b2 > b3 implies b4 = 0 )
by XREAL_1:182;
theorem Th3: :: SPRECT_3:3
canceled;
theorem Th4: :: SPRECT_3:4
canceled;
theorem Th5: :: SPRECT_3:5
for b
1, b
2, b
3 being
Nat holds
( b
1 -' b
2 <= b
3 implies b
1 <= b
3 + b
2 )
theorem Th6: :: SPRECT_3:6
for b
1, b
2, b
3 being
Nat holds
( b
1 <= b
2 + b
3 implies b
1 -' b
3 <= b
2 )
theorem Th7: :: SPRECT_3:7
for b
1, b
2, b
3 being
Nat holds
( b
1 <= b
2 -' b
3 & b
3 <= b
2 implies b
1 + b
3 <= b
2 )
theorem Th8: :: SPRECT_3:8
for b
1, b
2, b
3 being
Nat holds
( b
1 + b
2 <= b
3 implies b
2 <= b
3 -' b
1 )
theorem Th9: :: SPRECT_3:9
for b
1, b
2, b
3 being
Nat holds
not ( b
1 <= b
2 & b
2 < b
3 & not b
2 -' b
1 < b
3 -' b
1 )
theorem Th10: :: SPRECT_3:10
for b
1, b
2, b
3 being
Nat holds
not ( b
1 < b
2 & b
3 < b
2 & not b
1 -' b
3 < b
2 -' b
3 )
theorem Th11: :: SPRECT_3:11
theorem Th12: :: SPRECT_3:12
for b
1, b
2, b
3, b
4 being
set holds
Indices (b1,b2 ][ b3,b4) = {[1,1],[1,2],[2,1],[2,2]}
theorem Th13: :: SPRECT_3:13
for b
1 being
Natfor b
2, b
3 being
Point of
(TOP-REAL b1)for b
4 being
Real holds
( 0
< b
4 & b
2 = ((1 - b4) * b2) + (b4 * b3) implies b
2 = b
3 )
theorem Th14: :: SPRECT_3:14
for b
1 being
Natfor b
2, b
3 being
Point of
(TOP-REAL b1)for b
4 being
Real holds
( b
4 < 1 & b
2 = ((1 - b4) * b3) + (b4 * b2) implies b
2 = b
3 )
theorem Th15: :: SPRECT_3:15
for b
1 being
Natfor b
2, b
3 being
Point of
(TOP-REAL b1) holds
( b
2 = (1 / 2) * (b2 + b3) implies b
2 = b
3 )
theorem Th16: :: SPRECT_3:16
theorem Th17: :: SPRECT_3:17
theorem Th18: :: SPRECT_3:18
theorem Th19: :: SPRECT_3:19
theorem Th20: :: SPRECT_3:20
theorem Th21: :: SPRECT_3:21
theorem Th22: :: SPRECT_3:22
theorem Th23: :: SPRECT_3:23
theorem Th24: :: SPRECT_3:24
theorem Th25: :: SPRECT_3:25
theorem Th26: :: SPRECT_3:26
theorem Th27: :: SPRECT_3:27
theorem Th28: :: SPRECT_3:28
theorem Th29: :: SPRECT_3:29
theorem Th30: :: SPRECT_3:30
canceled;
theorem Th31: :: SPRECT_3:31
theorem Th32: :: SPRECT_3:32
canceled;
theorem Th33: :: SPRECT_3:33
theorem Th34: :: SPRECT_3:34
theorem Th35: :: SPRECT_3:35
theorem Th36: :: SPRECT_3:36
theorem Th37: :: SPRECT_3:37
canceled;
theorem Th38: :: SPRECT_3:38
theorem Th39: :: SPRECT_3:39
theorem Th40: :: SPRECT_3:40
theorem Th41: :: SPRECT_3:41
theorem Th42: :: SPRECT_3:42
theorem Th43: :: SPRECT_3:43
theorem Th44: :: SPRECT_3:44
theorem Th45: :: SPRECT_3:45
theorem Th46: :: SPRECT_3:46
theorem Th47: :: SPRECT_3:47
theorem Th48: :: SPRECT_3:48
theorem Th49: :: SPRECT_3:49
theorem Th50: :: SPRECT_3:50
theorem Th51: :: SPRECT_3:51
theorem Th52: :: SPRECT_3:52
theorem Th53: :: SPRECT_3:53
theorem Th54: :: SPRECT_3:54
theorem Th55: :: SPRECT_3:55
theorem Th56: :: SPRECT_3:56
theorem Th57: :: SPRECT_3:57
theorem Th58: :: SPRECT_3:58
theorem Th59: :: SPRECT_3:59
theorem Th60: :: SPRECT_3:60
theorem Th61: :: SPRECT_3:61
theorem Th62: :: SPRECT_3:62
theorem Th63: :: SPRECT_3:63
theorem Th64: :: SPRECT_3:64
theorem Th65: :: SPRECT_3:65
theorem Th66: :: SPRECT_3:66
theorem Th67: :: SPRECT_3:67
theorem Th68: :: SPRECT_3:68
theorem Th69: :: SPRECT_3:69
theorem Th70: :: SPRECT_3:70
theorem Th71: :: SPRECT_3:71
theorem Th72: :: SPRECT_3:72
theorem Th73: :: SPRECT_3:73