:: EUCLID_3 semantic presentation
:: deftheorem Def1 defines cpx2euc EUCLID_3:def 1 :
:: deftheorem Def2 defines euc2cpx EUCLID_3:def 2 :
theorem Th1: :: EUCLID_3:1
theorem Th2: :: EUCLID_3:2
theorem Th3: :: EUCLID_3:3
theorem Th4: :: EUCLID_3:4
theorem Th5: :: EUCLID_3:5
theorem Th6: :: EUCLID_3:6
theorem Th7: :: EUCLID_3:7
theorem Th8: :: EUCLID_3:8
theorem Th9: :: EUCLID_3:9
theorem Th10: :: EUCLID_3:10
theorem Th11: :: EUCLID_3:11
theorem Th12: :: EUCLID_3:12
theorem Th13: :: EUCLID_3:13
theorem Th14: :: EUCLID_3:14
theorem Th15: :: EUCLID_3:15
theorem Th16: :: EUCLID_3:16
theorem Th17: :: EUCLID_3:17
theorem Th18: :: EUCLID_3:18
theorem Th19: :: EUCLID_3:19
theorem Th20: :: EUCLID_3:20
theorem Th21: :: EUCLID_3:21
theorem Th22: :: EUCLID_3:22
theorem Th23: :: EUCLID_3:23
theorem Th24: :: EUCLID_3:24
theorem Th25: :: EUCLID_3:25
theorem Th26: :: EUCLID_3:26
theorem Th27: :: EUCLID_3:27
theorem Th28: :: EUCLID_3:28
theorem Th29: :: EUCLID_3:29
theorem Th30: :: EUCLID_3:30
theorem Th31: :: EUCLID_3:31
:: deftheorem Def3 defines Arg EUCLID_3:def 3 :
theorem Th32: :: EUCLID_3:32
theorem Th33: :: EUCLID_3:33
theorem Th34: :: EUCLID_3:34
theorem Th35: :: EUCLID_3:35
theorem Th36: :: EUCLID_3:36
theorem Th37: :: EUCLID_3:37
theorem Th38: :: EUCLID_3:38
theorem Th39: :: EUCLID_3:39
theorem Th40: :: EUCLID_3:40
theorem Th41: :: EUCLID_3:41
theorem Th42: :: EUCLID_3:42
:: deftheorem Def4 defines angle EUCLID_3:def 4 :
theorem Th43: :: EUCLID_3:43
theorem Th44: :: EUCLID_3:44
theorem Th45: :: EUCLID_3:45
theorem Th46: :: EUCLID_3:46
theorem Th47: :: EUCLID_3:47
theorem Th48: :: EUCLID_3:48
theorem Th49: :: EUCLID_3:49
theorem Th50: :: EUCLID_3:50
theorem Th51: :: EUCLID_3:51
theorem Th52: :: EUCLID_3:52
theorem Th53: :: EUCLID_3:53
theorem Th54: :: EUCLID_3:54
theorem Th55: :: EUCLID_3:55
theorem Th56: :: EUCLID_3:56
for
p1,
p2,
p3 being
Point of
(TOP-REAL 2) st
p2 <> p1 &
p1 <> p3 &
p3 <> p2 &
angle p2,
p1,
p3 < PI &
angle p1,
p3,
p2 < PI &
angle p3,
p2,
p1 < PI holds
((angle p2,p1,p3) + (angle p1,p3,p2)) + (angle p3,p2,p1) = PI
definition
let n be
Element of
NAT ;
let p1 be
Point of
(TOP-REAL n),
p2 be
Point of
(TOP-REAL n),
p3 be
Point of
(TOP-REAL n);
func Triangle c2,
c3,
c4 -> Subset of
(TOP-REAL a1) equals :: EUCLID_3:def 5
((LSeg p1,p2) \/ (LSeg p2,p3)) \/ (LSeg p3,p1);
correctness
coherence
((LSeg p1,p2) \/ (LSeg p2,p3)) \/ (LSeg p3,p1) is Subset of (TOP-REAL n);
;
end;
:: deftheorem Def5 defines Triangle EUCLID_3:def 5 :
:: deftheorem Def6 defines closed_inside_of_triangle EUCLID_3:def 6 :
definition
let n be
Element of
NAT ;
let p1 be
Point of
(TOP-REAL n),
p2 be
Point of
(TOP-REAL n),
p3 be
Point of
(TOP-REAL n);
func inside_of_triangle c2,
c3,
c4 -> Subset of
(TOP-REAL a1) equals :: EUCLID_3:def 7
(closed_inside_of_triangle p1,p2,p3) \ (Triangle p1,p2,p3);
correctness
coherence
(closed_inside_of_triangle p1,p2,p3) \ (Triangle p1,p2,p3) is Subset of (TOP-REAL n);
;
end;
:: deftheorem Def7 defines inside_of_triangle EUCLID_3:def 7 :
:: deftheorem Def8 defines outside_of_triangle EUCLID_3:def 8 :
definition
let n be
Element of
NAT ;
let p1 be
Point of
(TOP-REAL n),
p2 be
Point of
(TOP-REAL n),
p3 be
Point of
(TOP-REAL n);
func plane c2,
c3,
c4 -> Subset of
(TOP-REAL a1) equals :: EUCLID_3:def 9
(outside_of_triangle p1,p2,p3) \/ (closed_inside_of_triangle p1,p2,p3);
correctness
coherence
(outside_of_triangle p1,p2,p3) \/ (closed_inside_of_triangle p1,p2,p3) is Subset of (TOP-REAL n);
;
end;
:: deftheorem Def9 defines plane EUCLID_3:def 9 :
theorem Th57: :: EUCLID_3:57
theorem Th58: :: EUCLID_3:58
:: deftheorem Def10 defines are_lindependent2 EUCLID_3:def 10 :
theorem Th59: :: EUCLID_3:59
theorem Th60: :: EUCLID_3:60
for
n being
Element of
NAT for
p1,
p2,
p3,
p0 being
Point of
(TOP-REAL n) st
p2 - p1,
p3 - p1 are_lindependent2 &
p0 in plane p1,
p2,
p3 holds
ex
a1,
a2,
a3 being
Real st
(
p0 = ((a1 * p1) + (a2 * p2)) + (a3 * p3) &
(a1 + a2) + a3 = 1 & ( for
b1,
b2,
b3 being
Real st
p0 = ((b1 * p1) + (b2 * p2)) + (b3 * p3) &
(b1 + b2) + b3 = 1 holds
(
b1 = a1 &
b2 = a2 &
b3 = a3 ) ) )
theorem Th61: :: EUCLID_3:61
theorem Th62: :: EUCLID_3:62
theorem Th63: :: EUCLID_3:63
definition
let n be
Element of
NAT ;
let p1 be
Point of
(TOP-REAL n),
p2 be
Point of
(TOP-REAL n),
p3 be
Point of
(TOP-REAL n),
p be
Point of
(TOP-REAL n);
assume E41:
(
p2 - p1,
p3 - p1 are_lindependent2 &
p in plane p1,
p2,
p3 )
;
func tricord1 c2,
c3,
c4,
c5 -> Real means :
Def11:
:: EUCLID_3:def 11
ex
a2,
a3 being
Real st
(
(it + a2) + a3 = 1 &
p = ((it * p1) + (a2 * p2)) + (a3 * p3) );
existence
ex b1, a2, a3 being Real st
( (b1 + a2) + a3 = 1 & p = ((b1 * p1) + (a2 * p2)) + (a3 * p3) )
uniqueness
for b1, b2 being Real st ex a2, a3 being Real st
( (b1 + a2) + a3 = 1 & p = ((b1 * p1) + (a2 * p2)) + (a3 * p3) ) & ex a2, a3 being Real st
( (b2 + a2) + a3 = 1 & p = ((b2 * p1) + (a2 * p2)) + (a3 * p3) ) holds
b1 = b2
end;
:: deftheorem Def11 defines tricord1 EUCLID_3:def 11 :
for
n being
Element of
NAT for
p1,
p2,
p3,
p being
Point of
(TOP-REAL n) st
p2 - p1,
p3 - p1 are_lindependent2 &
p in plane p1,
p2,
p3 holds
for
b6 being
Real holds
(
b6 = tricord1 p1,
p2,
p3,
p iff ex
a2,
a3 being
Real st
(
(b6 + a2) + a3 = 1 &
p = ((b6 * p1) + (a2 * p2)) + (a3 * p3) ) );
definition
let n be
Element of
NAT ;
let p1 be
Point of
(TOP-REAL n),
p2 be
Point of
(TOP-REAL n),
p3 be
Point of
(TOP-REAL n),
p be
Point of
(TOP-REAL n);
assume E41:
(
p2 - p1,
p3 - p1 are_lindependent2 &
p in plane p1,
p2,
p3 )
;
func tricord2 c2,
c3,
c4,
c5 -> Real means :
Def12:
:: EUCLID_3:def 12
ex
a1,
a3 being
Real st
(
(a1 + it) + a3 = 1 &
p = ((a1 * p1) + (it * p2)) + (a3 * p3) );
existence
ex b1, a1, a3 being Real st
( (a1 + b1) + a3 = 1 & p = ((a1 * p1) + (b1 * p2)) + (a3 * p3) )
uniqueness
for b1, b2 being Real st ex a1, a3 being Real st
( (a1 + b1) + a3 = 1 & p = ((a1 * p1) + (b1 * p2)) + (a3 * p3) ) & ex a1, a3 being Real st
( (a1 + b2) + a3 = 1 & p = ((a1 * p1) + (b2 * p2)) + (a3 * p3) ) holds
b1 = b2
end;
:: deftheorem Def12 defines tricord2 EUCLID_3:def 12 :
for
n being
Element of
NAT for
p1,
p2,
p3,
p being
Point of
(TOP-REAL n) st
p2 - p1,
p3 - p1 are_lindependent2 &
p in plane p1,
p2,
p3 holds
for
b6 being
Real holds
(
b6 = tricord2 p1,
p2,
p3,
p iff ex
a1,
a3 being
Real st
(
(a1 + b6) + a3 = 1 &
p = ((a1 * p1) + (b6 * p2)) + (a3 * p3) ) );
definition
let n be
Element of
NAT ;
let p1 be
Point of
(TOP-REAL n),
p2 be
Point of
(TOP-REAL n),
p3 be
Point of
(TOP-REAL n),
p be
Point of
(TOP-REAL n);
assume E41:
(
p2 - p1,
p3 - p1 are_lindependent2 &
p in plane p1,
p2,
p3 )
;
func tricord3 c2,
c3,
c4,
c5 -> Real means :
Def13:
:: EUCLID_3:def 13
ex
a1,
a2 being
Real st
(
(a1 + a2) + it = 1 &
p = ((a1 * p1) + (a2 * p2)) + (it * p3) );
existence
ex b1, a1, a2 being Real st
( (a1 + a2) + b1 = 1 & p = ((a1 * p1) + (a2 * p2)) + (b1 * p3) )
uniqueness
for b1, b2 being Real st ex a1, a2 being Real st
( (a1 + a2) + b1 = 1 & p = ((a1 * p1) + (a2 * p2)) + (b1 * p3) ) & ex a1, a2 being Real st
( (a1 + a2) + b2 = 1 & p = ((a1 * p1) + (a2 * p2)) + (b2 * p3) ) holds
b1 = b2
end;
:: deftheorem Def13 defines tricord3 EUCLID_3:def 13 :
for
n being
Element of
NAT for
p1,
p2,
p3,
p being
Point of
(TOP-REAL n) st
p2 - p1,
p3 - p1 are_lindependent2 &
p in plane p1,
p2,
p3 holds
for
b6 being
Real holds
(
b6 = tricord3 p1,
p2,
p3,
p iff ex
a1,
a2 being
Real st
(
(a1 + a2) + b6 = 1 &
p = ((a1 * p1) + (a2 * p2)) + (b6 * p3) ) );
definition
let p1 be
Point of
(TOP-REAL 2);
let p2 be
Point of
(TOP-REAL 2);
let p3 be
Point of
(TOP-REAL 2);
func trcmap1 c1,
c2,
c3 -> Function of
(TOP-REAL 2),
R^1 means :: EUCLID_3:def 14
for
p being
Point of
(TOP-REAL 2) holds
it . p = tricord1 p1,
p2,
p3,
p;
existence
ex b1 being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b1 . p = tricord1 p1,p2,p3,p
uniqueness
for b1, b2 being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds b1 . p = tricord1 p1,p2,p3,p ) & ( for p being Point of (TOP-REAL 2) holds b2 . p = tricord1 p1,p2,p3,p ) holds
b1 = b2
end;
:: deftheorem Def14 defines trcmap1 EUCLID_3:def 14 :
definition
let p1 be
Point of
(TOP-REAL 2);
let p2 be
Point of
(TOP-REAL 2);
let p3 be
Point of
(TOP-REAL 2);
func trcmap2 c1,
c2,
c3 -> Function of
(TOP-REAL 2),
R^1 means :: EUCLID_3:def 15
for
p being
Point of
(TOP-REAL 2) holds
it . p = tricord2 p1,
p2,
p3,
p;
existence
ex b1 being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b1 . p = tricord2 p1,p2,p3,p
uniqueness
for b1, b2 being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds b1 . p = tricord2 p1,p2,p3,p ) & ( for p being Point of (TOP-REAL 2) holds b2 . p = tricord2 p1,p2,p3,p ) holds
b1 = b2
end;
:: deftheorem Def15 defines trcmap2 EUCLID_3:def 15 :
definition
let p1 be
Point of
(TOP-REAL 2);
let p2 be
Point of
(TOP-REAL 2);
let p3 be
Point of
(TOP-REAL 2);
func trcmap3 c1,
c2,
c3 -> Function of
(TOP-REAL 2),
R^1 means :: EUCLID_3:def 16
for
p being
Point of
(TOP-REAL 2) holds
it . p = tricord3 p1,
p2,
p3,
p;
existence
ex b1 being Function of (TOP-REAL 2),R^1 st
for p being Point of (TOP-REAL 2) holds b1 . p = tricord3 p1,p2,p3,p
uniqueness
for b1, b2 being Function of (TOP-REAL 2),R^1 st ( for p being Point of (TOP-REAL 2) holds b1 . p = tricord3 p1,p2,p3,p ) & ( for p being Point of (TOP-REAL 2) holds b2 . p = tricord3 p1,p2,p3,p ) holds
b1 = b2
end;
:: deftheorem Def16 defines trcmap3 EUCLID_3:def 16 :
theorem Th64: :: EUCLID_3:64
for
p1,
p2,
p3,
p being
Point of
(TOP-REAL 2) st
p2 - p1,
p3 - p1 are_lindependent2 holds
(
p in outside_of_triangle p1,
p2,
p3 iff (
tricord1 p1,
p2,
p3,
p < 0 or
tricord2 p1,
p2,
p3,
p < 0 or
tricord3 p1,
p2,
p3,
p < 0 ) )
theorem Th65: :: EUCLID_3:65
for
p1,
p2,
p3,
p being
Point of
(TOP-REAL 2) st
p2 - p1,
p3 - p1 are_lindependent2 holds
(
p in Triangle p1,
p2,
p3 iff (
tricord1 p1,
p2,
p3,
p >= 0 &
tricord2 p1,
p2,
p3,
p >= 0 &
tricord3 p1,
p2,
p3,
p >= 0 & (
tricord1 p1,
p2,
p3,
p = 0 or
tricord2 p1,
p2,
p3,
p = 0 or
tricord3 p1,
p2,
p3,
p = 0 ) ) )
theorem Th66: :: EUCLID_3:66
for
p1,
p2,
p3,
p being
Point of
(TOP-REAL 2) st
p2 - p1,
p3 - p1 are_lindependent2 holds
(
p in Triangle p1,
p2,
p3 iff ( (
tricord1 p1,
p2,
p3,
p = 0 &
tricord2 p1,
p2,
p3,
p >= 0 &
tricord3 p1,
p2,
p3,
p >= 0 ) or (
tricord1 p1,
p2,
p3,
p >= 0 &
tricord2 p1,
p2,
p3,
p = 0 &
tricord3 p1,
p2,
p3,
p >= 0 ) or (
tricord1 p1,
p2,
p3,
p >= 0 &
tricord2 p1,
p2,
p3,
p >= 0 &
tricord3 p1,
p2,
p3,
p = 0 ) ) )
by ;
theorem Th67: :: EUCLID_3:67
for
p1,
p2,
p3,
p being
Point of
(TOP-REAL 2) st
p2 - p1,
p3 - p1 are_lindependent2 holds
(
p in inside_of_triangle p1,
p2,
p3 iff (
tricord1 p1,
p2,
p3,
p > 0 &
tricord2 p1,
p2,
p3,
p > 0 &
tricord3 p1,
p2,
p3,
p > 0 ) )
theorem Th68: :: EUCLID_3:68