REAL is non empty non trivial non finite V60() V61() V62() V66() set
NAT is non empty non trivial ordinal non finite cardinal limit_cardinal V60() V61() V62() V63() V64() V65() V66() Element of bool REAL
bool REAL is non empty non trivial non finite set
INT is non empty non trivial non finite V60() V61() V62() V63() V64() V66() set
COMPLEX is non empty non trivial non finite V60() V66() set
RAT is non empty non trivial non finite V60() V61() V62() V63() V66() set
[:COMPLEX,COMPLEX:] is non empty non trivial non finite complex-valued set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite complex-valued set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:REAL,REAL:] is non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:REAL,REAL:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
[:RAT,RAT:] is RAT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is RAT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:NAT,NAT:] is RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[:NAT,NAT:],NAT:] is RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
omega is non empty non trivial ordinal non finite cardinal limit_cardinal V60() V61() V62() V63() V64() V65() V66() set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
{} is Function-like functional empty trivial V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() set
the Function-like functional empty trivial V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() set is Function-like functional empty trivial V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() set
2 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[:1,1:] is RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:1,1:] is non empty finite V39() set
[:[:1,1:],1:] is RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:[:1,1:],1:] is non empty finite V39() set
[:[:1,1:],REAL:] is non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:1,1:],REAL:] is non empty non trivial non finite set
[:2,2:] is RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
[:[:2,2:],REAL:] is non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:2,2:],REAL:] is non empty non trivial non finite set
TOP-REAL 2 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict V231() V232() RLTopStruct
the carrier of (TOP-REAL 2) is functional non empty set
F_Real is non empty non degenerated non trivial right_complementable almost_left_invertible strict Abelian add-associative right_zeroed V199() V201() V203() right-distributive left-distributive right_unital well-unital V211() left_unital doubleLoopStr
the carrier of F_Real is non empty non trivial V60() V61() V62() set
K637() is non empty strict multMagma
the carrier of K637() is non empty set
K642() is non empty strict V199() V200() V201() V203() V241() V242() V243() V244() V245() V246() multMagma
K643() is non empty strict V201() V203() V244() V245() V246() M25(K642())
K644() is non empty strict V199() V201() V203() V244() V245() V246() V247() M28(K642(),K643())
K646() is non empty strict V199() V201() V203() multMagma
K647() is non empty strict V199() V201() V203() V247() M25(K646())
0 is Function-like functional empty trivial V21() ordinal natural real V30() V31() ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() Element of NAT
K697(0,1,2) is non empty finite V60() V61() V62() V63() V64() V65() set
[:K697(0,1,2),K697(0,1,2):] is RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
[:[:K697(0,1,2),K697(0,1,2):],K697(0,1,2):] is RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:[:K697(0,1,2),K697(0,1,2):],K697(0,1,2):] is non empty finite V39() set
bool [:K697(0,1,2),K697(0,1,2):] is non empty finite V39() set
[:NAT,REAL:] is non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial non finite set
K860() is V283() TopStruct
the carrier of K860() is V60() V61() V62() set
RealSpace is strict V283() MetrStruct
R^1 is non empty strict TopSpace-like V283() TopStruct
Euclid 0 is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
TopSpaceMetr (Euclid 0) is trivial finite TopStruct
3 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
the carrier of R^1 is non empty V60() V61() V62() set
bool the carrier of R^1 is non empty set
-infty is non empty non real ext-real non positive negative set
REAL 0 is functional non empty FinSequence-membered FinSequenceSet of REAL
TOP-REAL 0 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict V231() V232() RLTopStruct
0. (TOP-REAL 0) is Relation-like NAT -defined REAL -valued Function-like one-to-one constant functional empty trivial V21() ordinal natural real ext-real non positive non negative finite finite-yielding V39() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing V60() V61() V62() V63() V64() V65() V66() zero Element of the carrier of (TOP-REAL 0)
the carrier of (TOP-REAL 0) is functional non empty set
the ZeroF of (TOP-REAL 0) is Relation-like NAT -defined REAL -valued Function-like one-to-one constant functional empty trivial V21() ordinal natural real ext-real non positive non negative finite finite-yielding V39() cardinal 0 -element FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued increasing decreasing non-decreasing non-increasing V60() V61() V62() V63() V64() V65() V66() Element of the carrier of (TOP-REAL 0)
{(0. (TOP-REAL 0))} is functional non empty trivial finite V39() 1 -element V60() V61() V62() V63() V64() V65() set
Seg 1 is non empty trivial finite 1 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= 1 ) } is set
{1} is non empty trivial finite V39() 1 -element V60() V61() V62() V63() V64() V65() set
Seg 2 is non empty finite 2 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= 2 ) } is set
{1,2} is non empty finite V39() V60() V61() V62() V63() V64() V65() set
TOP-REAL 1 is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict V231() V232() RLTopStruct
the carrier of (TOP-REAL 1) is functional non empty set
[: the carrier of (TOP-REAL 1), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL 1), the carrier of R^1:] is non empty set
the carrier of F_Real * is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Real
0. F_Real is V21() real ext-real zero Element of the carrier of F_Real
the ZeroF of F_Real is V21() real ext-real Element of the carrier of F_Real
- 1 is V21() real V30() V31() ext-real non positive Element of INT
K143() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
K145() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:[:REAL,REAL:],REAL:]
doubleLoopStr(# REAL,K143(),K145(),1,0 #) is strict doubleLoopStr
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict V231() V232() RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
Affn is Relation-like NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
@ Affn is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ Affn) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Affn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
n ^ Affn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
L is V21() real ext-real set
Intervals (n,L) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Intervals (Affn,L) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Intervals (n,L)) ^ (Intervals (Affn,L)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Intervals ((n ^ Affn),L) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (Intervals ((n ^ Affn),L)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (n ^ Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
len (n ^ Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len n is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(len n) + (len Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len ((Intervals (n,L)) ^ (Intervals (Affn,L))) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len (Intervals (n,L)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len (Intervals (Affn,L)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(len (Intervals (n,L))) + (len (Intervals (Affn,L))) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom (Intervals (n,L)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom n is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (Intervals (Affn,L)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom Affn is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom ((Intervals (n,L)) ^ (Intervals (Affn,L))) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
f is V21() ordinal natural real ext-real non negative finite cardinal set
(Intervals ((n ^ Affn),L)) . f is set
(n ^ Affn) . f is V21() real ext-real set
((n ^ Affn) . f) - L is V21() real ext-real Element of REAL
((n ^ Affn) . f) + L is V21() real ext-real Element of REAL
].(((n ^ Affn) . f) - L),(((n ^ Affn) . f) + L).[ is V60() V61() V62() Element of bool REAL
((Intervals (n,L)) ^ (Intervals (Affn,L))) . f is set
(Intervals (n,L)) . f is set
n . f is V21() real ext-real set
f is V21() ordinal natural real ext-real non negative finite cardinal set
(len (Intervals (n,L))) + f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
f is V21() ordinal natural real ext-real non negative finite cardinal set
(len (Intervals (n,L))) + f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((Intervals (n,L)) ^ (Intervals (Affn,L))) . f is set
(Intervals (Affn,L)) . f is set
Affn . f is V21() real ext-real set
n is set
Affn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
L is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Affn ^ L is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Affn ^ L) is set
product Affn is set
product L is set
len (Affn ^ L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(len Affn) + (len L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom Affn is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg (len Affn) is finite len Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len Affn ) } is set
(Affn ^ L) | (Seg (len Affn)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
dom (Affn ^ L) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
A is Relation-like Function-like set
dom A is set
Seg (len (Affn ^ L)) is finite len (Affn ^ L) -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len (Affn ^ L) ) } is set
L is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
L | (len Affn) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
L | (Seg (len Affn)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(L | (len Affn)) ^ f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
f ^ B is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len B is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(len f) + (len B) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom L is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom B is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k is set
Y is V21() ordinal natural real ext-real non negative finite cardinal set
Y + (len Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
L . (Y + (len Affn)) is set
(Affn ^ L) . (Y + (len Affn)) is set
L . Y is set
B . k is set
L . k is set
A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
L is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A ^ L is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
E is Relation-like Function-like set
dom E is set
E is set
f is V21() ordinal natural real ext-real non negative finite cardinal set
(Affn ^ L) . E is set
Affn . f is set
(A ^ L) . E is set
A . f is set
f is Relation-like Function-like set
dom f is set
f is Relation-like Function-like set
dom f is set
f is V21() ordinal natural real ext-real non negative finite cardinal set
f is V21() ordinal natural real ext-real non negative finite cardinal set
(len Affn) + f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
f is V21() ordinal natural real ext-real non negative finite cardinal set
(len Affn) + f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(Affn ^ L) . E is set
L . f is set
(A ^ L) . E is set
L . f is set
B is Relation-like Function-like set
dom B is set
B is Relation-like Function-like set
dom B is set
f is V21() ordinal natural real ext-real non negative finite cardinal set
len (A ^ L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(len A) + (len L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
E is Relation-like Function-like set
dom E is set
dom (A ^ L) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
{(0. n)} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of n
Affn is Element of bool the carrier of n
Lin Affn is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
Affn \ {(0. n)} is Element of bool the carrier of n
Lin (Affn \ {(0. n)}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
(0). n is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
the carrier of ((0). n) is non empty set
Lin {(0. n)} is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
(Affn \ {(0. n)}) \/ {(0. n)} is non empty Element of bool the carrier of n
(Lin (Affn \ {(0. n)})) + ((0). n) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
(Omega). n is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
the carrier of ((Omega). n) is non empty set
bool the carrier of ((Omega). n) is non empty set
L is finite Element of bool the carrier of ((Omega). n)
the carrier of n is non empty set
the ZeroF of n is Element of the carrier of n
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the Mult of n is Relation-like [:REAL, the carrier of n:] -defined the carrier of n -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of n:], the carrier of n:]
[:REAL, the carrier of n:] is non empty non trivial non finite set
[:[:REAL, the carrier of n:], the carrier of n:] is non empty non trivial non finite set
bool [:[:REAL, the carrier of n:], the carrier of n:] is non empty non trivial non finite set
RLSStruct(# the carrier of n, the ZeroF of n, the addF of n, the Mult of n #) is strict RLSStruct
bool the carrier of n is non empty set
Lin L is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of (Omega). n
TRn is finite Element of bool the carrier of n
Lin TRn is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of n
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is affinely-independent Element of bool the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
{(0. n)} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of n
L is Element of the carrier of n
- L is Element of the carrier of n
(- L) + Affn is affinely-independent Element of bool the carrier of n
((- L) + Affn) \ {(0. n)} is Element of bool the carrier of n
(((- L) + Affn) \ {(0. n)}) \/ {(0. n)} is non empty Element of bool the carrier of n
((- L) + Affn) \/ {(0. n)} is non empty Element of bool the carrier of n
card ((- L) + Affn) is ordinal cardinal set
card Affn is ordinal cardinal set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict V231() V232() RLTopStruct
Euclid n is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
TopSpaceMetr (Euclid n) is TopStruct
the carrier of (TOP-REAL n) is functional non empty set
the topology of (TOP-REAL n) is non empty Element of bool (bool the carrier of (TOP-REAL n))
bool the carrier of (TOP-REAL n) is non empty set
bool (bool the carrier of (TOP-REAL n)) is non empty set
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) is non empty strict TopSpace-like TopStruct
A is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
A + L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
A + L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
E is functional Element of bool the carrier of (TOP-REAL n)
the carrier of (Euclid n) is non empty set
the carrier of (TopSpaceMetr (Euclid n)) is set
bool the carrier of (TopSpaceMetr (Euclid n)) is non empty set
k is Element of bool the carrier of (TopSpaceMetr (Euclid n))
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
Y is V21() real ext-real set
Ball (B,Y) is Element of bool the carrier of (Euclid n)
bool the carrier of (Euclid n) is non empty set
Y / 2 is V21() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
Ball (f,(Y / 2)) is Element of bool the carrier of (Euclid n)
f is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
Ball (f,(Y / 2)) is Element of bool the carrier of (Euclid n)
Ek is functional Element of bool the carrier of (TOP-REAL n)
Pro is functional Element of bool the carrier of (TOP-REAL n)
Ek + Pro is functional Element of bool the carrier of (TOP-REAL n)
P is set
Ek + Pro is functional Element of bool the carrier of (TOP-REAL n)
{ (b1 + b2) where b1, b2 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ( b1 in Ek & b2 in Pro ) } is set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
B + y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
B + y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
yP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
dist (f,yP1) is V21() real ext-real Element of REAL
w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
MT is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
w - MT is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.(w - MT).| is V21() real ext-real non negative Element of REAL
yP is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
dist (f,yP) is V21() real ext-real Element of REAL
x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
x - MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.(x - MT1).| is V21() real ext-real non negative Element of REAL
(Y / 2) + (Y / 2) is V21() real ext-real Element of REAL
|.(x - MT1).| + |.(w - MT).| is V21() real ext-real non negative Element of REAL
x + w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
MT1 + MT is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(x + w) - (MT1 + MT) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
MT + MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
- (MT + MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(x + w) + (- (MT + MT1)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
- MT is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
- MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(- MT) + (- MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(x + w) + ((- MT) + (- MT1)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(x + w) + (- MT) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
((x + w) + (- MT)) + (- MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
w + (- MT) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(w + (- MT)) + x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
((w + (- MT)) + x) + (- MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
x + (- MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(w + (- MT)) + (x + (- MT1)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(w - MT) + (x + (- MT1)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(w - MT) + (x - MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
vP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
dist (B,vP1) is V21() real ext-real Element of REAL
(x - MT1) + (w - MT) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.((x - MT1) + (w - MT)).| is V21() real ext-real non negative Element of REAL
A is V21() real ext-real Element of REAL
L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
A * L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
A * L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
E is functional Element of bool the carrier of (TOP-REAL n)
the carrier of (Euclid n) is non empty set
the carrier of (TopSpaceMetr (Euclid n)) is set
bool the carrier of (TopSpaceMetr (Euclid n)) is non empty set
B is Element of bool the carrier of (TopSpaceMetr (Euclid n))
f is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
k is V21() real ext-real set
Ball (f,k) is Element of bool the carrier of (Euclid n)
bool the carrier of (Euclid n) is non empty set
k / 2 is V21() real ext-real Element of REAL
(k / 2) / 2 is V21() real ext-real Element of REAL
abs A is V21() real ext-real Element of REAL
(abs A) * 1 is V21() real ext-real Element of REAL
((k / 2) / 2) / (abs A) is V21() real ext-real Element of REAL
k is non empty V21() real ext-real positive non negative Element of REAL
(abs A) * k is V21() real ext-real Element of REAL
k is non empty V21() real ext-real positive non negative Element of REAL
(abs A) * k is V21() real ext-real Element of REAL
k is non empty V21() real ext-real positive non negative Element of REAL
(abs A) * k is V21() real ext-real Element of REAL
f is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
Ball (f,k) is non empty Element of bool the carrier of (Euclid n)
|.L.| is V21() real ext-real non negative Element of REAL
|.L.| + k is non empty V21() real ext-real positive non negative Element of REAL
(k / 2) / (|.L.| + k) is V21() real ext-real Element of REAL
Pro is non empty V21() real ext-real positive non negative Element of REAL
Ek is functional Element of bool the carrier of (TOP-REAL n)
P is V21() real ext-real Element of REAL
P - A is V21() real ext-real Element of REAL
abs (P - A) is V21() real ext-real Element of REAL
P * Ek is functional Element of bool the carrier of (TOP-REAL n)
P - A is V21() real ext-real Element of REAL
abs (P - A) is V21() real ext-real Element of REAL
B is set
{ (P * b1) where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : b1 in Ek } is set
y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
P * y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
P * y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
vP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
n |-> {} is Relation-like empty-yielding NAT -defined Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
MT1 - (n |-> {}) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
w - w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
MT1 - (w - w) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
MT1 - w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
(MT1 - w) + w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
|.x.| is V21() real ext-real non negative Element of REAL
x - vP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.(x - vP1).| is V21() real ext-real non negative Element of REAL
|.vP1.| is V21() real ext-real non negative Element of REAL
|.(x - vP1).| + |.vP1.| is V21() real ext-real non negative Element of REAL
yP is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
dist (f,yP) is V21() real ext-real Element of REAL
k + |.L.| is non empty V21() real ext-real positive non negative Element of REAL
Pro * |.x.| is V21() real ext-real non negative Element of REAL
Pro * (k + |.L.|) is V21() real ext-real non negative Element of REAL
A * vP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A * x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
- (A * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * vP1) + (- (A * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(- 1) * (A * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * vP1) + ((- 1) * (A * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(- 1) * A is V21() real ext-real Element of REAL
((- 1) * A) * x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * vP1) + (((- 1) * A) * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(- 1) * x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A * ((- 1) * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * vP1) + (A * ((- 1) * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
vP1 + ((- 1) * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A * (vP1 + ((- 1) * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
- x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
vP1 + (- x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A * (vP1 + (- x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
vP1 - x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A * (vP1 - x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.((A * vP1) + (- (A * x))).| is V21() real ext-real non negative Element of REAL
|.(vP1 - x).| is V21() real ext-real non negative Element of REAL
(abs A) * |.(vP1 - x).| is V21() real ext-real Element of REAL
P * x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
- (P * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * x) + (- (P * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(- 1) * (P * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * x) + ((- 1) * (P * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(- 1) * P is V21() real ext-real Element of REAL
((- 1) * P) * x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
(A * x) + (((- 1) * P) * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A + ((- 1) * P) is V21() real ext-real Element of REAL
(A + ((- 1) * P)) * x is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.((A * x) + (- (P * x))).| is V21() real ext-real non negative Element of REAL
A - P is V21() real ext-real Element of REAL
abs (A - P) is V21() real ext-real Element of REAL
(abs (A - P)) * |.x.| is V21() real ext-real Element of REAL
- (A - P) is V21() real ext-real Element of REAL
abs (- (A - P)) is V21() real ext-real Element of REAL
(abs (- (A - P))) * |.x.| is V21() real ext-real Element of REAL
Pro * (|.L.| + k) is V21() real ext-real non negative Element of REAL
((A * vP1) + (- (A * x))) + ((A * x) + (- (P * x))) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
|.(((A * vP1) + (- (A * x))) + ((A * x) + (- (P * x)))).| is V21() real ext-real non negative Element of REAL
|.((A * vP1) + (- (A * x))).| + |.((A * x) + (- (P * x))).| is V21() real ext-real non negative Element of REAL
(k / 2) + (k / 2) is V21() real ext-real Element of REAL
(A * vP1) - (P * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
A * w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
(A * w) - (n |-> {}) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
P * MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
((A * w) - (n |-> {})) - (P * MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
A * MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
(A * MT1) - (A * MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of n -tuples_on REAL
(A * vP1) - ((A * MT1) - (A * MT1)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A * vP1) - ((A * MT1) - (A * MT1))) - (P * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A * vP1) - (A * MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A * vP1) - (A * MT1)) + (A * MT1) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(((A * vP1) - (A * MT1)) + (A * MT1)) - (P * x) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(((A * vP1) - (A * MT1)) + (A * MT1)) + (- (P * x)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((A * vP1) - (A * MT1)) + ((A * x) + (- (P * x))) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
yP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
dist (f,yP1) is V21() real ext-real Element of REAL
the carrier of (TOP-REAL n) is functional non empty set
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the Mult of (TOP-REAL n) is Relation-like [:REAL, the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:REAL, the carrier of (TOP-REAL n):] is non empty non trivial non finite set
[:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty non trivial non finite set
bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty non trivial non finite set
RLSStruct(# the carrier of (TOP-REAL n), the ZeroF of (TOP-REAL n), the addF of (TOP-REAL n), the Mult of (TOP-REAL n) #) is strict RLSStruct
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
RealVectSpace (Seg n) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V231() V232() finite-dimensional RLSStruct
(Omega). (TOP-REAL n) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital Subspace of TOP-REAL n
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
the carrier of (TOP-REAL n) is functional non empty set
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
the addF of (TOP-REAL n) is Relation-like [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
[:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
bool [:[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty set
the Mult of (TOP-REAL n) is Relation-like [:REAL, the carrier of (TOP-REAL n):] -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):]
[:REAL, the carrier of (TOP-REAL n):] is non empty non trivial non finite set
[:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty non trivial non finite set
bool [:[:REAL, the carrier of (TOP-REAL n):], the carrier of (TOP-REAL n):] is non empty non trivial non finite set
RLSStruct(# the carrier of (TOP-REAL n), the ZeroF of (TOP-REAL n), the addF of (TOP-REAL n), the Mult of (TOP-REAL n) #) is strict RLSStruct
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
RealVectSpace (Seg n) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V231() V232() finite-dimensional RLSStruct
(Omega). (TOP-REAL n) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
dim ((Omega). (TOP-REAL n)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
dim n is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 + (dim n) is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affn is finite affinely-independent Element of bool the carrier of n
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
{(0. n)} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of n
L is Element of the carrier of n
- L is Element of the carrier of n
(- L) + Affn is finite affinely-independent Element of bool the carrier of n
((- L) + Affn) \ {(0. n)} is finite Element of bool the carrier of n
card {(0. n)} is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(((- L) + Affn) \ {(0. n)}) \/ {(0. n)} is non empty finite Element of bool the carrier of n
card ((((- L) + Affn) \ {(0. n)}) \/ {(0. n)}) is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
card (((- L) + Affn) \ {(0. n)}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card (((- L) + Affn) \ {(0. n)})) + (card {(0. n)}) is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card ((- L) + Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
A is finite set
A \ {(0. n)} is finite Element of bool A
bool A is non empty finite V39() set
card (A \ {(0. n)}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
Lin (((- L) + Affn) \ {(0. n)}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of n
dim (Lin (((- L) + Affn) \ {(0. n)})) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(A \ {(0. n)}) \/ {(0. n)} is non empty finite set
card ((A \ {(0. n)}) \/ {(0. n)}) is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
A \/ {(0. n)} is non empty finite set
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
dim n is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(dim n) + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] n is non empty non proper Element of bool the carrier of n
Affn is finite affinely-independent Element of bool the carrier of n
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
Affin Affn is Affine Element of bool the carrier of n
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
{(0. n)} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of n
L is Element of the carrier of n
- L is Element of the carrier of n
(- L) + Affn is finite affinely-independent Element of bool the carrier of n
((- L) + Affn) \ {(0. n)} is finite Element of bool the carrier of n
(- L) + L is Element of the carrier of n
{ ((- L) + b1) where b1 is Element of the carrier of n : b1 in Affn } is set
A is finite Element of bool the carrier of n
card A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
card {(0. n)} is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
A \ {(0. n)} is finite Element of bool the carrier of n
(A \ {(0. n)}) \/ {(0. n)} is non empty finite Element of bool the carrier of n
card (A \ {(0. n)}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card (A \ {(0. n)})) + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Lin (A \ {(0. n)}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of n
dim (Lin (A \ {(0. n)})) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(Omega). n is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of n
(Omega). (Lin (A \ {(0. n)})) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of Lin (A \ {(0. n)})
the addF of n is Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like quasi_total Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the Mult of n is Relation-like [:REAL, the carrier of n:] -defined the carrier of n -valued Function-like quasi_total Element of bool [:[:REAL, the carrier of n:], the carrier of n:]
[:REAL, the carrier of n:] is non empty non trivial non finite set
[:[:REAL, the carrier of n:], the carrier of n:] is non empty non trivial non finite set
bool [:[:REAL, the carrier of n:], the carrier of n:] is non empty non trivial non finite set
RLSStruct(# the carrier of n, the ZeroF of n, the addF of n, the Mult of n #) is strict RLSStruct
Lin A is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of n
the carrier of (Lin A) is non empty set
L + (Lin A) is Element of bool the carrier of n
Up (Lin A) is non empty Element of bool the carrier of n
L + (Up (Lin A)) is Element of bool the carrier of n
L is Element of the carrier of n
- L is Element of the carrier of n
(- L) + Affn is finite affinely-independent Element of bool the carrier of n
((- L) + Affn) \ {(0. n)} is finite Element of bool the carrier of n
A is finite Element of bool the carrier of n
Lin A is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of n
Up (Lin A) is non empty Element of bool the carrier of n
L + (Up (Lin A)) is Element of bool the carrier of n
L + (Lin A) is Element of bool the carrier of n
[#] (Lin A) is non empty non proper Element of bool the carrier of (Lin A)
the carrier of (Lin A) is non empty set
bool the carrier of (Lin A) is non empty set
the carrier of RLSStruct(# the carrier of n, the ZeroF of n, the addF of n, the Mult of n #) is set
the carrier of ((Omega). n) is non empty set
A \ {(0. n)} is finite Element of bool the carrier of n
Lin (A \ {(0. n)}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of n
(- L) + L is Element of the carrier of n
{ ((- L) + b1) where b1 is Element of the carrier of n : b1 in Affn } is set
(A \ {(0. n)}) \/ {(0. n)} is non empty finite Element of bool the carrier of n
card A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
card (A \ {(0. n)}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card (A \ {(0. n)})) + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
L is functional Element of bool the carrier of (TOP-REAL Affn)
TRn is functional Element of bool the carrier of (TOP-REAL n)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn) : b1 | n in TRn } is set
Affn - n is V21() real ext-real Element of REAL
the topology of (TOP-REAL Affn) is non empty Element of bool (bool the carrier of (TOP-REAL Affn))
bool (bool the carrier of (TOP-REAL Affn)) is non empty set
TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #) is non empty strict TopSpace-like TopStruct
Euclid Affn is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
TopSpaceMetr (Euclid Affn) is TopStruct
the carrier of (TopSpaceMetr (Euclid Affn)) is set
bool the carrier of (TopSpaceMetr (Euclid Affn)) is non empty set
the topology of (TOP-REAL n) is non empty Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) is non empty strict TopSpace-like TopStruct
Euclid n is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
TopSpaceMetr (Euclid n) is TopStruct
the carrier of (TopSpaceMetr (Euclid n)) is set
bool the carrier of (TopSpaceMetr (Euclid n)) is non empty set
L is Element of bool the carrier of (TopSpaceMetr (Euclid Affn))
the carrier of (Euclid n) is non empty set
E is Element of bool the carrier of (TopSpaceMetr (Euclid n))
A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A |-> {} is Relation-like empty-yielding NAT -defined Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len (A |-> {}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
@ (A |-> {}) is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ (A |-> {})) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Euclid A is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
the carrier of (Euclid A) is non empty set
f is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
f ^ (A |-> {}) is Relation-like NAT -defined Function-like finite K194(n,A) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K194(n,A) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
@ (f ^ (A |-> {})) is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ (f ^ (A |-> {}))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
f is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid A)
f ^ f is Relation-like NAT -defined Function-like finite K194(n,A) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len (f ^ f) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
the carrier of (Euclid Affn) is non empty set
len f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom f is finite n -element V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
k is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn)
k | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
k | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
B is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid Affn)
Y is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 / Y is V21() real V31() ext-real non negative Element of RAT
OpenHypercube (B,(1 / Y)) is open Element of bool the carrier of (TopSpaceMetr (Euclid Affn))
k is V21() real V31() ext-real non negative Element of RAT
OpenHypercube (f,k) is open Element of bool the carrier of (TopSpaceMetr (Euclid n))
Ek is set
Pro is Element of the carrier of (TopSpaceMetr (Euclid n))
Intervals (f,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Intervals (f,k)) is set
OpenHypercube (f,k) is open Element of bool the carrier of (TopSpaceMetr (Euclid A))
TopSpaceMetr (Euclid A) is TopStruct
the carrier of (TopSpaceMetr (Euclid A)) is set
bool the carrier of (TopSpaceMetr (Euclid A)) is non empty set
Intervals (f,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Intervals (f,k)) is set
(Intervals (f,k)) ^ (Intervals (f,k)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Intervals (B,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
P is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
P ^ f is Relation-like NAT -defined Function-like finite K194(n,A) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
product (Intervals (B,k)) is set
OpenHypercube (B,k) is open Element of bool the carrier of (TopSpaceMetr (Euclid Affn))
B is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn)
B | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len P is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom P is finite n -element V60() V61() V62() V63() V64() V65() Element of bool NAT
the carrier of (Euclid Affn) is non empty set
f is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid Affn)
f is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn)
f | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
f | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
len B is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
@ B is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
Y is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 / Y is V21() real V31() ext-real non negative Element of RAT
OpenHypercube (k,(1 / Y)) is open Element of bool the carrier of (TopSpaceMetr (Euclid n))
k is V21() real V31() ext-real non negative Element of RAT
OpenHypercube (f,k) is open Element of bool the carrier of (TopSpaceMetr (Euclid Affn))
Ek is set
Intervals (f,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Intervals (f,k)) is set
len f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
@ f is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ f) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B ^ P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
B is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len B is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(len B) + (len B) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Euclid A is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
the carrier of (Euclid A) is non empty set
Intervals (k,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
y is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid A)
Intervals (y,k) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Intervals (k,k)) ^ (Intervals (y,k)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Intervals (k,k)) is set
product (Intervals (y,k)) is set
yP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
yP1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
yP ^ yP1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len yP is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Pro is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn)
Pro | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Pro | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom yP is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Pro | (dom yP) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
n is V21() ordinal natural real ext-real non negative finite cardinal set
n -VectSp_over F_Real is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is non empty set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
n -tuples_on the carrier of F_Real is functional non empty FinSequence-membered FinSequenceSet of the carrier of F_Real
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
n + Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
TOP-REAL (n + Affn) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL (n + Affn)) is functional non empty set
bool the carrier of (TOP-REAL (n + Affn)) is non empty set
Affn |-> {} is Relation-like empty-yielding NAT -defined Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
{ (b1 ^ (Affn |-> {})) where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : verum } is set
L is functional Element of bool the carrier of (TOP-REAL n)
E is functional Element of bool the carrier of (TOP-REAL (n + Affn))
(TOP-REAL (n + Affn)) | E is strict TopSpace-like SubSpace of TOP-REAL (n + Affn)
the carrier of ((TOP-REAL (n + Affn)) | E) is set
bool the carrier of ((TOP-REAL (n + Affn)) | E) is non empty set
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (n + Affn)) : ( b1 | n in L & b1 in E ) } is set
the topology of (TOP-REAL n) is non empty Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
TopStruct(# the carrier of (TOP-REAL n), the topology of (TOP-REAL n) #) is non empty strict TopSpace-like TopStruct
Euclid n is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
TopSpaceMetr (Euclid n) is TopStruct
the carrier of (TopSpaceMetr (Euclid n)) is set
bool the carrier of (TopSpaceMetr (Euclid n)) is non empty set
B is Element of bool the carrier of ((TOP-REAL (n + Affn)) | E)
[#] ((TOP-REAL (n + Affn)) | E) is non proper open closed Element of bool the carrier of ((TOP-REAL (n + Affn)) | E)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (n + Affn)) : b1 | n in L } is set
[#] (TOP-REAL (n + Affn)) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL (n + Affn))
Y is set
k is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (n + Affn))
k | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
k | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
Y is functional Element of bool the carrier of (TOP-REAL (n + Affn))
Y /\ E is functional Element of bool the carrier of (TOP-REAL (n + Affn))
k is set
Ek is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (n + Affn))
Ek | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Ek | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
the topology of (TOP-REAL (n + Affn)) is non empty Element of bool (bool the carrier of (TOP-REAL (n + Affn)))
bool (bool the carrier of (TOP-REAL (n + Affn))) is non empty set
Y /\ ([#] ((TOP-REAL (n + Affn)) | E)) is Element of bool the carrier of ((TOP-REAL (n + Affn)) | E)
the topology of ((TOP-REAL (n + Affn)) | E) is non empty Element of bool (bool the carrier of ((TOP-REAL (n + Affn)) | E))
bool (bool the carrier of ((TOP-REAL (n + Affn)) | E)) is non empty set
k is set
Ek is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (n + Affn))
Ek | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Ek | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
k is functional Element of bool the carrier of (TOP-REAL (n + Affn))
k /\ ([#] ((TOP-REAL (n + Affn)) | E)) is Element of bool the carrier of ((TOP-REAL (n + Affn)) | E)
TopStruct(# the carrier of (TOP-REAL (n + Affn)), the topology of (TOP-REAL (n + Affn)) #) is non empty strict TopSpace-like TopStruct
Euclid (n + Affn) is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
TopSpaceMetr (Euclid (n + Affn)) is TopStruct
the carrier of (TopSpaceMetr (Euclid (n + Affn))) is set
bool the carrier of (TopSpaceMetr (Euclid (n + Affn))) is non empty set
Y is Element of bool the carrier of (TopSpaceMetr (Euclid (n + Affn)))
the carrier of (Euclid n) is non empty set
f is Element of bool the carrier of (TopSpaceMetr (Euclid n))
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid n)
len (Affn |-> {}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k ^ (Affn |-> {}) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
@ (k ^ (Affn |-> {})) is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ (k ^ (Affn |-> {}))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len k is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len (k ^ (Affn |-> {})) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
the carrier of (Euclid (n + Affn)) is non empty set
dom k is finite n -element V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
Ek is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid (n + Affn))
Ek | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Ek | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
@ (Affn |-> {}) is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
@ (@ (Affn |-> {})) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Euclid Affn is non empty strict Reflexive discerning V173() triangle V231() V232() MetrStruct
the carrier of (Euclid Affn) is non empty set
P is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 / P is V21() real V31() ext-real non negative Element of RAT
OpenHypercube (Ek,(1 / P)) is open Element of bool the carrier of (TopSpaceMetr (Euclid (n + Affn)))
OpenHypercube (k,(1 / P)) is open Element of bool the carrier of (TopSpaceMetr (Euclid n))
y is set
Pro is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (Euclid Affn)
OpenHypercube (Pro,(1 / P)) is open Element of bool the carrier of (TopSpaceMetr (Euclid Affn))
TopSpaceMetr (Euclid Affn) is TopStruct
the carrier of (TopSpaceMetr (Euclid Affn)) is set
bool the carrier of (TopSpaceMetr (Euclid Affn)) is non empty set
Intervals (Pro,(1 / P)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Intervals (Pro,(1 / P))) is set
Intervals (k,(1 / P)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Intervals (k,(1 / P))) ^ (Intervals (Pro,(1 / P))) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
k ^ Pro is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Intervals ((k ^ Pro),(1 / P)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
yP is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
product (Intervals (k,(1 / P))) is set
yP ^ Pro is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Intervals (Ek,(1 / P)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
product (Intervals (Ek,(1 / P))) is set
len yP is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(yP ^ Pro) | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(yP ^ Pro) | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom yP is finite n -element V60() V61() V62() V63() V64() V65() Element of bool NAT
(yP ^ Pro) | (dom yP) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
yP1 is Relation-like NAT -defined REAL -valued Function-like finite n + Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (n + Affn))
yP1 | n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
yP1 | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is affinely-independent Element of bool the carrier of n
conv Affn is convex Element of bool the carrier of n
L is Element of bool the carrier of n
Affin L is Affine Element of bool the carrier of n
(conv Affn) /\ (Affin L) is Element of bool the carrier of n
conv L is convex Element of bool the carrier of n
TRn is set
TRn |-- L is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
TRn |-- Affn is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
A is Element of the carrier of n
(TRn |-- L) . A is V21() real ext-real Element of REAL
n is V21() real ext-real Element of REAL
Affn is non empty RLSStruct
the carrier of Affn is non empty set
L is non empty set
[:L, the carrier of Affn:] is non empty set
bool [:L, the carrier of Affn:] is non empty set
TRn is Relation-like L -defined the carrier of Affn -valued Function-like Element of bool [:L, the carrier of Affn:]
n (#) TRn is Relation-like L -defined the carrier of Affn -valued Function-like Element of bool [:L, the carrier of Affn:]
A is set
(n (#) TRn) .: A is Element of bool the carrier of Affn
bool the carrier of Affn is non empty set
TRn .: A is Element of bool the carrier of Affn
n * (TRn .: A) is Element of bool the carrier of Affn
dom (n (#) TRn) is Element of bool L
bool L is non empty set
dom TRn is Element of bool L
E is set
f is set
(n (#) TRn) . f is set
(n (#) TRn) /. f is Element of the carrier of Affn
TRn /. f is Element of the carrier of Affn
n * (TRn /. f) is Element of the carrier of Affn
TRn . f is set
{ (n * b1) where b1 is Element of the carrier of Affn : b1 in TRn .: A } is set
E is set
f is Element of the carrier of Affn
n * f is Element of the carrier of Affn
f is set
TRn . f is set
TRn /. f is Element of the carrier of Affn
(n (#) TRn) /. f is Element of the carrier of Affn
(n (#) TRn) . f is set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
Affn is functional Element of bool the carrier of (TOP-REAL n)
Affin Affn is functional Affine Element of bool the carrier of (TOP-REAL n)
Lin Affn is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin Affn) is non empty non proper Element of bool the carrier of (Lin Affn)
the carrier of (Lin Affn) is non empty set
bool the carrier of (Lin Affn) is non empty set
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
- (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- (0. (TOP-REAL n))) + Affn is functional Element of bool the carrier of (TOP-REAL n)
Lin ((- (0. (TOP-REAL n))) + Affn) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
Up (Lin ((- (0. (TOP-REAL n))) + Affn)) is functional non empty Element of bool the carrier of (TOP-REAL n)
(0. (TOP-REAL n)) + (Up (Lin ((- (0. (TOP-REAL n))) + Affn))) is functional Element of bool the carrier of (TOP-REAL n)
(0. (TOP-REAL n)) + Affn is functional Element of bool the carrier of (TOP-REAL n)
Lin ((0. (TOP-REAL n)) + Affn) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
Up (Lin ((0. (TOP-REAL n)) + Affn)) is functional non empty Element of bool the carrier of (TOP-REAL n)
Up (Lin Affn) is functional non empty Element of bool the carrier of (TOP-REAL n)
n is non empty addLoopStr
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is finite Element of bool the carrier of n
L is Element of the carrier of n
L + Affn is Element of bool the carrier of n
{ H1(b1) where b1 is Element of the carrier of n : b1 in Affn } is set
card { H1(b1) where b1 is Element of the carrier of n : b1 in Affn } is ordinal cardinal set
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
card (L + Affn) is ordinal cardinal set
n is non empty RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is Element of bool the carrier of n
card Affn is ordinal cardinal set
L is V21() real ext-real set
L * Affn is Element of bool the carrier of n
card (L * Affn) is ordinal cardinal set
{ H1(b1) where b1 is Element of the carrier of n : b1 in Affn } is set
card { H1(b1) where b1 is Element of the carrier of n : b1 in Affn } is ordinal cardinal set
n is non empty RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is finite Element of bool the carrier of n
L is V21() real ext-real Element of REAL
L * Affn is Element of bool the carrier of n
card (L * Affn) is ordinal cardinal set
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
n is V21() real ext-real Element of REAL
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
L is Element of bool the carrier of Affn
card L is ordinal cardinal set
n * L is Element of bool the carrier of Affn
card (n * L) is ordinal cardinal set
0. Affn is zero Element of the carrier of Affn
the ZeroF of Affn is Element of the carrier of Affn
{(0. Affn)} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of Affn
card {(0. Affn)} is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
A is finite set
card A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
1 + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
TRn is finite set
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
1 * L is Element of bool the carrier of Affn
1 / n is V21() real ext-real Element of REAL
(1 / n) * n is V21() real ext-real Element of REAL
((1 / n) * n) * L is Element of bool the carrier of Affn
(1 / n) * (n * L) is Element of bool the carrier of Affn
TRn is set
{ (n * b1) where b1 is Element of the carrier of Affn : b1 in L } is set
A is Element of the carrier of Affn
n * A is Element of the carrier of Affn
TRn is set
{TRn} is non empty trivial finite 1 -element set
n is non empty RLSStruct
the carrier of n is non empty set
Affn is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
L is V21() real ext-real Element of REAL
L (#) Affn is Relation-like NAT -defined the carrier of n -valued Function-like Element of bool [:NAT, the carrier of n:]
[:NAT, the carrier of n:] is non empty non trivial non finite set
bool [:NAT, the carrier of n:] is non empty non trivial non finite set
dom (L (#) Affn) is V60() V61() V62() V63() V64() V65() Element of bool NAT
dom Affn is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
len Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Seg (len Affn) is finite len Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len Affn ) } is set
n is finite set
Affn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng Affn is finite set
n is 1-sorted
the carrier of n is set
bool the carrier of n is non empty set
Affn is finite Element of bool the carrier of n
L is Relation-like NAT -defined Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn)
rng L is finite set
n is non empty right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is finite Element of bool the carrier of n
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
L is Relation-like NAT -defined the carrier of n -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (n,Affn)
TRn is Element of the carrier of n
(card Affn) |-> TRn is Relation-like NAT -defined the carrier of n -valued Function-like finite card Affn -element FinSequence-like FinSubsequence-like Element of (card Affn) -tuples_on the carrier of n
(card Affn) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
L + ((card Affn) |-> TRn) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
TRn + Affn is finite Element of bool the carrier of n
rng L is finite Element of bool the carrier of n
len L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A is Relation-like NAT -defined the carrier of n -valued Function-like finite card Affn -element FinSequence-like FinSubsequence-like Element of (card Affn) -tuples_on the carrier of n
L is Relation-like NAT -defined the carrier of n -valued Function-like finite card Affn -element FinSequence-like FinSubsequence-like Element of (card Affn) -tuples_on the carrier of n
A + L is Relation-like NAT -defined the carrier of n -valued Function-like finite card Affn -element FinSequence-like FinSubsequence-like Element of (card Affn) -tuples_on the carrier of n
len (A + L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom (A + L) is finite card Affn -element V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg (card Affn) is finite card Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card Affn ) } is set
dom A is finite card Affn -element V60() V61() V62() V63() V64() V65() Element of bool NAT
rng (A + L) is finite Element of bool the carrier of n
E is set
f is set
(A + L) . f is set
f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A . f is set
rng A is finite Element of bool the carrier of n
L . f is set
B is Element of the carrier of n
B + TRn is Element of the carrier of n
TRn + B is Element of the carrier of n
{ (TRn + b1) where b1 is Element of the carrier of n : b1 in Affn } is set
E is set
{ (TRn + b1) where b1 is Element of the carrier of n : b1 in Affn } is set
f is Element of the carrier of n
TRn + f is Element of the carrier of n
dom L is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
f is set
L . f is set
B is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
L . B is set
(A + L) . B is set
f + TRn is Element of the carrier of n
card (TRn + Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
n is V21() real ext-real Element of REAL
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
L is finite Element of bool the carrier of Affn
n * L is finite Element of bool the carrier of Affn
TRn is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,L)
n (#) TRn is Relation-like NAT -defined the carrier of Affn -valued Function-like finite FinSequence-like FinSubsequence-like Element of bool [:NAT, the carrier of Affn:]
[:NAT, the carrier of Affn:] is non empty non trivial non finite set
bool [:NAT, the carrier of Affn:] is non empty non trivial non finite set
dom (n (#) TRn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom TRn is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
len (n (#) TRn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
rng TRn is finite Element of bool the carrier of Affn
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
rng (n (#) TRn) is finite Element of bool the carrier of Affn
(n (#) TRn) .: (dom TRn) is finite Element of bool the carrier of Affn
TRn .: (dom TRn) is finite Element of bool the carrier of Affn
n * (TRn .: (dom TRn)) is finite Element of bool the carrier of Affn
0. Affn is zero Element of the carrier of Affn
the ZeroF of Affn is Element of the carrier of Affn
{(0. Affn)} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of Affn
card {(0. Affn)} is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
card (n * L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
1 + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
L is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,Affn, the carrier of F_Real
the_rank_of L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Mx2Tran L is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL Affn) -valued Function-like quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL Affn):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL Affn):] is non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL Affn):] is non empty set
L is functional finite Element of bool the carrier of (TOP-REAL n)
(Mx2Tran L) .: L is functional finite Element of bool the carrier of (TOP-REAL Affn)
bool the carrier of (TOP-REAL Affn) is non empty set
E is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)
(Mx2Tran L) * E is Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL Affn)
rng E is functional finite Element of bool the carrier of (TOP-REAL n)
dom (Mx2Tran L) is functional Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
len ((Mx2Tran L) * E) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len E is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom ((Mx2Tran L) * E) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom E is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
rng ((Mx2Tran L) * E) is functional finite Element of bool the carrier of (TOP-REAL Affn)
((Mx2Tran L) * E) .: (dom ((Mx2Tran L) * E)) is functional finite Element of bool the carrier of (TOP-REAL Affn)
E .: (dom E) is functional finite Element of bool the carrier of (TOP-REAL n)
(Mx2Tran L) .: (E .: (dom E)) is functional finite Element of bool the carrier of (TOP-REAL Affn)
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is finite Element of bool the carrier of n
L is Relation-like NAT -defined the carrier of n -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (n,Affn)
TRn is set
TRn |-- Affn is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
(TRn |-- Affn) * L is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n is set
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
L is finite Element of bool the carrier of Affn
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
TRn is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,L)
(Affn,L,TRn,n) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n |-- L is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(n |-- L) * TRn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len (Affn,L,TRn,n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
rng TRn is finite Element of bool the carrier of Affn
len TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
Affn is Element of the carrier of n
L is Element of the carrier of n
Affn + L is Element of the carrier of n
TRn is finite affinely-independent Element of bool the carrier of n
Affn + TRn is finite affinely-independent Element of bool the carrier of n
Affin TRn is Affine Element of bool the carrier of n
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card TRn) |-> Affn is Relation-like NAT -defined the carrier of n -valued Function-like finite card TRn -element FinSequence-like FinSubsequence-like Element of (card TRn) -tuples_on the carrier of n
(card TRn) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
A is Relation-like NAT -defined the carrier of n -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (n,TRn)
A + ((card TRn) |-> Affn) is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,TRn,A,L) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
L |-- TRn is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(L |-- TRn) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
E is Relation-like NAT -defined the carrier of n -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (n,Affn + TRn)
(n,(Affn + TRn),E,(Affn + L)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Affn + L) |-- (Affn + TRn) is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn + TRn
((Affn + L) |-- (Affn + TRn)) * E is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
sum (L |-- TRn) is V21() real ext-real Element of REAL
{ (Affn + b1) where b1 is Element of the carrier of n : b1 in Affin TRn } is set
Affn + (Affin TRn) is Element of bool the carrier of n
rng A is finite Element of bool the carrier of n
len A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affin (Affn + TRn) is Affine Element of bool the carrier of n
1 * Affn is Element of the carrier of n
Affn + (L |-- TRn) is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of n
Carrier (Affn + (L |-- TRn)) is Element of bool the carrier of n
Carrier (L |-- TRn) is Element of bool the carrier of n
Affn + (Carrier (L |-- TRn)) is Element of bool the carrier of n
Sum (L |-- TRn) is Element of the carrier of n
k is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn + TRn
Sum k is Element of the carrier of n
(1 * Affn) + L is Element of the carrier of n
len (n,TRn,A,L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card (Affn + TRn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
len (n,(Affn + TRn),E,(Affn + L)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom (n,TRn,A,L) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (n,(Affn + TRn),E,(Affn + L)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
rng E is finite Element of bool the carrier of n
len E is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
sum k is V21() real ext-real Element of REAL
Y is V21() ordinal natural real ext-real non negative finite cardinal set
(n,TRn,A,L) . Y is V21() real ext-real set
A . Y is set
(L |-- TRn) . (A . Y) is V21() real ext-real set
dom A is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
A /. Y is Element of the carrier of n
Seg (card TRn) is finite card TRn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card TRn ) } is set
B is Relation-like NAT -defined the carrier of n -valued Function-like finite card TRn -element FinSequence-like FinSubsequence-like Element of (card TRn) -tuples_on the carrier of n
B . Y is set
(n,(Affn + TRn),E,(Affn + L)) . Y is V21() real ext-real set
E . Y is set
((Affn + L) |-- (Affn + TRn)) . (E . Y) is V21() real ext-real set
dom E is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
(A /. Y) + Affn is Element of the carrier of n
((A /. Y) + Affn) - Affn is Element of the carrier of n
(L |-- TRn) . (((A /. Y) + Affn) - Affn) is V21() real ext-real Element of REAL
Affn - Affn is Element of the carrier of n
(A /. Y) + (Affn - Affn) is Element of the carrier of n
(L |-- TRn) . ((A /. Y) + (Affn - Affn)) is V21() real ext-real Element of REAL
0. n is zero Element of the carrier of n
the ZeroF of n is Element of the carrier of n
(A /. Y) + (0. n) is Element of the carrier of n
(L |-- TRn) . ((A /. Y) + (0. n)) is V21() real ext-real Element of REAL
n is V21() real ext-real Element of REAL
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
L is Element of the carrier of Affn
n * L is Element of the carrier of Affn
TRn is finite affinely-independent Element of bool the carrier of Affn
n * TRn is finite affinely-independent Element of bool the carrier of Affn
Affin TRn is Affine Element of bool the carrier of Affn
A is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,TRn)
n (#) A is Relation-like NAT -defined the carrier of Affn -valued Function-like finite FinSequence-like FinSubsequence-like Element of bool [:NAT, the carrier of Affn:]
[:NAT, the carrier of Affn:] is non empty non trivial non finite set
bool [:NAT, the carrier of Affn:] is non empty non trivial non finite set
(Affn,TRn,A,L) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
L |-- TRn is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(L |-- TRn) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
E is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,n * TRn)
(Affn,(n * TRn),E,(n * L)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(n * L) |-- (n * TRn) is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of n * TRn
((n * L) |-- (n * TRn)) * E is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Carrier (L |-- TRn) is Element of bool the carrier of Affn
{ (n * b1) where b1 is Element of the carrier of Affn : b1 in Affin TRn } is set
dom E is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom A is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
n (*) (L |-- TRn) is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
Carrier (n (*) (L |-- TRn)) is Element of bool the carrier of Affn
n * (Carrier (L |-- TRn)) is Element of bool the carrier of Affn
sum (L |-- TRn) is V21() real ext-real Element of REAL
f is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of n * TRn
sum f is V21() real ext-real Element of REAL
Sum (L |-- TRn) is Element of the carrier of Affn
Sum f is Element of the carrier of Affn
len (Affn,(n * TRn),E,(n * L)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card (n * TRn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
len (Affn,TRn,A,L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
rng A is finite Element of bool the carrier of Affn
len A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom (Affn,TRn,A,L) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (Affn,(n * TRn),E,(n * L)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Affin (n * TRn) is Affine Element of bool the carrier of Affn
n * (Affin TRn) is Element of bool the carrier of Affn
B is V21() ordinal natural real ext-real non negative finite cardinal set
(Affn,TRn,A,L) . B is V21() real ext-real set
A . B is set
(L |-- TRn) . (A . B) is V21() real ext-real set
A /. B is Element of the carrier of Affn
E /. B is Element of the carrier of Affn
n * (A /. B) is Element of the carrier of Affn
(Affn,(n * TRn),E,(n * L)) . B is V21() real ext-real set
E . B is set
f . (E . B) is V21() real ext-real set
n " is V21() real ext-real Element of REAL
(n ") * (n * (A /. B)) is Element of the carrier of Affn
(L |-- TRn) . ((n ") * (n * (A /. B))) is V21() real ext-real Element of REAL
(n ") * n is V21() real ext-real Element of REAL
((n ") * n) * (A /. B) is Element of the carrier of Affn
(L |-- TRn) . (((n ") * n) * (A /. B)) is V21() real ext-real Element of REAL
1 * (A /. B) is Element of the carrier of Affn
(L |-- TRn) . (1 * (A /. B)) is V21() real ext-real Element of REAL
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
Affin L is functional Affine Element of bool the carrier of (TOP-REAL n)
TRn is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)
L is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,Affn, the carrier of F_Real
the_rank_of L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Mx2Tran L is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL Affn) -valued Function-like quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL Affn):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL Affn):] is non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL Affn):] is non empty set
(Mx2Tran L) .: L is functional finite Element of bool the carrier of (TOP-REAL Affn)
bool the carrier of (TOP-REAL Affn) is non empty set
(Mx2Tran L) * TRn is Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL Affn)
B is Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,(Mx2Tran L) .: L)
dom (Mx2Tran L) is functional Element of bool the carrier of (TOP-REAL n)
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
card ((Mx2Tran L) .: L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),L,TRn,k) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(k |-- L) * TRn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Mx2Tran L) . k is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn)
((TOP-REAL Affn),((Mx2Tran L) .: L),B,((Mx2Tran L) . k)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((Mx2Tran L) . k) |-- ((Mx2Tran L) .: L) is Relation-like the carrier of (TOP-REAL Affn) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (Mx2Tran L) .: L
(((Mx2Tran L) . k) |-- ((Mx2Tran L) .: L)) * B is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((TOP-REAL n),L,TRn,k) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len ((TOP-REAL Affn),((Mx2Tran L) .: L),B,((Mx2Tran L) . k)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k is V21() ordinal natural real ext-real non negative finite cardinal set
dom ((TOP-REAL Affn),((Mx2Tran L) .: L),B,((Mx2Tran L) . k)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom B is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom ((TOP-REAL n),L,TRn,k) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom TRn is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
TRn . k is Relation-like Function-like set
rng TRn is functional finite Element of bool the carrier of (TOP-REAL n)
((TOP-REAL n),L,TRn,k) . k is V21() real ext-real set
Ek is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(k |-- L) . Ek is V21() real ext-real Element of REAL
(Mx2Tran L) . Ek is Relation-like NAT -defined REAL -valued Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL Affn)
(((Mx2Tran L) . k) |-- ((Mx2Tran L) .: L)) . ((Mx2Tran L) . Ek) is V21() real ext-real Element of REAL
B . k is Relation-like Function-like set
(((Mx2Tran L) . k) |-- ((Mx2Tran L) .: L)) . (B . k) is V21() real ext-real set
((TOP-REAL Affn),((Mx2Tran L) .: L),B,((Mx2Tran L) . k)) . k is V21() real ext-real set
n is set
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
L is finite affinely-independent Element of bool the carrier of Affn
Affin L is Affine Element of bool the carrier of Affn
TRn is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,L)
(Affn,L,TRn,n) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n |-- L is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(n |-- L) * TRn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom (Affn,L,TRn,n) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
A is Element of bool the carrier of Affn
Affin A is Affine Element of bool the carrier of Affn
n |-- A is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
len (Affn,L,TRn,n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
rng TRn is finite Element of bool the carrier of Affn
len TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom TRn is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Carrier (n |-- A) is Element of bool the carrier of Affn
k is set
TRn . k is set
(Affn,L,TRn,n) . k is V21() real ext-real set
(n |-- L) . (TRn . k) is V21() real ext-real set
Y is Element of the carrier of Affn
k is set
L \ A is finite Element of bool the carrier of Affn
Y is set
TRn . Y is set
(Affn,L,TRn,n) . Y is V21() real ext-real set
(n |-- L) . k is V21() real ext-real set
L \ (L \ A) is finite Element of bool the carrier of Affn
L /\ A is finite Element of bool the carrier of Affn
n is set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
L is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of L is non empty set
bool the carrier of L is non empty set
TRn is finite affinely-independent Element of bool the carrier of L
Affin TRn is Affine Element of bool the carrier of L
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card TRn) -' Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((card TRn) -' Affn) |-> {} is Relation-like empty-yielding NAT -defined Function-like finite (card TRn) -' Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
A is Relation-like NAT -defined the carrier of L -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (L,TRn)
A .: (Seg Affn) is finite Element of bool the carrier of L
Affin (A .: (Seg Affn)) is Affine Element of bool the carrier of L
(L,TRn,A,n) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n |-- TRn is Relation-like the carrier of L -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(n |-- TRn) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(L,TRn,A,n) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(L,TRn,A,n) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
((L,TRn,A,n) | Affn) ^ (((card TRn) -' Affn) |-> {}) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
rng A is finite Element of bool the carrier of L
len (L,TRn,A,n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Seg (card TRn) is finite card TRn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card TRn ) } is set
dom A is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
(dom A) /\ (Seg Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
A .: (dom A) is finite Element of bool the carrier of L
(card TRn) - Affn is V21() real ext-real Element of REAL
len (((card TRn) -' Affn) |-> {}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len ((L,TRn,A,n) | Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(card TRn) - Affn is V21() real ext-real Element of REAL
len (((L,TRn,A,n) | Affn) ^ (((card TRn) -' Affn) |-> {})) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affn + ((card TRn) - Affn) is V21() real ext-real Element of REAL
k is V21() ordinal natural real ext-real non negative finite cardinal set
dom (L,TRn,A,n) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom A is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (((L,TRn,A,n) | Affn) ^ (((card TRn) -' Affn) |-> {})) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom ((L,TRn,A,n) | Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
(((L,TRn,A,n) | Affn) ^ (((card TRn) -' Affn) |-> {})) . k is V21() real ext-real set
((L,TRn,A,n) | Affn) . k is V21() real ext-real set
(L,TRn,A,n) . k is V21() real ext-real set
dom (((card TRn) -' Affn) |-> {}) is finite (card TRn) -' Affn -element V60() V61() V62() V63() V64() V65() Element of bool NAT
Ek is V21() ordinal natural real ext-real non negative finite cardinal set
Affn + Ek is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Ek is V21() ordinal natural real ext-real non negative finite cardinal set
Affn + Ek is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A . k is set
Pro is set
A . Pro is set
(((card TRn) -' Affn) |-> {}) . Ek is Function-like functional empty trivial V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() set
(((L,TRn,A,n) | Affn) ^ (((card TRn) -' Affn) |-> {})) . k is V21() real ext-real set
(L,TRn,A,n) . k is V21() real ext-real set
dom ((L,TRn,A,n) | Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (((card TRn) -' Affn) |-> {}) is finite (card TRn) -' Affn -element V60() V61() V62() V63() V64() V65() Element of bool NAT
k is set
A . k is set
Ek is V21() ordinal natural real ext-real non negative finite cardinal set
dom ((L,TRn,A,n) | Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (((card TRn) -' Affn) |-> {}) is finite (card TRn) -' Affn -element V60() V61() V62() V63() V64() V65() Element of bool NAT
Pro is V21() ordinal natural real ext-real non negative finite cardinal set
Affn + Pro is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(((card TRn) -' Affn) |-> {}) . Pro is Function-like functional empty trivial V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() set
(L,TRn,A,n) . k is V21() real ext-real set
n is set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
Affn |-> {} is Relation-like empty-yielding NAT -defined Function-like finite Affn -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
L is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of L is non empty set
bool the carrier of L is non empty set
TRn is finite affinely-independent Element of bool the carrier of L
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
Affin TRn is Affine Element of bool the carrier of L
A is Relation-like NAT -defined the carrier of L -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (L,TRn)
A .: (Seg Affn) is finite Element of bool the carrier of L
TRn \ (A .: (Seg Affn)) is finite Element of bool the carrier of L
Affin (TRn \ (A .: (Seg Affn))) is Affine Element of bool the carrier of L
(L,TRn,A,n) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n |-- TRn is Relation-like the carrier of L -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(n |-- TRn) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(L,TRn,A,n) /^ Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Affn |-> {}) ^ ((L,TRn,A,n) /^ Affn) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
(L,TRn,A,n) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(L,TRn,A,n) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
((L,TRn,A,n) | Affn) ^ ((L,TRn,A,n) /^ Affn) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg (card TRn) is finite card TRn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card TRn ) } is set
len (L,TRn,A,n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom (L,TRn,A,n) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
rng A is finite Element of bool the carrier of L
len A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom A is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
len (Affn |-> {}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len ((L,TRn,A,n) | Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k is V21() ordinal natural real ext-real non negative finite cardinal set
A . k is set
(L,TRn,A,n) . k is V21() real ext-real set
((L,TRn,A,n) | Affn) . k is V21() real ext-real set
(Affn |-> {}) . k is Function-like functional empty trivial V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() set
k is set
A . k is set
Ek is set
A . Ek is set
((L,TRn,A,n) | Affn) . k is V21() real ext-real set
(L,TRn,A,n) . k is V21() real ext-real set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
{(0* n)} is functional non empty trivial finite V39() 1 -element FinSequence-membered Element of bool (REAL n)
bool (REAL n) is non empty set
n -VectSp_over F_Real is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is non empty set
bool the carrier of (n -VectSp_over F_Real) is non empty set
Affn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card Affn) -' 1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affn \ {(0* n)} is functional finite Element of bool the carrier of (TOP-REAL n)
L is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn)
len L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
L . (len L) is Relation-like Function-like set
L | ((card Affn) -' 1) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
Seg ((card Affn) -' 1) is finite (card Affn) -' 1 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= (card Affn) -' 1 ) } is set
L | (Seg ((card Affn) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
rng (L | ((card Affn) -' 1)) is functional finite Element of bool the carrier of (TOP-REAL n)
rng L is functional finite Element of bool the carrier of (TOP-REAL n)
(card Affn) - 1 is V21() real V30() V31() ext-real Element of INT
((card Affn) -' 1) + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom L is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom (L | ((card Affn) -' 1)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
E is set
(L | ((card Affn) -' 1)) . E is Relation-like Function-like set
L . E is Relation-like Function-like set
<*(L . (len L))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,(L . (len L))] is non empty set
{1,(L . (len L))} is non empty finite set
{{1,(L . (len L))},{1}} is non empty finite V39() set
{[1,(L . (len L))]} is Function-like non empty trivial finite 1 -element set
(L | ((card Affn) -' 1)) ^ <*(L . (len L))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
rng <*(L . (len L))*> is trivial finite set
(rng (L | ((card Affn) -' 1))) \/ (rng <*(L . (len L))*>) is finite set
(rng (L | ((card Affn) -' 1))) \/ {(0* n)} is non empty finite set
E is Element of bool the carrier of (n -VectSp_over F_Real)
Lin E is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
the carrier of (Lin E) is non empty set
[#] (Lin E) is non empty non proper Element of bool the carrier of (Lin E)
bool the carrier of (Lin E) is non empty set
f is set
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
E \ {(0* n)} is Element of bool the carrier of (n -VectSp_over F_Real)
f is Relation-like NAT -defined the carrier of (Lin E) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Lin E)
f | ((card Affn) -' 1) is Relation-like NAT -defined the carrier of (Lin E) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (Lin E)
f | (Seg ((card Affn) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
rng (f | ((card Affn) -' 1)) is finite Element of bool the carrier of (Lin E)
Lin Affn is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin Affn) is non empty non proper Element of bool the carrier of (Lin Affn)
the carrier of (Lin Affn) is non empty set
bool the carrier of (Lin Affn) is non empty set
{(0. (TOP-REAL n))} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
Affn \ {(0. (TOP-REAL n))} is functional finite Element of bool the carrier of (TOP-REAL n)
Lin (Affn \ {(0. (TOP-REAL n))}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin (Affn \ {(0. (TOP-REAL n))})) is non empty non proper Element of bool the carrier of (Lin (Affn \ {(0. (TOP-REAL n))}))
the carrier of (Lin (Affn \ {(0. (TOP-REAL n))})) is non empty set
bool the carrier of (Lin (Affn \ {(0. (TOP-REAL n))})) is non empty set
Lin (Affn \ {(0* n)}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin (Affn \ {(0* n)})) is non empty non proper Element of bool the carrier of (Lin (Affn \ {(0* n)}))
the carrier of (Lin (Affn \ {(0* n)})) is non empty set
bool the carrier of (Lin (Affn \ {(0* n)})) is non empty set
Lin (E \ {(0* n)}) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
[#] (Lin (E \ {(0* n)})) is non empty non proper Element of bool the carrier of (Lin (E \ {(0* n)}))
the carrier of (Lin (E \ {(0* n)})) is non empty set
bool the carrier of (Lin (E \ {(0* n)})) is non empty set
Lin (rng (f | ((card Affn) -' 1))) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real , Lin E)
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
n -VectSp_over F_Real is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is non empty set
bool the carrier of (n -VectSp_over F_Real) is non empty set
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
Affn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
(card Affn) -' 1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
L is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn)
len L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
L . (len L) is Relation-like Function-like set
L | ((card Affn) -' 1) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
Seg ((card Affn) -' 1) is finite (card Affn) -' 1 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= (card Affn) -' 1 ) } is set
L | (Seg ((card Affn) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
f is Element of bool the carrier of (n -VectSp_over F_Real)
Lin f is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
the carrier of (Lin f) is non empty set
TRn is V21() real ext-real Element of REAL
Affin Affn is functional Affine Element of bool the carrier of (TOP-REAL n)
Lin Affn is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin Affn) is non empty non proper Element of bool the carrier of (Lin Affn)
the carrier of (Lin Affn) is non empty set
bool the carrier of (Lin Affn) is non empty set
k is Relation-like NAT -defined the carrier of (Lin f) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of Lin f
rng k is finite Element of bool the carrier of (Lin f)
bool the carrier of (Lin f) is non empty set
{(0* n)} is functional non empty trivial finite V39() 1 -element FinSequence-membered Element of bool (REAL n)
bool (REAL n) is non empty set
Affn \ {(0* n)} is functional finite Element of bool the carrier of (TOP-REAL n)
k is Element of the carrier of (Lin f)
k |-- k is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
((TOP-REAL n),Affn,L,k) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k |-- Affn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
(k |-- Affn) * L is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),Affn,L,k) | ((card Affn) -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),Affn,L,k) | (Seg ((card Affn) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (k |-- k) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Pro is Relation-like the carrier of (Lin f) -defined the carrier of F_Real -valued Function-like quasi_total Linear_Combination of Lin f
Sum Pro is Element of the carrier of (Lin f)
Carrier Pro is finite Element of bool the carrier of (Lin f)
(k |-- Affn) * (L | ((card Affn) -' 1)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
dom (k |-- Affn) is functional Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
Y is functional Element of bool the carrier of (TOP-REAL n)
len (((TOP-REAL n),Affn,L,k) | ((card Affn) -' 1)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len k is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] (Lin f) is non empty non proper Element of bool the carrier of (Lin f)
B is Relation-like the carrier of (Lin Affn) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Lin Affn
Carrier B is Element of bool the carrier of (Lin Affn)
Sum B is Element of the carrier of (Lin Affn)
y is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TOP-REAL n
Carrier y is functional Element of bool the carrier of (TOP-REAL n)
Sum y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
y | ([#] (Lin Affn)) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n),REAL:]
[: the carrier of (TOP-REAL n),REAL:] is non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n),REAL:] is non empty non trivial non finite set
yP is Relation-like the carrier of (Lin Affn) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Lin Affn
Carrier yP is Element of bool the carrier of (Lin Affn)
Sum yP is Element of the carrier of (Lin Affn)
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
yP1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
sum yP1 is V21() real ext-real Element of REAL
1 - (sum yP1) is V21() real ext-real Element of REAL
vP1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n),REAL:]
vP1 . (0. (TOP-REAL n)) is V21() real ext-real Element of REAL
Funcs ( the carrier of (TOP-REAL n),REAL) is non empty FUNCTION_DOMAIN of the carrier of (TOP-REAL n), REAL
w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
x is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of Funcs ( the carrier of (TOP-REAL n),REAL)
x . w is V21() real ext-real Element of REAL
w is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TOP-REAL n
Carrier w is functional Element of bool the carrier of (TOP-REAL n)
MT1 is set
w . MT1 is V21() real ext-real set
MT1 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {(0. (TOP-REAL n))}
Carrier MT1 is functional Element of bool the carrier of (TOP-REAL n)
P is Element of bool the carrier of (Lin Affn)
Sum MT1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
MT1 . (0. (TOP-REAL n)) is V21() real ext-real Element of REAL
(MT1 . (0. (TOP-REAL n))) * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(MT1 . (0. (TOP-REAL n))) * (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
MT is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
yP1 + MT is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TOP-REAL n
nc0 is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
sum nc0 is V21() real ext-real Element of REAL
sum MT is V21() real ext-real Element of REAL
(sum yP1) + (sum MT) is V21() real ext-real Element of REAL
(sum yP1) + (1 - (sum yP1)) is V21() real ext-real Element of REAL
Sum nc0 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
Sum yP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
Sum MT is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(Sum yP1) + (Sum MT) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(Sum yP1) + (Sum MT) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Vc1 is V21() ordinal natural real ext-real non negative finite cardinal set
k /. Vc1 is Element of the carrier of (Lin f)
(k |-- k) /. Vc1 is V21() real ext-real Element of the carrier of F_Real
MBCe is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
Pro . MBCe is set
dom (((TOP-REAL n),Affn,L,k) | ((card Affn) -' 1)) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
dom k is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k . Vc1 is set
Carrier MT is functional Element of bool the carrier of (TOP-REAL n)
MT . MBCe is V21() real ext-real Element of REAL
dom (k |-- k) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
(k |-- k) . Vc1 is V21() real ext-real set
nc0 . MBCe is V21() real ext-real Element of REAL
yP1 . MBCe is V21() real ext-real Element of REAL
(yP1 . MBCe) + (MT . MBCe) is V21() real ext-real Element of REAL
yP . MBCe is V21() real ext-real set
(((TOP-REAL n),Affn,L,k) | ((card Affn) -' 1)) . Vc1 is V21() real ext-real set
n is V21() ordinal natural real ext-real non negative finite cardinal set
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like add-continuous Mult-continuous RLTopStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
L is finite affinely-independent Element of bool the carrier of Affn
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
Affin L is Affine Element of bool the carrier of Affn
TRn is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,L)
A is Element of the carrier of Affn
A + L is finite affinely-independent Element of bool the carrier of Affn
(card L) |-> A is Relation-like NAT -defined the carrier of Affn -valued Function-like finite card L -element FinSequence-like FinSubsequence-like Element of (card L) -tuples_on the carrier of Affn
(card L) -tuples_on the carrier of Affn is functional non empty FinSequence-membered FinSequenceSet of the carrier of Affn
TRn + ((card L) |-> A) is Relation-like NAT -defined the carrier of Affn -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of Affn
transl (A,Affn) is Relation-like the carrier of Affn -defined the carrier of Affn -valued Function-like quasi_total being_homeomorphism Element of bool [: the carrier of Affn, the carrier of Affn:]
[: the carrier of Affn, the carrier of Affn:] is non empty set
bool [: the carrier of Affn, the carrier of Affn:] is non empty set
Affin (A + L) is Affine Element of bool the carrier of Affn
L is Relation-like NAT -defined the carrier of Affn -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like (Affn,A + L)
f is set
{ b1 where b1 is Element of the carrier of Affn : ( b1 in Affin L & (Affn,L,TRn,b1) | n in f ) } is set
(transl (A,Affn)) .: { b1 where b1 is Element of the carrier of Affn : ( b1 in Affin L & (Affn,L,TRn,b1) | n in f ) } is Element of bool the carrier of Affn
{ b1 where b1 is Element of the carrier of Affn : ( b1 in Affin (A + L) & (Affn,(A + L),L,b1) | n in f ) } is set
A + (Affin L) is Element of bool the carrier of Affn
k is set
dom (transl (A,Affn)) is Element of bool the carrier of Affn
Y is set
(transl (A,Affn)) . Y is set
k is Element of the carrier of Affn
(Affn,L,TRn,k) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k |-- L is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(k |-- L) * TRn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Affn,L,TRn,k) | n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
(Affn,L,TRn,k) | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
A + k is Element of the carrier of Affn
{ (A + b1) where b1 is Element of the carrier of Affn : b1 in Affin L } is set
(Affn,(A + L),L,(A + k)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(A + k) |-- (A + L) is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A + L
((A + k) |-- (A + L)) * L is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k is set
Y is Element of the carrier of Affn
(Affn,(A + L),L,Y) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Y |-- (A + L) is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A + L
(Y |-- (A + L)) * L is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Affn,(A + L),L,Y) | n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Affn,(A + L),L,Y) | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
k is Element of the carrier of Affn
A + k is Element of the carrier of Affn
(Affn,L,TRn,k) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k |-- L is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(k |-- L) * TRn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(transl (A,Affn)) . k is Element of the carrier of Affn
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
n -VectSp_over F_Real is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over F_Real
A is functional non empty finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card A is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
the carrier of (n -VectSp_over F_Real) is non empty set
bool the carrier of (n -VectSp_over F_Real) is non empty set
(card A) -' 1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affin A is functional non empty Affine Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
{(0* n)} is functional non empty trivial finite V39() 1 -element FinSequence-membered Element of bool (REAL n)
bool (REAL n) is non empty set
A \ {(0* n)} is functional finite Element of bool the carrier of (TOP-REAL n)
Lin (A \ {(0* n)}) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
Lin A is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
f is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,A)
len f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
f . (len f) is Relation-like Function-like set
rng f is functional finite Element of bool the carrier of (TOP-REAL n)
B is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over F_Real)
B | n is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over F_Real)
Seg n is finite n -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= n ) } is set
B | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
FinS2MX (B | n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of len (B | n),n, the carrier of F_Real
len (B | n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(card A) - 1 is V21() real V30() V31() ext-real Element of INT
k is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of F_Real
Mx2Tran k is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
k is functional Element of bool the carrier of (TOP-REAL Affn)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ((TOP-REAL n),A,f,b1) | Affn in k } is set
Ek is functional Element of bool the carrier of (TOP-REAL n)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : b1 | Affn in k } is set
P is set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
B | Affn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
B | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
[#] (Lin (A \ {(0* n)})) is non empty non proper Element of bool the carrier of (Lin (A \ {(0* n)}))
the carrier of (Lin (A \ {(0* n)})) is non empty set
bool the carrier of (Lin (A \ {(0* n)})) is non empty set
L is Element of bool the carrier of (n -VectSp_over F_Real)
L \ {(0* n)} is Element of bool the carrier of (n -VectSp_over F_Real)
Lin (L \ {(0* n)}) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
[#] (Lin (L \ {(0* n)})) is non empty non proper Element of bool the carrier of (Lin (L \ {(0* n)}))
the carrier of (Lin (L \ {(0* n)})) is non empty set
bool the carrier of (Lin (L \ {(0* n)})) is non empty set
f | ((card A) -' 1) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
Seg ((card A) -' 1) is finite (card A) -' 1 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= (card A) -' 1 ) } is set
f | (Seg ((card A) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
rng (f | ((card A) -' 1)) is functional finite Element of bool the carrier of (TOP-REAL n)
[#] (Lin A) is non empty non proper Element of bool the carrier of (Lin A)
the carrier of (Lin A) is non empty set
bool the carrier of (Lin A) is non empty set
Lin L is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
[#] (Lin L) is non empty non proper Element of bool the carrier of (Lin L)
the carrier of (Lin L) is non empty set
bool the carrier of (Lin L) is non empty set
lines k is finite Element of bool the carrier of (n -VectSp_over F_Real)
Lin (lines k) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
the carrier of (Lin (lines k)) is non empty set
B is Relation-like NAT -defined the carrier of (Lin (lines k)) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of Lin (lines k)
rng B is finite Element of bool the carrier of (Lin (lines k))
bool the carrier of (Lin (lines k)) is non empty set
the_rank_of k is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Det k is V21() real ext-real Element of the carrier of F_Real
rng (Mx2Tran k) is functional Element of bool the carrier of (TOP-REAL n)
P is functional Element of bool the carrier of (TOP-REAL n)
(Mx2Tran k) .: P is functional Element of bool the carrier of (TOP-REAL n)
y is set
yP is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),A,f,yP) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
yP |-- A is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A
(yP |-- A) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,yP) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),A,f,yP) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom (Mx2Tran k) is functional Element of bool the carrier of (TOP-REAL n)
yP1 is set
(Mx2Tran k) . yP1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
len ((TOP-REAL n),A,f,yP) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((TOP-REAL n),A,f,yP) | n is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,yP) | (Seg n) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (((TOP-REAL n),A,f,yP) | n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
(((TOP-REAL n),A,f,yP) | n) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(((TOP-REAL n),A,f,yP) | n) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
vP1 is Element of the carrier of (Lin (lines k))
vP1 |-- B is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
((TOP-REAL n),A,f,vP1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
vP1 |-- A is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A
(vP1 |-- A) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,vP1) | ((card A) -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,vP1) | (Seg ((card A) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
y is set
dom (Mx2Tran k) is functional Element of bool the carrier of (TOP-REAL n)
yP is set
(Mx2Tran k) . yP is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
yP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
yP1 | Affn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
yP1 | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
(Mx2Tran k) . yP1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
vP1 is Element of the carrier of (Lin (lines k))
vP1 |-- B is Relation-like NAT -defined the carrier of F_Real -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of the carrier of F_Real
((TOP-REAL n),A,f,vP1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
vP1 |-- A is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A
(vP1 |-- A) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,vP1) | ((card A) -' 1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,vP1) | (Seg ((card A) -' 1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
((TOP-REAL n),A,f,vP1) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,f,vP1) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
TRn is functional non empty finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card TRn is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
Affin TRn is functional non empty Affine Element of bool the carrier of (TOP-REAL n)
(TOP-REAL n) | (Affin TRn) is non empty strict TopSpace-like SubSpace of TOP-REAL n
the carrier of ((TOP-REAL n) | (Affin TRn)) is non empty set
bool the carrier of ((TOP-REAL n) | (Affin TRn)) is non empty set
(card TRn) - 1 is V21() real V30() V31() ext-real Element of INT
E is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn)
len E is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
E . (len E) is Relation-like Function-like set
rng E is functional finite Element of bool the carrier of (TOP-REAL n)
dom E is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
TRn \ {(0. (TOP-REAL n))} is functional finite Element of bool the carrier of (TOP-REAL n)
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 + (dim (TOP-REAL n)) is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
A + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
1 + n is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n -' A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
f is functional Element of bool the carrier of (TOP-REAL Affn)
{ b1 where b1 is Element of the carrier of ((TOP-REAL n) | (Affin TRn)) : ((TOP-REAL n),TRn,E,b1) | Affn in f } is set
B is Element of bool the carrier of ((TOP-REAL n) | (Affin TRn))
[#] ((TOP-REAL n) | (Affin TRn)) is non empty non proper open closed Element of bool the carrier of ((TOP-REAL n) | (Affin TRn))
k is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng k is finite set
n -VectSp_over F_Real is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional VectSpStr over F_Real
the carrier of (n -VectSp_over F_Real) is non empty set
1. (F_Real,n) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of F_Real
MX2FinS (1. (F_Real,n)) is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over F_Real)
k is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like finite FinSequence-like FinSubsequence-like OrdBasis of n -VectSp_over F_Real
len k is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k | A is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over F_Real)
Seg A is finite A -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= A ) } is set
k | (Seg A) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
FinS2MX (k | A) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of len (k | A),n, the carrier of F_Real
len (k | A) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Ek is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of A,n, the carrier of F_Real
Mx2Tran Ek is Relation-like the carrier of (TOP-REAL A) -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [: the carrier of (TOP-REAL A), the carrier of (TOP-REAL n):]
TOP-REAL A is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL A) is functional non empty set
[: the carrier of (TOP-REAL A), the carrier of (TOP-REAL n):] is non empty set
bool [: the carrier of (TOP-REAL A), the carrier of (TOP-REAL n):] is non empty set
len k is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card (TRn \ {(0. (TOP-REAL n))}) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
dom (Mx2Tran Ek) is functional Element of bool the carrier of (TOP-REAL A)
bool the carrier of (TOP-REAL A) is non empty set
[#] (TOP-REAL A) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL A)
0. (TOP-REAL A) is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL A)
the ZeroF of (TOP-REAL A) is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
(Mx2Tran Ek) . (0. (TOP-REAL A)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
rng k is finite Element of bool the carrier of (n -VectSp_over F_Real)
bool the carrier of (n -VectSp_over F_Real) is non empty set
rng (k | A) is finite Element of bool the carrier of (n -VectSp_over F_Real)
y is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
len y is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k | (len y) is Relation-like NAT -defined the carrier of (n -VectSp_over F_Real) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (n -VectSp_over F_Real)
Seg (len y) is finite len y -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len y ) } is set
k | (Seg (len y)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
rng (k | (len y)) is finite Element of bool the carrier of (n -VectSp_over F_Real)
Lin (rng (k | (len y))) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
[#] (Lin (rng (k | (len y)))) is non empty non proper Element of bool the carrier of (Lin (rng (k | (len y))))
the carrier of (Lin (rng (k | (len y)))) is non empty set
bool the carrier of (Lin (rng (k | (len y)))) is non empty set
Pro is Relation-like Function-like set
rng Pro is set
yP is functional Element of bool the carrier of (TOP-REAL n)
yP1 is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of F_Real
yP1 | (len y) is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of F_Real *
yP1 | (Seg (len y)) is Relation-like NAT -defined Function-like finite FinSubsequence-like set
yP1 ~ is Relation-like NAT -defined the carrier of F_Real * -valued Function-like finite FinSequence-like FinSubsequence-like tabular Matrix of n,n, the carrier of F_Real
Mx2Tran (yP1 ~) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
(TOP-REAL n) | yP is strict TopSpace-like SubSpace of TOP-REAL n
[#] ((TOP-REAL n) | yP) is non proper open closed Element of bool the carrier of ((TOP-REAL n) | yP)
the carrier of ((TOP-REAL n) | yP) is set
bool the carrier of ((TOP-REAL n) | yP) is non empty set
Det (yP1 ~) is V21() real ext-real Element of the carrier of F_Real
the_rank_of (yP1 ~) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(Mx2Tran (yP1 ~)) .: TRn is functional finite Element of bool the carrier of (TOP-REAL n)
(Mx2Tran (yP1 ~)) * E is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
x is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,(Mx2Tran (yP1 ~)) .: TRn)
rng x is functional finite Element of bool the carrier of (TOP-REAL n)
dom (Mx2Tran (yP1 ~)) is functional Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
card ((Mx2Tran (yP1 ~)) .: TRn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
len x is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom x is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
x . (len E) is Relation-like Function-like set
(Mx2Tran (yP1 ~)) . (0. (TOP-REAL n)) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
Mx2Tran yP1 is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
Lin TRn is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin TRn) is non empty non proper Element of bool the carrier of (Lin TRn)
the carrier of (Lin TRn) is non empty set
bool the carrier of (Lin TRn) is non empty set
rng y is functional finite Element of bool the carrier of (TOP-REAL n)
Lin (rng y) is non empty right_complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional Subspace of TOP-REAL n
[#] (Lin (rng y)) is non empty non proper Element of bool the carrier of (Lin (rng y))
the carrier of (Lin (rng y)) is non empty set
bool the carrier of (Lin (rng y)) is non empty set
(Mx2Tran yP1) .: yP is functional Element of bool the carrier of (TOP-REAL n)
(Mx2Tran yP1) | yP is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
rng ((Mx2Tran yP1) | yP) is functional Element of bool the carrier of (TOP-REAL n)
dom (Mx2Tran yP1) is functional Element of bool the carrier of (TOP-REAL n)
dom ((Mx2Tran yP1) | yP) is functional Element of bool the carrier of (TOP-REAL n)
[: the carrier of ((TOP-REAL n) | yP), the carrier of ((TOP-REAL n) | (Affin TRn)):] is set
bool [: the carrier of ((TOP-REAL n) | yP), the carrier of ((TOP-REAL n) | (Affin TRn)):] is non empty set
Det yP1 is V21() real ext-real Element of the carrier of F_Real
MT is Relation-like Function-like set
MT " is Relation-like Function-like set
MT1 is Relation-like the carrier of ((TOP-REAL n) | yP) -defined the carrier of ((TOP-REAL n) | (Affin TRn)) -valued Function-like quasi_total Element of bool [: the carrier of ((TOP-REAL n) | yP), the carrier of ((TOP-REAL n) | (Affin TRn)):]
MT1 " B is Element of bool the carrier of ((TOP-REAL n) | yP)
(n -' A) |-> {} is Relation-like empty-yielding NAT -defined Function-like finite n -' A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
{ (b1 ^ ((n -' A) |-> {})) where b1 is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A) : verum } is set
n - A is V21() real ext-real Element of REAL
A + (n -' A) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
MT " (Affin TRn) is set
MT " TRn is set
the_rank_of Ek is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Pro " is Relation-like Function-like set
dom (Pro ") is set
x (#) (Pro ") is Relation-like Function-like finite set
MBCe is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng MBCe is finite set
(Pro ") .: ((Mx2Tran (yP1 ~)) .: TRn) is finite set
(Mx2Tran Ek) " ((Mx2Tran (yP1 ~)) .: TRn) is functional Element of bool the carrier of (TOP-REAL A)
R is functional finite affinely-independent Element of bool the carrier of (TOP-REAL A)
(Mx2Tran Ek) .: R is functional finite Element of bool the carrier of (TOP-REAL n)
MBCe is Relation-like NAT -defined the carrier of (TOP-REAL A) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL A,R)
(Mx2Tran Ek) * MBCe is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
(Pro ") (#) Pro is Relation-like Function-like set
id (rng Pro) is Relation-like rng Pro -defined rng Pro -valued Function-like one-to-one total Element of bool [:(rng Pro),(rng Pro):]
[:(rng Pro),(rng Pro):] is set
bool [:(rng Pro),(rng Pro):] is non empty set
MBCeE is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,(Mx2Tran Ek) .: R)
(id (rng Pro)) * x is Relation-like NAT -defined rng Pro -valued Function-like one-to-one finite Element of bool [:NAT,(rng Pro):]
[:NAT,(rng Pro):] is set
bool [:NAT,(rng Pro):] is non empty set
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A) : ((TOP-REAL A),R,MBCe,b1) | Affn in f } is set
PPP is set
nPP is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
((TOP-REAL A),R,MBCe,nPP) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
nPP |-- R is Relation-like the carrier of (TOP-REAL A) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of R
(nPP |-- R) * MBCe is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL A),R,MBCe,nPP) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL A),R,MBCe,nPP) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
MT " B is set
card R is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
PPP is functional Element of bool the carrier of (TOP-REAL A)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ( b1 | A in PPP & b1 in yP ) } is set
dim (TOP-REAL A) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affin R is functional Affine Element of bool the carrier of (TOP-REAL A)
Pro .: ((Pro ") .: ((Mx2Tran (yP1 ~)) .: TRn)) is finite set
((Pro ") (#) Pro) .: ((Mx2Tran (yP1 ~)) .: TRn) is finite set
(id (rng Pro)) .: ((Mx2Tran (yP1 ~)) .: TRn) is finite Element of bool (rng Pro)
bool (rng Pro) is non empty set
x is set
MT . x is set
v1 is Element of the carrier of ((TOP-REAL n) | (Affin TRn))
((TOP-REAL n),TRn,E,v1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
v1 |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(v1 |-- TRn) * E is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),TRn,E,v1) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),TRn,E,v1) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom MT is set
(Mx2Tran (yP1 ~)) . v1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
v is Element of the carrier of ((TOP-REAL n) | yP)
w is set
(Mx2Tran Ek) . w is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
u is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
u | A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
u | (Seg A) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
w1 is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
((TOP-REAL A),R,MBCe,w1) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
w1 |-- R is Relation-like the carrier of (TOP-REAL A) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of R
(w1 |-- R) * MBCe is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),((Mx2Tran (yP1 ~)) .: TRn),x,v) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
v |-- ((Mx2Tran (yP1 ~)) .: TRn) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (Mx2Tran (yP1 ~)) .: TRn
(v |-- ((Mx2Tran (yP1 ~)) .: TRn)) * x is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
x is set
v is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
len v is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
v | A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
v | (Seg A) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (v | A) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Lin (rng (k | A)) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
(v | A) ^ ((n -' A) |-> {}) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
x is set
v is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
v ^ ((n -' A) |-> {}) is Relation-like NAT -defined Function-like finite K194(A,(n -' A)) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K194(A,(n -' A)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
len v is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
(v ^ ((n -' A) |-> {})) | A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(v ^ ((n -' A) |-> {})) | (Seg A) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom v is finite A -element V60() V61() V62() V63() V64() V65() Element of bool NAT
(v ^ ((n -' A) |-> {})) | (dom v) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
Lin (rng (k | A)) is non empty right_complementable Abelian add-associative right_zeroed strict vector-distributive scalar-distributive scalar-associative scalar-unital finite-dimensional M33( F_Real ,n -VectSp_over F_Real)
x is set
v is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
v | A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
v | (Seg A) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
v1 is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
v1 ^ ((n -' A) |-> {}) is Relation-like NAT -defined Function-like finite K194(A,(n -' A)) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
K194(A,(n -' A)) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
u is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
((TOP-REAL A),R,MBCe,u) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
u |-- R is Relation-like the carrier of (TOP-REAL A) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of R
(u |-- R) * MBCe is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL A),R,MBCe,u) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL A),R,MBCe,u) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
(Mx2Tran Ek) . u is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
MT . ((Mx2Tran Ek) . u) is set
(Mx2Tran (yP1 ~)) . (MT . ((Mx2Tran Ek) . u)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
((TOP-REAL n),((Mx2Tran Ek) .: R),MBCeE,((Mx2Tran Ek) . u)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((Mx2Tran Ek) . u) |-- ((Mx2Tran Ek) .: R) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (Mx2Tran Ek) .: R
(((Mx2Tran Ek) . u) |-- ((Mx2Tran Ek) .: R)) * MBCeE is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),TRn,E,(MT . ((Mx2Tran Ek) . u))) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(MT . ((Mx2Tran Ek) . u)) |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
((MT . ((Mx2Tran Ek) . u)) |-- TRn) * E is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
w1 is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL A)
w1 ^ ((n -' A) |-> {}) is Relation-like NAT -defined Function-like finite K194(A,(n -' A)) -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
dom w1 is finite A -element V60() V61() V62() V63() V64() V65() Element of bool NAT
((Mx2Tran Ek) . u) | (dom w1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len w1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Seg (len w1) is finite len w1 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len w1 ) } is set
((Mx2Tran Ek) . u) | (Seg (len w1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
((Mx2Tran Ek) . u) | A is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
((Mx2Tran Ek) . u) | (Seg A) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len v1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Seg (len v1) is finite len v1 -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len v1 ) } is set
v | (Seg (len v1)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
dom v1 is finite A -element V60() V61() V62() V63() V64() V65() Element of bool NAT
v | (dom v1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len MBCe is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom MBCe is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
MBCe . (len E) is Relation-like Function-like set
(Pro ") . (0. (TOP-REAL n)) is set
MBCe . (len MBCe) is Relation-like Function-like set
0* A is Relation-like NAT -defined REAL -valued Function-like finite A -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL A
REAL A is functional non empty FinSequence-membered FinSequenceSet of REAL
yP /\ (MT " B) is functional Element of bool the carrier of (TOP-REAL n)
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
TRn is functional Element of bool the carrier of (TOP-REAL Affn)
Affin L is functional Affine Element of bool the carrier of (TOP-REAL n)
f is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ((TOP-REAL n),L,f,b1) | Affn in TRn } is set
f is functional Element of bool the carrier of (TOP-REAL n)
rng f is functional finite Element of bool the carrier of (TOP-REAL n)
len f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
f . (len f) is Relation-like Function-like set
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
- k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
- k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(- k) + L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card ((- k) + L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ( b1 in Affin L & ((TOP-REAL n),L,f,b1) | Affn in TRn ) } is set
k is set
Ek is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),L,f,Ek) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Ek |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(Ek |-- L) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,f,Ek) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),L,f,Ek) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
(card L) |-> (- k) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite card L -element FinSequence-like FinSubsequence-like Element of (card L) -tuples_on the carrier of (TOP-REAL n)
(card L) -tuples_on the carrier of (TOP-REAL n) is functional non empty FinSequence-membered FinSequenceSet of the carrier of (TOP-REAL n)
f + ((card L) |-> (- k)) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
k is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,(- k) + L)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ((TOP-REAL n),((- k) + L),k,b1) | Affn in TRn } is set
transl ((- k),(TOP-REAL n)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
P is set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),L,f,B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
B |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(B |-- L) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,f,B) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),L,f,B) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
(transl ((- k),(TOP-REAL n))) .: f is functional Element of bool the carrier of (TOP-REAL n)
Affin ((- k) + L) is functional Affine Element of bool the carrier of (TOP-REAL n)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ( b1 in Affin ((- k) + L) & ((TOP-REAL n),((- k) + L),k,b1) | Affn in TRn ) } is set
P is set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),((- k) + L),k,B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
B |-- ((- k) + L) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (- k) + L
(B |-- ((- k) + L)) * k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),((- k) + L),k,B) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),((- k) + L),k,B) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
P is set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),((- k) + L),k,B) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
B |-- ((- k) + L) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (- k) + L
(B |-- ((- k) + L)) * k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),((- k) + L),k,B) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),((- k) + L),k,B) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
Seg (card L) is finite card L -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card L ) } is set
((card L) |-> (- k)) . (len f) is Relation-like Function-like set
rng k is functional finite Element of bool the carrier of (TOP-REAL n)
len k is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom k is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k . (len f) is Relation-like Function-like set
k + (- k) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
k + (- k) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
k . (len k) is Relation-like Function-like set
(transl ((- k),(TOP-REAL n))) .: (Affin L) is functional Element of bool the carrier of (TOP-REAL n)
(transl ((- k),(TOP-REAL n))) | (Affin L) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
rng ((transl ((- k),(TOP-REAL n))) | (Affin L)) is functional Element of bool the carrier of (TOP-REAL n)
dom (transl ((- k),(TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
dom ((transl ((- k),(TOP-REAL n))) | (Affin L)) is functional Element of bool the carrier of (TOP-REAL n)
(TOP-REAL n) | (Affin L) is strict TopSpace-like SubSpace of TOP-REAL n
[#] ((TOP-REAL n) | (Affin L)) is non proper open closed Element of bool the carrier of ((TOP-REAL n) | (Affin L))
the carrier of ((TOP-REAL n) | (Affin L)) is set
bool the carrier of ((TOP-REAL n) | (Affin L)) is non empty set
(TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L)) is strict TopSpace-like SubSpace of TOP-REAL n
[#] ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))) is non proper open closed Element of bool the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L)))
the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))) is set
bool the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))) is non empty set
[: the carrier of ((TOP-REAL n) | (Affin L)), the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))):] is set
bool [: the carrier of ((TOP-REAL n) | (Affin L)), the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))):] is non empty set
B is Relation-like the carrier of ((TOP-REAL n) | (Affin L)) -defined the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))) -valued Function-like quasi_total Element of bool [: the carrier of ((TOP-REAL n) | (Affin L)), the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L))):]
B .: f is Element of bool the carrier of ((TOP-REAL n) | ((transl ((- k),(TOP-REAL n))) .: (Affin L)))
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
L is functional Element of bool the carrier of (TOP-REAL n)
TRn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
A is functional Element of bool the carrier of (TOP-REAL Affn)
f is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ((TOP-REAL n),TRn,f,b1) | Affn in A } is set
A ` is functional Element of bool the carrier of (TOP-REAL Affn)
the carrier of (TOP-REAL Affn) \ A is set
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ((TOP-REAL n),TRn,f,b1) | Affn in A ` } is set
L ` is functional Element of bool the carrier of (TOP-REAL n)
the carrier of (TOP-REAL n) \ L is set
k is set
Y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),TRn,f,Y) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Y |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(Y |-- TRn) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((TOP-REAL n),TRn,f,Y) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((TOP-REAL n),TRn,f,Y) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),TRn,f,Y) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (((TOP-REAL n),TRn,f,Y) | Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k is set
Y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),TRn,f,Y) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Y |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(Y |-- TRn) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),TRn,f,Y) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),TRn,f,Y) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),TRn,f,k) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(k |-- TRn) * f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),TRn,f,k) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),TRn,f,k) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
L is functional Element of bool the carrier of (TOP-REAL n)
Affin L is functional Affine Element of bool the carrier of (TOP-REAL n)
{} (TOP-REAL n) is Function-like functional empty trivial proper V21() ordinal natural real ext-real non positive non negative finite V39() cardinal {} -element FinSequence-membered V60() V61() V62() V63() V64() V65() V66() open closed V179( TOP-REAL n) circled V186( TOP-REAL n) affinely-independent Element of bool the carrier of (TOP-REAL n)
TRn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
Affin TRn is functional Affine Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
Affin ([#] (TOP-REAL n)) is functional non empty Affine Element of bool the carrier of (TOP-REAL n)
A is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
Affin A is functional Affine Element of bool the carrier of (TOP-REAL n)
A \ TRn is functional finite Element of bool the carrier of (TOP-REAL n)
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
TOP-REAL (card L) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
0* (card L) is Relation-like NAT -defined REAL -valued Function-like finite card L -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL (card L)
REAL (card L) is functional non empty FinSequence-membered FinSequenceSet of REAL
0. (TOP-REAL (card L)) is Relation-like NAT -defined REAL -valued Function-like finite card L -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL (card L))
the carrier of (TOP-REAL (card L)) is functional non empty set
the ZeroF of (TOP-REAL (card L)) is Relation-like NAT -defined REAL -valued Function-like finite card L -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL (card L))
(card L) |-> {} is Relation-like empty-yielding NAT -defined Function-like finite card L -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
bool the carrier of (TOP-REAL (card L)) is non empty set
{((card L) |-> {})} is functional non empty trivial finite V39() 1 -element set
f is functional Element of bool the carrier of (TOP-REAL (card L))
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn)
rng the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is functional finite Element of bool the carrier of (TOP-REAL n)
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L) ^ the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
rng the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L) is functional finite Element of bool the carrier of (TOP-REAL n)
TRn \/ L is functional finite Element of bool the carrier of (TOP-REAL n)
TRn \/ A is functional finite Element of bool the carrier of (TOP-REAL n)
rng ( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L) ^ the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn)) is functional finite Element of bool the carrier of (TOP-REAL n)
len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
k is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,A)
Seg (card L) is finite card L -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card L ) } is set
k .: (Seg (card L)) is functional finite Element of bool the carrier of (TOP-REAL n)
dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k .: (dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)) is functional finite Element of bool the carrier of (TOP-REAL n)
k | (dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of (TOP-REAL n):]
[:NAT, the carrier of (TOP-REAL n):] is non empty non trivial non finite set
bool [:NAT, the carrier of (TOP-REAL n):] is non empty non trivial non finite set
rng (k | (dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L))) is functional finite Element of bool the carrier of (TOP-REAL n)
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ((TOP-REAL n),A,k,b1) | (card L) in f } is set
A \ L is functional finite Element of bool the carrier of (TOP-REAL n)
A /\ TRn is functional finite Element of bool the carrier of (TOP-REAL n)
Pro is set
P is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),A,k,P) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
P |-- A is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A
(P |-- A) * k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,k,P) | (card L) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,k,P) | (Seg (card L)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
((TOP-REAL n),A,k,P) /^ (card L) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(((TOP-REAL n),A,k,P) | (card L)) ^ (((TOP-REAL n),A,k,P) /^ (card L)) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((card L) |-> {}) ^ (((TOP-REAL n),A,k,P) /^ (card L)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
Pro is set
P is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),A,k,P) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
P |-- A is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of A
(P |-- A) * k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,k,P) | (card L) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),A,k,P) | (Seg (card L)) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
((TOP-REAL n),A,k,P) /^ (card L) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((card L) |-> {}) ^ (((TOP-REAL n),A,k,P) /^ (card L)) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
Affin L is functional closed Affine Element of bool the carrier of (TOP-REAL n)
(TOP-REAL n) | (Affin L) is strict TopSpace-like SubSpace of TOP-REAL n
the carrier of ((TOP-REAL n) | (Affin L)) is set
bool the carrier of ((TOP-REAL n) | (Affin L)) is non empty set
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
TRn is functional Element of bool the carrier of (TOP-REAL Affn)
A is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)
{ b1 where b1 is Element of the carrier of ((TOP-REAL n) | (Affin L)) : ((TOP-REAL n),L,A,b1) | Affn in TRn } is set
(card L) -' 1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
TOP-REAL ((card L) -' 1) is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
k is Element of bool the carrier of ((TOP-REAL n) | (Affin L))
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ( b1 in Affin L & ((TOP-REAL n),L,A,b1) | Affn in TRn ) } is set
[#] ((TOP-REAL n) | (Affin L)) is non proper open closed Element of bool the carrier of ((TOP-REAL n) | (Affin L))
Pro is set
P is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),L,A,P) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
P |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(P |-- L) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,A,P) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),L,A,P) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
Pro is set
P is Element of the carrier of ((TOP-REAL n) | (Affin L))
((TOP-REAL n),L,A,P) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
P |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(P |-- L) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,A,P) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),L,A,P) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
rng A is functional finite Element of bool the carrier of (TOP-REAL n)
len A is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom A is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
A . (len A) is Relation-like Function-like set
Seg (card L) is finite card L -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= card L ) } is set
Pro is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
- Pro is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
- Pro is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(card L) |-> (- Pro) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite card L -element FinSequence-like FinSubsequence-like Element of (card L) -tuples_on the carrier of (TOP-REAL n)
(card L) -tuples_on the carrier of (TOP-REAL n) is functional non empty FinSequence-membered FinSequenceSet of the carrier of (TOP-REAL n)
((card L) |-> (- Pro)) . (len A) is Relation-like Function-like set
(- Pro) + L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card ((- Pro) + L) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
transl ((- Pro),(TOP-REAL n)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like quasi_total being_homeomorphism Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
[: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] is non empty set
(transl ((- Pro),(TOP-REAL n))) .: (Affin L) is functional Element of bool the carrier of (TOP-REAL n)
(TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L)) is strict TopSpace-like SubSpace of TOP-REAL n
[#] ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))) is non proper open closed Element of bool the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L)))
the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))) is set
bool the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))) is non empty set
(transl ((- Pro),(TOP-REAL n))) | (Affin L) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):]
rng ((transl ((- Pro),(TOP-REAL n))) | (Affin L)) is functional Element of bool the carrier of (TOP-REAL n)
dom (transl ((- Pro),(TOP-REAL n))) is functional Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
dom ((transl ((- Pro),(TOP-REAL n))) | (Affin L)) is functional Element of bool the carrier of (TOP-REAL n)
[: the carrier of ((TOP-REAL n) | (Affin L)), the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))):] is set
bool [: the carrier of ((TOP-REAL n) | (Affin L)), the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))):] is non empty set
y is Relation-like the carrier of ((TOP-REAL n) | (Affin L)) -defined the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))) -valued Function-like quasi_total Element of bool [: the carrier of ((TOP-REAL n) | (Affin L)), the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))):]
y .: k is Element of bool the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L)))
A + ((card L) |-> (- Pro)) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (TOP-REAL n)
yP1 is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,(- Pro) + L)
rng yP1 is functional finite Element of bool the carrier of (TOP-REAL n)
len yP1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom yP1 is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
yP1 . (len A) is Relation-like Function-like set
Pro + (- Pro) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
Pro + (- Pro) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
0* n is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of REAL n
REAL n is functional non empty FinSequence-membered FinSequenceSet of REAL
yP1 . (len yP1) is Relation-like Function-like set
{ b1 where b1 is Element of the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L))) : ((TOP-REAL n),((- Pro) + L),yP1,b1) | Affn in TRn } is set
(- Pro) + (Affin L) is functional closed Element of bool the carrier of (TOP-REAL n)
Affin ((- Pro) + L) is functional closed Affine Element of bool the carrier of (TOP-REAL n)
(transl ((- Pro),(TOP-REAL n))) .: k is functional Element of bool the carrier of (TOP-REAL n)
yP is Element of bool the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L)))
{ b1 where b1 is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n) : ( b1 in Affin ((- Pro) + L) & ((TOP-REAL n),((- Pro) + L),yP1,b1) | Affn in TRn ) } is set
x is set
w is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),((- Pro) + L),yP1,w) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
w |-- ((- Pro) + L) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (- Pro) + L
(w |-- ((- Pro) + L)) * yP1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),((- Pro) + L),yP1,w) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),((- Pro) + L),yP1,w) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
x is set
w is Element of the carrier of ((TOP-REAL n) | ((transl ((- Pro),(TOP-REAL n))) .: (Affin L)))
((TOP-REAL n),((- Pro) + L),yP1,w) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
w |-- ((- Pro) + L) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of (- Pro) + L
(w |-- ((- Pro) + L)) * yP1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),((- Pro) + L),yP1,w) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),((- Pro) + L),yP1,w) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
Affin L is functional closed Affine Element of bool the carrier of (TOP-REAL n)
(TOP-REAL n) | (Affin L) is strict TopSpace-like SubSpace of TOP-REAL n
the carrier of ((TOP-REAL n) | (Affin L)) is set
bool the carrier of ((TOP-REAL n) | (Affin L)) is non empty set
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
TRn is functional Element of bool the carrier of (TOP-REAL Affn)
A is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,L)
{ b1 where b1 is Element of the carrier of ((TOP-REAL n) | (Affin L)) : ((TOP-REAL n),L,A,b1) | Affn in TRn } is set
f is Element of bool the carrier of ((TOP-REAL n) | (Affin L))
TRn ` is functional Element of bool the carrier of (TOP-REAL Affn)
the carrier of (TOP-REAL Affn) \ TRn is set
{ b1 where b1 is Element of the carrier of ((TOP-REAL n) | (Affin L)) : ((TOP-REAL n),L,A,b1) | Affn in TRn ` } is set
f ` is Element of bool the carrier of ((TOP-REAL n) | (Affin L))
the carrier of ((TOP-REAL n) | (Affin L)) \ f is set
k is set
Y is Element of the carrier of ((TOP-REAL n) | (Affin L))
((TOP-REAL n),L,A,Y) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Y |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(Y |-- L) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((TOP-REAL n),L,A,Y) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((TOP-REAL n),L,A,Y) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),L,A,Y) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (((TOP-REAL n),L,A,Y) | Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] (TOP-REAL Affn) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL Affn)
k is set
Y is Element of the carrier of ((TOP-REAL n) | (Affin L))
((TOP-REAL n),L,A,Y) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Y |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(Y |-- L) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,A,Y) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Seg Affn is finite Affn -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= Affn ) } is set
((TOP-REAL n),L,A,Y) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
k is Element of the carrier of ((TOP-REAL n) | (Affin L))
((TOP-REAL n),L,A,k) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
k |-- L is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(k |-- L) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,A,k) | Affn is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),L,A,k) | (Seg Affn) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
Affn is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
halfline (Affn,L) is functional Element of bool the carrier of (TOP-REAL n)
bool the carrier of (TOP-REAL n) is non empty set
{Affn,L} is functional non empty finite V39() affinely-independent Element of bool the carrier of (TOP-REAL n)
{Affn} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
<*Affn,L*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like set
<*Affn*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,Affn] is non empty set
{1,Affn} is non empty finite V39() set
{{1,Affn},{1}} is non empty finite V39() set
{[1,Affn]} is Function-like non empty trivial finite 1 -element set
<*L*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,L] is non empty set
{1,L} is non empty finite V39() set
{{1,L},{1}} is non empty finite V39() set
{[1,L]} is Function-like non empty trivial finite 1 -element set
<*Affn*> ^ <*L*> is Relation-like NAT -defined Function-like non empty finite K194(1,1) -element FinSequence-like FinSubsequence-like set
K194(1,1) is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
rng <*Affn,L*> is finite set
Affin {Affn,L} is functional non empty closed Affine Element of bool the carrier of (TOP-REAL n)
].-infty,1.] is V60() V61() V62() Element of bool REAL
A is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,{Affn,L})
A . 1 is Relation-like Function-like set
k is Relation-like the carrier of (TOP-REAL 1) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL 1), the carrier of R^1:]
(TOP-REAL n) | (Affin {Affn,L}) is non empty strict TopSpace-like SubSpace of TOP-REAL n
the carrier of ((TOP-REAL n) | (Affin {Affn,L})) is non empty set
B is V60() V61() V62() Element of bool the carrier of R^1
k " B is functional Element of bool the carrier of (TOP-REAL 1)
bool the carrier of (TOP-REAL 1) is non empty set
{ b1 where b1 is Element of the carrier of ((TOP-REAL n) | (Affin {Affn,L})) : ((TOP-REAL n),{Affn,L},A,b1) | 1 in k " B } is set
[#] ((TOP-REAL n) | (Affin {Affn,L})) is non empty non proper open closed Element of bool the carrier of ((TOP-REAL n) | (Affin {Affn,L}))
bool the carrier of ((TOP-REAL n) | (Affin {Affn,L})) is non empty set
k is set
Ek is Element of the carrier of ((TOP-REAL n) | (Affin {Affn,L}))
((TOP-REAL n),{Affn,L},A,Ek) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Ek |-- {Affn,L} is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {Affn,L}
(Ek |-- {Affn,L}) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),{Affn,L},A,Ek) | 1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),{Affn,L},A,Ek) | (Seg 1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
card {Affn,L} is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
dom k is functional Element of bool the carrier of (TOP-REAL 1)
[#] (TOP-REAL 1) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL 1)
k is Element of bool the carrier of ((TOP-REAL n) | (Affin {Affn,L}))
{L} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
L |-- {L} is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {L}
Carrier (L |-- {L}) is functional Element of bool the carrier of (TOP-REAL n)
(L |-- {L}) . Affn is V21() real ext-real Element of REAL
Affin {L} is functional non empty closed Affine Element of bool the carrier of (TOP-REAL n)
Ek is set
Ek |-- {Affn,L} is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {Affn,L}
((TOP-REAL n),{Affn,L},A,Ek) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
(Ek |-- {Affn,L}) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
B is V21() real ext-real set
1 - B is V21() real ext-real Element of REAL
(1 - B) * Affn is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(1 - B) * Affn is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
B * L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
B * L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((1 - B) * Affn) + (B * L) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((1 - B) * Affn) + (B * L) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
L |-- {Affn,L} is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {Affn,L}
(L |-- {Affn,L}) . Affn is V21() real ext-real Element of REAL
y is V21() real ext-real Element of REAL
y * (L |-- {Affn,L}) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TOP-REAL n
(y * (L |-- {Affn,L})) . Affn is V21() real ext-real Element of REAL
y * {} is V21() real ext-real Element of REAL
conv {Affn,L} is functional non empty convex Element of bool the carrier of (TOP-REAL n)
Affn |-- {Affn,L} is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {Affn,L}
(Affn |-- {Affn,L}) . Affn is V21() real ext-real Element of REAL
1 - y is V21() real ext-real Element of REAL
(1 - y) * (Affn |-- {Affn,L}) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TOP-REAL n
((1 - y) * (Affn |-- {Affn,L})) + (y * (L |-- {Affn,L})) is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TOP-REAL n
(Ek |-- {Affn,L}) . Affn is V21() real ext-real Element of REAL
((1 - y) * (Affn |-- {Affn,L})) . Affn is V21() real ext-real Element of REAL
(((1 - y) * (Affn |-- {Affn,L})) . Affn) + ((y * (L |-- {Affn,L})) . Affn) is V21() real ext-real Element of REAL
(1 - y) * ((Affn |-- {Affn,L}) . Affn) is V21() real ext-real Element of REAL
((1 - y) * ((Affn |-- {Affn,L}) . Affn)) + {} is V21() real ext-real Element of REAL
1 - {} is non empty V21() real ext-real positive non negative Element of REAL
(1 - y) + y is V21() real ext-real Element of REAL
len ((TOP-REAL n),{Affn,L},A,Ek) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((TOP-REAL n),{Affn,L},A,Ek) | 1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),{Affn,L},A,Ek) | (Seg 1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (((TOP-REAL n),{Affn,L},A,Ek) | 1) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
yP1 is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing Element of the carrier of (TOP-REAL 1)
dom yP1 is non empty trivial finite 1 -element V60() V61() V62() V63() V64() V65() Element of bool NAT
yP1 /. 1 is V21() real ext-real Element of REAL
yP1 . 1 is V21() real ext-real set
((TOP-REAL n),{Affn,L},A,Ek) . 1 is V21() real ext-real set
dom ((TOP-REAL n),{Affn,L},A,Ek) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k . yP1 is V21() real ext-real Element of the carrier of R^1
yP is Element of the carrier of ((TOP-REAL n) | (Affin {Affn,L}))
Ek is set
Pro is Element of the carrier of ((TOP-REAL n) | (Affin {Affn,L}))
((TOP-REAL n),{Affn,L},A,Pro) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
Pro |-- {Affn,L} is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of {Affn,L}
(Pro |-- {Affn,L}) * A is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),{Affn,L},A,Pro) | 1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL n),{Affn,L},A,Pro) | (Seg 1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
y is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing Element of the carrier of (TOP-REAL 1)
k . y is V21() real ext-real Element of the carrier of R^1
y /. 1 is V21() real ext-real Element of REAL
len y is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom y is non empty trivial finite 1 -element V60() V61() V62() V63() V64() V65() Element of bool NAT
y . 1 is V21() real ext-real set
((TOP-REAL n),{Affn,L},A,Pro) . 1 is V21() real ext-real set
dom ((TOP-REAL n),{Affn,L},A,Pro) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k . (((TOP-REAL n),{Affn,L},A,Pro) | 1) is V21() real ext-real set
(Pro |-- {Affn,L}) . Affn is V21() real ext-real Element of REAL
((Pro |-- {Affn,L}) . Affn) - ((Pro |-- {Affn,L}) . Affn) is V21() real ext-real Element of REAL
1 - ((Pro |-- {Affn,L}) . Affn) is V21() real ext-real Element of REAL
sum (Pro |-- {Affn,L}) is V21() real ext-real Element of REAL
Carrier (Pro |-- {Affn,L}) is functional Element of bool the carrier of (TOP-REAL n)
(Pro |-- {Affn,L}) . L is V21() real ext-real Element of REAL
((Pro |-- {Affn,L}) . Affn) + ((Pro |-- {Affn,L}) . L) is V21() real ext-real Element of REAL
Sum (Pro |-- {Affn,L}) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
1 - ((Pro |-- {Affn,L}) . L) is V21() real ext-real Element of REAL
(1 - ((Pro |-- {Affn,L}) . L)) * Affn is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(1 - ((Pro |-- {Affn,L}) . L)) * Affn is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((Pro |-- {Affn,L}) . L) * L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((Pro |-- {Affn,L}) . L) * L is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((1 - ((Pro |-- {Affn,L}) . L)) * Affn) + (((Pro |-- {Affn,L}) . L) * L) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((1 - ((Pro |-- {Affn,L}) . L)) * Affn) + (((Pro |-- {Affn,L}) . L) * L) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of n is non empty set
bool the carrier of n is non empty set
[: the carrier of n, the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of n, the carrier of R^1:] is non empty set
Affn is Element of bool the carrier of n
L is set
TRn is set
TRn |-- Affn is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
(TRn |-- Affn) . L is V21() real ext-real set
dom (TRn |-- Affn) is Element of bool the carrier of n
rng (TRn |-- Affn) is V60() V61() V62() Element of bool REAL
dom (TRn |-- Affn) is Element of bool the carrier of n
dom (TRn |-- Affn) is Element of bool the carrier of n
TRn is Relation-like the carrier of n -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of n, the carrier of R^1:]
A is Element of the carrier of n
TRn . A is V21() real ext-real Element of the carrier of R^1
A |-- Affn is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
(A |-- Affn) . L is V21() real ext-real set
TRn is Relation-like the carrier of n -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of n, the carrier of R^1:]
A is Relation-like the carrier of n -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of n, the carrier of R^1:]
L is set
TRn . L is V21() real ext-real set
L |-- Affn is Relation-like the carrier of n -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
(L |-- Affn) . L is V21() real ext-real set
A . L is V21() real ext-real set
{{}} is non empty trivial finite V39() 1 -element V60() V61() V62() V63() V64() V65() set
n is set
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
[#] Affn is non empty non proper Element of bool the carrier of Affn
([#] Affn) --> {} is Relation-like [#] Affn -defined {{}} -valued Function-like quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:([#] Affn),{{}}:]
[:([#] Affn),{{}}:] is INT -valued RAT -valued non empty complex-valued ext-real-valued real-valued natural-valued set
bool [:([#] Affn),{{}}:] is non empty set
L is Element of bool the carrier of Affn
(Affn,L,n) is Relation-like the carrier of Affn -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of Affn, the carrier of R^1:]
[: the carrier of Affn, the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of Affn, the carrier of R^1:] is non empty set
A is set
dom (Affn,L,n) is Element of bool the carrier of Affn
(Affn,L,n) . A is V21() real ext-real set
A |-- L is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(A |-- L) . n is V21() real ext-real set
Carrier (A |-- L) is Element of bool the carrier of Affn
dom (A |-- L) is Element of bool the carrier of Affn
n is set
Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital RLSStruct
the carrier of Affn is non empty set
bool the carrier of Affn is non empty set
[#] Affn is non empty non proper Element of bool the carrier of Affn
([#] Affn) --> {} is Relation-like [#] Affn -defined {{}} -valued Function-like quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:([#] Affn),{{}}:]
[:([#] Affn),{{}}:] is INT -valued RAT -valued non empty complex-valued ext-real-valued real-valued natural-valued set
bool [:([#] Affn),{{}}:] is non empty set
L is affinely-independent Element of bool the carrier of Affn
(Affn,L,n) is Relation-like the carrier of Affn -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of Affn, the carrier of R^1:]
[: the carrier of Affn, the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of Affn, the carrier of R^1:] is non empty set
conv L is convex Element of bool the carrier of Affn
(Affn,L,n) . n is V21() real ext-real set
n |-- L is Relation-like the carrier of Affn -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(n |-- L) . n is V21() real ext-real set
n is set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL Affn)
((TOP-REAL Affn),L,n) is Relation-like the carrier of (TOP-REAL Affn) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL Affn), the carrier of R^1:]
[: the carrier of (TOP-REAL Affn), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL Affn), the carrier of R^1:] is non empty set
Affin L is functional closed Affine Element of bool the carrier of (TOP-REAL Affn)
((TOP-REAL Affn),L,n) | (Affin L) is Relation-like the carrier of (TOP-REAL Affn) -defined the carrier of R^1 -valued Function-like complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL Affn), the carrier of R^1:]
(TOP-REAL Affn) | (Affin L) is strict TopSpace-like SubSpace of TOP-REAL Affn
the carrier of ((TOP-REAL Affn) | (Affin L)) is set
[: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:] is complex-valued ext-real-valued real-valued set
bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:] is non empty set
[#] (TOP-REAL Affn) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL Affn)
([#] (TOP-REAL Affn)) /\ (Affin L) is functional closed Element of bool the carrier of (TOP-REAL Affn)
[#] ((TOP-REAL Affn) | (Affin L)) is non proper open closed Element of bool the carrier of ((TOP-REAL Affn) | (Affin L))
bool the carrier of ((TOP-REAL Affn) | (Affin L)) is non empty set
f is Relation-like the carrier of ((TOP-REAL Affn) | (Affin L)) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:]
dom f is Element of bool the carrier of ((TOP-REAL Affn) | (Affin L))
TRn is V21() real ext-real Element of the carrier of R^1
([#] (TOP-REAL Affn)) --> TRn is Relation-like [#] (TOP-REAL Affn) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [:([#] (TOP-REAL Affn)), the carrier of R^1:]
[:([#] (TOP-REAL Affn)), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [:([#] (TOP-REAL Affn)), the carrier of R^1:] is non empty set
((TOP-REAL Affn) | (Affin L)) --> TRn is Relation-like the carrier of ((TOP-REAL Affn) | (Affin L)) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:]
the carrier of ((TOP-REAL Affn) | (Affin L)) --> TRn is Relation-like the carrier of ((TOP-REAL Affn) | (Affin L)) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:]
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
rng f is V60() V61() V62() Element of bool REAL
B is set
{B} is non empty trivial finite 1 -element set
f . n is V21() real ext-real set
{(f . n)} is non empty trivial finite 1 -element V60() V61() V62() set
k is V21() real ext-real Element of the carrier of R^1
((TOP-REAL Affn) | (Affin L)) --> k is Relation-like the carrier of ((TOP-REAL Affn) | (Affin L)) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:]
the carrier of ((TOP-REAL Affn) | (Affin L)) --> k is Relation-like the carrier of ((TOP-REAL Affn) | (Affin L)) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:]
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
{n} is non empty trivial finite 1 -element set
L \ {n} is functional finite Element of bool the carrier of (TOP-REAL Affn)
the Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L \ {n}) is Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L \ {n})
<*n*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
[1,n] is non empty set
{1,n} is non empty finite set
{{1,n},{1}} is non empty finite V39() set
{[1,n]} is Function-like non empty trivial finite 1 -element set
<*n*> ^ the Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L \ {n}) is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like set
rng <*n*> is trivial finite set
rng the Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L \ {n}) is functional finite Element of bool the carrier of (TOP-REAL Affn)
rng (<*n*> ^ the Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L \ {n})) is finite set
(rng <*n*>) \/ (rng the Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L \ {n})) is finite set
Pro is Relation-like the carrier of (TOP-REAL 1) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL 1), the carrier of R^1:]
dom <*n*> is non empty trivial finite 1 -element V60() V61() V62() V63() V64() V65() Element of bool NAT
k is Relation-like NAT -defined the carrier of (TOP-REAL Affn) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL Affn,L)
dom k is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
P is V60() V61() V62() Element of bool the carrier of R^1
Pro " P is functional Element of bool the carrier of (TOP-REAL 1)
bool the carrier of (TOP-REAL 1) is non empty set
{ b1 where b1 is Element of the carrier of ((TOP-REAL Affn) | (Affin L)) : ((TOP-REAL Affn),L,k,b1) | 1 in Pro " P } is set
{1} is non empty trivial finite V39() 1 -element V60() V61() V62() V63() V64() V65() Element of bool NAT
k . 1 is Relation-like Function-like set
<*n*> . 1 is set
f " P is Element of bool the carrier of ((TOP-REAL Affn) | (Affin L))
y is set
yP is Element of the carrier of ((TOP-REAL Affn) | (Affin L))
((TOP-REAL Affn),L,k,yP) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
yP |-- L is Relation-like the carrier of (TOP-REAL Affn) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(yP |-- L) * k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL Affn),L,k,yP) | 1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL Affn),L,k,yP) | (Seg 1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
vP1 is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing Element of the carrier of (TOP-REAL 1)
len vP1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom vP1 is non empty trivial finite 1 -element V60() V61() V62() V63() V64() V65() Element of bool NAT
dom ((TOP-REAL Affn),L,k,yP) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Pro . vP1 is V21() real ext-real Element of the carrier of R^1
vP1 /. 1 is V21() real ext-real Element of REAL
vP1 . 1 is V21() real ext-real set
((TOP-REAL Affn),L,k,yP) . 1 is V21() real ext-real set
(yP |-- L) . n is V21() real ext-real set
((TOP-REAL Affn),L,n) . yP is V21() real ext-real set
f . yP is V21() real ext-real set
dom Pro is functional Element of bool the carrier of (TOP-REAL 1)
[#] (TOP-REAL 1) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL 1)
y is set
((TOP-REAL Affn),L,k,y) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
y |-- L is Relation-like the carrier of (TOP-REAL Affn) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of L
(y |-- L) * k is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
len ((TOP-REAL Affn),L,k,y) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
((TOP-REAL Affn),L,k,y) | 1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence of REAL
((TOP-REAL Affn),L,k,y) | (Seg 1) is Relation-like NAT -defined Function-like finite FinSubsequence-like complex-valued ext-real-valued real-valued set
len (((TOP-REAL Affn),L,k,y) | 1) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
yP1 is Relation-like NAT -defined REAL -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued increasing decreasing non-decreasing non-increasing Element of the carrier of (TOP-REAL 1)
dom yP1 is non empty trivial finite 1 -element V60() V61() V62() V63() V64() V65() Element of bool NAT
f . y is V21() real ext-real set
((TOP-REAL Affn),L,n) . y is V21() real ext-real set
(y |-- L) . (k . 1) is V21() real ext-real set
((TOP-REAL Affn),L,k,y) . 1 is V21() real ext-real set
yP1 . 1 is V21() real ext-real set
yP1 /. 1 is V21() real ext-real Element of REAL
Pro . yP1 is V21() real ext-real Element of the carrier of R^1
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
n is set
Affn is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL Affn is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL Affn) is functional non empty set
bool the carrier of (TOP-REAL Affn) is non empty set
Affn + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
L is functional finite affinely-independent Element of bool the carrier of (TOP-REAL Affn)
card L is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
((TOP-REAL Affn),L,n) is Relation-like the carrier of (TOP-REAL Affn) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL Affn), the carrier of R^1:]
[: the carrier of (TOP-REAL Affn), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL Affn), the carrier of R^1:] is non empty set
Affin L is functional closed Affine Element of bool the carrier of (TOP-REAL Affn)
(TOP-REAL Affn) | (Affin L) is strict TopSpace-like SubSpace of TOP-REAL Affn
the carrier of ((TOP-REAL Affn) | (Affin L)) is set
[: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:] is complex-valued ext-real-valued real-valued set
bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:] is non empty set
((TOP-REAL Affn),L,n) | (Affin L) is Relation-like the carrier of (TOP-REAL Affn) -defined the carrier of R^1 -valued Function-like complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL Affn), the carrier of R^1:]
dim (TOP-REAL Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] (TOP-REAL Affn) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL Affn)
the topology of (TOP-REAL Affn) is non empty Element of bool (bool the carrier of (TOP-REAL Affn))
bool (bool the carrier of (TOP-REAL Affn)) is non empty set
TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #) is non empty strict TopSpace-like TopStruct
dom ((TOP-REAL Affn),L,n) is functional Element of bool the carrier of (TOP-REAL Affn)
f is V60() V61() V62() Element of bool the carrier of R^1
E is Relation-like the carrier of ((TOP-REAL Affn) | (Affin L)) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued continuous Element of bool [: the carrier of ((TOP-REAL Affn) | (Affin L)), the carrier of R^1:]
E " f is Element of bool the carrier of ((TOP-REAL Affn) | (Affin L))
bool the carrier of ((TOP-REAL Affn) | (Affin L)) is non empty set
(E " f) ` is Element of bool the carrier of ((TOP-REAL Affn) | (Affin L))
the carrier of ((TOP-REAL Affn) | (Affin L)) \ (E " f) is set
the topology of TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #) is non empty Element of bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #))
the carrier of TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #) is non empty set
bool the carrier of TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #) is non empty set
bool (bool the carrier of TopStruct(# the carrier of (TOP-REAL Affn), the topology of (TOP-REAL Affn) #)) is non empty set
((TOP-REAL Affn),L,n) " f is functional Element of bool the carrier of (TOP-REAL Affn)
(((TOP-REAL Affn),L,n) " f) ` is functional Element of bool the carrier of (TOP-REAL Affn)
the carrier of (TOP-REAL Affn) \ (((TOP-REAL Affn),L,n) " f) is set
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[.{},1.] is V60() V61() V62() Element of bool REAL
TRn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
conv TRn is functional convex Element of bool the carrier of (TOP-REAL n)
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn)
A is V60() V61() V62() Element of bool the carrier of R^1
E is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len E is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom E is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg (len E) is finite len E -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len E ) } is set
rng E is finite set
bool the carrier of (TOP-REAL n) is non empty Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
f is set
f is set
E . f is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . f is Relation-like Function-like set
((TOP-REAL n),TRn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . f)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),TRn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . f)) " A is functional Element of bool the carrier of (TOP-REAL n)
rng the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is functional finite Element of bool the carrier of (TOP-REAL n)
len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
f is Relation-like NAT -defined bool the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of bool the carrier of (TOP-REAL n)
rng f is finite Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
meet (rng f) is functional Element of bool the carrier of (TOP-REAL n)
f is set
B is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . k is Relation-like Function-like set
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
f . k is set
f |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(f |-- TRn) . B is V21() real ext-real Element of REAL
((TOP-REAL n),TRn,B) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),TRn,B) . f is V21() real ext-real set
((TOP-REAL n),TRn,B) " A is functional Element of bool the carrier of (TOP-REAL n)
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
Affin TRn is functional closed Affine Element of bool the carrier of (TOP-REAL n)
dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg (len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn)) is finite len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) ) } is set
f is set
B is set
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k is set
f . k is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . k is Relation-like Function-like set
Y is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
((TOP-REAL n),TRn,Y) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
dom ((TOP-REAL n),TRn,Y) is functional Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
f |-- TRn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of TRn
(f |-- TRn) . Y is V21() real ext-real Element of REAL
((TOP-REAL n),TRn,Y) . f is V21() real ext-real set
((TOP-REAL n),TRn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . k)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
((TOP-REAL n),TRn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . k)) " A is functional Element of bool the carrier of (TOP-REAL n)
f is functional Element of bool the carrier of (TOP-REAL n)
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
B is set
f . B is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . B is Relation-like Function-like set
((TOP-REAL n),TRn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . B)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),TRn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,TRn) . B)) " A is functional Element of bool the carrier of (TOP-REAL n)
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
Affn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
conv Affn is functional convex Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
Affin ([#] (TOP-REAL n)) is functional non empty closed Affine Element of bool the carrier of (TOP-REAL n)
TRn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
Affin TRn is functional closed Affine Element of bool the carrier of (TOP-REAL n)
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
card TRn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
conv TRn is functional convex Element of bool the carrier of (TOP-REAL n)
Affin Affn is functional closed Affine Element of bool the carrier of (TOP-REAL n)
(conv TRn) /\ (Affin Affn) is functional Element of bool the carrier of (TOP-REAL n)
n is V21() ordinal natural real ext-real non negative finite cardinal set
TOP-REAL n is non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital TopSpace-like T_0 T_1 T_2 strict add-continuous Mult-continuous V231() V232() finite-dimensional RLTopStruct
the carrier of (TOP-REAL n) is functional non empty set
bool the carrier of (TOP-REAL n) is non empty set
n + 1 is non empty V21() ordinal natural real V30() V31() ext-real positive non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
Affn is functional finite affinely-independent Element of bool the carrier of (TOP-REAL n)
card Affn is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of omega
Int Affn is functional Element of bool the carrier of (TOP-REAL n)
].{},1.[ is V60() V61() V62() Element of bool REAL
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) is Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn)
L is V60() V61() V62() Element of bool the carrier of R^1
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len f is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg (len f) is finite len f -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len f ) } is set
rng f is finite set
bool the carrier of (TOP-REAL n) is non empty Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
f is set
B is set
f . B is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . B is Relation-like Function-like set
((TOP-REAL n),Affn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . B)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),Affn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . B)) " L is functional Element of bool the carrier of (TOP-REAL n)
rng the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) is functional finite Element of bool the carrier of (TOP-REAL n)
len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
f is Relation-like NAT -defined bool the carrier of (TOP-REAL n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of bool the carrier of (TOP-REAL n)
rng f is finite Element of bool (bool the carrier of (TOP-REAL n))
bool (bool the carrier of (TOP-REAL n)) is non empty set
meet (rng f) is functional Element of bool the carrier of (TOP-REAL n)
B is set
dim (TOP-REAL n) is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
Affin Affn is functional closed Affine Element of bool the carrier of (TOP-REAL n)
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Y is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . Y is Relation-like Function-like set
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
f . Y is set
B |-- Affn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
(B |-- Affn) . k is V21() real ext-real Element of REAL
((TOP-REAL n),Affn,k) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),Affn,k) . B is V21() real ext-real set
((TOP-REAL n),Affn,k) " L is functional Element of bool the carrier of (TOP-REAL n)
Carrier (B |-- Affn) is functional Element of bool the carrier of (TOP-REAL n)
k is set
(B |-- Affn) . k is V21() real ext-real set
k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(B |-- Affn) . k is V21() real ext-real Element of REAL
conv Affn is functional convex convex closed Element of bool the carrier of (TOP-REAL n)
Sum (B |-- Affn) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
conv Affn is functional convex convex closed Element of bool the carrier of (TOP-REAL n)
Affin Affn is functional closed Affine Element of bool the carrier of (TOP-REAL n)
dom the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
Seg (len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn)) is finite len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) -element V60() V61() V62() V63() V64() V65() with_non-empty_elements Element of bool NAT
{ b1 where b1 is V21() ordinal natural real V30() V31() ext-real non negative finite cardinal V60() V61() V62() V63() V64() V65() Element of NAT : ( 1 <= b1 & b1 <= len the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) ) } is set
B is set
k is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
Sum k is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
sum k is V21() real ext-real Element of REAL
B |-- Affn is Relation-like the carrier of (TOP-REAL n) -defined REAL -valued Function-like quasi_total complex-valued ext-real-valued real-valued Linear_Combination of Affn
Carrier k is functional Element of bool the carrier of (TOP-REAL n)
Y is set
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k is set
f . k is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . k is Relation-like Function-like set
Ek is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
(B |-- Affn) . Ek is V21() real ext-real Element of REAL
{Ek} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
((TOP-REAL n),Affn,Ek) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),Affn,Ek) . B is V21() real ext-real set
dom ((TOP-REAL n),Affn,Ek) is functional Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)
((TOP-REAL n),Affn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . k)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
((TOP-REAL n),Affn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . k)) " L is functional Element of bool the carrier of (TOP-REAL n)
[#] R^1 is non empty non proper V60() V61() V62() open closed Element of bool the carrier of R^1
B is functional Element of bool the carrier of (TOP-REAL n)
dom f is finite V60() V61() V62() V63() V64() V65() Element of bool NAT
k is set
f . k is set
the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . k is Relation-like Function-like set
((TOP-REAL n),Affn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . k)) is Relation-like the carrier of (TOP-REAL n) -defined the carrier of R^1 -valued Function-like quasi_total complex-valued ext-real-valued real-valued Element of bool [: the carrier of (TOP-REAL n), the carrier of R^1:]
[: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty complex-valued ext-real-valued real-valued set
bool [: the carrier of (TOP-REAL n), the carrier of R^1:] is non empty set
((TOP-REAL n),Affn,( the Relation-like NAT -defined the carrier of (TOP-REAL n) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like ( TOP-REAL n,Affn) . k)) " L is functional Element of bool the carrier of (TOP-REAL n)
0. (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued zero Element of the carrier of (TOP-REAL n)
the ZeroF of (TOP-REAL n) is Relation-like NAT -defined REAL -valued Function-like finite n -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued Element of the carrier of (TOP-REAL n)
{(0. (TOP-REAL n))} is functional non empty trivial finite V39() 1 -element affinely-independent Element of bool the carrier of (TOP-REAL n)
[#] (TOP-REAL n) is functional non empty non proper open closed Element of bool the carrier of (TOP-REAL n)