:: GEOMTRAP semantic presentation
:: deftheorem Def1 defines '||' GEOMTRAP:def 1 :
theorem Th1: :: GEOMTRAP:1
theorem Th2: :: GEOMTRAP:2
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1for b
6, b
7, b
8, b
9 being
Element of
(OASpace b1) holds
( b
6 = b
2 & b
7 = b
3 & b
8 = b
4 & b
9 = b
5 implies ( b
6,b
7 // b
8,b
9 iff b
2,b
3 // b
4,b
5 ) )
theorem Th3: :: GEOMTRAP:3
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
8, b
9, b
10, b
11 being
Element of the
carrier of
(Lambda (OASpace b1)) holds
( b
8 = b
4 & b
9 = b
5 & b
10 = b
6 & b
11 = b
7 implies ( b
8,b
9 // b
10,b
11 iff b
4,b
5 '||' b
6,b
7 ) ) )
theorem Th4: :: GEOMTRAP:4
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1for b
8, b
9, b
10, b
11 being
Element of the
carrier of
(AMSpace b1,b2,b3) holds
( b
8 = b
4 & b
9 = b
5 & b
10 = b
6 & b
11 = b
7 implies ( b
8,b
9 // b
10,b
11 iff b
4,b
5 '||' b
6,b
7 ) )
:: deftheorem Def2 defines # GEOMTRAP:def 2 :
theorem Th5: :: GEOMTRAP:5
canceled;
theorem Th6: :: GEOMTRAP:6
canceled;
theorem Th7: :: GEOMTRAP:7
theorem Th8: :: GEOMTRAP:8
theorem Th9: :: GEOMTRAP:9
theorem Th10: :: GEOMTRAP:10
theorem Th11: :: GEOMTRAP:11
theorem Th12: :: GEOMTRAP:12
theorem Th13: :: GEOMTRAP:13
theorem Th14: :: GEOMTRAP:14
Lemma14:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds
( b2,b3 // b3,b4 implies ( b4,b3 // b3,b2 & b2,b3 // b2,b4 & b3,b4 // b2,b4 ) )
theorem Th15: :: GEOMTRAP:15
theorem Th16: :: GEOMTRAP:16
Lemma17:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1 holds
( b2,b3 // b4,b5 implies b2,b3 '||' b2 # b5,b3 # b4 )
Lemma18:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1 holds
( b2 # b3 = b4 # b5 implies b4 - b2 = - (b5 - b3) )
Lemma19:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7 being VECTOR of b1 holds
( Gen b2,b3 & b4,b5,b6,b7 are_Ort_wrt b2,b3 implies ( b6,b7,b4,b5 are_Ort_wrt b2,b3 & b4,b5,b7,b6 are_Ort_wrt b2,b3 ) )
Lemma20:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being VECTOR of b1 holds
( Gen b2,b3 implies b4,b5,b6,b6 are_Ort_wrt b2,b3 )
Lemma21:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7, b8 being VECTOR of b1 holds
( Gen b2,b3 & b4,b5,b6,b7 are_Ort_wrt b2,b3 & b4,b5,b6,b8 are_Ort_wrt b2,b3 implies b4,b5,b7,b8 are_Ort_wrt b2,b3 )
Lemma22:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1 holds
( Gen b2,b3 & b4,b5,b4,b5 are_Ort_wrt b2,b3 implies b4 = b5 )
Lemma23:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6 being VECTOR of b1 holds
( Gen b2,b3 implies ( b4,b5,b6,b6 are_Ort_wrt b2,b3 & b6,b6,b4,b5 are_Ort_wrt b2,b3 ) )
Lemma24:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7, b8, b9 being VECTOR of b1 holds
( Gen b2,b3 & ( b4,b5 '||' b6,b7 or b6,b7 '||' b4,b5 ) & ( b8,b9,b6,b7 are_Ort_wrt b2,b3 or b6,b7,b8,b9 are_Ort_wrt b2,b3 ) & b6 <> b7 implies ( b4,b5,b8,b9 are_Ort_wrt b2,b3 & b8,b9,b4,b5 are_Ort_wrt b2,b3 ) )
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5, c
6, c
7 be
VECTOR of c
1;
pred c
4,c
5,c
6,c
7 are_DTr_wrt c
2,c
3 means :
Def3:
:: GEOMTRAP:def 3
( a
4,a
5 // a
6,a
7 & a
4,a
5,a
4 # a
5,a
6 # a
7 are_Ort_wrt a
2,a
3 & a
6,a
7,a
4 # a
5,a
6 # a
7 are_Ort_wrt a
2,a
3 );
end;
:: deftheorem Def3 defines are_DTr_wrt GEOMTRAP:def 3 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
( b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 iff ( b
4,b
5 // b
6,b
7 & b
4,b
5,b
4 # b
5,b
6 # b
7 are_Ort_wrt b
2,b
3 & b
6,b
7,b
4 # b
5,b
6 # b
7 are_Ort_wrt b
2,b
3 ) );
theorem Th17: :: GEOMTRAP:17
theorem Th18: :: GEOMTRAP:18
theorem Th19: :: GEOMTRAP:19
theorem Th20: :: GEOMTRAP:20
theorem Th21: :: GEOMTRAP:21
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 & b
4,b
5,b
8,b
9 are_DTr_wrt b
2,b
3 & b
4 <> b
5 implies b
6,b
7,b
8,b
9 are_DTr_wrt b
2,b
3 )
theorem Th22: :: GEOMTRAP:22
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & ( for b
7 being
VECTOR of b
1 holds
( not b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 & not b
4,b
5,b
7,b
6 are_DTr_wrt b
2,b
3 ) ) )
theorem Th23: :: GEOMTRAP:23
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 implies b
6,b
7,b
4,b
5 are_DTr_wrt b
2,b
3 )
theorem Th24: :: GEOMTRAP:24
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 implies b
5,b
4,b
7,b
6 are_DTr_wrt b
2,b
3 )
Lemma32:
for b1 being RealLinearSpace
for b2, b3, b4, b5, b6, b7, b8, b9 being VECTOR of b1 holds
( Gen b2,b3 & b4 <> b5 & b4,b5 '||' b4,b6 & b4,b5 '||' b4,b7 & b4,b5 '||' b4,b8 & b4,b5 '||' b4,b9 implies b6,b7 '||' b8,b9 )
theorem Th25: :: GEOMTRAP:25
theorem Th26: :: GEOMTRAP:26
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 & b
4,b
5,b
6,b
8 are_DTr_wrt b
2,b
3 & not b
4 = b
5 & not b
7 = b
8 )
theorem Th27: :: GEOMTRAP:27
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4 <> b
5 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 & ( b
4,b
5,b
6,b
8 are_DTr_wrt b
2,b
3 or b
4,b
5,b
8,b
6 are_DTr_wrt b
2,b
3 ) implies b
7 = b
8 )
theorem Th28: :: GEOMTRAP:28
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 implies b
4,b
5,b
4 # b
6,b
5 # b
7 are_DTr_wrt b
2,b
3 )
theorem Th29: :: GEOMTRAP:29
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
not (
Gen b
2,b
3 & b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 & not b
4,b
5,b
4 # b
7,b
5 # b
6 are_DTr_wrt b
2,b
3 & not b
4,b
5,b
5 # b
6,b
4 # b
7 are_DTr_wrt b
2,b
3 )
theorem Th30: :: GEOMTRAP:30
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4 = b
5 # b
9 & b
4 = b
6 # b
10 & b
4 = b
7 # b
11 & b
4 = b
8 # b
12 & b
5,b
6,b
7,b
8 are_DTr_wrt b
2,b
3 implies b
9,b
10,b
11,b
12 are_DTr_wrt b
2,b
3 )
Lemma39:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1
for b6, b7, b8, b9 being Real holds
( b2 = (b6 * b3) + (b7 * b4) & b5 = (b8 * b3) + (b9 * b4) implies ( b2 + b5 = ((b6 + b8) * b3) + ((b7 + b9) * b4) & b2 - b5 = ((b6 - b8) * b3) + ((b7 - b9) * b4) ) )
Lemma40:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1
for b5, b6, b7 being Real holds
( b2 = (b5 * b3) + (b6 * b4) implies b7 * b2 = ((b7 * b5) * b3) + ((b7 * b6) * b4) )
Lemma41:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Real holds
( Gen b2,b3 & (b4 * b2) + (b5 * b3) = (b6 * b2) + (b7 * b3) implies ( b4 = b6 & b5 = b7 ) )
:: deftheorem Def4 defines pr1 GEOMTRAP:def 4 :
:: deftheorem Def5 defines pr2 GEOMTRAP:def 5 :
Lemma44:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1 holds
( Gen b2,b3 implies b4 = ((pr1 b2,b3,b4) * b2) + ((pr2 b2,b3,b4) * b3) )
Lemma45:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1
for b5, b6 being Real holds
( Gen b2,b3 & b4 = (b5 * b2) + (b6 * b3) implies ( b5 = pr1 b2,b3,b4 & b6 = pr2 b2,b3,b4 ) )
Lemma46:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1
for b6 being Real holds
( Gen b2,b3 implies ( pr1 b2,b3,(b4 + b5) = (pr1 b2,b3,b4) + (pr1 b2,b3,b5) & pr2 b2,b3,(b4 + b5) = (pr2 b2,b3,b4) + (pr2 b2,b3,b5) & pr1 b2,b3,(b4 - b5) = (pr1 b2,b3,b4) - (pr1 b2,b3,b5) & pr2 b2,b3,(b4 - b5) = (pr2 b2,b3,b4) - (pr2 b2,b3,b5) & pr1 b2,b3,(b6 * b4) = b6 * (pr1 b2,b3,b4) & pr2 b2,b3,(b6 * b4) = b6 * (pr2 b2,b3,b4) ) )
Lemma47:
for b1 being RealLinearSpace
for b2, b3, b4, b5 being VECTOR of b1 holds
( Gen b2,b3 implies ( 2 * (pr1 b2,b3,(b4 # b5)) = (pr1 b2,b3,b4) + (pr1 b2,b3,b5) & 2 * (pr2 b2,b3,(b4 # b5)) = (pr2 b2,b3,b4) + (pr2 b2,b3,b5) ) )
Lemma48:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1 holds (- b2) + (- b3) = - (b2 + b3)
Lemma49:
for b1 being RealLinearSpace
for b2, b3, b4 being VECTOR of b1
for b5, b6 being Real holds (b2 + (b5 * b3)) - (b4 + (b6 * b3)) = (b2 - b4) + ((b5 - b6) * b3)
definition
let c
1 be
RealLinearSpace;
let c
2, c
3, c
4, c
5 be
VECTOR of c
1;
func PProJ c
2,c
3,c
4,c
5 -> Real equals :: GEOMTRAP:def 6
((pr1 a2,a3,a4) * (pr1 a2,a3,a5)) + ((pr2 a2,a3,a4) * (pr2 a2,a3,a5));
correctness
coherence
((pr1 c2,c3,c4) * (pr1 c2,c3,c5)) + ((pr2 c2,c3,c4) * (pr2 c2,c3,c5)) is Real;
;
end;
:: deftheorem Def6 defines PProJ GEOMTRAP:def 6 :
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5 being
VECTOR of b
1 holds
PProJ b
2,b
3,b
4,b
5 = ((pr1 b2,b3,b4) * (pr1 b2,b3,b5)) + ((pr2 b2,b3,b4) * (pr2 b2,b3,b5));
theorem Th31: :: GEOMTRAP:31
theorem Th32: :: GEOMTRAP:32
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6 being
VECTOR of b
1 holds
(
PProJ b
2,b
3,b
4,
(b5 + b6) = (PProJ b2,b3,b4,b5) + (PProJ b2,b3,b4,b6) &
PProJ b
2,b
3,b
4,
(b5 - b6) = (PProJ b2,b3,b4,b5) - (PProJ b2,b3,b4,b6) &
PProJ b
2,b
3,
(b5 - b6),b
4 = (PProJ b2,b3,b5,b4) - (PProJ b2,b3,b6,b4) &
PProJ b
2,b
3,
(b5 + b6),b
4 = (PProJ b2,b3,b5,b4) + (PProJ b2,b3,b6,b4) ) )
theorem Th33: :: GEOMTRAP:33
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5 being
VECTOR of b
1for b
6 being
Real holds
(
PProJ b
2,b
3,
(b6 * b4),b
5 = b
6 * (PProJ b2,b3,b4,b5) &
PProJ b
2,b
3,b
4,
(b6 * b5) = b
6 * (PProJ b2,b3,b4,b5) &
PProJ b
2,b
3,
(b6 * b4),b
5 = (PProJ b2,b3,b4,b5) * b
6 &
PProJ b
2,b
3,b
4,
(b6 * b5) = (PProJ b2,b3,b4,b5) * b
6 ) )
theorem Th34: :: GEOMTRAP:34
theorem Th35: :: GEOMTRAP:35
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
( b
4,b
5,b
6,b
7 are_Ort_wrt b
2,b
3 iff
PProJ b
2,b
3,
(b5 - b4),
(b7 - b6) = 0 ) )
theorem Th36: :: GEOMTRAP:36
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6 being
VECTOR of b
1 holds 2
* (PProJ b2,b3,b4,(b5 # b6)) = (PProJ b2,b3,b4,b5) + (PProJ b2,b3,b4,b6) )
theorem Th37: :: GEOMTRAP:37
theorem Th38: :: GEOMTRAP:38
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8 being
VECTOR of b
1for b
9 being
Real holds
( b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 & b
4 <> b
5 & b
9 = ((PProJ b2,b3,(b4 - b5),(b4 + b5)) - (2 * (PProJ b2,b3,(b4 - b5),b6))) * ((PProJ b2,b3,(b4 - b5),(b4 - b5)) " ) & b
8 = b
6 + (b9 * (b4 - b5)) implies b
7 = b
8 ) )
Lemma57:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1 holds
( Gen b2,b3 implies for b4, b5, b6, b7, b8, b9 being VECTOR of b1
for b10, b11 being Real holds
( b4 <> b5 & b10 = ((PProJ b2,b3,(b4 - b5),(b4 + b5)) - (2 * (PProJ b2,b3,(b4 - b5),b6))) * ((PProJ b2,b3,(b4 - b5),(b4 - b5)) " ) & b11 = ((PProJ b2,b3,(b4 - b5),(b4 + b5)) - (2 * (PProJ b2,b3,(b4 - b5),b7))) * ((PProJ b2,b3,(b4 - b5),(b4 - b5)) " ) & b8 = b6 + (b10 * (b4 - b5)) & b9 = b7 + (b11 * (b4 - b5)) implies ( b9 - b8 = (b7 - b6) + ((b11 - b10) * (b4 - b5)) & b11 - b10 = ((- 2) * (PProJ b2,b3,(b4 - b5),(b7 - b6))) * ((PProJ b2,b3,(b4 - b5),(b4 - b5)) " ) ) ) )
theorem Th39: :: GEOMTRAP:39
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13 being
VECTOR of b
1 holds
( b
4 <> b
5 & b
4,b
5,b
6,b
10 are_DTr_wrt b
2,b
3 & b
4,b
5,b
7,b
11 are_DTr_wrt b
2,b
3 & b
4,b
5,b
8,b
12 are_DTr_wrt b
2,b
3 & b
4,b
5,b
9,b
13 are_DTr_wrt b
2,b
3 & b
6,b
7 // b
8,b
9 implies b
10,b
11 // b
12,b
13 ) )
theorem Th40: :: GEOMTRAP:40
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
VECTOR of b
1 holds
( b
4 <> b
5 & b
4,b
5,b
6,b
9 are_DTr_wrt b
2,b
3 & b
4,b
5,b
7,b
10 are_DTr_wrt b
2,b
3 & b
4,b
5,b
8,b
11 are_DTr_wrt b
2,b
3 & b
8 = b
6 # b
7 implies b
11 = b
9 # b
10 ) )
theorem Th41: :: GEOMTRAP:41
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8, b
9 being
VECTOR of b
1 holds
( b
4 <> b
5 & b
4,b
5,b
6,b
8 are_DTr_wrt b
2,b
3 & b
4,b
5,b
7,b
9 are_DTr_wrt b
2,b
3 implies b
4,b
5,b
6 # b
7,b
8 # b
9 are_DTr_wrt b
2,b
3 ) )
theorem Th42: :: GEOMTRAP:42
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1 holds
(
Gen b
2,b
3 implies for b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13 being
VECTOR of b
1 holds
( b
4 <> b
5 & b
4,b
5,b
6,b
10 are_DTr_wrt b
2,b
3 & b
4,b
5,b
7,b
11 are_DTr_wrt b
2,b
3 & b
4,b
5,b
8,b
12 are_DTr_wrt b
2,b
3 & b
4,b
5,b
9,b
13 are_DTr_wrt b
2,b
3 & b
6,b
7,b
8,b
9 are_Ort_wrt b
2,b
3 implies b
10,b
11,b
12,b
13 are_Ort_wrt b
2,b
3 ) )
theorem Th43: :: GEOMTRAP:43
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11, b
12, b
13 being
VECTOR of b
1 holds
(
Gen b
2,b
3 & b
4 <> b
5 & b
4,b
5,b
6,b
10 are_DTr_wrt b
2,b
3 & b
4,b
5,b
7,b
11 are_DTr_wrt b
2,b
3 & b
4,b
5,b
8,b
12 are_DTr_wrt b
2,b
3 & b
4,b
5,b
9,b
13 are_DTr_wrt b
2,b
3 & b
6,b
7,b
8,b
9 are_DTr_wrt b
2,b
3 implies b
10,b
11,b
12,b
13 are_DTr_wrt b
2,b
3 )
definition
let c
1 be
RealLinearSpace;
let c
2, c
3 be
VECTOR of c
1;
func DTrapezium c
1,c
2,c
3 -> Relation of
[:the carrier of a1,the carrier of a1:] means :
Def7:
:: GEOMTRAP:def 7
for b
1, b
2 being
set holds
(
[b1,b2] in a
4 iff ex b
3, b
4, b
5, b
6 being
VECTOR of a
1 st
( b
1 = [b3,b4] & b
2 = [b5,b6] & b
3,b
4,b
5,b
6 are_DTr_wrt a
2,a
3 ) );
existence
ex b1 being Relation of [:the carrier of c1,the carrier of c1:] st
for b2, b3 being set holds
( [b2,b3] in b1 iff ex b4, b5, b6, b7 being VECTOR of c1 st
( b2 = [b4,b5] & b3 = [b6,b7] & b4,b5,b6,b7 are_DTr_wrt c2,c3 ) )
uniqueness
for b1, b2 being Relation of [:the carrier of c1,the carrier of c1:] holds
( ( for b3, b4 being set holds
( [b3,b4] in b1 iff ex b5, b6, b7, b8 being VECTOR of c1 st
( b3 = [b5,b6] & b4 = [b7,b8] & b5,b6,b7,b8 are_DTr_wrt c2,c3 ) ) ) & ( for b3, b4 being set holds
( [b3,b4] in b2 iff ex b5, b6, b7, b8 being VECTOR of c1 st
( b3 = [b5,b6] & b4 = [b7,b8] & b5,b6,b7,b8 are_DTr_wrt c2,c3 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines DTrapezium GEOMTRAP:def 7 :
for b
1 being
RealLinearSpacefor b
2, b
3 being
VECTOR of b
1for b
4 being
Relation of
[:the carrier of b1,the carrier of b1:] holds
( b
4 = DTrapezium b
1,b
2,b
3 iff for b
5, b
6 being
set holds
(
[b5,b6] in b
4 iff ex b
7, b
8, b
9, b
10 being
VECTOR of b
1 st
( b
5 = [b7,b8] & b
6 = [b9,b10] & b
7,b
8,b
9,b
10 are_DTr_wrt b
2,b
3 ) ) );
theorem Th44: :: GEOMTRAP:44
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1 holds
(
[[b2,b3],[b4,b5]] in DTrapezium b
1,b
6,b
7 iff b
2,b
3,b
4,b
5 are_DTr_wrt b
6,b
7 )
:: deftheorem Def8 defines MidPoint GEOMTRAP:def 8 :
definition
let c
1 be
RealLinearSpace;
let c
2, c
3 be
VECTOR of c
1;
func DTrSpace c
1,c
2,c
3 -> strict AfMidStruct equals :: GEOMTRAP:def 9
AfMidStruct(# the
carrier of a
1,
(MidPoint a1),
(DTrapezium a1,a2,a3) #);
correctness
coherence
AfMidStruct(# the carrier of c1,(MidPoint c1),(DTrapezium c1,c2,c3) #) is strict AfMidStruct ;
;
end;
:: deftheorem Def9 defines DTrSpace GEOMTRAP:def 9 :
:: deftheorem Def10 defines Af GEOMTRAP:def 10 :
:: deftheorem Def11 GEOMTRAP:def 11 :
canceled;
:: deftheorem Def12 defines # GEOMTRAP:def 12 :
theorem Th45: :: GEOMTRAP:45
canceled;
theorem Th46: :: GEOMTRAP:46
theorem Th47: :: GEOMTRAP:47
for b
1 being
RealLinearSpacefor b
2, b
3, b
4, b
5, b
6, b
7 being
VECTOR of b
1for b
8, b
9, b
10, b
11 being
Element of
(DTrSpace b1,b2,b3) holds
(
Gen b
2,b
3 & b
4 = b
8 & b
5 = b
9 & b
6 = b
10 & b
7 = b
11 implies ( b
8,b
9 // b
10,b
11 iff b
4,b
5,b
6,b
7 are_DTr_wrt b
2,b
3 ) )
theorem Th48: :: GEOMTRAP:48
Lemma67:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4,b5 // b5,b6 implies ( b4 = b5 & b5 = b6 ) )
Lemma68:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7, b8, b9 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4,b5 // b6,b7 & b4,b5 // b8,b9 & b4 <> b5 implies b6,b7 // b8,b9 )
Lemma69:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4,b5 // b6,b7 implies ( b6,b7 // b4,b5 & b5,b4 // b7,b6 ) )
Lemma70:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6 being Element of (DTrSpace b1,b2,b3) holds
not ( Gen b2,b3 & ( for b7 being Element of (DTrSpace b1,b2,b3) holds
( not b4,b5 // b6,b7 & not b4,b5 // b7,b6 ) ) )
Lemma71:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7, b8 being Element of (DTrSpace b1,b2,b3) holds
not ( Gen b2,b3 & b4,b5 // b6,b7 & b4,b5 // b6,b8 & not b4 = b5 & not b7 = b8 )
Lemma72:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 implies b4 # b5 = b5 # b4 )
Lemma73:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 implies b4 # b4 = b4 )
Lemma74:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 implies (b4 # b5) # (b6 # b7) = (b4 # b6) # (b5 # b7) )
Lemma75:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5 being Element of (DTrSpace b1,b2,b3) holds
not ( Gen b2,b3 & ( for b6 being Element of (DTrSpace b1,b2,b3) holds
not b6 # b4 = b5 ) )
Lemma76:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4 # b5 = b4 # b6 implies b5 = b6 )
Lemma77:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4,b5 // b6,b7 implies b4,b5 // b4 # b6,b5 # b7 )
Lemma78:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7 being Element of (DTrSpace b1,b2,b3) holds
not ( Gen b2,b3 & b4,b5 // b6,b7 & not b4,b5 // b4 # b7,b5 # b6 & not b4,b5 // b5 # b6,b4 # b7 )
Lemma79:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7, b8, b9, b10, b11, b12 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4,b5 // b6,b7 & b4 # b8 = b9 & b5 # b10 = b9 & b6 # b11 = b9 & b7 # b12 = b9 implies b8,b10 // b11,b12 )
Lemma80:
for b1 being RealLinearSpace
for b2, b3 being VECTOR of b1
for b4, b5, b6, b7, b8, b9, b10, b11, b12, b13 being Element of (DTrSpace b1,b2,b3) holds
( Gen b2,b3 & b4 <> b5 & b4,b5 // b6,b7 & b4,b5 // b8,b9 & b4,b5 // b10,b11 & b4,b5 // b12,b13 & b6,b8 // b10,b12 implies b7,b9 // b11,b13 )
definition
let c
1 be non
empty AfMidStruct ;
attr a
1 is
MidOrdTrapSpace-like means :
Def13:
:: GEOMTRAP:def 13
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
1 # b
2 = b
2 # b
1 & b
1 # b
1 = b
1 &
(b1 # b2) # (b3 # b4) = (b1 # b3) # (b2 # b4) & ex b
11 being
Element of a
1 st b
11 # b
1 = b
2 & ( b
1 # b
2 = b
1 # b
3 implies b
2 = b
3 ) & ( b
1,b
2 // b
3,b
4 implies b
1,b
2 // b
1 # b
3,b
2 # b
4 ) & not ( b
1,b
2 // b
3,b
4 & not b
1,b
2 // b
1 # b
4,b
2 # b
3 & not b
1,b
2 // b
2 # b
3,b
1 # b
4 ) & ( b
1,b
2 // b
3,b
4 & b
1 # b
5 = b
9 & b
2 # b
6 = b
9 & b
3 # b
7 = b
9 & b
4 # b
8 = b
9 implies b
5,b
6 // b
7,b
8 ) & ( b
9 <> b
10 & b
9,b
10 // b
1,b
5 & b
9,b
10 // b
2,b
6 & b
9,b
10 // b
3,b
7 & b
9,b
10 // b
4,b
8 & b
1,b
2 // b
3,b
4 implies b
5,b
6 // b
7,b
8 ) & ( b
1,b
2 // b
2,b
3 implies ( b
1 = b
2 & b
2 = b
3 ) ) & ( b
1,b
2 // b
5,b
6 & b
1,b
2 // b
7,b
8 & b
1 <> b
2 implies b
5,b
6 // b
7,b
8 ) & ( b
1,b
2 // b
3,b
4 implies ( b
3,b
4 // b
1,b
2 & b
2,b
1 // b
4,b
3 ) ) & not for b
11 being
Element of a
1 holds
( not b
1,b
2 // b
3,b
11 & not b
1,b
2 // b
11,b
3 ) & not ( b
1,b
2 // b
3,b
9 & b
1,b
2 // b
3,b
10 & not b
1 = b
2 & not b
9 = b
10 ) );
end;
:: deftheorem Def13 defines MidOrdTrapSpace-like GEOMTRAP:def 13 :
for b
1 being non
empty AfMidStruct holds
( b
1 is
MidOrdTrapSpace-like iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
2 # b
3 = b
3 # b
2 & b
2 # b
2 = b
2 &
(b2 # b3) # (b4 # b5) = (b2 # b4) # (b3 # b5) & ex b
12 being
Element of b
1 st b
12 # b
2 = b
3 & ( b
2 # b
3 = b
2 # b
4 implies b
3 = b
4 ) & ( b
2,b
3 // b
4,b
5 implies b
2,b
3 // b
2 # b
4,b
3 # b
5 ) & not ( b
2,b
3 // b
4,b
5 & not b
2,b
3 // b
2 # b
5,b
3 # b
4 & not b
2,b
3 // b
3 # b
4,b
2 # b
5 ) & ( b
2,b
3 // b
4,b
5 & b
2 # b
6 = b
10 & b
3 # b
7 = b
10 & b
4 # b
8 = b
10 & b
5 # b
9 = b
10 implies b
6,b
7 // b
8,b
9 ) & ( b
10 <> b
11 & b
10,b
11 // b
2,b
6 & b
10,b
11 // b
3,b
7 & b
10,b
11 // b
4,b
8 & b
10,b
11 // b
5,b
9 & b
2,b
3 // b
4,b
5 implies b
6,b
7 // b
8,b
9 ) & ( b
2,b
3 // b
3,b
4 implies ( b
2 = b
3 & b
3 = b
4 ) ) & ( b
2,b
3 // b
6,b
7 & b
2,b
3 // b
8,b
9 & b
2 <> b
3 implies b
6,b
7 // b
8,b
9 ) & ( b
2,b
3 // b
4,b
5 implies ( b
4,b
5 // b
2,b
3 & b
3,b
2 // b
5,b
4 ) ) & not for b
12 being
Element of b
1 holds
( not b
2,b
3 // b
4,b
12 & not b
2,b
3 // b
12,b
4 ) & not ( b
2,b
3 // b
4,b
10 & b
2,b
3 // b
4,b
11 & not b
2 = b
3 & not b
10 = b
11 ) ) );
theorem Th49: :: GEOMTRAP:49
consider c1 being MidOrdTrapSpace;
set c2 = Af c1;
E82:
now
let c
3, c
4, c
5, c
6, c
7, c
8, c
9, c
10, c
11, c
12 be
Element of
(Af c1);
E83:
now
let c
13, c
14, c
15, c
16 be
Element of
(Af c1);
let c
17, c
18, c
19, c
20 be
Element of the
carrier of c
1;
assume E84:
( c
13 = c
17 & c
14 = c
18 & c
15 = c
19 & c
16 = c
20 )
;
hence
( c
13,c
14 // c
15,c
16 iff c
17,c
18 // c
19,c
20 )
by E85;
end;
reconsider c
13 = c
3, c
14 = c
4, c
15 = c
5, c
16 = c
6, c
17 = c
7, c
18 = c
8, c
19 = c
9, c
20 = c
10, c
21 = c
11, c
22 = c
12 as
Element of c
1 ;
E84:
now
assume
c
3,c
4 // c
4,c
5
;
then
c
13,c
14 // c
14,c
15
by E83;
hence
( c
3 = c
4 & c
4 = c
5 )
by Def13;
end;
E85:
now
assume
( c
3,c
4 // c
7,c
8 & c
3,c
4 // c
9,c
10 & c
3 <> c
4 )
;
then
( c
13,c
14 // c
17,c
18 & c
13,c
14 // c
19,c
20 & c
13 <> c
14 )
by E83;
then
c
17,c
18 // c
19,c
20
by Def13;
hence
c
7,c
8 // c
9,c
10
by E83;
end;
E86:
now
assume
c
3,c
4 // c
5,c
6
;
then
c
13,c
14 // c
15,c
16
by E83;
then
( c
15,c
16 // c
13,c
14 & c
14,c
13 // c
16,c
15 )
by Def13;
hence
( c
5,c
6 // c
3,c
4 & c
4,c
3 // c
6,c
5 )
by E83;
end;
E87:
not for b
1 being
Element of
(Af c1) holds
( not c
3,c
4 // c
5,b
1 & not c
3,c
4 // b
1,c
5 )
proof
consider c
23 being
Element of c
1 such that E88:
( c
13,c
14 // c
15,c
23 or c
13,c
14 // c
23,c
15 )
by Def13;
reconsider c
24 = c
23 as
Element of
(Af c1) ;
take
c
24
;
thus
not ( not c
3,c
4 // c
5,c
24 & not c
3,c
4 // c
24,c
5 )
by E83, E88;
end;
now
assume
( c
3,c
4 // c
5,c
11 & c
3,c
4 // c
5,c
12 )
;
then
( c
13,c
14 // c
15,c
21 & c
13,c
14 // c
15,c
22 )
by E83;
hence
( c
3 = c
4 or c
11 = c
12 )
by Def13;
end;
hence
( ( c
3,c
4 // c
4,c
5 implies ( c
3 = c
4 & c
4 = c
5 ) ) & ( c
3,c
4 // c
7,c
8 & c
3,c
4 // c
9,c
10 & c
3 <> c
4 implies c
7,c
8 // c
9,c
10 ) & ( c
3,c
4 // c
5,c
6 implies ( c
5,c
6 // c
3,c
4 & c
4,c
3 // c
6,c
5 ) ) & not for b
1 being
Element of
(Af c1) holds
( not c
3,c
4 // c
5,b
1 & not c
3,c
4 // b
1,c
5 ) & not ( c
3,c
4 // c
5,c
11 & c
3,c
4 // c
5,c
12 & not c
3 = c
4 & not c
11 = c
12 ) )
by E84, E85, E86, E87;
end;
definition
let c
3 be non
empty AffinStruct ;
attr a
1 is
OrdTrapSpace-like means :
Def14:
:: GEOMTRAP:def 14
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( ( b
1,b
2 // b
2,b
3 implies ( b
1 = b
2 & b
2 = b
3 ) ) & ( b
1,b
2 // b
5,b
6 & b
1,b
2 // b
7,b
8 & b
1 <> b
2 implies b
5,b
6 // b
7,b
8 ) & ( b
1,b
2 // b
3,b
4 implies ( b
3,b
4 // b
1,b
2 & b
2,b
1 // b
4,b
3 ) ) & not for b
11 being
Element of a
1 holds
( not b
1,b
2 // b
3,b
11 & not b
1,b
2 // b
11,b
3 ) & not ( b
1,b
2 // b
3,b
9 & b
1,b
2 // b
3,b
10 & not b
1 = b
2 & not b
9 = b
10 ) );
end;
:: deftheorem Def14 defines OrdTrapSpace-like GEOMTRAP:def 14 :
for b
1 being non
empty AffinStruct holds
( b
1 is
OrdTrapSpace-like iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( ( b
2,b
3 // b
3,b
4 implies ( b
2 = b
3 & b
3 = b
4 ) ) & ( b
2,b
3 // b
6,b
7 & b
2,b
3 // b
8,b
9 & b
2 <> b
3 implies b
6,b
7 // b
8,b
9 ) & ( b
2,b
3 // b
4,b
5 implies ( b
4,b
5 // b
2,b
3 & b
3,b
2 // b
5,b
4 ) ) & not for b
12 being
Element of b
1 holds
( not b
2,b
3 // b
4,b
12 & not b
2,b
3 // b
12,b
4 ) & not ( b
2,b
3 // b
4,b
10 & b
2,b
3 // b
4,b
11 & not b
2 = b
3 & not b
10 = b
11 ) ) );
theorem Th50: :: GEOMTRAP:50
theorem Th51: :: GEOMTRAP:51
for b
1 being
OrdTrapSpacefor b
2, b
3, b
4, b
5 being
Element of b
1for b
6, b
7, b
8, b
9 being
Element of
(Lambda b1) holds
( b
2 = b
6 & b
3 = b
7 & b
4 = b
8 & b
5 = b
9 implies ( b
6,b
7 // b
8,b
9 iff ( b
2,b
3 // b
4,b
5 or b
2,b
3 // b
5,b
4 ) ) )
Lemma86:
for b1 being OrdTrapSpace
for b2, b3, b4 being Element of (Lambda b1) holds
ex b5 being Element of (Lambda b1) st b2,b3 // b4,b5
Lemma87:
for b1 being OrdTrapSpace
for b2, b3, b4, b5 being Element of (Lambda b1) holds
( b2,b3 // b4,b5 implies b4,b5 // b2,b3 )
Lemma88:
for b1 being OrdTrapSpace
for b2, b3, b4, b5, b6, b7 being Element of (Lambda b1) holds
( b2 <> b3 & b2,b3 // b4,b5 & b2,b3 // b6,b7 implies b4,b5 // b6,b7 )
Lemma89:
for b1 being OrdTrapSpace
for b2, b3, b4, b5, b6 being Element of (Lambda b1) holds
not ( b2,b3 // b4,b5 & b2,b3 // b4,b6 & not b2 = b3 & not b5 = b6 )
Lemma90:
for b1 being OrdTrapSpace
for b2, b3 being Element of b1 holds b2,b3 // b2,b3
Lemma91:
for b1 being OrdTrapSpace
for b2, b3 being Element of (Lambda b1) holds b2,b3 // b3,b2
definition
let c
3 be non
empty AffinStruct ;
attr a
1 is
TrapSpace-like means :
Def15:
:: GEOMTRAP:def 15
for b
1, b
2, b
3, b
4, b
5, b
6 being
Element of a
1 holds
( b
1,b
2 // b
2,b
1 & not ( b
1,b
2 // b
3,b
4 & b
1,b
2 // b
3,b
6 & not b
1 = b
2 & not b
4 = b
6 ) & ( b
5 <> b
6 & b
5,b
6 // b
1,b
2 & b
5,b
6 // b
3,b
4 implies b
1,b
2 // b
3,b
4 ) & ( b
1,b
2 // b
3,b
4 implies b
3,b
4 // b
1,b
2 ) & ex b
7 being
Element of a
1 st b
1,b
2 // b
3,b
7 );
end;
:: deftheorem Def15 defines TrapSpace-like GEOMTRAP:def 15 :
for b
1 being non
empty AffinStruct holds
( b
1 is
TrapSpace-like iff for b
2, b
3, b
4, b
5, b
6, b
7 being
Element of b
1 holds
( b
2,b
3 // b
3,b
2 & not ( b
2,b
3 // b
4,b
5 & b
2,b
3 // b
4,b
7 & not b
2 = b
3 & not b
5 = b
7 ) & ( b
6 <> b
7 & b
6,b
7 // b
2,b
3 & b
6,b
7 // b
4,b
5 implies b
2,b
3 // b
4,b
5 ) & ( b
2,b
3 // b
4,b
5 implies b
4,b
5 // b
2,b
3 ) & ex b
8 being
Element of b
1 st b
2,b
3 // b
4,b
8 ) );
definition
let c
3 be non
empty AffinStruct ;
attr a
1 is
Regular means :
Def16:
:: GEOMTRAP:def 16
for b
1, b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10 being
Element of a
1 holds
( b
1 <> b
2 & b
1,b
2 // b
3,b
4 & b
1,b
2 // b
5,b
6 & b
1,b
2 // b
7,b
8 & b
1,b
2 // b
9,b
10 & b
3,b
5 // b
7,b
9 implies b
4,b
6 // b
8,b
10 );
end;
:: deftheorem Def16 defines Regular GEOMTRAP:def 16 :
for b
1 being non
empty AffinStruct holds
( b
1 is
Regular iff for b
2, b
3, b
4, b
5, b
6, b
7, b
8, b
9, b
10, b
11 being
Element of b
1 holds
( b
2 <> b
3 & b
2,b
3 // b
4,b
5 & b
2,b
3 // b
6,b
7 & b
2,b
3 // b
8,b
9 & b
2,b
3 // b
10,b
11 & b
4,b
6 // b
8,b
10 implies b
5,b
7 // b
9,b
11 ) );