:: EUCLIDLP semantic presentation
theorem Th1: :: EUCLIDLP:1
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5 being
Element of
REAL b
4 holds
(
(b1 / b2) * (b3 * b5) = ((b1 * b3) / b2) * b
5 &
(1 / b2) * (b3 * b5) = (b3 / b2) * b
5 )
theorem Th2: :: EUCLIDLP:2
theorem Th3: :: EUCLIDLP:3
theorem Th4: :: EUCLIDLP:4
theorem Th5: :: EUCLIDLP:5
theorem Th6: :: EUCLIDLP:6
theorem Th7: :: EUCLIDLP:7
Lemma8:
for b1 being Nat
for b2 being Element of REAL b1 holds (- 1) * b2 = - b2
theorem Th8: :: EUCLIDLP:8
theorem Th9: :: EUCLIDLP:9
theorem Th10: :: EUCLIDLP:10
theorem Th11: :: EUCLIDLP:11
for b
1 being
Natfor b
2, b
3, b
4 being
Element of
REAL b
1 holds
( b
2 = b
3 + b
4 iff b
3 = b
2 - b
4 )
theorem Th12: :: EUCLIDLP:12
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
Element of
REAL b
1 holds
( b
2 = (b3 + b4) + b
5 iff b
2 - b
3 = b
4 + b
5 )
theorem Th13: :: EUCLIDLP:13
Lemma15:
for b1 being Nat
for b2, b3 being Element of REAL b1 holds
not ( b2 <> b3 & not |.(b2 - b3).| <> 0 )
theorem Th14: :: EUCLIDLP:14
theorem Th15: :: EUCLIDLP:15
for b
1 being
Realfor b
2 being
Natfor b
3, b
4, b
5 being
Element of
REAL b
2 holds
not ( b
3 - b
4 = b
1 * b
5 & b
3 <> b
4 & not b
1 <> 0 )
theorem Th16: :: EUCLIDLP:16
for b
1, b
2 being
Realfor b
3 being
Natfor b
4 being
Element of
REAL b
3 holds
(
(b1 - b2) * b
4 = (b1 * b4) + ((- b2) * b4) &
(b1 - b2) * b
4 = (b1 * b4) + (- (b2 * b4)) &
(b1 - b2) * b
4 = (b1 * b4) - (b2 * b4) )
theorem Th17: :: EUCLIDLP:17
for b
1 being
Realfor b
2 being
Natfor b
3, b
4 being
Element of
REAL b
2 holds
( b
1 * (b3 - b4) = (b1 * b3) + (- (b1 * b4)) & b
1 * (b3 - b4) = (b1 * b3) + ((- b1) * b4) & b
1 * (b3 - b4) = (b1 * b3) - (b1 * b4) )
theorem Th18: :: EUCLIDLP:18
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5 being
Element of
REAL b
4 holds
((b1 - b2) - b3) * b
5 = ((b1 * b5) - (b2 * b5)) - (b3 * b5)
theorem Th19: :: EUCLIDLP:19
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5, b
6, b
7, b
8 being
Element of
REAL b
4 holds b
5 - (((b1 * b6) + (b2 * b7)) + (b3 * b8)) = b
5 + ((((- b1) * b6) + ((- b2) * b7)) + ((- b3) * b8))
theorem Th20: :: EUCLIDLP:20
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5, b
6 being
Element of
REAL b
4 holds b
5 - (((b1 + b2) + b3) * b6) = ((b5 + ((- b1) * b6)) + ((- b2) * b6)) + ((- b3) * b6)
theorem Th21: :: EUCLIDLP:21
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
Element of
REAL b
1 holds
(b2 + b3) + (b4 + b5) = (b2 + b4) + (b3 + b5)
theorem Th22: :: EUCLIDLP:22
for b
1 being
Natfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
REAL b
1 holds
((b2 + b3) + b4) + ((b5 + b6) + b7) = ((b2 + b5) + (b3 + b6)) + (b4 + b7)
theorem Th23: :: EUCLIDLP:23
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
Element of
REAL b
1 holds
(b2 + b3) - (b4 + b5) = (b2 - b4) + (b3 - b5)
theorem Th24: :: EUCLIDLP:24
for b
1 being
Natfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
REAL b
1 holds
((b2 + b3) + b4) - ((b5 + b6) + b7) = ((b2 - b5) + (b3 - b6)) + (b4 - b7)
theorem Th25: :: EUCLIDLP:25
for b
1 being
Realfor b
2 being
Natfor b
3, b
4, b
5 being
Element of
REAL b
2 holds b
1 * ((b3 + b4) + b5) = ((b1 * b3) + (b1 * b4)) + (b1 * b5)
theorem Th26: :: EUCLIDLP:26
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5, b
6 being
Element of
REAL b
4 holds b
1 * ((b2 * b5) + (b3 * b6)) = ((b1 * b2) * b5) + ((b1 * b3) * b6)
theorem Th27: :: EUCLIDLP:27
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Natfor b
6, b
7, b
8 being
Element of
REAL b
5 holds b
1 * (((b2 * b6) + (b3 * b7)) + (b4 * b8)) = (((b1 * b2) * b6) + ((b1 * b3) * b7)) + ((b1 * b4) * b8)
theorem Th28: :: EUCLIDLP:28
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Natfor b
6, b
7 being
Element of
REAL b
5 holds
((b1 * b6) + (b2 * b7)) + ((b3 * b6) + (b4 * b7)) = ((b1 + b3) * b6) + ((b2 + b4) * b7)
theorem Th29: :: EUCLIDLP:29
for b
1, b
2, b
3, b
4, b
5, b
6 being
Realfor b
7 being
Natfor b
8, b
9, b
10 being
Element of
REAL b
7 holds
(((b1 * b8) + (b2 * b9)) + (b3 * b10)) + (((b4 * b8) + (b5 * b9)) + (b6 * b10)) = (((b1 + b4) * b8) + ((b2 + b5) * b9)) + ((b3 + b6) * b10)
theorem Th30: :: EUCLIDLP:30
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Natfor b
6, b
7 being
Element of
REAL b
5 holds
((b1 * b6) + (b2 * b7)) - ((b3 * b6) + (b4 * b7)) = ((b1 - b3) * b6) + ((b2 - b4) * b7)
theorem Th31: :: EUCLIDLP:31
for b
1, b
2, b
3, b
4, b
5, b
6 being
Realfor b
7 being
Natfor b
8, b
9, b
10 being
Element of
REAL b
7 holds
(((b1 * b8) + (b2 * b9)) + (b3 * b10)) - (((b4 * b8) + (b5 * b9)) + (b6 * b10)) = (((b1 - b4) * b8) + ((b2 - b5) * b9)) + ((b3 - b6) * b10)
theorem Th32: :: EUCLIDLP:32
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5, b
6, b
7 being
Element of
REAL b
4 holds
(
(b1 + b2) + b
3 = 1 implies
((b1 * b5) + (b2 * b6)) + (b3 * b7) = (b5 + (b2 * (b6 - b5))) + (b3 * (b7 - b5)) )
theorem Th33: :: EUCLIDLP:33
for b
1, b
2 being
Realfor b
3 being
Natfor b
4, b
5, b
6, b
7 being
Element of
REAL b
3 holds
not ( b
4 = (b5 + (b1 * (b6 - b5))) + (b2 * (b7 - b5)) & ( for b
8 being
Real holds
not ( b
4 = ((b8 * b5) + (b1 * b6)) + (b2 * b7) &
(b8 + b1) + b
2 = 1 ) ) )
theorem Th34: :: EUCLIDLP:34
for b
1 being
Nat holds
not ( b
1 >= 1 & not
1* b
1 <> 0* b
1 )
theorem Th35: :: EUCLIDLP:35
theorem Th36: :: EUCLIDLP:36
:: deftheorem Def1 defines // EUCLIDLP:def 1 :
theorem Th37: :: EUCLIDLP:37
for b
1 being
Natfor b
2, b
3 being
Element of
REAL b
1 holds
not ( b
2 // b
3 & ( for b
4 being
Real holds
not ( b
4 <> 0 & b
2 = b
4 * b
3 ) ) )
theorem Th38: :: EUCLIDLP:38
for b
1 being
Natfor b
2, b
3, b
4 being
Element of
REAL b
1 holds
( b
2 // b
3 & b
3 // b
4 implies b
2 // b
4 )
:: deftheorem Def2 defines are_lindependent2 EUCLIDLP:def 2 :
Lemma40:
for b1 being Nat
for b2, b3 being Element of REAL b1 holds
not ( b2,b3 are_lindependent2 & not b2 <> 0* b1 )
theorem Th39: :: EUCLIDLP:39
theorem Th40: :: EUCLIDLP:40
theorem Th41: :: EUCLIDLP:41
for b
1, b
2, b
3, b
4 being
Realfor b
5 being
Natfor b
6, b
7, b
8, b
9, b
10 being
Element of
REAL b
5 holds
not ( b
10,b
6 are_lindependent2 & b
10 = (b1 * b7) + (b2 * b8) & b
6 = (b3 * b7) + (b4 * b8) & ( for b
11, b
12, b
13, b
14 being
Real holds
not ( b
7 = (b11 * b10) + (b12 * b6) & b
8 = (b13 * b10) + (b14 * b6) ) ) )
theorem Th42: :: EUCLIDLP:42
theorem Th43: :: EUCLIDLP:43
theorem Th44: :: EUCLIDLP:44
theorem Th45: :: EUCLIDLP:45
for b
1 being
Natfor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
Element of
REAL b
1 holds
( b
2 - b
3,b
4 - b
5 are_lindependent2 & b
6 in Line b
3,b
2 & b
7 in Line b
3,b
2 & b
6 <> b
7 & b
8 in Line b
5,b
4 & b
9 in Line b
5,b
4 & b
8 <> b
9 implies b
7 - b
6,b
9 - b
8 are_lindependent2 )
Lemma45:
for b1 being Nat
for b2, b3 being Element of REAL b1 holds
not ( b2 // b3 & not b2,b3 are_ldependent2 )
theorem Th46: :: EUCLIDLP:46
Lemma46:
for b1 being Nat
for b2, b3 being Element of REAL b1 holds
( b2,b3 are_ldependent2 & b2 <> 0* b1 & b3 <> 0* b1 implies b2 // b3 )
theorem Th47: :: EUCLIDLP:47
theorem Th48: :: EUCLIDLP:48
:: deftheorem Def3 defines _|_ EUCLIDLP:def 3 :
theorem Th49: :: EUCLIDLP:49
theorem Th50: :: EUCLIDLP:50
theorem Th51: :: EUCLIDLP:51
:: deftheorem Def4 defines line_of_REAL EUCLIDLP:def 4 :
theorem Th52: :: EUCLIDLP:52
theorem Th53: :: EUCLIDLP:53
theorem Th54: :: EUCLIDLP:54
theorem Th55: :: EUCLIDLP:55
theorem Th56: :: EUCLIDLP:56
theorem Th57: :: EUCLIDLP:57
theorem Th58: :: EUCLIDLP:58
theorem Th59: :: EUCLIDLP:59
theorem Th60: :: EUCLIDLP:60
:: deftheorem Def5 defines dist_v EUCLIDLP:def 5 :
:: deftheorem Def6 defines dist EUCLIDLP:def 6 :
theorem Th61: :: EUCLIDLP:61
theorem Th62: :: EUCLIDLP:62
theorem Th63: :: EUCLIDLP:63
theorem Th64: :: EUCLIDLP:64
definition
let c
1 be
Nat;
let c
2, c
3 be
Element of
line_of_REAL c
1;
pred c
2 // c
3 means :
Def7:
:: EUCLIDLP:def 7
ex b
1, b
2, b
3, b
4 being
Element of
REAL a
1 st
( a
2 = Line b
1,b
2 & a
3 = Line b
3,b
4 & b
2 - b
1 // b
4 - b
3 );
symmetry
for b1, b2 being Element of line_of_REAL c1 holds
not ( ex b3, b4, b5, b6 being Element of REAL c1 st
( b1 = Line b3,b4 & b2 = Line b5,b6 & b4 - b3 // b6 - b5 ) & ( for b3, b4, b5, b6 being Element of REAL c1 holds
not ( b2 = Line b3,b4 & b1 = Line b5,b6 & b4 - b3 // b6 - b5 ) ) )
;
end;
:: deftheorem Def7 defines // EUCLIDLP:def 7 :
theorem Th65: :: EUCLIDLP:65
definition
let c
1 be
Nat;
let c
2, c
3 be
Element of
line_of_REAL c
1;
pred c
2 _|_ c
3 means :
Def8:
:: EUCLIDLP:def 8
ex b
1, b
2, b
3, b
4 being
Element of
REAL a
1 st
( a
2 = Line b
1,b
2 & a
3 = Line b
3,b
4 & b
2 - b
1 _|_ b
4 - b
3 );
symmetry
for b1, b2 being Element of line_of_REAL c1 holds
not ( ex b3, b4, b5, b6 being Element of REAL c1 st
( b1 = Line b3,b4 & b2 = Line b5,b6 & b4 - b3 _|_ b6 - b5 ) & ( for b3, b4, b5, b6 being Element of REAL c1 holds
not ( b2 = Line b3,b4 & b1 = Line b5,b6 & b4 - b3 _|_ b6 - b5 ) ) )
;
end;
:: deftheorem Def8 defines _|_ EUCLIDLP:def 8 :
theorem Th66: :: EUCLIDLP:66
theorem Th67: :: EUCLIDLP:67
theorem Th68: :: EUCLIDLP:68
theorem Th69: :: EUCLIDLP:69
theorem Th70: :: EUCLIDLP:70
theorem Th71: :: EUCLIDLP:71
theorem Th72: :: EUCLIDLP:72
theorem Th73: :: EUCLIDLP:73
theorem Th74: :: EUCLIDLP:74
theorem Th75: :: EUCLIDLP:75
theorem Th76: :: EUCLIDLP:76
theorem Th77: :: EUCLIDLP:77
theorem Th78: :: EUCLIDLP:78
theorem Th79: :: EUCLIDLP:79
theorem Th80: :: EUCLIDLP:80
theorem Th81: :: EUCLIDLP:81
theorem Th82: :: EUCLIDLP:82
theorem Th83: :: EUCLIDLP:83
for b
1 being
Natfor b
2, b
3, b
4, b
5, b
6 being
Element of
REAL b
1for b
7, b
8 being
Element of
line_of_REAL b
1 holds
( b
2 - b
3,b
4 - b
3 are_lindependent2 & b
5 in Line b
3,b
2 & b
6 in Line b
3,b
4 & b
7 = Line b
2,b
4 & b
8 = Line b
5,b
6 implies ( b
7 // b
8 iff ex b
9 being
Real st
( b
9 <> 0 & b
5 - b
3 = b
9 * (b2 - b3) & b
6 - b
3 = b
9 * (b4 - b3) ) ) )
theorem Th84: :: EUCLIDLP:84
theorem Th85: :: EUCLIDLP:85
theorem Th86: :: EUCLIDLP:86
:: deftheorem Def9 defines plane EUCLIDLP:def 9 :
:: deftheorem Def10 defines being_plane EUCLIDLP:def 10 :
theorem Th87: :: EUCLIDLP:87
theorem Th88: :: EUCLIDLP:88
for b
1 being
Natfor b
2, b
3, b
4, b
5, b
6, b
7 being
Element of
REAL b
1 holds
( b
2 in plane b
3,b
4,b
5 & b
6 in plane b
3,b
4,b
5 & b
7 in plane b
3,b
4,b
5 implies
plane b
2,b
6,b
7 c= plane b
3,b
4,b
5 )
theorem Th89: :: EUCLIDLP:89
theorem Th90: :: EUCLIDLP:90
for b
1 being
Natfor b
2, b
3, b
4, b
5, b
6 being
Element of
REAL b
1 holds
( b
2 in plane b
3,b
4,b
5 & b
6 in plane b
3,b
4,b
5 implies
Line b
2,b
6 c= plane b
3,b
4,b
5 )
theorem Th91: :: EUCLIDLP:91
theorem Th92: :: EUCLIDLP:92
for b
1, b
2, b
3 being
Realfor b
4 being
Natfor b
5, b
6, b
7, b
8 being
Element of
REAL b
4 holds
not ( b
5 - b
5,b
6 - b
5 are_lindependent2 & b
7 in plane b
5,b
8,b
6 & b
7 = ((b1 * b5) + (b2 * b8)) + (b3 * b6) & not
(b1 + b2) + b
3 = 1 & not
0* b
4 in plane b
5,b
8,b
6 )
theorem Th93: :: EUCLIDLP:93
for b
1 being
Natfor b
2, b
3, b
4, b
5 being
Element of
REAL b
1 holds
( b
2 in plane b
3,b
4,b
5 iff ex b
6, b
7, b
8 being
Real st
(
(b6 + b7) + b
8 = 1 & b
2 = ((b6 * b3) + (b7 * b4)) + (b8 * b5) ) )
theorem Th94: :: EUCLIDLP:94
for b
1, b
2, b
3, b
4, b
5, b
6 being
Realfor b
7 being
Natfor b
8, b
9, b
10, b
11 being
Element of
REAL b
7 holds
( b
8 - b
9,b
10 - b
9 are_lindependent2 & b
11 in plane b
9,b
8,b
10 &
(b1 + b2) + b
3 = 1 & b
11 = ((b1 * b9) + (b2 * b8)) + (b3 * b10) &
(b4 + b5) + b
6 = 1 & b
11 = ((b4 * b9) + (b5 * b8)) + (b6 * b10) implies ( b
1 = b
4 & b
2 = b
5 & b
3 = b
6 ) )
:: deftheorem Def11 defines plane_of_REAL EUCLIDLP:def 11 :
theorem Th95: :: EUCLIDLP:95
theorem Th96: :: EUCLIDLP:96
theorem Th97: :: EUCLIDLP:97
theorem Th98: :: EUCLIDLP:98
theorem Th99: :: EUCLIDLP:99
for b
1 being
Natfor b
2, b
3, b
4 being
Element of
REAL b
1 holds
(
Line b
2,b
3 c= plane b
2,b
3,b
4 &
Line b
3,b
4 c= plane b
2,b
3,b
4 &
Line b
4,b
2 c= plane b
2,b
3,b
4 )
theorem Th100: :: EUCLIDLP:100
:: deftheorem Def12 defines are_coplane EUCLIDLP:def 12 :
theorem Th101: :: EUCLIDLP:101
theorem Th102: :: EUCLIDLP:102
theorem Th103: :: EUCLIDLP:103
theorem Th104: :: EUCLIDLP:104
theorem Th105: :: EUCLIDLP:105
theorem Th106: :: EUCLIDLP:106
theorem Th107: :: EUCLIDLP:107
theorem Th108: :: EUCLIDLP:108
theorem Th109: :: EUCLIDLP:109
theorem Th110: :: EUCLIDLP:110
theorem Th111: :: EUCLIDLP:111
theorem Th112: :: EUCLIDLP:112
theorem Th113: :: EUCLIDLP:113
theorem Th114: :: EUCLIDLP:114
theorem Th115: :: EUCLIDLP:115
theorem Th116: :: EUCLIDLP:116
theorem Th117: :: EUCLIDLP:117
theorem Th118: :: EUCLIDLP:118
theorem Th119: :: EUCLIDLP:119