:: GEOMTRAP semantic presentation
:: deftheorem Def1 defines '||' GEOMTRAP:def 1 :
theorem Th1: :: GEOMTRAP:1
theorem Th2: :: GEOMTRAP:2
for
V being
RealLinearSpace for
u,
v,
u1,
v1 being
VECTOR of
V for
p,
q,
p1,
q1 being
Element of
(OASpace V) st
p = u &
q = v &
p1 = u1 &
q1 = v1 holds
(
p,
q // p1,
q1 iff
u,
v // u1,
v1 )
theorem Th3: :: GEOMTRAP:3
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V st
Gen w,
y holds
for
p,
q,
p1,
q1 being
Element of the
carrier of
(Lambda (OASpace V)) st
p = u &
q = v &
p1 = u1 &
q1 = v1 holds
(
p,
q // p1,
q1 iff
u,
v '||' u1,
v1 )
theorem Th4: :: GEOMTRAP:4
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V for
p,
q,
p1,
q1 being
Element of the
carrier of
(AMSpace V,w,y) st
p = u &
q = v &
p1 = u1 &
q1 = v1 holds
(
p,
q // p1,
q1 iff
u,
v '||' u1,
v1 )
:: 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
Lemma65:
for V being RealLinearSpace
for u, y, v being VECTOR of V st u,y // y,v holds
( v,y // y,u & u,y // u,v & y,v // u,v )
theorem Th15: :: GEOMTRAP:15
theorem Th16: :: GEOMTRAP:16
Lemma69:
for V being RealLinearSpace
for u, v, u1, v1 being VECTOR of V st u,v // u1,v1 holds
u,v '||' u # v1,v # u1
Lemma74:
for V being RealLinearSpace
for u1, u2, v1, v2 being VECTOR of V st u1 # u2 = v1 # v2 holds
v1 - u1 = - (v2 - u2)
Lemma75:
for V being RealLinearSpace
for w, y, u, v, u1, v1 being VECTOR of V st Gen w,y & u,v,u1,v1 are_Ort_wrt w,y holds
( u1,v1,u,v are_Ort_wrt w,y & u,v,v1,u1 are_Ort_wrt w,y )
Lemma76:
for V being RealLinearSpace
for w, y, u, v, u1 being VECTOR of V st Gen w,y holds
u,v,u1,u1 are_Ort_wrt w,y
Lemma77:
for V being RealLinearSpace
for w, y, u, v, w1, v1, v2 being VECTOR of V st Gen w,y & u,v,w1,v1 are_Ort_wrt w,y & u,v,w1,v2 are_Ort_wrt w,y holds
u,v,v1,v2 are_Ort_wrt w,y
Lemma78:
for V being RealLinearSpace
for w, y, u, v being VECTOR of V st Gen w,y & u,v,u,v are_Ort_wrt w,y holds
u = v
Lemma79:
for V being RealLinearSpace
for w, y, u, v, u1 being VECTOR of V st Gen w,y holds
( u,v,u1,u1 are_Ort_wrt w,y & u1,u1,u,v are_Ort_wrt w,y )
Lemma80:
for V being RealLinearSpace
for w, y, u1, v1, u, v, u2, v2 being VECTOR of V st Gen w,y & ( u1,v1 '||' u,v or u,v '||' u1,v1 ) & ( u2,v2,u,v are_Ort_wrt w,y or u,v,u2,v2 are_Ort_wrt w,y ) & u <> v holds
( u1,v1,u2,v2 are_Ort_wrt w,y & u2,v2,u1,v1 are_Ort_wrt w,y )
definition
let V be
RealLinearSpace;
let w be
VECTOR of
V;
let y be
VECTOR of
V;
let u be
VECTOR of
V;
let u1 be
VECTOR of
V;
let v be
VECTOR of
V;
let v1 be
VECTOR of
V;
pred c4,
c5,
c6,
c7 are_DTr_wrt c2,
c3 means :
Def3:
:: GEOMTRAP:def 3
(
u,
u1 // v,
v1 &
u,
u1,
u # u1,
v # v1 are_Ort_wrt w,
y &
v,
v1,
u # u1,
v # v1 are_Ort_wrt w,
y );
end;
:: deftheorem Def3 defines are_DTr_wrt GEOMTRAP:def 3 :
for
V being
RealLinearSpace for
w,
y,
u,
u1,
v,
v1 being
VECTOR of
V holds
(
u,
u1,
v,
v1 are_DTr_wrt w,
y iff (
u,
u1 // v,
v1 &
u,
u1,
u # u1,
v # v1 are_Ort_wrt w,
y &
v,
v1,
u # u1,
v # v1 are_Ort_wrt w,
y ) );
theorem Th17: :: GEOMTRAP:17
theorem Th18: :: GEOMTRAP:18
theorem Th19: :: GEOMTRAP:19
theorem Th20: :: GEOMTRAP:20
theorem Th21: :: GEOMTRAP:21
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1,
u2,
v2 being
VECTOR of
V st
Gen w,
y &
u,
v,
u1,
v1 are_DTr_wrt w,
y &
u,
v,
u2,
v2 are_DTr_wrt w,
y &
u <> v holds
u1,
v1,
u2,
v2 are_DTr_wrt w,
y
theorem Th22: :: GEOMTRAP:22
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1 being
VECTOR of
V st
Gen w,
y holds
ex
t being
VECTOR of
V st
(
u,
v,
u1,
t are_DTr_wrt w,
y or
u,
v,
t,
u1 are_DTr_wrt w,
y )
theorem Th23: :: GEOMTRAP:23
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V st
Gen w,
y &
u,
v,
u1,
v1 are_DTr_wrt w,
y holds
u1,
v1,
u,
v are_DTr_wrt w,
y
theorem Th24: :: GEOMTRAP:24
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V st
Gen w,
y &
u,
v,
u1,
v1 are_DTr_wrt w,
y holds
v,
u,
v1,
u1 are_DTr_wrt w,
y
Lemma108:
for V being RealLinearSpace
for w, y, u, v, u1, v1, u2, v2 being VECTOR of V st Gen w,y & u <> v & u,v '||' u,u1 & u,v '||' u,v1 & u,v '||' u,u2 & u,v '||' u,v2 holds
u1,v1 '||' u2,v2
theorem Th25: :: GEOMTRAP:25
theorem Th26: :: GEOMTRAP:26
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1,
v2 being
VECTOR of
V st
Gen w,
y &
u,
v,
u1,
v1 are_DTr_wrt w,
y &
u,
v,
u1,
v2 are_DTr_wrt w,
y & not
u = v holds
v1 = v2
theorem Th27: :: GEOMTRAP:27
for
V being
RealLinearSpace for
w,
y,
u,
u1,
v,
v1,
v2 being
VECTOR of
V st
Gen w,
y &
u <> u1 &
u,
u1,
v,
v1 are_DTr_wrt w,
y & (
u,
u1,
v,
v2 are_DTr_wrt w,
y or
u,
u1,
v2,
v are_DTr_wrt w,
y ) holds
v1 = v2
theorem Th28: :: GEOMTRAP:28
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V st
Gen w,
y &
u,
v,
u1,
v1 are_DTr_wrt w,
y holds
u,
v,
u # u1,
v # v1 are_DTr_wrt w,
y
theorem Th29: :: GEOMTRAP:29
for
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V st
Gen w,
y &
u,
v,
u1,
v1 are_DTr_wrt w,
y & not
u,
v,
u # v1,
v # u1 are_DTr_wrt w,
y holds
u,
v,
v # u1,
u # v1 are_DTr_wrt w,
y
theorem Th30: :: GEOMTRAP:30
for
V being
RealLinearSpace for
w,
y,
u,
u1,
u2,
v1,
v2,
t1,
t2,
w1,
w2 being
VECTOR of
V st
Gen w,
y &
u = u1 # t1 &
u = u2 # t2 &
u = v1 # w1 &
u = v2 # w2 &
u1,
u2,
v1,
v2 are_DTr_wrt w,
y holds
t1,
t2,
w1,
w2 are_DTr_wrt w,
y
Lemma120:
for V being RealLinearSpace
for v1, w, y, v2 being VECTOR of V
for b1, b2, c1, c2 being Real st v1 = (b1 * w) + (b2 * y) & v2 = (c1 * w) + (c2 * y) holds
( v1 + v2 = ((b1 + c1) * w) + ((b2 + c2) * y) & v1 - v2 = ((b1 - c1) * w) + ((b2 - c2) * y) )
Lemma121:
for V being RealLinearSpace
for v, w, y being VECTOR of V
for b1, b2, a being Real st v = (b1 * w) + (b2 * y) holds
a * v = ((a * b1) * w) + ((a * b2) * y)
Lemma122:
for V being RealLinearSpace
for w, y being VECTOR of V
for a1, a2, b1, b2 being Real st Gen w,y & (a1 * w) + (a2 * y) = (b1 * w) + (b2 * y) holds
( a1 = b1 & a2 = b2 )
:: deftheorem Def4 defines pr1 GEOMTRAP:def 4 :
:: deftheorem Def5 defines pr2 GEOMTRAP:def 5 :
Lemma125:
for V being RealLinearSpace
for w, y, u being VECTOR of V st Gen w,y holds
u = ((pr1 w,y,u) * w) + ((pr2 w,y,u) * y)
Lemma126:
for V being RealLinearSpace
for w, y, u being VECTOR of V
for a, b being Real st Gen w,y & u = (a * w) + (b * y) holds
( a = pr1 w,y,u & b = pr2 w,y,u )
Lemma127:
for V being RealLinearSpace
for w, y, u, v being VECTOR of V
for a being Real st Gen w,y holds
( pr1 w,y,(u + v) = (pr1 w,y,u) + (pr1 w,y,v) & pr2 w,y,(u + v) = (pr2 w,y,u) + (pr2 w,y,v) & pr1 w,y,(u - v) = (pr1 w,y,u) - (pr1 w,y,v) & pr2 w,y,(u - v) = (pr2 w,y,u) - (pr2 w,y,v) & pr1 w,y,(a * u) = a * (pr1 w,y,u) & pr2 w,y,(a * u) = a * (pr2 w,y,u) )
Lemma132:
for V being RealLinearSpace
for w, y, u, v being VECTOR of V st Gen w,y holds
( 2 * (pr1 w,y,(u # v)) = (pr1 w,y,u) + (pr1 w,y,v) & 2 * (pr2 w,y,(u # v)) = (pr2 w,y,u) + (pr2 w,y,v) )
Lemma133:
for V being RealLinearSpace
for u, v being VECTOR of V holds (- u) + (- v) = - (u + v)
Lemma134:
for V being RealLinearSpace
for u2, v, u1 being VECTOR of V
for a2, a1 being Real holds (u2 + (a2 * v)) - (u1 + (a1 * v)) = (u2 - u1) + ((a2 - a1) * v)
definition
let V be
RealLinearSpace;
let w be
VECTOR of
V;
let y be
VECTOR of
V;
let u be
VECTOR of
V;
let v be
VECTOR of
V;
func PProJ c2,
c3,
c4,
c5 -> Real equals :: GEOMTRAP:def 6
((pr1 w,y,u) * (pr1 w,y,v)) + ((pr2 w,y,u) * (pr2 w,y,v));
correctness
coherence
((pr1 w,y,u) * (pr1 w,y,v)) + ((pr2 w,y,u) * (pr2 w,y,v)) is Real;
;
end;
:: deftheorem Def6 defines PProJ GEOMTRAP:def 6 :
for
V being
RealLinearSpace for
w,
y,
u,
v being
VECTOR of
V holds
PProJ w,
y,
u,
v = ((pr1 w,y,u) * (pr1 w,y,v)) + ((pr2 w,y,u) * (pr2 w,y,v));
theorem Th31: :: GEOMTRAP:31
theorem Th32: :: GEOMTRAP:32
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
v,
v1 being
VECTOR of
V holds
(
PProJ w,
y,
u,
(v + v1) = (PProJ w,y,u,v) + (PProJ w,y,u,v1) &
PProJ w,
y,
u,
(v - v1) = (PProJ w,y,u,v) - (PProJ w,y,u,v1) &
PProJ w,
y,
(v - v1),
u = (PProJ w,y,v,u) - (PProJ w,y,v1,u) &
PProJ w,
y,
(v + v1),
u = (PProJ w,y,v,u) + (PProJ w,y,v1,u) )
theorem Th33: :: GEOMTRAP:33
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
v being
VECTOR of
V for
a being
Real holds
(
PProJ w,
y,
(a * u),
v = a * (PProJ w,y,u,v) &
PProJ w,
y,
u,
(a * v) = a * (PProJ w,y,u,v) &
PProJ w,
y,
(a * u),
v = (PProJ w,y,u,v) * a &
PProJ w,
y,
u,
(a * v) = (PProJ w,y,u,v) * a )
theorem Th34: :: GEOMTRAP:34
theorem Th35: :: GEOMTRAP:35
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
v,
u1,
v1 being
VECTOR of
V holds
(
u,
v,
u1,
v1 are_Ort_wrt w,
y iff
PProJ w,
y,
(v - u),
(v1 - u1) = 0 )
theorem Th36: :: GEOMTRAP:36
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
v,
v1 being
VECTOR of
V holds 2
* (PProJ w,y,u,(v # v1)) = (PProJ w,y,u,v) + (PProJ w,y,u,v1)
theorem Th37: :: GEOMTRAP:37
theorem Th38: :: GEOMTRAP:38
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
p,
q,
u,
v,
v' being
VECTOR of
V for
A being
Real st
p,
q,
u,
v are_DTr_wrt w,
y &
p <> q &
A = ((PProJ w,y,(p - q),(p + q)) - (2 * (PProJ w,y,(p - q),u))) * ((PProJ w,y,(p - q),(p - q)) " ) &
v' = u + (A * (p - q)) holds
v = v'
Lemma144:
for V being RealLinearSpace
for w, y being VECTOR of V st Gen w,y holds
for u, u', u1, u2, t1, t2 being VECTOR of V
for A1, A2 being Real st u <> u' & A1 = ((PProJ w,y,(u - u'),(u + u')) - (2 * (PProJ w,y,(u - u'),u1))) * ((PProJ w,y,(u - u'),(u - u')) " ) & A2 = ((PProJ w,y,(u - u'),(u + u')) - (2 * (PProJ w,y,(u - u'),u2))) * ((PProJ w,y,(u - u'),(u - u')) " ) & t1 = u1 + (A1 * (u - u')) & t2 = u2 + (A2 * (u - u')) holds
( t2 - t1 = (u2 - u1) + ((A2 - A1) * (u - u')) & A2 - A1 = ((- 2) * (PProJ w,y,(u - u'),(u2 - u1))) * ((PProJ w,y,(u - u'),(u - u')) " ) )
theorem Th39: :: GEOMTRAP:39
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
u',
u1,
u2,
v1,
v2,
t1,
t2,
w1,
w2 being
VECTOR of
V st
u <> u' &
u,
u',
u1,
t1 are_DTr_wrt w,
y &
u,
u',
u2,
t2 are_DTr_wrt w,
y &
u,
u',
v1,
w1 are_DTr_wrt w,
y &
u,
u',
v2,
w2 are_DTr_wrt w,
y &
u1,
u2 // v1,
v2 holds
t1,
t2 // w1,
w2
theorem Th40: :: GEOMTRAP:40
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
u',
u1,
u2,
v1,
t1,
t2,
w1 being
VECTOR of
V st
u <> u' &
u,
u',
u1,
t1 are_DTr_wrt w,
y &
u,
u',
u2,
t2 are_DTr_wrt w,
y &
u,
u',
v1,
w1 are_DTr_wrt w,
y &
v1 = u1 # u2 holds
w1 = t1 # t2
theorem Th41: :: GEOMTRAP:41
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
u',
u1,
u2,
t1,
t2 being
VECTOR of
V st
u <> u' &
u,
u',
u1,
t1 are_DTr_wrt w,
y &
u,
u',
u2,
t2 are_DTr_wrt w,
y holds
u,
u',
u1 # u2,
t1 # t2 are_DTr_wrt w,
y
theorem Th42: :: GEOMTRAP:42
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V st
Gen w,
y holds
for
u,
u',
u1,
u2,
v1,
v2,
t1,
t2,
w1,
w2 being
VECTOR of
V st
u <> u' &
u,
u',
u1,
t1 are_DTr_wrt w,
y &
u,
u',
u2,
t2 are_DTr_wrt w,
y &
u,
u',
v1,
w1 are_DTr_wrt w,
y &
u,
u',
v2,
w2 are_DTr_wrt w,
y &
u1,
u2,
v1,
v2 are_Ort_wrt w,
y holds
t1,
t2,
w1,
w2 are_Ort_wrt w,
y
theorem Th43: :: GEOMTRAP:43
for
V being
RealLinearSpace for
w,
y,
u,
u',
u1,
u2,
v1,
v2,
t1,
t2,
w1,
w2 being
VECTOR of
V st
Gen w,
y &
u <> u' &
u,
u',
u1,
t1 are_DTr_wrt w,
y &
u,
u',
u2,
t2 are_DTr_wrt w,
y &
u,
u',
v1,
w1 are_DTr_wrt w,
y &
u,
u',
v2,
w2 are_DTr_wrt w,
y &
u1,
u2,
v1,
v2 are_DTr_wrt w,
y holds
t1,
t2,
w1,
w2 are_DTr_wrt w,
y
definition
let V be
RealLinearSpace;
let w be
VECTOR of
V;
let y be
VECTOR of
V;
func DTrapezium c1,
c2,
c3 -> Relation of
[:the carrier of a1,the carrier of a1:] means :
Def7:
:: GEOMTRAP:def 7
for
x,
z being
set holds
(
[x,z] in it iff ex
u,
u1,
v,
v1 being
VECTOR of
V st
(
x = [u,u1] &
z = [v,v1] &
u,
u1,
v,
v1 are_DTr_wrt w,
y ) );
existence
ex b1 being Relation of [:the carrier of V,the carrier of V:] st
for x, z being set holds
( [x,z] in b1 iff ex u, u1, v, v1 being VECTOR of V st
( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) )
uniqueness
for b1, b2 being Relation of [:the carrier of V,the carrier of V:] st ( for x, z being set holds
( [x,z] in b1 iff ex u, u1, v, v1 being VECTOR of V st
( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ) & ( for x, z being set holds
( [x,z] in b2 iff ex u, u1, v, v1 being VECTOR of V st
( x = [u,u1] & z = [v,v1] & u,u1,v,v1 are_DTr_wrt w,y ) ) ) holds
b1 = b2
end;
:: deftheorem Def7 defines DTrapezium GEOMTRAP:def 7 :
for
V being
RealLinearSpace for
w,
y being
VECTOR of
V for
b4 being
Relation of
[:the carrier of V,the carrier of V:] holds
(
b4 = DTrapezium V,
w,
y iff for
x,
z being
set holds
(
[x,z] in b4 iff ex
u,
u1,
v,
v1 being
VECTOR of
V st
(
x = [u,u1] &
z = [v,v1] &
u,
u1,
v,
v1 are_DTr_wrt w,
y ) ) );
theorem Th44: :: GEOMTRAP:44
for
V being
RealLinearSpace for
u,
v,
u1,
v1,
w,
y being
VECTOR of
V holds
(
[[u,v],[u1,v1]] in DTrapezium V,
w,
y iff
u,
v,
u1,
v1 are_DTr_wrt w,
y )
:: deftheorem Def8 defines MidPoint GEOMTRAP:def 8 :
definition
let V be
RealLinearSpace;
let w be
VECTOR of
V;
let y be
VECTOR of
V;
func DTrSpace c1,
c2,
c3 -> strict AfMidStruct equals :: GEOMTRAP:def 9
AfMidStruct(# the
carrier of
V,
(MidPoint V),
(DTrapezium V,w,y) #);
correctness
coherence
AfMidStruct(# the carrier of V,(MidPoint V),(DTrapezium V,w,y) #) 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
V being
RealLinearSpace for
w,
y,
u,
v,
u1,
v1 being
VECTOR of
V for
a,
b,
a1,
b1 being
Element of
(DTrSpace V,w,y) st
Gen w,
y &
u = a &
v = b &
u1 = a1 &
v1 = b1 holds
(
a,
b // a1,
b1 iff
u,
v,
u1,
v1 are_DTr_wrt w,
y )
theorem Th48: :: GEOMTRAP:48
Lemma182:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c being Element of (DTrSpace V,w,y) st Gen w,y & a,b // b,c holds
( a = b & b = c )
Lemma183:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, a1, b1, c1, d1 being Element of (DTrSpace V,w,y) st Gen w,y & a,b // a1,b1 & a,b // c1,d1 & a <> b holds
a1,b1 // c1,d1
Lemma184:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c, d being Element of (DTrSpace V,w,y) st Gen w,y & a,b // c,d holds
( c,d // a,b & b,a // d,c )
Lemma185:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c being Element of (DTrSpace V,w,y) st Gen w,y holds
ex d being Element of (DTrSpace V,w,y) st
( a,b // c,d or a,b // d,c )
Lemma186:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c, d1, d2 being Element of (DTrSpace V,w,y) st Gen w,y & a,b // c,d1 & a,b // c,d2 & not a = b holds
d1 = d2
Lemma187:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b being Element of (DTrSpace V,w,y) st Gen w,y holds
a # b = b # a
Lemma188:
for V being RealLinearSpace
for w, y being VECTOR of V
for a being Element of (DTrSpace V,w,y) st Gen w,y holds
a # a = a
Lemma189:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c, d being Element of (DTrSpace V,w,y) st Gen w,y holds
(a # b) # (c # d) = (a # c) # (b # d)
Lemma198:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b being Element of (DTrSpace V,w,y) st Gen w,y holds
ex p being Element of (DTrSpace V,w,y) st p # a = b
Lemma199:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, a1, a2 being Element of (DTrSpace V,w,y) st Gen w,y & a # a1 = a # a2 holds
a1 = a2
Lemma200:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c, d being Element of (DTrSpace V,w,y) st Gen w,y & a,b // c,d holds
a,b // a # c,b # d
Lemma201:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c, d being Element of (DTrSpace V,w,y) st Gen w,y & a,b // c,d & not a,b // a # d,b # c holds
a,b // b # c,a # d
Lemma202:
for V being RealLinearSpace
for w, y being VECTOR of V
for a, b, c, d, a1, p, b1, c1, d1 being Element of (DTrSpace V,w,y) st Gen w,y & a,b // c,d & a # a1 = p & b # b1 = p & c # c1 = p & d # d1 = p holds
a1,b1 // c1,d1
Lemma203:
for V being RealLinearSpace
for w, y being VECTOR of V
for p, q, a, a1, b, b1, c, c1, d, d1 being Element of (DTrSpace V,w,y) st Gen w,y & p <> q & p,q // a,a1 & p,q // b,b1 & p,q // c,c1 & p,q // d,d1 & a,b // c,d holds
a1,b1 // c1,d1
definition
let IT be non
empty AfMidStruct ;
attr a1 is
MidOrdTrapSpace-like means :
Def13:
:: GEOMTRAP:def 13
for
a,
b,
c,
d,
a1,
b1,
c1,
d1,
p,
q being
Element of
IT holds
(
a # b = b # a &
a # a = a &
(a # b) # (c # d) = (a # c) # (b # d) & ex
p being
Element of
IT st
p # a = b & (
a # b = a # c implies
b = c ) & (
a,
b // c,
d implies
a,
b // a # c,
b # d ) & ( not
a,
b // c,
d or
a,
b // a # d,
b # c or
a,
b // b # c,
a # d ) & (
a,
b // c,
d &
a # a1 = p &
b # b1 = p &
c # c1 = p &
d # d1 = p implies
a1,
b1 // c1,
d1 ) & (
p <> q &
p,
q // a,
a1 &
p,
q // b,
b1 &
p,
q // c,
c1 &
p,
q // d,
d1 &
a,
b // c,
d implies
a1,
b1 // c1,
d1 ) & (
a,
b // b,
c implies (
a = b &
b = c ) ) & (
a,
b // a1,
b1 &
a,
b // c1,
d1 &
a <> b implies
a1,
b1 // c1,
d1 ) & (
a,
b // c,
d implies (
c,
d // a,
b &
b,
a // d,
c ) ) & ex
d being
Element of
IT st
(
a,
b // c,
d or
a,
b // d,
c ) & (
a,
b // c,
p &
a,
b // c,
q & not
a = b implies
p = q ) );
end;
:: deftheorem Def13 defines MidOrdTrapSpace-like GEOMTRAP:def 13 :
for
IT being non
empty AfMidStruct holds
(
IT is
MidOrdTrapSpace-like iff for
a,
b,
c,
d,
a1,
b1,
c1,
d1,
p,
q being
Element of
IT holds
(
a # b = b # a &
a # a = a &
(a # b) # (c # d) = (a # c) # (b # d) & ex
p being
Element of
IT st
p # a = b & (
a # b = a # c implies
b = c ) & (
a,
b // c,
d implies
a,
b // a # c,
b # d ) & ( not
a,
b // c,
d or
a,
b // a # d,
b # c or
a,
b // b # c,
a # d ) & (
a,
b // c,
d &
a # a1 = p &
b # b1 = p &
c # c1 = p &
d # d1 = p implies
a1,
b1 // c1,
d1 ) & (
p <> q &
p,
q // a,
a1 &
p,
q // b,
b1 &
p,
q // c,
c1 &
p,
q // d,
d1 &
a,
b // c,
d implies
a1,
b1 // c1,
d1 ) & (
a,
b // b,
c implies (
a = b &
b = c ) ) & (
a,
b // a1,
b1 &
a,
b // c1,
d1 &
a <> b implies
a1,
b1 // c1,
d1 ) & (
a,
b // c,
d implies (
c,
d // a,
b &
b,
a // d,
c ) ) & ex
d being
Element of
IT st
(
a,
b // c,
d or
a,
b // d,
c ) & (
a,
b // c,
p &
a,
b // c,
q & not
a = b implies
p = q ) ) );
theorem Th49: :: GEOMTRAP:49
consider MOS being MidOrdTrapSpace;
set X = Af MOS;
E207:
now
let a be
Element of
(Af MOS),
b be
Element of
(Af MOS),
c be
Element of
(Af MOS),
d be
Element of
(Af MOS),
a1 be
Element of
(Af MOS),
b1 be
Element of
(Af MOS),
c1 be
Element of
(Af MOS),
d1 be
Element of
(Af MOS),
p be
Element of
(Af MOS),
q be
Element of
(Af MOS);
E45:
now
let a be
Element of
(Af MOS),
b be
Element of
(Af MOS),
c be
Element of
(Af MOS),
d be
Element of
(Af MOS);
let a' be
Element of the
carrier of
MOS,
b' be
Element of the
carrier of
MOS,
c' be
Element of the
carrier of
MOS,
d' be
Element of the
carrier of
MOS;
assume E46:
(
a = a' &
b = b' &
c = c' &
d = d' )
;
hence
(
a,
b // c,
d iff
a',
b' // c',
d' )
by Def3;
end;
reconsider a' =
a,
b' =
b,
c' =
c,
d' =
d,
a1' =
a1,
b1' =
b1,
c1' =
c1,
d1' =
d1,
p' =
p,
q' =
q as
Element of
MOS ;
E67:
now
assume
a,
b // b,
c
;
then
a',
b' // b',
c'
by ;
hence
(
a = b &
b = c )
by ;
end;
E72:
now
assume
(
a,
b // a1,
b1 &
a,
b // c1,
d1 &
a <> b )
;
then
(
a',
b' // a1',
b1' &
a',
b' // c1',
d1' &
a' <> b' )
by ;
then
a1',
b1' // c1',
d1'
by ;
hence
a1,
b1 // c1,
d1
by ;
end;
E73:
now
assume
a,
b // c,
d
;
then
a',
b' // c',
d'
by ;
then
(
c',
d' // a',
b' &
b',
a' // d',
c' )
by ;
hence
(
c,
d // a,
b &
b,
a // d,
c )
by ;
end;
E88:
ex
d being
Element of
(Af MOS) st
(
a,
b // c,
d or
a,
b // d,
c )
proof
consider x' being
Element of
MOS such that E89:
(
a',
b' // c',
x' or
a',
b' // x',
c' )
by ;
reconsider x =
x' as
Element of
(Af MOS) ;
take
x
;
thus
(
a,
b // c,
x or
a,
b // x,
c )
by , E88;
end;
now
assume
(
a,
b // c,
p &
a,
b // c,
q )
;
then
(
a',
b' // c',
p' &
a',
b' // c',
q' )
by ;
hence
(
a = b or
p = q )
by ;
end;
hence
( (
a,
b // b,
c implies (
a = b &
b = c ) ) & (
a,
b // a1,
b1 &
a,
b // c1,
d1 &
a <> b implies
a1,
b1 // c1,
d1 ) & (
a,
b // c,
d implies (
c,
d // a,
b &
b,
a // d,
c ) ) & ex
d being
Element of
(Af MOS) st
(
a,
b // c,
d or
a,
b // d,
c ) & (
a,
b // c,
p &
a,
b // c,
q & not
a = b implies
p = q ) )
by , Def3, Th19, Th20;
end;
definition
let IT be non
empty AffinStruct ;
attr a1 is
OrdTrapSpace-like means :
Def14:
:: GEOMTRAP:def 14
for
a,
b,
c,
d,
a1,
b1,
c1,
d1,
p,
q being
Element of
MOS holds
( (
a,
b // b,
c implies (
a = b &
b = c ) ) & (
a,
b // a1,
b1 &
a,
b // c1,
d1 &
a <> b implies
a1,
b1 // c1,
d1 ) & (
a,
b // c,
d implies (
c,
d // a,
b &
b,
a // d,
c ) ) & ex
d being
Element of
MOS st
(
a,
b // c,
d or
a,
b // d,
c ) & (
a,
b // c,
p &
a,
b // c,
q & not
a = b implies
p = q ) );
end;
:: deftheorem Def14 defines OrdTrapSpace-like GEOMTRAP:def 14 :
for
IT being non
empty AffinStruct holds
(
IT is
OrdTrapSpace-like iff for
a,
b,
c,
d,
a1,
b1,
c1,
d1,
p,
q being
Element of
IT holds
( (
a,
b // b,
c implies (
a = b &
b = c ) ) & (
a,
b // a1,
b1 &
a,
b // c1,
d1 &
a <> b implies
a1,
b1 // c1,
d1 ) & (
a,
b // c,
d implies (
c,
d // a,
b &
b,
a // d,
c ) ) & ex
d being
Element of
IT st
(
a,
b // c,
d or
a,
b // d,
c ) & (
a,
b // c,
p &
a,
b // c,
q & not
a = b implies
p = q ) ) );
theorem Th50: :: GEOMTRAP:50
theorem Th51: :: GEOMTRAP:51
for
OTS being
OrdTrapSpace for
a,
b,
c,
d being
Element of
OTS for
a',
b',
c',
d' being
Element of
(Lambda OTS) st
a = a' &
b = b' &
c = c' &
d = d' holds
(
a',
b' // c',
d' iff (
a,
b // c,
d or
a,
b // d,
c ) )
Lemma221:
for OTS being OrdTrapSpace
for a', b', c' being Element of (Lambda OTS) ex d' being Element of (Lambda OTS) st a',b' // c',d'
Lemma222:
for OTS being OrdTrapSpace
for a', b', c', d' being Element of (Lambda OTS) st a',b' // c',d' holds
c',d' // a',b'
Lemma223:
for OTS being OrdTrapSpace
for a1', b1', a', b', c', d' being Element of (Lambda OTS) st a1' <> b1' & a1',b1' // a',b' & a1',b1' // c',d' holds
a',b' // c',d'
Lemma224:
for OTS being OrdTrapSpace
for a', b', c', d', d1' being Element of (Lambda OTS) st a',b' // c',d' & a',b' // c',d1' & not a' = b' holds
d' = d1'
Lemma225:
for OTS being OrdTrapSpace
for a, b being Element of OTS holds a,b // a,b
Lemma226:
for OTS being OrdTrapSpace
for a', b' being Element of (Lambda OTS) holds a',b' // b',a'
definition
let IT be non
empty AffinStruct ;
attr a1 is
TrapSpace-like means :
Def15:
:: GEOMTRAP:def 15
for
a',
b',
c',
d',
p',
q' being
Element of
MOS holds
(
a',
b' // b',
a' & (
a',
b' // c',
d' &
a',
b' // c',
q' & not
a' = b' implies
d' = q' ) & (
p' <> q' &
p',
q' // a',
b' &
p',
q' // c',
d' implies
a',
b' // c',
d' ) & (
a',
b' // c',
d' implies
c',
d' // a',
b' ) & ex
x' being
Element of
MOS st
a',
b' // c',
x' );
end;
:: deftheorem Def15 defines TrapSpace-like GEOMTRAP:def 15 :
for
IT being non
empty AffinStruct holds
(
IT is
TrapSpace-like iff for
a',
b',
c',
d',
p',
q' being
Element of
IT holds
(
a',
b' // b',
a' & (
a',
b' // c',
d' &
a',
b' // c',
q' & not
a' = b' implies
d' = q' ) & (
p' <> q' &
p',
q' // a',
b' &
p',
q' // c',
d' implies
a',
b' // c',
d' ) & (
a',
b' // c',
d' implies
c',
d' // a',
b' ) & ex
x' being
Element of
IT st
a',
b' // c',
x' ) );
definition
let IT be non
empty AffinStruct ;
attr a1 is
Regular means :
Def16:
:: GEOMTRAP:def 16
for
p,
q,
a,
a1,
b,
b1,
c,
c1,
d,
d1 being
Element of
MOS st
p <> q &
p,
q // a,
a1 &
p,
q // b,
b1 &
p,
q // c,
c1 &
p,
q // d,
d1 &
a,
b // c,
d holds
a1,
b1 // c1,
d1;
end;
:: deftheorem Def16 defines Regular GEOMTRAP:def 16 :
for
IT being non
empty AffinStruct holds
(
IT is
Regular iff for
p,
q,
a,
a1,
b,
b1,
c,
c1,
d,
d1 being
Element of
IT st
p <> q &
p,
q // a,
a1 &
p,
q // b,
b1 &
p,
q // c,
c1 &
p,
q // d,
d1 &
a,
b // c,
d holds
a1,
b1 // c1,
d1 );