:: RLAFFIN1 semantic presentation
begin
registration
let
RLS
be ( ( non
empty
) ( non
empty
)
RLSStruct
) ;
let
A
be ( (
empty
) (
Relation-like
non-empty
empty-yielding
RAT
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V68
()
V71
() )
set
)
-valued
Function-like
one-to-one
constant
functional
empty
proper
V24
()
ordinal
natural
V32
()
ext-real
non
positive
non
negative
finite
finite-yielding
V44
()
cardinal
{}
: ( ( ) ( )
set
)
-element
V55
()
V56
()
V57
()
V58
()
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Subset
of ) ;
cluster
conv
A
: ( (
empty
) (
Relation-like
non-empty
empty-yielding
RAT
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V68
()
V71
() )
set
)
-valued
Function-like
one-to-one
constant
functional
empty
proper
V24
()
ordinal
natural
V32
()
ext-real
non
positive
non
negative
finite
finite-yielding
V44
()
cardinal
{}
: ( ( ) ( )
set
)
-element
V55
()
V56
()
V57
()
V58
()
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
->
empty
convex
;
end;
registration
let
RLS
be ( ( non
empty
) ( non
empty
)
RLSStruct
) ;
let
A
be ( ( non
empty
) ( non
empty
)
Subset
of ) ;
cluster
conv
A
: ( ( non
empty
) ( non
empty
)
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
->
non
empty
convex
;
end;
theorem
:: RLAFFIN1:1
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
v
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
conv
{
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( (
convex
) ( non
empty
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
{
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:2
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
being ( ( ) ( )
Subset
of ) holds
A
: ( ( ) ( )
Subset
of )
c=
conv
A
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:3
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) holds
conv
A
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
conv
B
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:4
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
S
,
A
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
conv
S
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) holds
conv
S
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
conv
(
S
: ( ( ) ( )
Subset
of )
\/
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:5
for
V
being ( ( non
empty
add-associative
) ( non
empty
add-associative
)
addLoopStr
)
for
A
being ( ( ) ( )
Subset
of )
for
v
,
w
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
(
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
w
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
add-associative
) ( non
empty
add-associative
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
add-associative
) ( non
empty
add-associative
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
(
w
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
add-associative
) ( non
empty
add-associative
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
add-associative
) ( non
empty
add-associative
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:6
for
V
being ( ( non
empty
Abelian
right_zeroed
) ( non
empty
Abelian
right_zeroed
V135
() )
addLoopStr
)
for
A
being ( ( ) ( )
Subset
of ) holds
(
0.
V
: ( ( non
empty
Abelian
right_zeroed
) ( non
empty
Abelian
right_zeroed
V135
() )
addLoopStr
)
)
: ( ( ) (
V84
(
b
1
: ( ( non
empty
Abelian
right_zeroed
) ( non
empty
Abelian
right_zeroed
V135
() )
addLoopStr
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
Abelian
right_zeroed
) ( non
empty
Abelian
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
Abelian
right_zeroed
) ( non
empty
Abelian
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
A
: ( ( ) ( )
Subset
of ) ;
theorem
:: RLAFFIN1:7
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
g
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) holds
card
A
: ( ( ) ( )
Subset
of ) : ( (
cardinal
) (
ordinal
cardinal
)
set
)
=
card
(
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( (
cardinal
) (
ordinal
cardinal
)
set
) ;
theorem
:: RLAFFIN1:8
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
v
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
(
{}
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
)
)
: ( ( ) (
Relation-like
non-empty
empty-yielding
RAT
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V68
()
V71
() )
set
)
-valued
Function-like
one-to-one
constant
functional
empty
proper
V24
()
ordinal
natural
V32
()
ext-real
non
positive
non
negative
finite
finite-yielding
V44
()
cardinal
{}
: ( ( ) ( )
set
)
-element
V55
()
V56
()
V57
()
V58
()
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
{}
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) (
Relation-like
non-empty
empty-yielding
RAT
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V68
()
V71
() )
set
)
-valued
Function-like
one-to-one
constant
functional
empty
proper
V24
()
ordinal
natural
V32
()
ext-real
non
positive
non
negative
finite
finite-yielding
V44
()
cardinal
{}
: ( ( ) ( )
set
)
-element
V55
()
V56
()
V57
()
V58
()
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:9
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
A
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
B
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:10
for
r
,
s
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
AR
being ( ( ) ( )
Subset
of ) holds
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
AR
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
AR
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:11
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
AR
being ( ( ) ( )
Subset
of ) holds 1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
*
AR
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
AR
: ( ( ) ( )
Subset
of ) ;
theorem
:: RLAFFIN1:12
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
*
A
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
{
(
0.
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
)
: ( ( ) (
V84
(
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:13
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
LS1
,
LS2
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
for
F
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
(
LS1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
+
LS2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
(
LS1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
+
(
LS2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:14
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
for
F
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) holds
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:15
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of ) is
linearly-independent
&
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) &
Lin
B
: ( ( ) ( )
Subset
of ) : ( (
strict
) ( non
empty
left_complementable
right_complementable
strict
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
Subspace
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
=
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) holds
ex
I
being ( (
linearly-independent
) (
linearly-independent
)
Subset
of ) st
(
A
: ( ( ) ( )
Subset
of )
c=
I
: ( (
linearly-independent
) (
linearly-independent
)
Subset
of ) &
I
: ( (
linearly-independent
) (
linearly-independent
)
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) &
Lin
I
: ( (
linearly-independent
) (
linearly-independent
)
Subset
of ) : ( (
strict
) ( non
empty
left_complementable
right_complementable
strict
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
Subspace
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
=
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) ;
begin
definition
let
G
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ;
let
LG
be ( ( ) (
Relation-like
the
carrier
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) ;
let
g
be ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) ;
func
g
+
LG
->
( ( ) (
Relation-like
the
carrier
of
G
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( ) ( )
set
) )
means
:: RLAFFIN1:def 1
for
h
being ( ( ) ( )
Element
of ( ( ) ( )
set
) ) holds
it
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
G
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
G
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
G
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
G
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
.
h
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
LG
: ( ( ) ( )
set
)
.
(
h
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
-
g
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
G
: ( ( ) ( )
set
) ,
G
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
G
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
G
: ( ( ) ( )
set
) ,
G
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
G
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
G
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
end;
theorem
:: RLAFFIN1:16
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
LG
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
for
g
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
Carrier
(
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
(
Carrier
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:17
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
LG1
,
LG2
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
for
g
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
(
LG1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
+
LG2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
=
(
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
LG1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
+
(
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
LG2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) ;
theorem
:: RLAFFIN1:18
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) ;
theorem
:: RLAFFIN1:19
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
LG
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
for
g
,
h
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
(
h
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
=
(
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
h
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) )
+
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) ;
theorem
:: RLAFFIN1:20
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
g
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
(
ZeroLC
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
=
ZeroLC
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) ;
theorem
:: RLAFFIN1:21
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
LG
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) holds
(
0.
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
)
: ( ( ) (
V84
(
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
) )
+
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
=
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) ;
definition
let
R
be ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ;
let
LR
be ( ( ) (
Relation-like
the
carrier
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) ;
let
r
be ( ( ) (
V24
()
V32
()
ext-real
)
Real
) ;
func
r
(*)
LR
->
( ( ) (
Relation-like
the
carrier
of
R
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( ) ( )
set
) )
means
:: RLAFFIN1:def 2
for
v
being ( ( ) ( )
Element
of ( ( ) ( )
set
) ) holds
it
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
R
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
.
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
LR
: ( ( ) ( )
set
)
.
(
(
r
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
R
: ( ( ) ( )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
R
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
R
: ( ( ) ( )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
"
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
R
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
if
r
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
R
: ( ( ) ( )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
R
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
R
: ( ( ) ( )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
<>
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
otherwise
it
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
R
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
R
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
ZeroLC
R
: ( ( ) ( )
set
) : ( ( ) (
Relation-like
the
carrier
of
R
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( ) ( )
set
) ) ;
end;
theorem
:: RLAFFIN1:22
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
LR
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) holds
Carrier
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
LR
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
Carrier
LR
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:23
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
LR
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) st
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
<>
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
Carrier
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
LR
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
Carrier
LR
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:24
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
LR1
,
LR2
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
(
LR1
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
+
LR2
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
=
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
LR1
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
+
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
LR2
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) ;
theorem
:: RLAFFIN1:25
for
r
,
s
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
L
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
=
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
L
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) ;
theorem
:: RLAFFIN1:26
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
(
ZeroLC
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
=
ZeroLC
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) ;
theorem
:: RLAFFIN1:27
for
r
,
s
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
LR
being ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
(
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
LR
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
=
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
(*)
LR
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) ;
theorem
:: RLAFFIN1:28
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
LR
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) holds 1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
(*)
LR
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
=
LR
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) ;
begin
definition
let
S
be ( ( non
empty
) ( non
empty
)
addLoopStr
) ;
let
LS
be ( ( ) (
Relation-like
the
carrier
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) ;
func
sum
LS
->
( ( ) (
V24
()
V32
()
ext-real
)
Real
)
means
:: RLAFFIN1:def 3
ex
F
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
S
: ( ( ) ( )
set
) : ( ( ) ( )
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( )
set
) ) st
(
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) is
one-to-one
&
rng
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
set
)
=
Carrier
LS
: ( ( ) ( )
set
) : ( ( ) ( )
Element
of
K19
( the
carrier
of
S
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) &
it
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
S
: ( ( ) ( )
set
) ,
S
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
)
-defined
S
: ( ( ) ( )
set
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
S
: ( ( ) ( )
set
) ,
S
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ,
S
: ( ( ) ( )
set
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
Sum
(
LS
: ( ( ) ( )
set
)
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) );
end;
theorem
:: RLAFFIN1:29
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
LS
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
for
F
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) st
Carrier
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
misses
rng
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
set
) holds
Sum
(
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) ;
theorem
:: RLAFFIN1:30
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
LS
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
for
F
being ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) st
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) is
one-to-one
&
Carrier
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
rng
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
set
) holds
sum
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
Sum
(
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
*
F
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-valued
Function-like
FinSequence-like
)
FinSequence
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
Relation-like
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) )
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
FinSequence-like
V55
()
V56
()
V57
() )
FinSequence
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:31
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
) holds
sum
(
ZeroLC
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
)
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) ;
theorem
:: RLAFFIN1:32
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
LS
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
for
v
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) st
Carrier
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
{
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) holds
sum
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
.
v
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:33
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
LS
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
for
v1
,
v2
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) st
Carrier
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
{
v1
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) ,
v2
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) &
v1
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
<>
v2
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
sum
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
(
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
.
v1
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
+
(
LS
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
.
v2
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:34
for
S
being ( ( non
empty
) ( non
empty
)
addLoopStr
)
for
LS1
,
LS2
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
S
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) holds
sum
(
LS1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
+
LS2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
(
sum
LS1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
+
(
sum
LS2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
) ( non
empty
)
addLoopStr
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:35
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
sum
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:36
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L1
,
L2
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
sum
(
L1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
-
L2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
(
sum
L1
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
-
(
sum
L2
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:37
for
G
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
)
for
LG
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
G
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
for
g
being ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) ) holds
sum
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
sum
(
g
: ( ( ) ( )
Element
of ( ( ) ( non
empty
)
set
) )
+
LG
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
V135
() )
addLoopStr
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
) ;
theorem
:: RLAFFIN1:38
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
LR
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
R
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) st
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
<>
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
sum
LR
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
sum
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
LR
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
) ;
theorem
:: RLAFFIN1:39
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
Sum
(
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
=
(
(
sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
(
Sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:40
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
Sum
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
(*)
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
Sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) ;
begin
definition
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
let
A
be ( ( ) ( )
Subset
of ) ;
attr
A
is
affinely-independent
means
:: RLAFFIN1:def 4
(
A
: ( ( ) ( )
set
) is
empty
or ex
v
being ( ( ) ( )
VECTOR
of ( ( ) ( )
set
) ) st
(
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
A
: ( ( ) ( )
set
) &
(
(
-
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
V
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
+
A
: ( ( ) ( )
set
)
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) )
\
{
(
0.
V
: ( ( ) ( )
set
)
)
: ( ( ) ( )
Element
of the
carrier
of
V
: ( ( ) ( )
set
) : ( ( ) ( )
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
Element
of
K19
( the
carrier
of
V
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) is
linearly-independent
) );
end;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
cluster
empty
->
affinely-independent
for ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( ) ( )
set
) : ( ( ) ( )
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
let
v
be ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) ;
cluster
{
v
: ( ( ) ( )
Element
of the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
set
)
->
affinely-independent
for ( ( ) ( )
Subset
of ) ;
let
w
be ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) ;
cluster
{
v
: ( ( ) ( )
Element
of the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) ,
w
: ( ( ) ( )
Element
of the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) (
finite
)
set
)
->
affinely-independent
for ( ( ) ( )
Subset
of ) ;
end;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
cluster
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
affinely-independent
for ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: RLAFFIN1:41
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
(
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
iff for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) st
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
A
: ( ( ) ( )
Subset
of ) holds
(
(
-
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
\
{
(
0.
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
)
: ( ( ) (
V84
(
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
affinely-independent
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
linearly-independent
) ;
theorem
:: RLAFFIN1:42
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
(
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
iff for
L
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
A
: ( ( ) ( )
Subset
of ) ) st
Sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( ) ( )
Subset
of ) ) : ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
=
0.
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) (
V84
(
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) &
sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( ) ( )
Subset
of ) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
Carrier
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( ) ( )
Subset
of ) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
{}
: ( ( ) ( )
set
) ) ;
theorem
:: RLAFFIN1:43
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
&
B
: ( ( ) ( )
Subset
of )
c=
A
: ( ( ) ( )
Subset
of ) holds
B
: ( ( ) ( )
Subset
of ) is
affinely-independent
;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
cluster
linearly-independent
->
affinely-independent
for ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
end;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
let
I
be ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) ;
let
v
be ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) ;
cluster
v
: ( ( ) ( )
Element
of the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) )
+
I
: ( (
affinely-independent
) (
affinely-independent
)
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
->
affinely-independent
;
end;
theorem
:: RLAFFIN1:44
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) st
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
affinely-independent
holds
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
let
I
be ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) ;
let
r
be ( ( ) (
V24
()
V32
()
ext-real
)
Real
) ;
cluster
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
I
: ( (
affinely-independent
) (
affinely-independent
)
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
->
affinely-independent
;
end;
theorem
:: RLAFFIN1:45
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) st
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
A
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
affinely-independent
&
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
<>
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
;
theorem
:: RLAFFIN1:46
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) st
0.
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) (
V84
(
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
in
A
: ( ( ) ( )
Subset
of ) holds
(
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
iff
A
: ( ( ) ( )
Subset
of )
\
{
(
0.
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
)
: ( ( ) (
V84
(
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
affinely-independent
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
linearly-independent
) ;
definition
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
let
F
be ( ( ) ( )
Subset-Family
of ) ;
attr
F
is
affinely-independent
means
:: RLAFFIN1:def 5
for
A
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
in
F
: ( ( ) ( )
set
) holds
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
;
end;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
cluster
empty
->
affinely-independent
for ( ( ) ( )
Element
of
K19
(
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
let
I
be ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) ;
cluster
{
I
: ( (
affinely-independent
) (
affinely-independent
)
Element
of
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
set
)
->
affinely-independent
for ( ( ) ( )
Subset-Family
of ) ;
end;
registration
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
cluster
empty
affinely-independent
for ( ( ) ( )
Element
of
K19
(
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
cluster
non
empty
affinely-independent
for ( ( ) ( )
Element
of
K19
(
K19
( the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
end;
theorem
:: RLAFFIN1:47
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
F1
,
F2
being ( ( ) ( )
Subset-Family
of ) st
F1
: ( ( ) ( )
Subset-Family
of ) is
affinely-independent
&
F2
: ( ( ) ( )
Subset-Family
of ) is
affinely-independent
holds
F1
: ( ( ) ( )
Subset-Family
of )
\/
F2
: ( ( ) ( )
Subset-Family
of ) : ( ( ) ( )
Element
of
K19
(
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
affinely-independent
;
theorem
:: RLAFFIN1:48
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
F1
,
F2
being ( ( ) ( )
Subset-Family
of ) st
F1
: ( ( ) ( )
Subset-Family
of )
c=
F2
: ( ( ) ( )
Subset-Family
of ) &
F2
: ( ( ) ( )
Subset-Family
of ) is
affinely-independent
holds
F1
: ( ( ) ( )
Subset-Family
of ) is
affinely-independent
;
begin
definition
let
RLS
be ( ( non
empty
) ( non
empty
)
RLSStruct
) ;
let
A
be ( ( ) ( )
Subset
of ) ;
func
Affin
A
->
( ( ) ( )
Subset
of )
equals
:: RLAFFIN1:def 6
meet
{
B
: ( (
Affine
) (
Affine
)
Subset
of ) where
B
is ( (
Affine
) (
Affine
)
Subset
of ) :
A
: ( ( ) ( )
set
)
c=
B
: ( (
Affine
) (
Affine
)
Subset
of )
}
: ( ( ) ( )
set
) ;
end;
registration
let
RLS
be ( ( non
empty
) ( non
empty
)
RLSStruct
) ;
let
A
be ( ( ) ( )
Subset
of ) ;
cluster
Affin
A
: ( ( ) ( )
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Subset
of )
->
Affine
;
end;
registration
let
RLS
be ( ( non
empty
) ( non
empty
)
RLSStruct
) ;
let
A
be ( (
empty
) (
Relation-like
non-empty
empty-yielding
RAT
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V68
()
V71
() )
set
)
-valued
Function-like
one-to-one
constant
functional
empty
proper
V24
()
ordinal
natural
V32
()
ext-real
non
positive
non
negative
finite
finite-yielding
V44
()
cardinal
{}
: ( ( ) ( )
set
)
-element
V55
()
V56
()
V57
()
V58
()
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Subset
of ) ;
cluster
Affin
A
: ( (
empty
) (
Relation-like
non-empty
empty-yielding
RAT
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V68
()
V71
() )
set
)
-valued
Function-like
one-to-one
constant
functional
empty
proper
V24
()
ordinal
natural
V32
()
ext-real
non
positive
non
negative
finite
finite-yielding
V44
()
cardinal
{}
: ( ( ) ( )
set
)
-element
V55
()
V56
()
V57
()
V58
()
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
->
empty
;
end;
registration
let
RLS
be ( ( non
empty
) ( non
empty
)
RLSStruct
) ;
let
A
be ( ( non
empty
) ( non
empty
)
Subset
of ) ;
cluster
Affin
A
: ( ( non
empty
) ( non
empty
)
Element
of
K19
( the
carrier
of
RLS
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
->
non
empty
;
end;
theorem
:: RLAFFIN1:49
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
being ( ( ) ( )
Subset
of ) holds
A
: ( ( ) ( )
Subset
of )
c=
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:50
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
being ( (
Affine
) (
Affine
)
Subset
of ) holds
A
: ( (
Affine
) (
Affine
)
Subset
of )
=
Affin
A
: ( (
Affine
) (
Affine
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:51
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) &
B
: ( ( ) ( )
Subset
of ) is
Affine
holds
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) ;
theorem
:: RLAFFIN1:52
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
B
: ( ( ) ( )
Subset
of ) holds
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
c=
Affin
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:53
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) holds
Affin
(
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
=
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
(
Affin
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
Affine
)
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:54
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
AR
being ( ( ) ( )
Subset
of ) st
AR
: ( ( ) ( )
Subset
of ) is
Affine
holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
AR
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
Affine
;
theorem
:: RLAFFIN1:55
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
R
being ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
)
for
AR
being ( ( ) ( )
Subset
of ) st
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
<>
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
Affin
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
AR
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
Affin
AR
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
Affine
)
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:56
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
Affin
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
=
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
Affin
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) (
Affine
)
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:57
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) st
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) holds
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
=
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
+
(
Up
(
Lin
(
(
-
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
A
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
)
: ( (
strict
) ( non
empty
left_complementable
right_complementable
strict
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
Subspace
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( non
empty
) ( non
empty
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:58
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
(
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
iff for
B
being ( ( ) ( )
Subset
of ) st
B
: ( ( ) ( )
Subset
of )
c=
A
: ( ( ) ( )
Subset
of ) &
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
=
Affin
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) holds
A
: ( ( ) ( )
Subset
of )
=
B
: ( ( ) ( )
Subset
of ) ) ;
theorem
:: RLAFFIN1:59
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
=
{
(
Sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( ) ( )
Subset
of ) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) where
L
is ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
A
: ( ( ) ( )
Subset
of ) ) :
sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( ) ( )
Subset
of ) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
}
;
theorem
:: RLAFFIN1:60
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of )
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
c=
A
: ( ( ) ( )
Subset
of ) holds
ex
Ia
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
(
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
c=
Ia
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) &
Ia
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
c=
A
: ( ( ) ( )
Subset
of ) &
Affin
Ia
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
=
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ) ;
theorem
:: RLAFFIN1:61
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
,
B
being ( (
finite
) (
finite
)
Subset
of ) st
A
: ( (
finite
) (
finite
)
Subset
of ) is
affinely-independent
&
Affin
A
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
=
Affin
B
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
card
B
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
omega
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
set
) )
<=
card
A
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
omega
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
set
) ) holds
B
: ( (
finite
) (
finite
)
Subset
of ) is
affinely-independent
;
theorem
:: RLAFFIN1:62
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) holds
(
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) is
convex
iff (
sum
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) & ( for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) holds
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
<=
L
: ( ( ) (
Relation-like
the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ) ) ) ;
theorem
:: RLAFFIN1:63
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) st
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) is
convex
holds
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
x
: ( ( ) ( )
set
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
<=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) ;
theorem
:: RLAFFIN1:64
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
L
being ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) st
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) is
convex
&
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
x
: ( ( ) ( )
set
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
Carrier
L
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
finite
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
=
{
x
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
set
) ;
theorem
:: RLAFFIN1:65
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
conv
A
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:66
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) st
x
: ( ( ) ( )
set
)
in
conv
A
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) &
(
conv
A
: ( ( ) ( )
Subset
of )
)
: ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
\
{
x
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
set
) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
convex
holds
x
: ( ( ) ( )
set
)
in
A
: ( ( ) ( )
Subset
of ) ;
theorem
:: RLAFFIN1:67
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
being ( ( ) ( )
Subset
of ) holds
Affin
(
conv
A
: ( ( ) ( )
Subset
of )
)
: ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
=
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:68
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
conv
A
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
c=
conv
B
: ( ( ) ( )
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) holds
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
c=
Affin
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:69
for
RLS
being ( ( non
empty
) ( non
empty
)
RLSStruct
)
for
A
,
B
being ( ( ) ( )
Subset
of ) st
A
: ( ( ) ( )
Subset
of )
c=
Affin
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) holds
Affin
(
A
: ( ( ) ( )
Subset
of )
\/
B
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
) ( non
empty
)
RLSStruct
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of )
=
Affin
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
begin
definition
let
V
be ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ;
let
A
be ( ( ) ( )
Subset
of ) ;
assume
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
;
let
x
be ( ( ) ( )
set
) ;
assume
x
: ( ( ) ( )
set
)
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
func
x
|--
A
->
( ( ) (
Relation-like
the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
A
: ( ( ) ( )
set
) )
means
:: RLAFFIN1:def 7
(
Sum
it
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) ) : ( ( ) (
Relation-like
)
set
)
-defined
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) ) : ( ( ) (
Relation-like
)
set
) ,
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) : ( ( ) ( non
empty
)
set
) )
=
x
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) &
sum
it
: ( (
Function-like
quasi_total
) (
Relation-like
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) ) : ( ( ) (
Relation-like
)
set
)
-defined
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
)
-valued
Function-like
quasi_total
)
Element
of
K19
(
K20
(
K20
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ,
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) ) : ( ( ) (
Relation-like
)
set
) ,
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RLSStruct
) ) : ( ( ) (
Relation-like
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) );
end;
theorem
:: RLAFFIN1:70
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v1
,
v2
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) holds
(
(
(
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
5
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
=
(
(
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
(
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
5
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
+
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
(
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
5
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) ;
theorem
:: RLAFFIN1:71
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
x
: ( ( ) ( )
set
)
in
conv
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) holds
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
4
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) ) is
convex
&
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
<=
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
4
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) &
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
4
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
<=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) ) ;
theorem
:: RLAFFIN1:72
for
x
,
y
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
x
: ( ( ) ( )
set
)
in
conv
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) holds
(
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
4
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
y
: ( ( ) ( )
set
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) iff (
x
: ( ( ) ( )
set
)
=
y
: ( ( ) ( )
set
) &
x
: ( ( ) ( )
set
)
in
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) ) ) ;
theorem
:: RLAFFIN1:73
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
x
: ( ( ) ( )
set
)
in
Affin
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) & ( for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) st
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) holds
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
<=
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ) holds
x
: ( ( ) ( )
set
)
in
conv
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:74
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
x
: ( ( ) ( )
set
)
in
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) holds
(
conv
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
\
{
x
: ( ( ) ( )
set
)
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
)
set
) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
convex
;
theorem
:: RLAFFIN1:75
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
for
B
being ( ( ) ( )
Subset
of ) st
x
: ( ( ) ( )
set
)
in
Affin
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) & ( for
y
being ( ( ) ( )
set
) st
y
: ( ( ) ( )
set
)
in
B
: ( ( ) ( )
Subset
of ) holds
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
y
: ( ( ) ( )
set
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) ) holds
(
x
: ( ( ) ( )
set
)
in
Affin
(
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
\
B
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of ) &
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
=
x
: ( ( ) ( )
set
)
|--
(
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
\
B
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
\
b
4
: ( ( ) ( )
Subset
of ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ) ) ;
theorem
:: RLAFFIN1:76
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
for
B
being ( ( ) ( )
Subset
of ) st
x
: ( ( ) ( )
set
)
in
conv
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) & ( for
y
being ( ( ) ( )
set
) st
y
: ( ( ) ( )
set
)
in
B
: ( ( ) ( )
Subset
of ) holds
(
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
y
: ( ( ) ( )
set
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
0
: ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) ) holds
x
: ( ( ) ( )
set
)
in
conv
(
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
\
B
: ( ( ) ( )
Subset
of )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( (
convex
) (
convex
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) ;
theorem
:: RLAFFIN1:77
for
x
being ( ( ) ( )
set
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
B
being ( ( ) ( )
Subset
of )
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
B
: ( ( ) ( )
Subset
of )
c=
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) &
x
: ( ( ) ( )
set
)
in
Affin
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) holds
x
: ( ( ) ( )
set
)
|--
B
: ( ( ) ( )
Subset
of ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( ) ( )
Subset
of ) )
=
x
: ( ( ) ( )
set
)
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
4
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) ) ;
theorem
:: RLAFFIN1:78
for
r
,
s
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v1
,
v2
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) st
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
+
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) holds
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
(
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ;
theorem
:: RLAFFIN1:79
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
,
B
being ( (
finite
) (
finite
)
Subset
of ) st
A
: ( (
finite
) (
finite
)
Subset
of ) is
affinely-independent
&
Affin
A
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
c=
Affin
B
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) holds
card
A
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
omega
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
set
) )
<=
card
B
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
omega
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
set
) ) ;
theorem
:: RLAFFIN1:80
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
A
,
B
being ( (
finite
) (
finite
)
Subset
of ) st
A
: ( (
finite
) (
finite
)
Subset
of ) is
affinely-independent
&
Affin
A
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of )
c=
Affin
B
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
card
A
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
omega
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
set
) )
=
card
B
: ( (
finite
) (
finite
)
Subset
of ) : ( ( ) (
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
omega
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
set
) ) holds
B
: ( (
finite
) (
finite
)
Subset
of ) is
affinely-independent
;
theorem
:: RLAFFIN1:81
for
r
,
s
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
L1
,
L2
being ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
V
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) ) st
L1
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
<>
L2
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) holds
(
(
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
L1
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
+
(
(
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
L2
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
)
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
=
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
) iff
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
(
(
L2
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
-
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
/
(
(
L2
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
-
(
L1
: ( ( ) (
Relation-like
the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ) ;
theorem
:: RLAFFIN1:82
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) holds
(
A
: ( ( ) ( )
Subset
of )
\/
{
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
affinely-independent
)
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of
K19
( the
carrier
of
b
1
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) is
affinely-independent
iff (
A
: ( ( ) ( )
Subset
of ) is
affinely-independent
& (
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
A
: ( ( ) ( )
Subset
of ) or not
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) ) ) ) ;
theorem
:: RLAFFIN1:83
for
r
,
s
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
w
,
v1
,
v2
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
A
being ( ( ) ( )
Subset
of ) st not
w
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
A
: ( ( ) ( )
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
A
: ( ( ) ( )
Subset
of ) &
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
A
: ( ( ) ( )
Subset
of ) &
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
<>
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) ) &
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
w
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
(
(
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
=
(
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
w
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
(
(
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
3
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) holds
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
s
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
) &
v1
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
=
v2
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) ) ;
theorem
:: RLAFFIN1:84
for
r
being ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
for
V
being ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
)
for
v
,
w
,
p
being ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
for
I
being ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) st
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) &
w
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) : ( ( ) (
Affine
)
Subset
of ) &
p
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
in
Affin
(
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
\
{
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
}
: ( ( ) ( non
empty
trivial
finite
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-element
affinely-independent
)
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of
K19
( the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( non
empty
)
set
) ) : ( ( ) (
Affine
)
Subset
of ) &
w
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
=
(
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
*
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) )
+
(
(
1 : ( ( ) ( non
empty
V24
()
ordinal
natural
V32
()
V33
()
V34
()
ext-real
positive
non
negative
finite
cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
() )
Element
of
NAT
: ( ( ) ( non
trivial
ordinal
non
finite
cardinal
limit_cardinal
V65
()
V66
()
V67
()
V68
()
V69
()
V70
()
V71
() )
Element
of
K19
(
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) : ( ( ) ( non
empty
non
trivial
non
finite
)
set
) ) )
-
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
)
: ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) )
*
p
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
)
: ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) : ( ( ) ( )
Element
of the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
) ) holds
r
: ( ( ) (
V24
()
V32
()
ext-real
)
Real
)
=
(
w
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) )
|--
I
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of )
)
: ( ( ) (
Relation-like
the
carrier
of
b
2
: ( ( non
empty
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
) ( non
empty
left_complementable
right_complementable
Abelian
add-associative
right_zeroed
vector-distributive
scalar-distributive
scalar-associative
scalar-unital
V135
() )
RealLinearSpace
) : ( ( ) ( non
empty
)
set
)
-defined
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
)
-valued
Function-like
total
quasi_total
V55
()
V56
()
V57
() )
Linear_Combination
of
b
6
: ( (
affinely-independent
) (
affinely-independent
)
Subset
of ) )
.
v
: ( ( ) ( )
VECTOR
of ( ( ) ( non
empty
)
set
) ) : ( ( ) (
V24
()
V32
()
ext-real
)
Element
of
REAL
: ( ( ) ( non
empty
non
trivial
non
finite
V65
()
V66
()
V67
()
V71
() )
set
) ) ;